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COSMOS FloWorks Fundamentals

VIEWS: 2,588 PAGES: 238

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                                                                                                      Contents




       Solving Engineering Tasks
          Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-1
          Simulating Engineering Tasks with COSMOSFloWorks . . . . . . . . . . . . . . . . . . . . . .1-4
          Solving Engineering Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6
          Frequent Errors and Improper Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11
       Advanced Knowledge
          Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1
          Mesh Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1
            Types of Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1
            Mesh Construction Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3
            Basic Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-4
            Control Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5
            Resolving Small Features by Using the Control Planes . . . . . . . . . . . . . . . . . . . . 2-5
            Contracting the Basic Mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-6
            Resolving Small Solid Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-7
            Curvature Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-7
            Tolerance Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-9

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         Narrow Channel Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9
         Local Mesh Settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12
         Recommendations for Creating the Computational Mesh. . . . . . . . . . . . . . . . . 2-13
      Mesh-associated Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13
         Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13
         Visualizing the Basic Mesh Before Constructing the Initial Mesh . . . . . . . . . . 2-14
         Enhanced Capabilities of the Results Loading . . . . . . . . . . . . . . . . . . . . . . . . . 2-14
         Viewing the Initial Computational Mesh Saved in the .cpt Files . . . . . . . . . . . 2-15
         Viewing the Computational Mesh Cells with the Mesh Option . . . . . . . . . . . . 2-15
         Visualizing the Real Computational Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 2-16
         Switching off the Interpolation and Extrapolation of Calculation Results . . . . 2-18
         Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20
      Calculation Control Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21
         Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21
         Finishing the Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21
         Refinement of the Computational Mesh During Calculation . . . . . . . . . . . . . . 2-23
      Flow Freezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-25
         What is Flow Freezing? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-25
         How It Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-26
         Flow Freezing in a Permanent Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-26
         Flow Freezing in a Periodic Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-28
      Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-29
         Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-29
         Limitations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-30
         Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-30
         Examples of use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-31
         Rotating impeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-31
         Hydrofoil in a tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-32
         Ball valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-32
         Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-33
      Steam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-33
         Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-33
         Limitations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-33
         Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-34
         Example of use. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35
         Heat exchanger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-35
         Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35
      Humidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35
         Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35

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            Limitations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-36
            Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-36
            Example of use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-38
            Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-38
            Recommendations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-38
          Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-39
            Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-39
            Limitations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-40
            Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-40
            Example of use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-42
            Joule-Thomson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-42
            Recommendations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-43
            References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-43
        Meshing – Additional Insight
          Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1
          Initial Mesh Generation Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2
          Refinements at Interfaces Between Substances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
          Local Mesh Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12
          Irregular Cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13
          The "Optimize thin walls resolution" option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13
          Postamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14
          Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14
        Validation Examples
          Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1
          Flow through a Cone Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3
          Laminar Flows Between Two Parallel Plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7
          Laminar and Turbulent Flows in Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-17
          Flows Over Smooth and Rough Flat Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-23
          Flow in a 90-degree Bend Square Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-27
          Flows in 2D Channels with Bilateral and Unilateral Sudden Expansions . . . . . . . . 4-31
          Flow over a Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-35
          Supersonic Flow in a 2D Convergent-Divergent Channel. . . . . . . . . . . . . . . . . . . . 4-39
          Supersonic Flow over a Segmental Conic Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-43

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      Flow over a Heated Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-49
      Convection and Radiation in an Annular Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-55
      Heat Transfer from a Pin-fin Heat Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-61
      Unsteady Heat Conduction in a Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-65
      Tube with Hot Laminar Flow and Outer Heat Transfer . . . . . . . . . . . . . . . . . . . . . 4-69
      Flow over a Heated Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-73
      Natural Convection in a Square Cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-77
      Particles Trajectories in Uniform Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-83
      Porous Screen in a Non-uniform Stream. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-87
      Lid-driven Flows in Triangular and Trapezoidal Cavities . . . . . . . . . . . . . . . . . . . 4-93
      Flow in a Cylindrical Vessel with a Rotating Cover . . . . . . . . . . . . . . . . . . . . . . . . 4-99
      Flow in an Impeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-103
      Cavitation on a hydrofoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-109
      Thermoelectric Cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-113
      References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-117

    Technical Reference
      Physical Capabilities of COSMOSFloWorks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1
      Governing Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2
        The Navier-Stokes Equations for Laminar and Turbulent Fluid Flows. . . . . . . . 5-2
        Laminar/turbulent Boundary Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-6
        Constitutive Laws and Thermophysical Properties . . . . . . . . . . . . . . . . . . . . . . . .5-6
        Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-7
        Compressible Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10
        Non-Newtonian Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-10
        Equilibrium volume condensation of water from steam . . . . . . . . . . . . . . . . . . . 5-11
        Conjugate Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12
        Thermoelectric Coolers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-13
        Radiation Heat Transfer Between Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-14
        General Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-14
        Ray Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15
        View Factor Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-17
        Environment and Solar Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-18
        Radiative Surface Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-18
        Viewing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19


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             Global Rotating Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19
             Local rotating regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20
             Mass Transfer in Fluid Mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21
             Flows in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21
             Two-phase (fluid + particles) Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-23
             Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-25
             Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-26
             Internal Flow Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-26
             External Flow Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-27
             Wall Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28
             Internal Fluid Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28
             Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-29
          Numerical Solution Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-29
             Computational Mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-29
             Spatial Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-31
             Spatial Approximations at the Solid/fluid Interface . . . . . . . . . . . . . . . . . . . . . . 5-31
             Temporal Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-32
             Form of the Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-33
             Methods to Resolve Linear Algebraic Systems . . . . . . . . . . . . . . . . . . . . . . . . . 5-34
             Iterative Methods for Nonsymmetrical Problems. . . . . . . . . . . . . . . . . . . . . . . . 5-34
             Iterative Methods for Symmetric Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34
             Multigrid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34
          References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-35




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                                                                                                      1
                                                    Solving Engineering Tasks




       Introduction

              Engineering problems are problems connected with designing various objects or systems.
              There are three general approaches to solving engineering problems:
                 • an experimental approach: a hardware rig or prototype, i.e., the full-scale object
                   and/or its model, is manufactured and the experiments needed for designing the
                   object are conducted with this hardware;
                 • a computational approach: the computations needed for designing the object are
                   performed and their results are directly used for designing the object, without
                   conducting any experiments;
                 • a computational-experimental approach combines computations and
                   experiments (with the manufactured full-scale object and/or its model) needed for
                   designing the object, their sequence and contents depending on the solved problem,
                   e.g. iterative procedures may be run.
              Each of the first two approaches has advantages and disadvantages.
              The purely experimental approach, being properly conducted, does not require additional
              validations of the obtained results, but it is very expensive, even if it is realized on the
              object models, since testing facilities and hardware are required anyway. Moreover, if the
              object models are tested, the obtained results must be scaled to the full-scale object, so
              some computations are still required.




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    Chapter Introduction


            The purely computational approach, being properly performed, is substantially less
            expensive than the experimental one, both in finances and in time, but it requires
            assurance in adequacy of the obtained computational results. Naturally, such assurance
            must be based on numerous verifications and validations of the used computational codes,
            both from mathematical and physical viewpoints, i.e., both on the mathematical accuracy
            of the obtained results (the results’ adequacy to the used mathematical model) and on the
            adequacy of the used mathematical model to the governing physical processes, that is
            validated by comparing the computations with the available experimental data.
            The third approach, if it reasonably combines experiments and computations, joins the
            advantages of both of the first two above-mentioned approaches and avoids their
            disadvantages. Complex engineering problems are solved mainly in this way. A
            computational code validated on available experimental data allows of quickly selecting
            the optimal object design and/or its optimal operating mode. Then necessary experiments
            are conducted to verify the selection.
            When selecting from the world market a computational code that is most suitable for
            solving your problems, it is necessary to take into account the following suggestions. Any
            computational code is based, firstly, on a mathematical model of the governing physical
            processes, expressed, as a rule, in the form of a set of differential and/or integral equations
            derived from physical laws, and include, if necessary, semi-empirical and empirical
            constants and/or relationships. Secondly, a method of solving these equations is required.
            Since, as a rule, the equations of the mathematical model cannot be solved analytically,
            they are solved in a discrete form on a computational mesh, so the solution of the
            mathematical problem is obtained with a certain degree of accuracy. Naturally, the
            accuracy of solution of the mathematical problem depends both on the method of
            discretising the differential and/or integral equations and on the method of solving the
            obtained discrete equations. Once these methods have been selected, the accuracy of
            solution of the mathematical problem depends on how well the computational mesh
            resolves the problem regions of non-linear behavior. To provide good accuracy, the mesh
            must be rather fine in these regions. Moreover, a usual way of estimating the accuracy of
            solution of the mathematical problem consists of obtaining solutions on several different
            meshes, from coarser to finer. So, if beginning from some mesh in this set, the difference
            in the interesting physical parameters between the solutions obtained on the finer and
            coarser meshes becomes negligible from the viewpoint of the engineering problem, i.e.,
            the solution flattens, then the accuracy of solution of the mathematical problem required
            for solving this engineering problem is considered to be attained, since the so-called
            solution mesh convergence is attained. Naturally, the solution of the mathematical
            problem can differ from the experimental values (i.e., from the solution of the physical
            problem, if it is known), and this difference depends, firstly, from the conformity of the
            mathematical model and the simulated physical processes, and, secondly, on the errors
            with which these experimental values have been measured and which, as a rule, are known
            and tend to decrease upon increasing the number of tests in which they are measured.




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              Correspondingly, the computational codes presented on the market differ from each other
              not only in their cost, but also in accuracy of mathematical simulation of the physical
              problems, as well as in the procedure of specifying the initial data, in the amount of user’s
              time needed for this specification, in the procedure of solving a problem and the computer
              memory and CPU time needed for obtaining a solution of the required accuracy, and at last
              in the procedures of processing and visualization of the obtained results and the user’s
              time needed for that.
              Naturally, a highly accurate solution requires a fine computational mesh, and consequently
              rather substantial computer memory and CPU time, as well as, in some cases, increased
              user time and efforts for specifying the initial data for the calculation. As a result, if the
              time needed to solve an engineering problem with a computational code exceeds some
              threshold time, then either the engineering problem becomes irrelevant, e.g. because your
              competitors have out-distanced you by this time, or alternative approaches, which may be
              not so accurate, but are surely faster, are used instead in order to solve this problem at
              given time span.
              Before getting acquainted with the recommended procedure of obtaining a reliable and
              rather accurate solution of an engineering problem with COSMOSFloWorks, it is
              expedient to consider COSMOSFloWorks’ features governing the below-described
              strategy of solving engineering problems with COSMOSFloWorks.
              Since COSMOSFloWorks is based on solving time-dependent Navier-Stokes equations,
              steady-state problems are solved through a steady-state approach. To more quickly obtain
              the steady-state solution, a method of local (over the computational domain) time steps is
              employed. A multigrid method is used for accelerating the solution convergence and
              suppressing parasitic oscillations. The computational domain is designed as a
              parallelepiped enveloping the model with planes orthogonal to the axes of the SolidWorks
              model’s Cartesian Global coordinate system. The computational mesh is built by dividing
              the computational domain into parallelepiped cells whose sides are orthogonal to the
              Global coordinate system axes. (The cells lying outside the fluid-filled regions and outside
              solids with heat conduction inside do not participate in the subsequent calculations).
              Procedures of the computational mesh refinement (splitting) are used to better resolve the
              model features, such as high-curvature surfaces in contact with fluid, thin walls
              surrounded by fluid, narrow flow passages (gaps), and the specified insulators’
              boundaries. During the subsequent calculations during the solving of the problem the
              computational mesh can be refined additionally (if that is allowed by the user-defined
              settings) to better resolve the high-gradient flow and solid regions revealed in these
              calculations (Solution-Adaptive Meshing).




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    Chapter Simulating Engineering Tasks with COSMOSFloWorks


            Since steady-state problems are solved in COSMOSFloWorks through the steady-state
            approach, it is necessary to properly select the termination moment for the calculation. If
            the calculation is finished too early, i.e., when the steady state solution has not been
            attained yet, then the obtained solution can depend on the specified initial conditions and
            so be not very reliable. On the contrary, if the calculation is finished too late, then some
            time has been wasted uselessly. To optimize the termination moment for the calculation
            and to determine more accurately physical parameters of interest which oscillate in
            iterations (e.g. a force acting on a model surface, or a model hydraulic resistance), you
            may specify physical parameters of interest as the calculation goals.
            The way to simulate an engineering problem with SolidWorks+COSMOSFloWorks
            correctly and adequately from the physical viewpoint, i.e. to state the corresponding
            model problem, and to solve this model problem properly and reliably with
            COSMOSFloWorks, is described in the chapters Simulating Engineering Tasks with
            COSMOSFloWorks and Solving Engineering Tasks.


    1    Simulating Engineering Tasks with COSMOSFloWorks

            It is necessary to remember that a fast but inaccurate beginning will cost you much efforts
            and time spent uselessly not only for specifying the initial data, but, even worse, for the
            subsequent calculations, until they will finally become reliable. Therefore, we strongly
            recommend that you carefully read this section.

    1.1 Selecting Geometrical and Physical Features of the Task
            Before you start to create a SolidWorks model and a COSMOSFloWorks project, it is
            necessary to select the engineering problem’s geometrical and physical features that most
            substantially influence this problem’s solution - first of all, those of them which are
            important for estimating the possibility of solving this problem with COSMOSFloWorks.
            For example,
               • if the problem contains movable parts, then it is necessary to estimate the
                 importance of taking their motions into account when solving the problem, and, if
                 these motions are important, then to estimate a possibility of solving this problem
                 with a quasi-stationary approach, since model parts’ motions during a calculation
                 are not considered in COSMOSFloWorks (however, you may specify a translational
                 and/or rotational motion of the specific wall),
               • if the problem includes several fluids, or fluid and solid, then it is necessary to
                 estimate the importance of chemical reactions between them for the problem’s
                 solution, and, if the reactions are important, i.e., the reactions rates are rather high
                 and the reacting fluids are intensely mixed with each other under the problem’s
                 conditions, then to estimate a possibility of introducing the reaction products as an
                 additional fluid when solving this problem, since chemical reactions are not
                 considered in COSMOSFloWorks,
               • if the problem includes fluids of different types (for example, a gas and a liquid),
                 and there is an interface between them or these fluids are mixing, then it is

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                    necessary to estimate the importance of taking it into account, since
                    COSMOSFloWorks can not consider a free fluid surface, or mixing of fluids of
                    different types.
              We can present other examples of an clear impossibility of solving some engineering
              problems with COSMOSFloWorks, as well as of simplifying the engineering problems for
              solving them with COSMOSFloWorks, but it is impossible to envision and describe all the
              possible situations in the present document, so that on each particular case you will have
              to make decision by yourself.

       1.2 Creating the Model and the COSMOSFloWorks Project
              If the SolidWorks model has already been created when designing the object, i.e. it is fully
              adequate to the object, then, to solve the engineering problem with COSMOSFloWorks, it
              may be required:
                 • to simplify the model by removing the parts which do not influence the problem’s
                   solution, but consume computer resources, i.e. memory and CPU time. For
                   example, a corrugated model surface which will result in an exceedingly large
                   number of mesh cells required to resolve it can be specified instead as smooth
                   surface with equivalent wall roughness. If a model has thin solid protrusions or
                   narrow fluid-filled blind holes whose influence on the overall flow pattern is, by
                   rough estimate, barely perceptible, it would be better to remove these features in
                   order to avoid the excessive mesh splitting around them.
                 • to add auxiliary parts to the model, e.g. inlet and outlet tubes for stabilization of the
                   flow, lids to cover the inlet and outlet openings, and parts to denote rotating regions,
                   local initial meshes or other areas where special conditions are applied.
              Both these actions, being executed properly, can be very pivotal in obtaining a reliable and
              accurate solution. Naturally, adding the auxiliary parts to a model will inevitably cause an
              increase of the computational mesh cells and, consequently, the required computer
              memory and CPU time, therefore these parts’ dimensions must be adequate to the stated
              problem.
              If a model has not been created yet, it is expedient to take all the above-mentioned factors
              into account when creating it.
              If all effects of these actions are not clear enough, it may be worthwhile to vary the model
              parts and/or their dimensions in a series of calculations in order to determine their effects
              on the obtained solution.
              Then, in accordance with the problem’s physical features revealed and adapted to
              COSMOSFloWorks capabilities, the basic part of the COSMOSFloWorks project is
              specified, i.e., the problem type (internal or external), fluids and solids involved in the
              problem, physical features taken into account (e.g. heat conduction in solids,
              time-dependent analysis, gravitational effects, etc.), boundaries of the calculation domain,
              initial and boundary conditions, and, if necessary, fluid subdomains, rotating regions,
              volume and/or surface heat sources, fans and other features and conditions.


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    Chapter Solving Engineering Tasks


            The specified boundary conditions, as well as heat sources, fans, and other conditions and
            features must correspond to the problem’s physical statement and must not conflict with
            each other.
            Eventually, you specify the physical parameters of interest as the COSMOSFloWorks
            project goals. They can be local or integral, defined within the whole computational domain
            or on some model surfaces, or in some volumes (local parameters are determined over some
            region in the form of their minimum, or maximum, average, or bulk average values). This
            will allow you to substantially increase reliability and accuracy of determination of these
            physical parameters, since their behavior is saved on each iteration during the calculation
            and can be analyzed later. On the contrary, the convergence behavior of all other parameters
            can not be analyzed afterwards, as they are saved only at the last iteration and, probably, at
            some user-specified iterations, whose number is restricted by the disk space limit.

    2    Solving Engineering Tasks

            As soon as you have specified the basic part of the COSMOSFloWorks project that is
            unlikely to be changed in the subsequent calculations, the next step is to select the strategy
            of solving the engineering problem with COSMOSFloWorks, i.e., obtaining the reliable
            and accurate solution of the problem.

    2.1 Strategy of Solving the Engineering Tasks
            As it has been mentioned in Introduction, by performing a series of calculations on a set of
            computational meshes ranging from coarser to finer ones, we can estimate the accuracy of
            solution of the mathematical problem. As soon as the calculation on a finer mesh does not
            yield a noticeably different (from the engineering problem’s viewpoint) solution, i.e. the
            solution flattens with respect to the mesh cells’ number, we can conclude that the solution
            of the mathematical problem has achieved mesh convergence, i.e., the required
            mathematical solution accuracy is attained. Naturally, first you must determine the
            threshold for a solution-vs.-mesh change, so that the change smaller than this threshold
            will be considered as negligible. Since the determination of this threshold is possible only
            in relation with some physical parameter, it is natural to connect it with the physical
            parameters of interest of the engineering problem in question, in particular, with the
            admissible determination errors of these physical parameters. Moreover, since steady-state
            problems are solved with COSMOSFloWorks through the steady-state approach, the
            supervision for a behavior of the calculation goals during the calculation (i.e., in
            iterations) can serve two purposes. Firstly, if these parameters oscillate during the
            solution, it will allow you to determine their values and observation errors more accurately
            by averaging them over a number of iterations and determining their deviation from this
            average value. Secondly, you may want to intervene in the calculation process by finishing
            the calculation manually if you see that either the calculation is unacceptable for you by
            some reasons, or, vice versa, if the solution has actually already converged, so that there is
            no reason to calculate any further.



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              Therefore, the strategy of solving an engineering problem with COSMOSFloWorks
              consists, first of all, in performing several calculations on the same basic project (i.e., with
              the same model, inside the same computational domain, and with similar boundary and
              initial conditions) varying only the computational mesh. Since the computational mesh is
              built automatically in COSMOSFloWorks, it may be varied by varying the project
              parameters that govern its design (the initial computational mesh on which the calculation
              starts, and maybe its refinement during the calculation): Result Resolution Level,
              Minimum Gap Size, Minimum Wall Thickness.
              An additional item of this strategy of solving an engineering problem with
              COSMOSFloWorks consists in varying the auxiliary elements added to the model as
              needed to solve the problem with COSMOSFloWorks (e.g. inlet and outlet tubes attached
              to the inlet and outlet openings, for internal problems), whose dimensions are questionable
              from the viewpoint of their necessity and sufficiency. Those physical parameters of the
              engineering problem whose values are not known exactly and which, in your opinion, can
              influence the problem solution, must be varied also. When performing these calculations,
              there is no need to investigate the solution-vs.-mesh convergence again, since it has been
              already achieved before. It is enough to just perform these calculations with the project
              mesh settings that provided the solution with satisfactory accuracy during the
              solution-vs.-mesh convergence investigation. The same applies also to the parametric
              engineering calculations while you are changing the model parts and/or flow parameters.
              However, you must keep in mind the potential necessity for checking the
              solution-vs.-mesh convergence, because in doubtful cases it must be checked again.
              In spite of the apparent simplicity of the proposed strategy, its full realization is usually
              troublesome due to the substantial difficulties including, first of all, a dramatic increase of
              the requirements for computer memory and CPU time when you are substantially
              increasing the number of cells in the computational mesh. Since both the computer
              memory and the time for which the engineering problem must be solved are usually
              restricted, the specific realization of this strategy eventually governs the accuracy of the
              problem solution, whether it will be satisfactory or not. Perhaps, a further simplification of
              the model and/or reducing the computational domain will be required.
              Some specific description of this strategy are presented in the next sections of this
              document.

       2.2 Settings for Resolving the Geometrical Features of the Model and for
           Obtaining the Required Solution Accuracy
              The computational mesh variation described in Section 2.1 is the key item of the proposed
              strategy of solving engineering problems with COSMOSFloWorks.
              The result resolution level specified in the Wizard governs the number of basic mesh cells,
              the criteria for refinement (splitting) of the basic mesh to resolve the model geometry, i.e.,
              creating the initial mesh, as well as the criteria for refinement (splitting) of the initial mesh
              during the problem solution. The Result Resolution specified in the Wizard defines the
              following parameters in the created project: the Level of initial mesh and the Results
              resolution level. The Level of initial mesh governs only the initial mesh and is accessible
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    Chapter Solving Engineering Tasks


            (after the Wizard has been finished) from the Initial Mesh dialog box. The Results
            resolution level is accessible from the Calculation Control Options dialog box and
            controls the refinement of computational mesh during calculation and the calculation
            finishing conditions. The Geometry Resolution options that also influence the initial mesh
            may be changed on the Automatic Settings tab of the the Initial Mesh dialog box. Their
            effects can be altered on the other tabs of the Initial Mesh dialog box or in the Local Initial
            Mesh dialog box.
            Before creating the initial mesh, COSMOSFloWorks automatically determines the
            minimum gap size and the minimum wall thickness for the walls whose are in contact with
            a fluid on both sides. That is required for resolving the geometrical features of the model
            with COSMOSFloWorks computational mesh. So, when creating the initial mesh, it is
            taken into account that the number of the mesh cells along the normal to the model surface
            must not be less than a certain number if the distance along this normal from this surface
            to the opposite wall is not less than the minimum gap size. Depending on the mesh cell
            arrangement, the model flow passages not resolved with the computational mesh either are
            automatically replaced with a wall, or increased up to the mesh cell size. In the automatic
            mode these mesh parameters are determined from dimensions of the surfaces on which
            boundary conditions have been specified, e.g. the model inlet and outlet openings in an
            internal analysis, as well as those surfaces and volumes on or in which heat sources, local
            initial conditions, surface and/or volume goals and some of the other conditions and
            features have been specified. Before the calculation, you can see the minimum gap size
            and the minimum wall thickness determined in such a way. If these values cannot provide
            an adequate resolution of the model geometry, you can specify them manually. At that, it
            is necessary to take into account that the number of the computational mesh cells
            generated to resolve the model geometrical features depends on the specified result
            resolution level.
            Evidently, when creating a COSMOSFloWorks project it is necessary to make sure that
            both the minimum gap size and the minimum wall thickness are relevant to the model
            geometry. However, if the model geometry is complicated (e.g. there are non-circular flow
            passages, sharp edges protruding into the stream, etc.), it can be difficult to determine
            these parameters unambiguously. In this case it may be useful to perform several
            calculations by varying these parameters within a reasonable range in order to reveal their
            influence on the problem solution. In accordance with the strategy of solving engineering
            problems, these calculations must be performed at different result resolution levels.
            The initial mesh created at result resolution levels of 3…5 is not changed during the
            solving of a problem, i.e. is not adapted to the solution being obtained. Result resolution
            levels of 5…7 yield the same initial mesh, but at result resolution levels of 6 and 7 the
            mesh is refined during the calculations in the regions of increased physical parameters
            gradients. At level 8, a finer initial mesh is generated and refinements during calculation
            takes place.
            It makes sense to perform calculations at the result resolution level of 3 if both the model
            geometry and the flow field are relatively smooth. For more complex problems we
            recommend first of all to perform the calculation at the result resolution level of 4 or,

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              better, 5 (naturally, with specifying explicitly the minimum gap size and minimum wall
              thickness). After that, if the calculation at the result resolution level of 5 has been
              performed, we recommend, in order to ascertain the mesh convergence, to perform the
              calculation at the result resolution level of 7 and, if the computer resources allow you to do
              this, at the result resolution level of 8.

       2.3 Monitoring the Calculation
              Monitoring the calculation, i.e., at least, monitoring behavior of the physical parameters
              specified by you as the project goals (you can inspect also physical parameters fields at the
              specified planar cross-sections) is useful for the following reasons:
                 • you can intervene in the process of calculation, i.e., manually finish the calculation
                   before it finishes automatically, if you see that either the calculation is unacceptable
                   for you for some reasons (e.g. if COSMOSFloWorks has generated warnings
                   making clear that the sequential calculation is senseless), or, vice versa, when
                   solving a steady-state problem (that concerns some time-dependent problems also),
                   the solution has already converged, so that there is no reason to continue the
                   calculation;
                 • if a steady-state problem is solved, and the physical parameters specified by you as
                   the project goals oscillate during the iterations, then inspecting these parameters’
                   behavior during the calculation will allow you to determine their values and
                   determination errors more accurately by averaging their values over the iterations
                   and determining their deviations from these average values;
                 • if the physical parameters of interest do not change substantially during the
                   calculation, you can obtain their intermediate (preliminary) values beforehand, and
                   in the subsequent iterations they will be refined finally;
                 • if you solve a time-dependent problem, you can immediately see the calculation
                   results before the calculation is finished.
              The first above-mentioned reason is especially useful since it allows you to substantially
              reduce the CPU time in some cases. For example, if you do not specify the high Mach
              number gas flow in the project settings, whereas in fact the flow becomes supersonic, or if
              COSMOSFloWorks warns you about a vortex at the model outlet, that substantially
              reduces the calculation accuracy, making it necessary to change some of the problem
              settings (i.e. specify high Mach number flow for the first case or lengthen the model outlet
              tube for the second one). If you solve a steady-state problem at the result resolution level
              of 7 or 8 and you see that the computational mesh refinements performed during the
              calculation do not increase the number of cells in the mesh and, therefore, do not
              noticeably improve the problem solution (the values of the project goals does not change),
              you can finish the calculation relatively early (say, after 1…2 travels have been
              performed).




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    Chapter Solving Engineering Tasks


    2.4 Viewing and Analyzing the Obtained Solution
            When viewing and analyzing the obtained solution after finishing the calculation, it is
            recommended first of all, in accordance with the above-mentioned suggestions, to plot the
            evolutions of the project goals during the calculation, if you did not monitor them directly
            as the calculation went on. If a steady-state problem is solved, and you have specified the
            physical parameter of interest as the project goal, then, if this parameter has oscillated
            during the calculation, you can determine its value more accurately by averaging it over
            the last iterations interval in which its steady-state oscillation is seen. By that you also
            determine the variance of this goal, i.e., its deviation from the average value, that
            characterizes the goal determination error in the obtained solution.
            It is also useful to check for vortices at the model outlet, as well as to see the flow pattern
            in the model and, if heat transfer in solids has been calculated, the temperature distribution
            over the solid parts of the model. Naturally, first of all it is expedient to see the obtained
            field of the physical parameter you are interesting in, not only in the region of interest, but
            also in a broader area, in order to check this field for apparently incosistent results.
            It is also worthwhile to examine the obtained fields of other physical parameters related to
            the one you are interested in. For example, if you are interested in the total pressure loss,
            you may want to see the velocity field, whereas if you are interested in the temperature of
            solid, a picture of the fluid-to-solid heat flux field is also useful.

    2.5 Estimating the Reliability and Adequacy of the Obtained Solution
            In accordance with the general approach to estimating reliability and accuracy of the
            engineering problem solution obtained with a computational code, this estimation consists
            of the following two parts: an estimation of how accurate is the solution of the
            mathematical problem corresponding to the mathematical model of the physical process,
            and an estimation of accuracy of simulating the physical process with the given
            mathematical model.
            The accuracy of solution of the mathematical problem is determined by mathematical
            methods, independently of the consistency of the model to the physical process under
            consideration. In our case, this accuracy estimation is based on analyzing the mesh
            convergence of the problem solutions obtained on different computational meshes (See
            Section 2.2). Then, since steady-state problems are solved with COSMOSFloWorks via a
            steady-state approach by employing local time steps, it is useful to verify additionally the
            accuracy of the obtained solution by solving the similar time-dependent problem not
            employing local time steps.
            As soon as the mathematical problem solution of a satisfactory accuracy has been
            obtained, the next step consists of estimating the accuracy of simulating the physical
            process under consideration with the mathematical model employed in the computational
            code. To do this, the obtained solution is compared with the available experimental data
            (taking into account their errors which consist of measurement errors and experimental
            errors arising from possible spurious influences). Naturally, since experimental data are
            always restricted, for this validation it is desirable to select the data which are as close to

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              the engineering problem being solved as possible. To validate the computational code on
              the available experimental data, you have to solve the corresponding test problem in
              addition to the engineering problem being solved (preferably before you start to solve the
              engineering problem following the above-mentioned strategy), but this operation
              increases the reliability of estimating the obtained solution of the engineering problem so
              substantially that the required additional time and efforts will be fully paid back later on,
              in particular when solving similar engineering problems.
              If after solving the test problem you see that accuracy of its solution obtained with
              COSMOSFloWorks is not satisfactory from your viewpoint, check to see that you have
              properly specified the COSMOSFloWorks project, that all substantial features of the
              engineering problem have been taken into account, and, finally, that COSMOSFloWorks
              restrictions do not impede solving this engineering problem.

       3    Frequent Errors and Improper Actions

              Let us consider errors and improper actions frequently done when solving engineering
              problems with COSMOSFloWorks.
                 When Specifying Initial Data:
                    • not taking into account physical features which are important for the engineering
                      problem under consideration: e.g. high Mach number gas flow (should be taken
                      into account if M>3 for steady-state or M>1 for transient tasks or supersonic flow
                      occurs in about a half of the computational domain or greater), gravitational
                      effects (must be taken into account if either the fluid velocity is small, the fluid
                      density is temperature-dependent, and a heat source is considered, or several
                      fluids having substantially different densities are considered in a gravitational
                      field), necessity of the time-dependent analysis (e.g. at the moderate Reynolds
                      numbers, when unsteady vortices are generated);
                    • incorrectly specifying symmetry planes as the computational domain boundaries
                      (e.g. at the moderate Reynolds numbers, when unsteady vortices are generated;
                      you should keep in mind that the symmetry of model geometry and initial and
                      boundary conditions does not guarantee you the symmetry of flow field);
                    • if symmetry planes have been specified and you click Reset at the Size tab of the
                      Computational Domain dialog box, please do not forget to replace Symmetry
                      by Default at the Boundary Condition tab;
                    • if you have specified symmetry planes and intend to specify mass or volume flow
                      rates at a model inlet or outlet openings, please do not forget to specify only their
                      parts falling into the computational domain instead of the total flow rates at these
                      openings;
                    • if you specify integral boundary or volume conditions (heat transfer rates, heat
                      generation rates, etc.), please remember that their values specified in the
                      COSMOSFloWorks dialog boxes correspond to the area or volume's part falling
                      into the computational domain;

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    Chapter Frequent Errors and Improper Actions


                  • if you specify a flow swirl on a model inlet or outlet openings (in the Fans or
                    Boundary Conditions dialog boxes), please do not forget to specify properly
                    their swirl axes and the proper coordinate system for that in the Definition tab;
                  • if you specify a Unidirectional or Orthotropic porous medium, please do not
                    forget to specify their directions;
                  • please do not forget that the specified boundary conditions must not conflict with
                    each other. For example, if you deal with gas flows and the model inlet flow is
                    subsonic, whereas the flow inside the model becomes supersonic, it is incorrect
                    to specify flow velocity or volume flow rate as a boundary condition at the model
                    inlet, since they are fully determined by the geometry of the model flow passage
                    and the fluid’s specific heat ratio;
                  • if you solve a time-dependent problem, and this problem has cyclic-in-time
                    boundary conditions, thus leading to a steady-state cyclic-in-time solution, to
                    obtain which you have to calculate the flow several times in cycle, every time
                    specifying the solution from the previous calculation as the initial condition for
                    the next calculation, there is no need to specify the boundary conditions for
                    several cycles. Instead it is more convenient to specify them for a cycle and
                    perform a series of calculations, running each calculation with selected Take
                    previous results check box in the Run dialog box;
                  • when specifying Surface Goals, Volume Goals, Equation Goals, it is better to
                    give them sensible names to identify these goals unambiguously, instead of
                    selecting them in the tree and looking for the respective places at the model in the
                    SolidWorks graphics area;
                  • if you want to monitor the intermediate calculation results at certain sections of
                    the model during the calculation, it is better to determine these sections’ positions
                    in the Global coordinate system beforehand, i.e. before actually running the
                    calculation, since during the calculation it is a bit more difficult and you may be
                    literally ’late’ in terms of the problem’s physical time;
               When Monitoring a Calculation:
                  • when monitoring intermediate calculation results during a calculation, please do
                    not forget the spatial nature of the problem being solved (of course, if the
                    problem itself is not 2D). To take a look at the full pattern it is expedient to see
                    the results at least in 2 or 3 intersecting planes;
               When Viewing the Obtained Solution after Finishing a Calculation:
                  • please take into account that all settings made in the View Settings dialog box
                    concern all Cut Plots, 3D Plots, Surface Plots, Flow Trajectories, Isosurfaces,
                    which are active in the SolidWorks graphics area, therefore:
                     • your will not be able to open the Flow Trajectories dialog box if a parameter
                       defined only on wall surfaces has been selected on the Contours tab and the
                       Use from contours option has been selected at the Flow Trajectories tab of
                       the View Settings dialog box;


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                     • to view different result features in different panes simultaneously, it is
                       necessary to split the SolidWorks graphics area into 2 or 4 panes and build
                       different result features in different graphical areas through their individual
                       Cut Plots, 3D Plots, Surface Plots, Flow Trajectories, Isosurfaces defined
                       in these areas;
                   • if you intend to see integral physical parameters (e.g. area, mass or volume flow
                     rates, heat generation rates, forces, etc.) with the Surface Parameters dialog
                     box, please remember that
                     • their shown values are determined over the parts of the surface that belong to
                       the computational domain;
                     • their determination errors include errors of representing these surfaces in
                       SolidWorks and COSMOSFloWorks, the latter depends on the computational
                       mesh;
                   • if you want to see a computational mesh in Cut Plots and/or Surface Plots,
                     please do not forget to select Display mesh under Tools, Options, Third Party
                     Options, otherwise the Mesh button in the Cut Plots and Surface Plots
                     PropertyManagers will be absent, so you can not view a computational mesh.




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    Chapter Frequent Errors and Improper Actions




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                                                                                                  2
                                                              Advanced Knowledge




       Introduction

              The present document supplies you with our experience of employing the advanced
              COSMOSFloWorks capabilities, organized in the following topics:
                 Manual adjustment of the initial computational mesh settings
                 Mesh-associated tools (viewing the mesh before and after the calculation and
                 advanced post-processing tools)
                 Calculation control options (refinement of the computational mesh during calculation,
                 conditions of finishing the calculation)
                 Flow freezing

       1    Mesh Introduction

              This chapter provides the fundamentals of working with the COSMOSFloWorks
              computational mesh, describes the mesh generation procedure, and explains the use of
              parameters governing both automatically and manually controlled meshes.
              First, let us introduce a set of definitions.

       1.1 Types of Cells
              Any COSMOSFloWorks calculation is performed in a rectangular parallelepiped-shaped
              computational domain whose boundaries are orthogonal to the axes of the Cartesian
              Global Coordinate System. A computational mesh splits the computational domain with a
              set of planes orthogonal to the Cartesian Global Coordinate System's axes to form
              rectangular parallelepipeds called cells. The resulting computational mesh consists of
              cells of the following four types:
                 • Fluid cells are the cells located entirely in the fluid.

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               • Solid cells are the cells located entirely in the solid.
               • Partial cells are the cells which are partly in the solid and partly in the fluid. For
                  each partial cells the following information is kept: coordinates of intersections of
                  the cell edges with the solid surface and normal to the solid surface within the cell.
               • Irregular cells are partial cells for which the normal to the solid surface cannot be
          determined. Cells of this type are never generated with the modern version of
          COSMOSFloWorks, however, such cells may be found in the meshes built with the previous
          versions of COSMOSFloWorks)
            As an illustration let us look at the original model (Fig.1.1) and the generated
            computational mesh (Fig.1.2).




                            Fig.1.1    The original model.



                                           Fluid cell                      First level cell




             Partial cell
                                                                                               Partial cell




                                                                                              Solid cell




                                                        Zero level cell (basic cell)

                        Fig.1.2 The computational mesh cells of different types




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                          Optimize thin walls resolution. In the early versions of
                          COSMOSFloWorks refinement of the mesh within model's walls
                          was needed to resolve thin walls properly, but it could also lead to
                          increase in number of cells in adjacent fluid regions, especially in
                          narrow channels between walls. If this additional mesh refinement is
                          critical for obtaining the proper results and you want to perform
                          calculation on the same mesh as in the earlier version of
                          COSMOSFloWorks, clear the Optimize thin walls resolution check
                          box. In this case the mesh will be almost the same as in that earlier
                          version, with the main difference of absence of irregular cells.

       1.2 Mesh Construction Stages
              Refinement is a process of splitting a rectangular computational mesh cell into eight cells
              by three orthogonal planes that divide the cell's edges in halves. The non-split initial cells
              that compose the basic mesh are called basic cells or zero level cells. Cells obtained by the
              first splitting of the basic cells are called first level cells, the next splitting produces
              second level cells, and so on. The maximum level of splitting is seven. A seventh level cell
              is 87 times smaller in volume than the basic cell.

                          During the solution-adaptive meshing the cells can be refined and
                          merged. See ”Refinement of the Computational Mesh During
                          Calculation’ on page 23.
              The following rule is applied to the processes of refinement and merging: the levels of two
              neighboring cells can only be the same or differ by one, so that, say, a fifth level cell can
              have only neighboring cells of fourth, fifth, or sixth level.
              The mesh is constructed in the following steps:
                 Construction of the basic mesh taking into account the Control Planes and the
                 respective values of cells number and cell size ratios.
                 Resolving of the interface between substances, including refinement of the basic mesh
                 at the solid/fluid and solid/solid boundaries to resolve the relatively small solid features
                 and solid/solid interface, tolerance and curvature refinement of the mesh at a
                 solid/fluid, solid/porous and a fluid/porous boundaries to resolve the interface
                 curvature (e.g. small-radius surfaces of revolution, etc).

                          If you switch on or off heat conduction in solids, or add/move
                          insulators, you should rebuild the mesh.

                 Narrow channels refinement, that is the refinement of the mesh in narrow channels
                 taking into account the respective user-specified settings.
                 Refinement of all fluid, and/or solid, and/or partial mesh cells up to the user-specified
                 level.
                 Mesh conservation, i.e. a set of control procedures, including check for the difference
                 in area of cell facets common for the adjacent cells of different levels.
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            After each of these stages is passed, the number of cells is increased to some extent.
            In COSMOSFloWorks you can control the following parameters and options which
            govern the computational mesh:
            1 Nx, the number of basic mesh cells (zero level cells) along the X axis of the Global
               Coordinate System. 1 ≤ Nx ≤ 1000
            2 Ny, the number of basic mesh cells (zero level cells) along the Y axis of the Global
               Coordinate System. 1 ≤ Ny ≤ 1000.
            3 Nz, the number of basic mesh cells (zero level cells) along the Z axis of the Global
               Coordinate System. 1 ≤ Nz ≤ 1000.
            4 Control planes. By adding and relocating them you can contract and/or stretch the
               basic mesh in the specified directions and regions. Six control planes coincident with
               the computational domain's boundaries are always present in any project.
            5 Small solid features refinement level (Lb). 0 ≤ Lb ≤ 7.
            6 Curvature refinement level (Lcur). 0 ≤ Lcur ≤ 7.
            7 Curvature refinement criterion (Ccur). 0 ≤ Ccur ≤ π.
            8 Tolerance refinement level (Ltol). 0 ≤ Ltol ≤ 7.
            9 Tolerance refinement criterion (Ctol). 0 ≤ C tol.
            10 Narrow channels refinement: Characteristic number of cells across a narrow channel,
               Narrow channels refinement level, The minimum and maximum height of narrow
               channels to be refined.
            These options are described in more detail below in this chapter.

    1.3 Basic Mesh
            The basic mesh is a mesh of zero level cells. In case of 2D calculation (i.e. if you select the
            2D plane flow option in the Computational Domain dialog box) only one basic mesh cell
            is generated automatically along the eliminated direction. By default COSMOSFloWorks
            constructs each cell as close to cubic shape as possible.

                        The number of basic mesh cells could be one or two less than the
                        user-defined number (Nx, Ny, Nz). There is no limitation on a cell
                        oblongness or aspect ratio, but you should carefully check the
                        calculation results in all cases for the absence of too oblong or
                        stretched cells.




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                          a) 10x12x1                                          b) 40x36x1

                                       Fig.1.3 Basic mesh examples.


       1.4 Control Planes
              The Control Planes option is a powerful tool for creating an optimal computational mesh,
              and the user should certainly become acquainted with this tool if he is interested in
              optimal meshes resulting in higher accuracy and decreasing the CPU time and required
              computer memory. Control planes allow you to resolve small features, contract the basic
              mesh locally to resolve a particular region by a denser mesh and stretch the basic mesh to
              avoid excessively dense meshes.

       1.5 Resolving Small Features by Using the Control Planes
              If the level of splitting is not high enough, small solid features may be not resolved
              properly. In this case, two methods can be used to improve the mesh:
                 • increase the level of splitting. However, this may result in unnecessary increase of
                   the number of cells in other regions, creating a non-optimal mesh, or
                 • set a control plane crossing the relevant small feature (e.g. a solid's sharp edge).
                   This will allow you to resolve this feature better without creating an excessively
                   dense mesh elsewhere. It is especially convenient in cases of sharp edges oriented
                   along the Global Coordinate System axes.

                         It is recommended that you place a control plane slightly submerged
                         into the solid, and avoid placing it coincident with the solid surface.




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    1.6 Contracting the Basic Mesh
            Using control planes you may contract the basic mesh in the regions of interest. To do this,
            you need to set control planes surrounding the region and assign the proper Ratio values to
            the respective intervals. The cell sizes on the interval are changed gradually so that the
            proportion between the first and the last cells of the interval is close (but not necessarily
            equal) to the entered Ratio value. Negative values of the ratio correspond to the reverse
            order of cell size increase. Alternatively, you may explicitly set the Number of cells for
            each interval, in which case the Ratio value becomes mandatory. For example, assume
            that there are two control planes Plane1 and Plane2 (see Fig.1.4) and the ratio on the
            interval between them is set to 2. Then the basic mesh cells adjacent to the Plane1 will be
            approximately two times longer than the basic mesh cells adjacent to the Plane2.




                                Fig.1.4 Specifying custom control planes.




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              Use of control planes is especially recommended for external analyses, where the
              computational domain may be substantially larger than the model.




               Fig.1.5   Default control planes.             Fig.1.6 Two custom control planes.
              In the Fig.1.6 two custom control planes are set through the center of the body with the
              ratio set to 5 and -5, respectively, on the intervals to the both sides of each plane.

       1.7 Resolving Small Solid Features
              The procedure of resolving small solid features refines only the cells where the solid/fluid
              (solid/solid, solid/porous as well as fluid/porous) interface curvature is too high: the
              maximum angle between the normals to a solid surface inside the cell exceeds 120°, i.e.
              the solid surface has a protrusion within the cell.
              Such cells are split until the the Small solid features refinement level of splitting mesh
              cells is achieved.

       1.8 Curvature Refinement
              The curvature refinement level is the maximum level to which the cells will be split during
              refinement of the computational mesh until the curvature of the solid/fluid or fluid/porous
              interface within the cell becomes lower than the specified curvature criterion (Ccur).
              The curvature refinement procedure has the following stages:
              1 Each solid surface is triangulated: COSMOSFloWorks gets triangles that make up the
                 SolidWorks surfaces.

                          The performance settings do not govern the triangulation
                          performance.

              2 A local (for each cell) interface curvature is determined as the maximum angle
                 between the normals to the triangles within the cell.
              3 If this angle exceeds the specified Ccur, and the curvature refinement level is not
                 reached then the cell is split.

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    Chapter Mesh Introduction


            The curvature refinement is a powerful tool, so that the competent usage of it allows you
            to obtain proper and optimal computational mesh. Look at the following illustrations to
            the curvature refinement by the example of a sphere.




      Fig.1.7 Curvature refinement level is 0;         Fig.1.8 Curvature refinement level is 1;
      Total number of cells is 64.                     Total number of cells is 120.




     Fig.1.9 Curvature refinement level is 2;           Fig.1.10 Curvature refinement level is 2;
     Curvature criterion is 0.317;                      Curvature criterion is 0.1;
     Total number of cells is 120.                      Total number of cells is 148.




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       1.9 Tolerance Refinement
              Tolerance refinement allows you to control how well (with what tolerance) mesh polygons
              approximate the real interface. The tolerance refinement may affect the same cells that
              were affected by the small solid features refinement and the curvature refinement. It
              resolves the interface's curvature more effectively than the small solid features refinement,
              and, in contrast to the curvature refinement, discerns small and large features of equal
              curvature, thus avoiding refinements in regions of less importance (see images below).
              Any surface is approximated by a set of polygons whose vertices are surface's intersection
              points with the cells' edges. This approach accurately represents flat faces though
              curvature surfaces are approximated with some deviations (e.g. a circle can be
              approximated by a polygon). The tolerance refinement criterion controls this deviation. A
              cell will be split if the distance (h, see below) between the outermost interface's point
              within the cell and the polygon approximating this interface is larger than the specified
              criterion value.

       1.10 Narrow Channel Refinement
              The narrow channel refinement is applied to each flow passage within the computational
              domain (or a region, in case that local mesh settings are specified) unless you specify for
              COSMOSFloWorks to ignore passages of a specified height. The Narrow Channels term
              is conventional and used for the definition of the flow passages of the model in the
              direction normal to the solid/fluid interface.
              The basic concept of narrow channel refinement is to resolve the narrow channels with a
              sufficient number of cells to provide a reasonable level of solution accuracy. It is
              especially important to have narrow channels resolved in analyses of low Reynolds
              numbers or analyses with long channels, i.e. in such analyses where the boundary layer
              thickness becomes comparable to the size of the partial cells where the layer is developed.
              The narrow channel settings available in COSMOSFloWorks are the following:
                 • Narrow channels refinement level – the maximum level of cells refinement in
                   narrow channels with respect to the basic mesh cell.
                 • Characteristic number of cell across a narrow channel – the number of cells
                   (including partial cells) that COSMOSFloWorks will attempt to set across the model
                   flow passages in the direction normal to the solid/fluid interface. If possible, the
                   number of cells across narrow channels will be equal to the specified characteristic
                   number, otherwise it will be as close to it as possible. The Characteristic number
                   of cells across a narrow channel (let us denote it as Nc) and the Narrow channels
                   refinement level (let us denote it as L) both influence the mesh in narrow channels
                   in the following manner: the basic mesh in narrow channels will be split to have Nc
                   number per channel, if the resulting cells satisfy the specified L. In other words,
                   whatever the specified Nc, the smallest possible cell in a narrow channel is 8L times
                   smaller in volume (or 2L times smaller in each linear dimension) than the basic
                   mesh cell. This is necessary to avoid undesirable mesh splitting in very fine
                   channels that may cause the number of cells to increase to an unreasonable value.

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    Chapter Mesh Introduction


               • The minimum height of narrow channels, The maximum height of narrow
                 channels – the minimum and maximum bounds for the height outside of which a
                 flow passage will not be considered as a narrow channel and thus will not be refined
                 by the narrow channel resolution procedure.
            For example, if you specify the minimum and maximum height of narrow channels, the
            cells will be split only in those fluid regions where the distance between the opposite walls
            of the flow passage in the direction normal to wall lies between the specified minimum
            and maximum heights.
            The narrow channel refinement operates as follows: the normal to the solid surface for
            each partial cell is extended up to the next solid surface, which will be considered to be the
            opposite wall of the flow passage. If the number of cells per this normal-to-wall direction
            is less than the specified Nc, the cells will be split to satisfy the narrow channel settings as
            described above.
            Although the settings that produce an optimal mesh depends on a particular task, here are
            some ’rule-of-thumb’ recommendations for narrow channel settings:
            1 Set the number of cells across narrow channel to a minimum of 5.
            2 Use the minimum and maximum heights of narrow channels to concentrate on the
               regions of interest.
            3 If possible, avoid setting high values for the narrow channels refinement level, since it
               may cause a significant increase in the number of cells where it is not necessary.




          Fig.1.11 Curvature refinement level is 3; Small solid features refinement level is 3; Narrow channel
          refinement is disabled; Total number of cells is 6476.




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            Fig.1.12 Curvature refinement level is 3; Small solid features refinement level is 3; Narrow channel
            refinement is on: 5 cells across narrow channels, Narrow channels refinement level is 2; Total number of
            cells is 8457.




            Fig.1.13 Curvature refinement level is 3; Small solid features refinement level is 3; Narrow channel
            refinement is on: 5 cells across narrow channels, Narrow channels refinement level is 7; Total number of
            cells is 33293.




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    Chapter Mesh Introduction


    1.11 Local Mesh Settings
            The local mesh settings option is one more tool to help create an optimal mesh. Use of
            local mesh settings is especially beneficial if you are interested in resolving a particular
            region within a complex model.
            The local mesh settings can be applied to a component, face, edge or vertex. You can
            apply local mesh settings to fluid regions and solid bodies. To apply the local mesh
            settings to a fluid region you need to specify this region as a solid part or subassembly and
            then disable this component in the Component Control dialog box. The local mesh settings
            are applied to the cells intersected with the selected component, face, edge, or a cell
            enclosing the selected vertex. However, cells adjacent to the cell of the local region may
            be also affected due to the refinement rules described in the Mesh Construction Stages
            chapter.




        Fig.1.14 The local mesh settings used: The rhombic channel is refined into 4th level cells, and two narrow
        channels are refined to have 10 cells across each channel.




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       1.12 Recommendations for Creating the Computational Mesh
              1 At the beginning create the mesh using the default (automatic) mesh settings. Start
                with the Level of initial mesh of 3. On this stage it is important to recognize the
                appropriate values of the minimum gap size and minimum wall thickness which will
                provide you with the suitable mesh. The default values of the minimum gap size and
                minimum wall thickness are calculated using information about the overall model
                 dimensions, the Computational Domain size, and area of surfaces where conditions
                 (boundary conditions, sources, etc.) and goals are specified. Don't switch off the
                 Optimize thin walls resolution option, since it allows you to resolve the model's thin
                 walls without the excessive mesh refinement.
              2 Closely analyze the obtained automatic mesh, paying attention to the total numbers of
                 cells, resolution of the regions of interest and narrow channels. If the automatic mesh
                 does not satisfy you and changing of the minimum gap size and minimum wall
                 thickness values do not give the desired effect you can proceed with the custom mesh.
              3 Start to create your custom mesh with the disabled narrow channel refinement, while
                the Small solid features refinement level and the Curvature refinement level are
                 both set to 0. This will produce only zero level cells (basic mesh only). Use control
                 planes to optimize the basic mesh.
              4 Next, adjust the basic mesh by step-by-step increase of the Small solid features
                refinement level and the Curvature refinement level. Then, enable the narrow
                 channels refinement.
              5 Finally, try to use the local mesh settings.


       2    Mesh-associated Tools

       2.1 Introduction
              Since the mesh settings tool is an indirect way of constructing the computational mesh, to
              better visualize the resulting mesh various post-processing tools are offered by
              COSMOSFloWorks. In particular, these tools allow to visualize the mesh in detail before
              the calculation, substantially reducing the CPU and user time.
              The computational mesh constructed by COSMOSFloWorks or other CFD codes cannot
              resolve the model geometry at the mesh cell level exactly. A discrepancy can lead to
              prediction errors. To facilitate an analysis of these errors and/or to avoid their appearance,
              COSMOSFloWorks offers various options for visualizing the real computational
              geometry corresponding to the computational mesh used in the analysis.




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    Chapter Mesh-associated Tools


            Since the numerical solution is obtained inevitably in the discrete form, i.e., in the centers
            of computational mesh cells, it is interpolated and extrapolated by the post-processor to
            present the results in a smooth form, which is typically more convenient to the user. As a
            result, some prediction errors can stem from these interpolations and extrapolations. To
            facilitate an analysis of such errors and/or to prevent their appearance,
            COSMOSFloWorks offers an option to visualize the physical parameters’ values
            calculated at the centers of computational mesh cells, so that when presenting results by
            coloring an area with a palette, the results are considered constant within each cell.

    2.2 Visualizing the Basic Mesh Before Constructing the Initial Mesh
            Using this option the user can inspect the Basic mesh and its Control planes corresponding
            to the mesh settings, which can be made manually or retained by default. The plot appears
            as soon as these settings have been made or changed, so you immediately see the resulting
            mesh. (See Help or User’s Guide defining the Basic mesh and its Control planes).
            To enable this option, select the Show basic mesh option in the FloWorks, Project menu,
            or in the Initial Mesh dialog box. The option is accessible both before and after the
            calculation.
            Using this option, you may shifting the Control planes to desired positions to assure that
            certain features of the model geometry are captured by the computational mesh.




                  Fig.2.1 The Basic mesh (left) and the Initial mesh (right).


    2.3 Enhanced Capabilities of the Results Loading
            COSMOSFloWorks allows to view not only the calculation results and the current
            computational mesh on which they have been obtained, but also the initial (i.e., on which
            the calculation begins) computational mesh separately. The latter can be viewed either
            before or after the calculation, allowing the user to compare the initial and current (i.e.,
            refined during the calculation) computational meshes.




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              To view various meshes, you must open the corresponding file via the Load results dialog
              box. The calculation results, including the current computational mesh, are saved in the
              .fld files, whereas the initial computational mesh is saved separately in the .cpt file. All
              these files are saved in the project folder, whose name (a numeric string) is formed by
              COSMOSFloWorks and must not be changed. The .cpt files and the final (i.e., with the
              solution obtained at the last iteration) .fld files have the name similar to that of the project
              folder, whereas the solutions obtained during the calculation at the previous iterations
              (corresponding to certain physical time moments, if the problem is time-dependent) are
              saved in the .fld files with names “r_<iteration number>”, e.g. the project initial data are
              saved in the r_000000.fld file.

                          Do not try to load the calculation results obtained in another project
                          with a different geometry; the effect is unpredictable.




                                      Fig.2.2 The Load Results dialog box.


       2.4 Viewing the Initial Computational Mesh Saved in the .cpt Files
              To optimize the process of solving an engineering problem and to save time, in some cases
              it may be useful to view the initial computational mesh before performing the calculation,
              particularly to be sure that the model features are resolved well by this mesh. To view the
              initial computational mesh after loading the .cpt file, COSMOSFloWorks offers you Cut
              Plots, Surface Plots , and the Mesh option (see below), which are also used for viewing
              the calculation results.

       2.5 Viewing the Computational Mesh Cells with the Mesh Option
              To view fluid cells of the computational mesh cells (i.e. the cells lying fully in the fluid),
              solid cells (lying fully in the solid), and partial cells lying partly in the fluid and partly in
              the solid, COSMOSFloWorks offers you the Mesh option.
              Different colors can be used to better differentiate between the computational mesh cells
              of each of the above-mentioned types. To see the cells in a certain parallelepiped region,
              the user must specify the coordinates of the region boundaries in the Global Coordinate
              System.

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    Chapter Mesh-associated Tools



                           Visualization of a large amount of computational mesh cells (e.g. all
                           fluid cells in the whole computational domain) may be impractical,
                           since it could require substantial time and memory, and even then
                           you might not be able to see all the visualized cells because the
                           majority of them will likely be screened from view by other cells.
            Using the Mesh option, you can also save the information concerning the mesh cells,
            including the physical parameters values obtained in their centers, in ASCII or Excel files.

    2.6 Visualizing the Real Computational Geometry
            Since the SolidWorks model geometry, especially its high-curvature parts, cannot be
            resolved exactly at the cell level by the rectangular (parallelepiped) computational mesh,
            the real computational geometry corresponding to the computational mesh used in the
            analysis can be viewed after the calculation to avoid or estimate the prediction errors
            stemming from this discrepancy. If no solution-adaptive meshing occurs during the
            calculation, the real computational geometry can be viewed just after the mesh generation.
            This option is employed by clearing the Use CAD geometry check box in Cut Plots, 3D
            Plots, Surface Plots, Flow Trajectories, Point Parameters and XY Plots. The result is
            especially clear when colored Contours are used to visualize a physical parameter values
            (see Fig.2.3).




           Fig.2.3 Surface Plots on a SolidWorks model inner surface (left) and on its computational
           realization (right).

            This capability is especially useful for revealing the model surface regions which are
            inadequately resolved by the computational mesh. Let us consider Fig.2.4 with the
            temperature Cut Plots as an example. The white detail in the bottom part is an insulator, so
            the heat transfer within it is not considered. It contains a small closed cavity which is
            omitted in the analysis and therefore is invisible on the cut plot, too. However, the mesh
            resolution of the triple border between the insulator, the cavity, and the heat-conducting
            solid body leads to the formation of cog-shaped artifacts.




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           Fig.2.4 Cut plots using CAD geometry (left) and meshed geometry (right).

               On the other hand, this option may be useful when creating Surface Plots for SolidWorks
               models containing rippled surfaces whose ripples have not been resolved by the
               computational mesh and are not essential from the problem solution viewpoint, since the
               coloring of the simplified solid/fluid interface instead of coloring the actual SolidWorks
               model faces allows you to substantially reduce the CPU time and memory requirements.

                            If the computational mesh has resolved the SolidWorks model well,
                            so the obtained computational results are adequate, then enable the
                            Use CAD geometry option before performing the final Cut Plots and
                            Surface Plots to obtain smooth pictures which are more convenient
                            for the analysis.
                            When creating a Surface Plot with the Use CAD geometry option
                            switched off, only the solid/fluid interfaces of partial cells within the
                            computational mesh are colored. When a Surface Plot is created in
                            the Use all faces mode, solid/fluid interfaces of all partial cells are
                            colored. However, when a Surface Plot is created on a selected
                            surface, the solid/fluid interfaces are colored only in the partial cells
                            intersected by the SolidWorks model surface approximated by
                            triangles inside SolidWorks, which may differ from the
                            mesh-approximated surface of the model. As a result, there may exist
                            some partial cells which are not intersected by the triangulated
                            surface, and therefore their solid/fluid interfaces would not be
                            colored (see gray strips in Fig.2.5). Naturally, this circumstance
                            concerns the picture only and does not affect the calculation results.




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    Chapter Mesh-associated Tools




                                                  Fig.2.5


    2.7 Switching off the Interpolation and Extrapolation of Calculation Results
            Since the numerical solution is obtained inevitably in the discrete form, i.e., in the form of
            values in the centers of the computational mesh cells in COSMOSFloWorks, it is
            interpolated and extrapolated by the post-processor to present the results in a smooth form,
            which is typically more convenient to the user. As a result, prediction errors can stem from
            and/or be hidden by such interpolation and extrapolation that smoothens the calculation
            results. To facilitate the revealing, analysis, and elimination of such errors,
            COSMOSFloWorks offers an option to visualize the physical parameter values ’as is’, i.e.
            without interpolation, when presenting calculation results in Cut Plots and Surface Plots
            (other result features, namely, isolines, isosurfaces, flow streamlines and particle
            trajectories can not be built at all without interpolation), so when coloring a surface with a
            palette, the results are considered constant within the mesh cells (see Fig.2.6).

                        Since the mesh cells’ centers used in coloring the surface can lie at
                        different distances from the surface, this can introduce an additional
                        variegation into the picture, if the value of the displayed parameter
                        depends noticeably on this distance (see Fig.2.6).




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             Fig.2.6 The fluid velocity Surface Plots created with (left) and without (right) interpolating and
             extrapolating the calculation results.




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    Chapter Mesh-associated Tools


    2.8 Conclusion
            The presented mesh-associated tools of COSMOSFloWorks are additional tools for
            obtaining reliable and accurate results with this code. These tools are summarized in the
            table:


                                             Application

                                    Basic   Initial    After the
                  Option            mesh    mesh      calculation                Reason

            Visualizing the          +        +            +          To inspect the Basic mesh and
            Basic mesh                                                setting its Control planes

            Widened                           +            +          To view the Initial mesh and
            capabilities of                                           the calculation results
            loading the results

            Viewing the Initial               +            +          To analyze the Initial mesh
            mesh

            Viewing mesh                      +            +          To view mesh cells and save
            cells of different                                        the respective physical
            type                                                      parameters values

            Visualizing the                   +            +          For analysis of inadequate
            computational                                             results and quick
            geometry                                                  post-processing of the results
                                                                      of complicated models

            Switching off the                              +          For analysis of inadequate
            interpolation of                                          results
            results




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       3    Calculation Control Options

       3.1 Introduction
              The Calculation Control Options dialog box introduced into COSMOSFloWorks allows
              you to control:
                 • conditions of finishing the calculation,
                 • saving of the results during the calculation,
                 • refinement of the computational mesh during the calculation,
                 • freezing the flow calculation,
                 • time step for a time-dependent analysis,
                 • number of rays traced from the surface if radiating heat transfer is enabled.
              This dialog box is accessible both before the calculation and during the calculation. In the
              last case the new-made settings are applied to the current calculation starting from the next
              iteration.
              The main information on employing the options of Finishing the calculation and
              Refining the computational mesh during calculation is presented in this document.

       3.2 Finishing the Calculation
              COSMOSFloWorks solves the time-dependent set of equations for all problems, including
              steady-state cases. For such cases it is necessary to recognize the moment when a
              steady-state solution is attained and therefore the calculation should be finished. A set of
              independent finishing conditions offered by COSMOSFloWorks allow the user to select
              the most appropriate conditions and criteria on when to stop the calculation. The following
              finishing conditions are offered by COSMOSFloWorks:
                 • maximum number of refinements;
                 • maximum number of iterations;
                 • maximum physical time (for time-dependent problems only);
                 • maximum CPU time;
                 • maximum number of travels;
                 • convergence of the Goals.

                          Travel is the number of iterations required for the propagation of a
                          disturbance over the whole computational domain. Current number
                          of iterations per one travel is presented in the Info box of the
                          Calculation monitor.
              In COSMOSFloWorks you can select the finishing conditions that are most appropriate
              from your viewpoint to solve the problem under consideration, and specify their values.
              For the latter two conditions (i.e., for the maximum number of travels and the Goals

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    Chapter Calculation Control Options


            convergence settings) COSMOSFloWorks presents their default values (details are
            described below). You can also select the superposition mode for multiple finishing
            conditions in the Finish Conditions value cell: either to finish the calculation when all the
            selected finishing conditions are satisfied or when at least one of them is satisfied.
            In any case, information on the finishing conditions due to which the calculation has
            finished is shown in the Monitor’s Log box.
            The Goals convergence finishing condition is complex since it consists of satisfying all the
            specified Goals criteria. A specified Goal criterion includes a specified dispersion, which
            is the difference between the maximum and minimum values of the Goal, and a specified
            analysis interval over which this difference (i.e., the dispersion) is determined. The
            interval is taken from the last iteration rearwards and is the same for all specified Goals.
            The analysis interval is applied after an automatically specified initial calculation period
            (in travels), and, if refinement of the computational mesh during calculation is enabled,
            after an automatically or manually specified relaxation period (in travels or in iterations)
            since the last mesh refinement is reached. As soon as the Goal dispersion obtained in the
            calculation becomes lower than the specified dispersion, the Goal is considered
            converged. As soon as all Goals included in the Goals convergence finishing condition (by
            selecting them in the On/Off column) have converged, this condition is considered
            satisfied. The Goals not included into the Goals convergence finishing condition are used
            for information only, i.e., with no influence on the calculation finishing conditions.
            Let us consider the COSMOSFloWorks default values for the maximum number of travels
            and the Goals convergence settings in detail. These default (recommended by
            COSMOSFloWorks) values depend on the Result resolution level either specified in the
            Wizard or changed by pressing the Reset button in the Calculation Control Options dialog
            box. For higher Result resolution levels the finishing conditions are tighter.
            The default maximum number of travels depends on
                • the type of the specified Goal (i.e., dynamic or diffusive, see below);
                • the specified Result resolution level;
                • the problem's type (i.e., incompressible liquid or compressible gas, low or high
                  Mach number gas flow, time-dependent or steady-state).

                        The Dynamic goals are: Static Pressure, Dynamic Pressure, Total
                        Pressure, Mass Flow Rate, Forces, Volume Flow Rate, and Velocity.
                        The Diffusive goals are: Temperature, Density, Mass in Volume,
                        Heat flux, Heat transfer rate, Concentrations, Mass Flow Rate of
                        species, and Volume Flow Rate of species.




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              The default Goals convergence settings are the default analysis interval, which is shown
              in the Finish tab of the Calculation Control Options dialog box, and the default Goals
              criterion dispersion values, which are not shown in the Calculation Control Options dialog
              box, but, instead, are shown in the Monitor’s Goal Table or Goal Plot table (in the Criteria
              column), since they depend on the values of the Goal physical parameter calculated in the
              computational domain, and therefore are not known before the calculation and, moreover,
              can change during it. In contrast, the Goals criterion dispersion values specified manually
              do not change during the calculation.
              As for the automatically specified initial calculation period (measured in travels), it
              depends on the problem type, the Goal type, and the specified Result resolution level.
                            • the manually specified analysis interval for the Goals
                              convergence finishing criteria must be substantially longer than
                              the typical period of the flow field oscillation (if it occurs);
                            • the Goals determined on solid/fluid interfaces or model
                              openings, as well as the Post-processor Surface Parameters, yield
                              the most accurate and correct numerical information on flow or
                              solid parameters, especially integral ones;
                            • Global Goals yield the most reliable information on flow or solid
                              parameters, although they may be too general;
                            • the CPU time depends slightly on the number of the specified
                              Goals, but, in some cases, vary substantially in the case of
                              presence of a Surface Goal;
                            • Surface and Volume Goals provide exactly the same information
                              that may be obtained via the Surface and Volume Parameters
                              Post-processor features, respectively.

       3.3 Refinement of the Computational Mesh During Calculation
              Refinement of the computational mesh during calculation is a process of splitting or
              merging of the computational mesh cells in high-gradient flow areas. This option has the
              following governing parameters:
                 • refinement level,
                 • splitting/merging criteria (also named refinement/unrefinement criteria,
                   respectively),
                 • permission to refine cells in fluid and/or solid regions,
                 • approximate maximum number of cells,
                 • strategy of refinements during the calculation.
              The first four parameters are described in COSMOSFloWorks Help and User’s Guide.
              Here, let us consider the Refinement Strategy in detail. The following three strategies are
              available:
                 • Periodic refinement;


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    Chapter Calculation Control Options


                • Tabular Refinement;
                • Manual Only refinement.
            In the first two strategies the refinement moment is known beforehand. The solution
            gradients are analyzed over iterations belonging to the Relaxation interval, which is
            calculated from the current moment rearwards. As a result, only steady-state gradients are
            taken into account. The default length of the Relaxation interval can be adjusted manually.
            On the other hand, the analysis must not continue with the same relaxation interval
            defined from the start of the calculation, in order to avoid taking into account the initial
            highly unsteady period. Therefore, a period of at least two relaxation intervals is
            recommended before the first refinement. If the first assigned refinement is scheduled in a
            shorter term from the beginning, the period over which the gradients are analyzed is
            shortened accordingly, so that in the extreme case it can be as short as one current
            iteration. If you initiate a refinement manually within this period, the gradients are
            analyzed in one current iteration only. Naturally, such a short period give not very reliable
            gradients and hence may result in an inadequate solution or excessive CPU time and
            memory requirements.
            The figure below illustrates these concepts. Here, the letter r denotes the Relaxation
            interval. This figure involves both Periodic and Tabular refinements. Case 1 is the
            recommended normal approach. In the Case 2 the first refinement is too close to the
            starting point of the calculation, so the gradients are analyzed over the shorter interval
            (which could even be reduced to only one current iteration in the extreme case). Case 3 is
            a particular case when a refinement is initiated manually just before a previously assigned
            refinement. As a result, the manual refinement is well-defined, since the gradients have




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              been analyzed over almost the entire relaxation interval, but on the other hand, the
              previously assigned refinement is performed on the substantially shorter interval, and
              therefore its action can be incorrect. Case 3 demonstrates the possible error of performing
              manual and previously assigned refinements concurrently.

                                                     Collecting of the statistics is prohibited
                                                     Statistics are collected
                                                     Refinement

                                                        Case 1
                    r                           r


               0                                    Ref. point 1                 Ref. point 2

                        Case 2                                             Case 3

                   r        r                                          r        r


               0                 Ref. point 1                      Manual ref. Auto ref.
                                     Fig.3.1 Refinement strategy.




       4    Flow Freezing

       4.1 What is Flow Freezing?
              Sometimes it is necessary to solve a problem that deals with different processes
              developing at substantially different rates. If the difference in rates is substantial (10 times
              or higher) then the CPU time required to solve the problem is governed almost exclusively
              by the slower process. To reduce the CPU time, a reasonable approach is to stop the
              calculation of the fastest process (which is fully developed by that time and does not
              change further) and use its results to continue the calculation of the slower processes. Such
              an approach is called “freezing”.
              In the case of problems solved with COSMOSFloWorks the processes of convective mass,
              momentum, and energy transport are the fastest processes to develop and to converge,
              whereas the processes of mass, momentum, and energy transfer by diffusion are the
              slowest ones. Accordingly, COSMOSFloWorks offers the “Flow Freezing” option that
              allow you to freeze, or fix, the pressure and velocity field while continuing the calculation
              of temperature and composition. This option is especially useful in solving steady-state


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    Chapter Flow Freezing


            problems involving diffusion processes that are important from the user’s viewpoint, e.g.
            species or heat propagation in dead zones of the flow. Time-dependent analyses with
            nearly steady-state velocity fields and diffusion processes developing with time are also
            examples of this class of problems. As a result, the CPU time for solving such problems
            can be substantially reduced by applying the Flow Freezing option.
            COSMOSFloWorks treats Flow Freezing for the High Mach number flows differently. All
            flow parameters are frozen, but the temperature of the solid is calculated using these fixed
            parameters at the outer of the boundary layer and user defined time step. Temperature
            change on the solid's surface and relevant variation of the heat flows are accounted in the
            boundary layer only. It is impossible (and makes no sense) to use Flow Freezing for
            calculation of concentration propagation in the High Mach number flow. If custom time
            step is not specified, the steady-state temperature of solid will be reached in one time step
            assumed to be infinite.

    4.2 How It Works
            To access the Flow Freezing option, open the Calculation Control Options dialog box,
            then the Advanced tab. This option has three modes: Disabled (by default), Periodic, and
            Permanent.

        Flow Freezing in a Permanent Mode
            As an example of applying the Flow Freezing option, let us consider a plane flow (2D)
            problem of heating the vortex core in a vessel (Fig.4.1).




                                Fig.4.1 Heating the vortex core in a vessel.

            At the beginning the entire fluid region is filled with a cold (T=300 K) liquid. A hot
            (T=400K) liquid enters the vessel through the lower channel (the upper channel is the
            exit). As a result, a vortex with a cold core is developed in the vessel. The vortex core
            temperature is changed mainly due to heat diffusion. To measure it, a small body is placed
            at the vortex center and disabled in the Component Control dialog box, so that it is treated
            by COSMOSFloWorks as a fluid region. Its minimum temperature (i.e., the minimum
            fluid temperature in this region) is the Volume Goal of the calculation.




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              First of all, let us consider Flow Freezing operating in the Permanent mode. The only
              user-specified parameter in Permanent mode is the starting moment of enabling the Flow
              Freezing option. Until this moment the calculation runs in a usual manner. After this
              moment the fluid velocity field becomes frozen, i.e., it is no longer calculated, but is taken
              from the last iteration performed just before the Flow Freezing Start moment. For the
              remainder of the run only the equations’ terms concerning heat conduction and diffusion
              are calculated. As a result, the CPU time required per iteration is reduced.
              The starting moment of the Flow Freezing option should be set not too early in order to let
              the flow field to fully develop. As a rule, an initial period of not less than 0.25 travels is
              required to satisfy this condition. In most problems the 0.5 travel initial period is
              sufficient, but there are problems that require a longer initial period.

                          The Flow Freezing Start moment, as well as other parameters of the
                          Calculation Control Options dialog box can be changed during a
                          calculation.

                          As soon as the Flow Freezing option is invoked, only the slowest
                          processes are calculated. As a result, the convergence and finishing
                          criteria can become non-optimal. Therefore, to avoid obtaining
                          incorrect results when enabling the Flow Freezing option, it is
                          recommended to increase the maximum number of travels specified
                          at the Finish tab of the Calculation Control Options dialog box by
                          1.5…5 times compared to the number that was set automatically or
                          required for the calculation performed without the Flow Freezing
                          option.
              When first solving the problem under consideration we set the maximum number of
              travels to 10. The calculation performed without applying the Flow Freezing option then
              required about 10 travels (the CPU time of 13 min. 20 s on a 500 MHz PIII computer) to
              reach the convergence of the project Goal (the steady-state minimum fluid temperature in
              the vortex core). However, the steady-state fluid velocity field was reached in about 0.5
              travels, i.e., substantially earlier. So, by applying the Flow Freezing option in the
              Permanent mode (just after 0.5 travels) the same calculation only required a CPU time of 7
              min. 25 s on the same computer to reach the convergence of the project Goal.
              Convergence histories of the both Goal are plotted in Fig.4.2.
              If it is necessary to perform several calculations with the same fluid velocity field, but
              different temperatures and/or species concentrations, it is expedient to first calculate this
              fluid velocity field without applying the Flow Freezing option. Then, clone the
              COSMOSFloWorks project into several projects (including copying the calculation
              results), make the required changes to these projects, and perform the remaining
              calculations for these projects using the calculated results as initial conditions and
              applying the Flow Freezing option in the Permanent mode with a zero Start period.




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    Chapter Flow Freezing




          Fig.4.2 The convergence history of the Goal (which is the minimum fluid temperature in the vortex
          core) with and without applying the Flow Freezing option.

                         If you forget to use the calculated results as initial conditions, then
                         the saved fluid velocity field will be lost in the cloned project, so the
                         project must be created again. To use the calculated results as initial
                         conditions for the current project, select the Transferred type of
                         Parameter definition for the initial conditions in the General Settings
                         dialog box.

        Flow Freezing in a Periodic Mode
            In some problems the flow field depends on temperature (or species concentrations), so
            both the velocity and the temperature (concentrations) change simultaneously throughout
            the calculation. Nevertheless, since they change in a different manner, i.e., the velocity
            field changes faster than the temperature (concentrations) field, therefore approaching its
            steady state solution earlier, the Flow Freezing option can be used in a Periodic mode to
            reduce the CPU time required for solving such problems. The Periodic mode of the Flow
            Freezing option consists of calculating the velocity field not in each of the iterations (time
            steps), but periodically for a number of iterations specified in No freezing (iterations)
            after a period of freezing specified in the Freezing (iterations) (see Fig.4.3) The
            temperatures and concentrations are calculated in each iteration. Examples include
            channel flows with specified mass flow rates and pressures, so the fluid density and,
            therefore, velocity depend on the fluid temperature, or flows involving free convection,
            where due to the buoyancy the hot fluid rises, so the velocity field depends on the fluid
            temperature.




                                                       Fig.4.3


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              As an example, let us consider a 3D external problem of an air jet outflow from a body
              face into still air (see Fig.4.4, in which the jet outflow face is marked by a red line). Here,
              the wire frame is the computational domain. The other body seen in this figure is
              introduced and disabled in the Component Control dialog box (so it is a fluid region) in
              order to see the air temperature averaged over its face (the project Goal), depending on the
              air temperature specified at the jet outflow face.




                    Fig.4.4 Air jet outflow from a body face into a still air.

              This problem is solved in several stages. At the first stage, the calculation is performed for
              the cold (T = 300 K, which is equal to the environment temperature) air jet. This
              calculation requires a CPU time of 12 min. 14 s. Then we clone the project including
              copying the results. Next, we set the outlet air temperature to T = 400 K, specify the
              Periodic mode of the Flow Freezing option by its Start moment of 0.25 travels (in order
              for the heat to have time to propagate along the jet to the measuring face) and under
              Duration specify 10 as both the Freezing (iterations) and No freezing (iterations) values.
              Then perform the calculation on the same computational mesh with the Take previous
              results option in the Run box. As you can notice, the calculation with flow freezing takes
              less CPU time than the similar calculation without the Flow Freezing option enabled.

       5    Cavitation

       5.1 Physical model
              Cavitation is a common problem for many engineering devices dealing with liquid flows.
              The deleterious effects of cavitation include: lowered performance, load asymmetry,
              erosion and pitting of blade surfaces, vibration and noise, and reduction of the overall
              machine life. Cavitation models used today range from rather crude approximations to
              sophisticated bubble dynamics models. Details about bubble generation, growth and
              collapse are important for the prediction of a solid surfaces erosion, but are not necessary
              to estimate the performance of a pump, valve or other equipment. In COSMOSFloWorks
              an engineering model of cavitation is employed to predict the extent of cavitation and its
              influence on the performance of the analyzed device.



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    Chapter Cavitation


        Limitations and Assumptions
            The model has the following limitations and/or assumptions:
               • Cavitation is currently available only for incompressible water (when defining the
                 project fluids you should select Water from the list of Pre-Defined liquids);
                 cavitation in mixtures of different liquids cannot be calculated.
               • The properties of the dissolved non-condensable gas are set to be equal to those of
                 air.
               • Thermodynamic parameters in the phase transition areas should be contained within
                 the following bounds:
                  277.15 < T < 583.15 K, 800 < P < 107 Pa.
               • The model does not describe the detailed structure of the cavitation area, i.e
                 parameters of individual vapor bubbles are not considered.
               • The parameters of the flow at the inlet boundary conditions must be such that the
                 volume fraction of liquid water in the inlet flow would be at least 0.1.

    5.2 Interface
            Cavitation option in COSMOSFloWorks is
            switched on by checking the Cavitation
            check box either in Wizard or in the
            General Settings window. Since
            COSMOSFloWorks may consider
            cavitation only in incompressible water, the
            selection of any fluid type other than liquid
            or of any fluid other than water renders this
            check box unavailable.




            Cavitation option for a fluid subdomain is
            switched on in a manner similar to that for the whole project.
            Once enabled, the Cavitation option requires you to specify the
            Dissolved gas mass fraction. The default value of this parameter is
            0.00001. This value is typical for air dissolved in water under normal
            conditions and therefore is appropriate for most cases. If needed, you
            can specify a different value of the Dissolved gas mass fraction in the
            range of 10-4...10-8.




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              Cavitation is represented in the calculation results via the following
              parameters: Water Mass Fraction, Water Volume Fraction, Vapour
              Mass Fraction, and Vapour Volume Fraction, which describe the local
              fraction of the fluid components (water and vapour) by mass and by
              volume. Note that you may need to check some of those parameters in
              the Parameter list to enable their selection in the View Settings
              window.




       5.3 Examples of use

           Rotating impeller
              Water flows through a rotating impeller with five blades of curved shape, as shown on the
              picture. The aim of simulation is to predict the impeller characteristics.
              Due to the pressure drop on suction side of the impeller blades, a cavitation may develop
              in these areas, which cannot but affect the impeller performance.
              The appearance of calculated cavitation area in the form of isosurfaces is shown below on
              Figure 2.1.




                               Fig. 2.1. Isosurfaces for vapour volume fraction of 10%.




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    Chapter Cavitation


        Hydrofoil in a tunnel
            A symmetric hydrofoil is placed in a sufficiently wide water-filled tunnel with a non-zero
            angle of attack. Obviously, water flow develops some pressure drop on the leeward side of
            the hydrofoil, which at certain conditions can lead to cavitation.
            Figure 2.2 contains a representation of the
            calculated cavitation area visualized in terms of
            vapour volume fraction.




        Ball valve
            Water flows inside an about half-opened ball valve Fig. 2.2. Calculated cavitation area.
            (see Figure 2.3) at the relatively low pressure and
            high velocity producing cavitation.




                                     Fig. 2.3. Model of the ball valve.

            The results visualized in the form of Cut plot with Vapour volume fraction as displayed
            parameter are presented on Figure 2.4. It is clearly seen that sudden expansion of the flow
            produce an area of strong cavitation.




                          Fig. 2.4. Distribution of the vapour volume fraction.



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       5.4 Recommendations
                 • If your analyze a flow of water in some points of which the local static pressure may
                   fall below the liquid's vapour pressure at the local temperature causing cavitation or
                   if a localized boiling of water may occur in the water flow due to intense heating, it
                   is recommended to select the Cavitation option in the Fluids dialog box of the
                   Wizard or General Settings.
                 • Cavitation area growths slowly during calculation and there is a risk that the
                   calculation will stop before the cavitation area develops completely. To avoid this,
                   always specify Global Goal of Average Density and increase the Analysis interval
                   on the Finish tab of the Calculation Control Options dialog box up to 2.5 travels.
                   Also make sure that the other finish conditions do not cause the calculation to stop
                   before goals are converged. The easiest way to ensure that is to select If all are
                   satisfied in the Value cell for the Finish conditions on the Finish tab of the
                   Calculation Control Options dialog box.
                 • The Cavitation option should not be selected if you calculate a water flow in the
                   model without flow openings (inlet and outlet).
                 • The fluid region where cavitation occurs should be well resolved by the
                   computational mesh.
                 • To see the cavitation areas, you may select, for example, Vapour Volume Fraction
                   or Density (the latter one is probably the best choice) as the parameter for
                   visualization.

       6    Steam

       6.1 Physical model
              COSMOSFloWorks allows you to consider water steam among the project fluids. Like
              Humidity, the Steam option may be used to analyze engineering problems concerning
              water vapour and its volume condensation, along with the corresponding changes in the
              physical properties of the project fluid. Steam option in COSMOSFloWorks describes the
              behavior of pure water steam (for which, say, a description as "air with 100% humidity at
              100°C", although nominally correct, would sound a bit weird) or its mixtures with other
              gases.

           Limitations and Assumptions
              The model has the following limitations and/or assumptions:
                 • COSMOSFloWorks project may include pure Steam or its mixture with Gases (but
                   not with Real gases).
                 • Thermodynamic parameters of steam should be contained within the following
                   bounds:
                    283 < T < 610 K, P < 107 Pa.
                 • The volume fraction of condensed water should never exceed 5%.
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    Chapter Steam


              • Steam option is incompatible with the High Mach number flow option, i.e. the two
                can not be employed simultaneously.
              • The employed model of condensation is fully equilibrium and considers only
                volume condensation.

    6.2 Interface
           Steam is treated by COSMOSFloWorks as
           a special kind of fluid and may be selected
           from the Engineering Database just like
           any other fluid.
           Steam may be assigned for a fluid
           subdomain as well as for the whole project.




           Steam may be mixed with any regular
           Gases (but not with Real gases). In this case, its concentration in a form
           of mass or volume fraction must be specified in Initial conditions, as
           well as in all boundary conditions.


           Steam content in the mixtures of water steam with other gases is
           represented in the calculation results via Steam Mass Fraction, Steam
           Volume Fraction (that represent mass and volume fractions of water,
           respectively) and Relative Humidity (which is the ratio of the local
           partial density of water to the density of saturated water vapor under
           current conditions). The content of particular form of water, i.e. vapor or
           liquid, is represented via Condensate Mass Fraction (that represents
           mass fraction of condensed steam in the fluid) and Moisture Content
           (that represents the fraction of condensed steam with respect to the
           overall content of steam). Note that you may need to check some of
           those parameters in the Parameter list to enable their selection in the
           View Settings window.




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       6.3 Example of use

           Heat exchanger
              COSMOSFloWorks calculates the equilibrium condensation in water steam as steam
              flows through a cooled tube of a heat exchanger. Figure 2.5 shows cut plot of the
              condensate mass fraction parameter.




                          Fig. 2.5. Cut plot showing the condensate mass fraction.


       6.4 Recommendations
                 • To avoid the risk of finishing the calculation before the condensation develops
                   completely, always specify some goal strongly dependent on condensation, for
                   example Global Goal of Average Density, and make sure that the calculation will
                   not stop before this goal is converged.
                 • To see the condensation areas, you may use Relative Humidity or the Condensate
                   Mass Fraction as the parameter for visualization.


       7    Humidity

       7.1 Physical model
              COSMOSFloWorks allows you to consider the relative humidity of the gas or mixture of
              gases. This allows you to analyze engineering problems where the condensation of water
              vapor contained in the air (or other gas), or, more generally speaking, where any
              differences in physical properties of wet and dry air play an important role. Examples may
              include air conditioning systems (especially in wet climate or in the places where relative
              humidity is very important, e.g. libraries, art museums, etc.), tank steamers, steam turbines
              and other kinds of industrial equipment. COSMOSFloWorks can calculate equilibrium
              volume (but not surface) condensation of steam into water. As a result, the local fractions
              of gaseous and condensed steam are determined. In addition, the corresponding changes
              of the fluid temperature, density, enthalpy, specific heat, and sonic velocity are determined
              and taken into account.




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    Chapter Humidity


        Limitations and Assumptions
            The model has the following limitations and/or assumptions:
               • Humidity is currently available only in Gases (both in individual gases and in
                 mixtures), but not in Real gases.
               • Thermodynamic parameters in the fluid areas where humidity is considered should
                 be contained within the following bounds:
                  283 < T < 610 K, P < 10 7 Pa.
               • The volume fraction of condensed water should never exceed 5%.
               • Humidity option is incompatible with the High Mach number flow option, i.e. the
                 two can not be employed simultaneously.
               • The model does not describe the condensation process in as subtle detail as the
                 parameters of individual liquid droplets.
               • Surface condensation, i.e. the formation of dew on solid surfaces, is not considered.
               • The condensed steam has no history, since the employed condensation model is
                 fully equilibrium. In other words, the state of condensed steam at given point is
                 governed solely by the local conditions at this point.

    7.2 Interface
            Humidity option in COSMOSFloWorks is
            switched on by checking the Humidity
            check box either in Wizard or in the
            General Settings window. This check box
            is present only if the current fluid type is
            set to Gases.




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              Once Humidity is switched on, the relative
              humidity of the gas becomes available to
              specify in the Initial conditions window.
              The relative humidity is defined as the
              ratio of the current water vapor density to
              that of saturated water vapor under current
              conditions.




              Humidity can be assigned for a fluid
              subdomain as well as for the whole project by selecting the check box
              of the same name, and, once assigned, becomes available to specify in
              the Humidity Parameters group box.

              The relative humidity must be specified within all boundary and initial
              conditions in contact with the fluid region for which the calculation of
              relative humidity is performed.


              Together with the humidity value for boundary and initial
              conditions you must also specify the values of Humidity
              reference pressure and Humidity reference
              temperature that describe the conditions under which the
              relative humidity has been determined, since these values
              may differ from the current pressure and temperature.




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    Chapter Humidity


            Humidity is represented in the calculation results via the following
            parameters: Steam Mass Fraction, Steam Volume Fraction (that
            represent mass and volume fractions of water, respectively) and Relative
            Humidity (which is the ratio of the local partial density of water to the
            density of saturated water vapor under current conditions). The content
            of particular form of water, i.e. vapor or liquid, is represented via
            Condensate Mass Fraction (that represents mass fraction of water
            condensate in the fluid) and Moisture Content (that represents the
            fraction of condensed water with respect to the overall content of water).
            Note that you may need to check some of those parameters in the
            Parameter list to enable their selection in the View Settings window.

    7.3 Example of use

        Aircraft
            An air flow around an aircraft model can be simulated with the Humidity option selected.
            The examination of relative humidity distribution (Figure 2.6) reveals broad areas of more
            than 80% relative humidity from above of both wings. Naturally, these areas (together
            with smaller zones near the cockpit and the tail unit) are enriched with water condensate,
            as it may be seen on Figure 2.7.




        Fig. 2.6. Flow trajectories colored in             Fig. 2.7. Isosurfaces of condensate mass
         accordance with relative humidity.                            fraction = 0.00015


    7.4 Recommendations
               • If your analyze a flow of gas containing some amount of water vapor and the
                 conditions are likely to get over the dew point, it is recommended to consider
                 humidity in the calculation as described in this chapter.



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                 • To avoid the risk of finishing the calculation before the condensation develops
                   completely, always specify some goal strongly dependent on condensation, for
                   example Global Goal of Average Density, and make sure that the calculation will
                   not stop before this goal is converged.
                 • To see the condensation areas, you may use Relative Humidity or the Condensate
                   Mass Fraction as the parameter for visualization.


       8    Real Gases

       8.1 Physical model
              COSMOSFloWorks has an ability to consider real gases. A wide choice of predefined real
              gases is presented. The user may also create user-defined real gases by specifying their
              parameters. This option may be useful in the engineering problems concerning gases at
              nearly-condensation temperatures and/or at nearly-critical and supercritical (that is to say,
              very high) pressures, i.e. at conditions where the behavior of the gas can no longer be
              represented adequately by the ideal-gas state equation.
              The model of real gas implemented in COSMOSFloWorks employs a custom
              modification of the Redlich-Kwong state equation. Naturally, the equation unavoidably
              has certain bounds of applicability, which are explained on the picture below:




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    Chapter Real Gases


            The area of validity of the model includes zones 10, 11 and 12. (Each predefined real gas
            has its own values of Pmin, Pmax, Tmin, and Tmax, and those are also to be specified for a
            user-defined real gas.) If the calculated pressure and/or temperature fall outside of this
            area, COSMOSFloWorks issues a warning. The warning for zones 1 - 8 is: Real gas
            parameters (pressure and/or temperature) are outside the definitional domain of
            substance properties, with comment: P < Pmin, P > P max, T < Tmin, or T > Tmax,
            depending on what has actually happen. The warning for zone 9 is: Phase transition in
            the Real gas may occur.

        Limitations and Assumptions

            The model has the following limitations and/or assumptions:
               • Real gas may be used in a COSMOSFloWorks project as pure fluid or in mixture
                 with Gases (but not with other Real gases).
               • Pressure and temperature of real gas should be contained within certain limits (those
                 are specified individually for each of the predefined real gases).
               • Real gas should not be put under conditions that cause its condensation into liquid.
               • The use of real gas is incompatible with the High Mach number flow option.
               • The precision of calculation of thermodynamic properties at nearly-critical
                 temperatures and supercritical pressures may be lowered to some extent in
                 comparison to other parameter ranges. The calculations involving user-defined real
                 gases at supercritical pressures are not recommended.
               • The copying of pre-defined real gases to user-defined folder is impossible since the
                 employed models are not exactly similar.

    8.2 Interface
            Real gases are a special type of fluids and
            may be selected from the Engineering
            Database along with other fluids.
            Real gas may be assigned for a fluid
            subdomain as well as for the whole project.




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              Real gases may be mixed with regular Gases (though not with each
              other). In this case, substance concentrations in a form of mass or
              volume fractions must be specified in Initial conditions, as well as in all
              boundary conditions.


              To create a user-defined real gas,
              the user must create a new item
              in the corresponding folder in the
              Engineering Database and
              specify the following parameters:




                 • Molar mass;
                 • Critical pressure pc;
                 • Critical temperature Tc;
                 • Critical compressibility factor Zc;
                 • Redlich-Kwong equation type that should be used, i.e. the
                   original one or its modifications by Wilson, Barnes-King, or
                   Soave;
                 • Acentric factor ω (if applicable);
                 • Minimum temperature, i.e. the lower margin of validity of the model;
                 • Maximum temperature, i.e. the corresponding upper margin;
                 • Order of ideal gas heat capacity polynomial, i.e. the order of polynomial function
                   of temperature that defines the "ideal-gas" constituent of the real gas specific heat at
                   constant pressure;
                 • Coefficients of ideal gas heat capacity polynomial, i.e. the coefficients of the
                   aforementioned polynomial;
                 • Polarity (check if the gas in question has polar molecules);
                 • Vapor viscosity dependence on temperature, i.e. the coefficients a and n in the
                   equation describing vapor viscosity as η = a·Tn;
                 • Vapor thermal conductivity
                   dependence on temperature, which
                   includes the coefficients a and n and the
                   choice of dependency type between linear λ = a+n·T and power-law λ = a·Tn
                   forms;




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    Chapter Real Gases


               • Liquid viscosity dependence on temperature, which includes the coefficients a
                 and n and the choice of dependency type between power-law η = a·Tn and
                 exponential η = 10a(1/T-1/n) forms;
               • Liquid thermal conductivity dependence on temperature, which includes the
                 coefficients a and n and the choice of dependency type between linear λ = a+n·T
                 and power-law λ = a·Tn forms;
            The coefficients of the user-specified dependencies for thermophysical properties should
            be entered only in SI unit system, except those for the exponential form of dynamic
            viscosity of the liquid, which should be taken exclusively from Ref. 1.
            Note that the foregoing dependencies for the specific heat and transport properties cover
            only the ’ideal-gas’ constituents of the corresponding properties, i.e. their values at
            low-pressure limit, and the actual formulae contain pressure-dependent corrections which
            are calculated automatically.
            The post-processor display parameters
            concerning real gas includes its mass and
            volume fractions in a mixture (if it is not a sole
            component of the fluid) and the Real Gas
            State. The latter parameter represents the local
            phase state of real gas, which may be Vapor,
            Liquid, Supercritical, or Out of range. Once
            selected, it renders inaccessible the Palette and
            Min/Max settings within the View settings
            window and replaces the Color bar with the
            schematic phase diagram that provides an
            explanation of meaning of particular colors, as
            shown on the picture.




    8.3 Example of use

        Joule-Thomson effect

            A flow of nitrogen through a tube containing narrow restriction is simulated. To reduce
            computation time, the tube was split in halves by a symmetry plane and Symmetry
            condition was applied to the corresponding boundary of the Computational Domain.




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              The calculation within ideal gas approximation, i.e. with nitrogen selected from Gases as
              the project fluid, results in the temperature distribution shown on Figure 2.1. It is clearly
              seen that the temperature of the gas, after undergoing a noticeable drop while passing
              through the hole, later reinstates its initial value. This is an expected behavior of an ideal
              gas, as its enthalpy does not depend on pressure.




                            Fig. 2.8. Field of temperatures for a flow of ideal gas.

              The calculation was repeated with fluid changed to nitrogen selected from Real Gases and
              all other conditions similar. Now the gas temperature at outlet is different from that at inlet
              (see Figure 2.9).




                            Fig. 2.9. Field of temperatures for a flow of real gas.

              Hence we may conclude that the real gas reveals a nonzero Joule-Thomson effect, as
              expected.

       8.4 Recommendations
                 • Minimum temperature for user-defined real gas should be set at least 5...10 K higher
                   than the triple point of the actual substance.
                 • Maximum temperature for user-defined real gas should be set so as to keep away
                   from the area of dissociation of the gas.
                 • The user-specified dependencies for the specific heat and transport properties of the
                   user-defined real gases should be valid in the whole temperature range from Tmin to
                   Tmax (or, as for liquid, in the whole temperature range where the liquid exists).

       8.5 References
              1 R.C. Reid, J.M. Prausnitz, B.E. Poling. The properties of gases and liquids, 4th edition,
                 McGraw-Hill Inc., NY, USA, 1987.



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    Chapter Real Gases




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                                                                                                      3
                                                Meshing – Additional Insight




        Introduction

              COSMOSFloWorks considers the real model designed in SolidWorks and generates a
              rectangular computational mesh automatically distinguishing the fluid and solid domains.
              The corresponding computational domain is generated in the form of a rectangular
              parallelepiped enclosing the model. In the mesh generation process, the computational
              domain is divided into uniform rectangular parallelepiped-shaped cells, which form a
              so-called basic mesh. Then, using information about the model geometry,
              COSMOSFloWorks further constructs the mesh by means of various refinements, i.e.
              splitting of the basic mesh cells into smaller rectangular parallelepiped-shaped cells, thus
              better representing the model and fluid regions. The mesh from which the calculation
              starts, so-called initial mesh, is fully defined by the generated basic mesh and the
              refinement settings.
              Each refinement has its criterion and level. The refinement criterion denotes which cells
              have to be split, and the refinement level denotes the smallest size to which the cells can
              be split. Regardless of the refinement considered, the smallest cell size is always defined
              with respect to the basic mesh cell size so the constructed basic mesh is of great
              importance for the resulting computational mesh.
              The main types of refinements are:
                 Small Solid Features Refinement
                 Curvature Refinement
                 Tolerance Refinement
                 Narrow Channel Refinement
                 Square Difference Refinement
              In addition, the following two types of refinements can be invoked locally:
                 Cell Type Refinement
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    Chapter Initial Mesh Generation Stages


                Solid Boundary Refinement
            During the calculation, the initial mesh can be refined further using the
                Solution-Adaptive Refinement.
            Though it depends on a refinement which criterion or level is available for user control,
            we will consider all of them (except for the Solution-Adaptive Refinement) to give you a
            comprehensive understanding of how the COSMOSFloWorks meshing works.
            In the chapter below the most important conclusions are marked with the blue italic font.
            For abbreviation list refer to the Glossary paragraph.

    1    Initial Mesh Generation Stages

    1.1 Basic Mesh Generation and Resolving the Interface
            1.1.1) Create basic mesh cells whose sizes are governed by the computational domain
            size, the user-specified Control Planes and the number of the basic mesh cells. [Nx, Ny,
            Nz, Control Planes. Parameters which act on each stage are summarized in square
            brackets at the end of the stage.]
            1.1.2) Analyze triangulation in each basic mesh cell at the interfaces between different
            substances (such as solid/fluid, solid/porous, solid/solid and porous/fluid interfaces) in
            order to find the maximum angle between normals to the triangles which compose the
            interface within the cell.
            1.1.3) Depending on the maximum angle found, the decision whether to split the cell or
            not is made in accordance with the specified Small solid features refinement level
            (SSFRL), Narrow channel refinement level (NCRL), Curvature refinement level (CRL)
            and Curvature criterion (CRC), Tolerance refinement level (TRL) and Tolerance
            Refinement Criterion (TRC) (see the Refinements at Interfaces Between Substances
            paragraph). [SSFRL, NCRL, CRL and CRC]

                        If a cell belongs to a local initial mesh area, then the corresponding
                        local refinement levels will be applied (see the Local Mesh Settings
                        paragraph).
            1.1.4) If a basic mesh cell is split, the resulting child cells are analyzed as described in
            1.1.2 and 1.1.3, and split further, if necessary. The cell splitting will proceed until the
            interface resolution satisfies the specified SSFR criterion, CRC and TRC, or the
            corresponding level of splitting reaches its specified value.

                        The specified levels of splitting denote the maximum admissible
                        splitting, i.e. they show to which level a basic mesh cell can be split
                        if it is required for resolving the solid/fluid interface within the cell.
            1.1.5) The operations 1.1.2 to 1.1.4 are applied for the next basic mesh cell and so on,
            taking into account the following Cell Mating rule: two neighboring cells (i.e. cells having
            a common face) can be only cells whose levels are similar or differ by one. This rule has
            the highest priority as it is necessary for simplifying numerical algorithm in solver.
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               The fourth-level red cells appearing
               after resolving the cog cause the
               neighboring cells to be split up to
               third level (yellow cells), that, in
               turn, causes the subsequent
               refinement producing second level
               cells (green cells) and first level
               cells (blue cells). The white zero
               level cell (basic mesh cell) remains
               unsplit since it borders on first level
               cells only, thus satisfying the rule.
                                                                               Fig.1.1
                                                           Fluid cell refinement due to the Cell Mating rule.




                          The Cell Mating rule is strict and has higher priority than the other
                          cell operations. The rule is also enforced for the cells that are entirely
                          in a solid.
              The mesh at this stage is called the primary mesh. The primary mesh implies the complete
              basic mesh with the resolution of the solid/fluid (as well as solid/solid, solid/porous, etc.)
              interface by the small solid features refinements and the curvature refinement also taking
              into account the local mesh settings.

       1.2 Narrow Channel Refinement
              After the primary mesh has been created, the narrow channel refinement is put in action.
              The Narrow Channels term is conventional and used for the definition of the model flow
              passages which are ’narrow’ in the direction normal to the solid/fluid interface.
              Regardless of the real solid curvature, the mesh approximation is that the solid boundary is
              always represented by a set of flat elements, whose nodes are the points where the model
              intersects with the cell edges. Thus, whatever the model geometry, there is always a flat
              element within a partial cell and the normal to this element denotes the direction normal to
              the solid/fluid interface for this partial cell. See the Irregular Cells paragraph for details.
              The narrow channel refinement operates as follows:
              1.2.1) For each partial cell COSMOSFloWorks calculates the “local” narrow channel
              width as the distance between this partial cell and the next partial cell found on the line
              normal to the solid/fluid interface of this cell (i.e. normal to the flat surface element
              located in the cell).

                          If the line normal to the solid/fluid interface crosses a local initial
                          mesh area, then the corresponding local narrow channel refinement
                          settings is applied to the cells in this direction.


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    Chapter Initial Mesh Generation Stages


            1.2.2) If the distance value falls within the range defined by the Minimum height of
            narrow channel (NCHmin) and Maximum height of narrow channel (NCHmax) options,
            the number of cells per this interval is calculated including both partial cells and taking
            into account which portion of each partial cell is in fluid. [NCHmin, NCHmax]
            1.2.3) More precisely, the number of cells across the channel (i.e. on the interval between
            the two partial cells) is calculated as N = Nf + np1 + np2 , where Nf is the number of fluid
            cells on the interval, and np1 and np2 are the fluid portions of the both partial cells. This
            value is compared with the specified Characteristic number of cells across a narrow
            channel (CNC). If N is less than the specified CNC then the cells on this interval are split.
            For example, on Fig.1.2 Nf = 2, np1 = n p2 = 0.4, and N = 2+0.4+0.4 = 2.8 which is less than
            the criterion. On Fig.1.3 the partial cells are split, so that the fluid portions of the
            newly-formed partial cells are np1 = np2 = 9/10, and the criterion is satisfied (N > CNC).




                               Fig.1.2                                        Fig.1.3
                          NCRL = 2; CNC = 3;                            NCRL = 3; CNC = 3;
                            N = 2.8 < CNC                                 N = 3.8 > CNC

                        Like in the other refinements, the Narrow channel refinement level
                        (NCRL) denotes the maximum level to which the cells can be split to
                        satisfy the CNC criterion. The NCRL has higher priority than the
                        CNC, so the refinement will proceed until the CNC criterion is
                        satisfied or all the cells reach the Narrow channel resolution level.
            The narrow channel refinement is symmetrical with respect to the midpoint of the interval
            and proceeds from the both ending partial cells towards the midpoint. [CNC, NCRL].




                               Fig.1.4                                        Fig.1.5
                          CNC = 5; NCRL = 1                              CNC = 5; NCRL = 3

            On Fig.1.4 the specified Characteristic number of cells across a channel is 5 but only two
            cells were generated since the maximum refinement level of one allows only basic mesh
            cells and first-level cells to be generated.


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              On Fig.1.5 the specified Narrow channel refinement level is high enough to allow five
              cells to be placed across the channel.
              Note that (see Fig.1.6) the partial cells near the channel’s dead end and the orifice were not
              split like the other partial cells along the channel. This is due to the fact that the right angle
              was approximated by the flat element sloping to the both sides of the channel. Therefore
              the normal to the solid\fluid interface determined in these corner cells, unlike the other
              partial cells, is not perpendicular to the channel, so the number of cells per this direction
              satisfies the criterion without further splitting.
              1.2.4) Refinement on Openings This refinement is intended to force the splitting of the
              partial cells which were not refined due to the “chamfer” approximation of the right angles
              (Fig.1.6), if these cells are at the boundary condition surface.




                                    Fig.1.6                                         Fig.1.7
                 Normal to the solid\fluid interface direction in   CNC = 5; NCRL = 3; Inlet velocity at the
                               the corner cells.                              channel end-wall.

              On Fig.1.7 the boundary condition specified on the wall in the end of the channel causes
              the Refinement on Opening procedure that splits the red partial cells to the next level.
              1.2.5) Next, for all the fluid cells within the entire computational domain the following
              Fluid Cell Leveling procedure is applied: if a fluid cell is located between two cells of
              higher level, it is split to be equalized with the level of neighboring smaller cells.




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    Chapter Initial Mesh Generation Stages


    1.3 Thin walls resolution
            In contrast to the narrow channels, thin walls can be resolved without the mesh refinement
            inside the wall, since the both sides of the thin wall may reside in the same cell. Therefore,
            the amount of cells needed to resolve a thin wall is generally lower than the number of
            cells needed to properly resolve a channel of the same width. See Fig.1.8 - 1.10 illustrating
            the thin walls resolution technology and its limitations.



                            Solid 2                               Fluid 1




                                                                                  Solid 1
                                                Fluid 2


                                                        Fig.1.8
                One mesh cell can contain more than one fluid and/or solid volume; during calculation
                each volume has an individual set of parameters depending on its type (fluid or solid).




                                                          Fig.1.9
                 If the wall thickness is greater than the basic mesh cell's size across the wall or if the
                wall creates only one fluid volume in the cell, then the opposite sides of the wall will not
                     lay within the same cell. Such walls are resolved with two or more cells across.




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                     Model geometry                                Meshed geometry


                                                                                                    Trimmed edge




                                                                                                     Trimmed cell




                                                     Fig.1.10
            The edges of thin walls ending within a mesh cell are trimmed. These mesh cells are called
                                                 Trimmed cells.



       1.4 Square Difference Refinement
              The Square Difference Refinement checks the neighboring partial cells of different levels
              for the difference between their fluid passage areas. If the difference between the fluid
              passage area of the higher-level cell and the total fluid passage areas of the adjacent
              lower-lever cells exceeds the Square Difference Refinement Criterion (SDRC) then the
              greater-level cell is split to the level of adjacent cells in order to equalize the fluid passage
              areas (see Fig.1.11). The Square Difference Refinement is always enabled and cannot be
              disabled since it is a strict solver requirement. As with the Cell Mating rule, this is another
              condition imposed by the solver to provide stability for the convergence processes.
              Though you cannot turn off the Square Difference Refinement, you can control its
              criterion, which is directly proportional to the Curvature refinement criterion. [CRC].




                            Fig.1.11                                          Fig.1.12
                   Two adjacent partial cells of        Cut plot of the cylinder. The concerned cells are blue.
                  different levels at the cylinder         SSFRL = 2; CRL = 0; CRC = 3.14; NCRL = 1.
                              surface.




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    Chapter Initial Mesh Generation Stages


            Fig.1.11 shows neighboring partial cells of different levels at the cylinder's solid/fluid
            interface. The fluid passage area of the higher-level cell is the ABDE polygon. The total
            fluid passage area of the lower-level cells is the ABCDE polygon, so the difference
            between the fluid passages is the yellow BCD triangle. In this example we have increased
            the curvature refinement criterion to π, thereby increasing the Square Difference
            Refinement Criterion so that the fluid passage difference (BCD) is smaller than the
            criterion, and thus, there is no need to split the higher-level cell.
            Note that the Square Difference Refinement may cause a domino effect when one splitting
            produces cells which become lower-level cells for the next adjacent cell causing it to split
            too, and so on, resulting in an increased number of cells.




                               Fig.1.13                                        Fig.1.14
                    SSFRL = 3, CRL = 0; CRC = 0.45                 SSFRL = 3, CRL = 2; CRC = 0.50;
                         Total cells = 49391.                           Total cells = 41376.

            In the Fig.1.13 the total number of cells is nearly 20% more than in the Fig.1.14 in spite of
            the fact that the Curvature refinement is disabled (CRL = 0) in the first case. Here, the
            model geometry is similar and before the Square Difference Refinement the mesh is
            practically the same in both cases and mostly governed by the Small Solid Features
            Refinement when the SSFRL exceeds the CRL, i.e. changing the CRL from 0 to 3 would
            not change substantially the number of cells. However, in the first case the curvature
            criterion is lower, resulting in a more stringent criterion of the Square Difference
            Refinement. So the smaller Square Difference Refinement criterion leads to a greater
            number of cells subject to the Square Difference Refinement. In the Fig.1.13 you can see a
            stripe of the third level cells along the cylinder. This is the result of the Square Difference
            Refinement and the domino effect when a cell on the cylinder edge involves the
            neighboring cell in the refinement procedure and so forth along the cylinder.

                        Increase of the curvature criterion will increase the Square
                        Difference Refinement Criterion, and, in turn, decrease the number
                        of cells in both cases.
            If in the first case we specify the same CRC as in the second case (0.5054 rad), the total
            number of cells decreases to 40963.


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       1.5 Mesh Diagnostic
              The mesh diagnostic is intended to inspect the resulting initial mesh but not to change the
              total number of cells.

       2   Refinements at Interfaces Between Substances

              Different interface types (solid/fluid, solid1/solid2, solid/porous or porous/fluid) are
              checked on different refinement criteria, namely: small solid features criterion, curvature
              refinement criterion, tolerance refinement criterion and narrow channel refinement
              criterion for solid/fluid and solid/porous interfaces; small solid features criterion for
              solid1/solid2 interfaces; small solid features criterion and curvature refinement criterion
              for porous/fluid interfaces. Whereas the specified refinement levels are equally applied to
              any interface type.

       2.1 Small Solid Features Refinement
              The small solid features refinement acts on the cells where the maximum angle between
              normals to the surface-forming triangles is strictly greater than 120°. To make this
              120-degree criterion easier to understand, let us consider simple small solid features of
              planar faces only. The normal to triangles that form the planar face is normal to the planar
              face too. Therefore, instead of considering the normals to the triangles we can consider
              normals to faces, or better the angle between faces.




                                                       Fig.2.1
                                            SSFRL = 1, CRL = 0, NCRL = 0

              In Fig.2.1 the cells with the cogs of 150 and 60 degrees were not split by the small solid
              features refinement because the maximum angles between the faces (i.e. between normals
              to the triangles enclosed by the cell) are 30° and 120°, respectively. If the angle between
              the normals becomes greater than 120° (121° for the 59°-cog) then the cell is split. The
              cell with the square spike surely has to be split because the lateral faces of the spike have
              their normals at the angle of 180°, thus satisfying the 120-degree criterion.
              Note that rectangular corners (like in the rightmost cell) do not satisfy the criterion and
              therefore will not be resolved by the small solid features refinement.




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    Chapter Refinements at Interfaces Between Substances



                           Remember that if the Narrow channel refinement is enabled, the
                           maximum level to which the small solid features refinement can split
                           the cells is set as the maximum level from the specified SSFRL and
                           Narrow channel refinement level (NCRL). In other words, if the
                           Narrow channel refinement is enabled, the SSFRL has no effect if it
                           is smaller than the NCRL.




                             Fig.2.2 SSFRL = 0, CRL = 0, NCRL = 1

            From Fig.2.2 it is clear that the cells are split by the 120-degree criterion up to the first
            level, as defined by the narrow channel refinement level.
            For the information about how the NCRL influences the narrow channel refinement see
            the Narrow Channel Refinement paragraph.

    2.2 Curvature Refinement
            The curvature refinement works in the same manner as the small solid features refinement
            with the difference that the critical angle between the normals can be specified by the user
            (in radians) as curvature refinement criterion (CRC). Here, the smaller the criterion, the
            better resolution of the solid curvature. To give more precise and descriptive explanation,
            the following table presents several CRC values together with the corresponding angles
            between normals and the angles between planar faces.

            Table 2.1: Influence of the curvature criterion on the solid curvature resolution.

          Curvature
          criterion, rad          0.3176   0.4510   0.5548   0.6435    1.0472    1.5708   2.0944    3.1416

          α' between               >19      >25      >31       >36      >60       >90      >120      180
          normals, [degrees]

          α between faces,         <161     <154     <148     <143      <120      <90       <60        0
          [degrees]




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              The table states that if the CRC is equal to 0.4510 rad, then all the cells where the angle
              between normals to the surface-forming triangles is more than 25 degrees will be split.




               Fig.2.3 CRL = 1, CRC = 0.5548,                 Fig.2.4 CRL = 1, CRC = 0.451,
               SSFRL = 0, NCRL = 0                            SSFRL = 0, NCRL = 0

              You can see that the curvature criterion set to 0.4510 rad splits the cells with the
              150-degrees cog.

                          Note that the curvature refinement works exactly as the small solid
                          features refinement when the curvature criterion is equal to 2.0944
                          rad (2/3π).
              However, the default curvature criterion values are small enough to resolve obtuse angles
              and curvature well. Increasing the curvature criterion is reasonable if you want to avoid
              superfluous refinement but it is recommended that you try different criteria to find the
              most appropriate one.
              The curvature criterion also denotes the criterion of the Square Difference Refinement.
              The square difference refinement criterion is directly proportional to the CRC, so the
              smaller CRC, the smaller square difference refinement criterion, resulting in a greater
              number of cells appearing after the Square Difference Refinement.

       2.3 SSFRL or CRL
              Why is it necessary to have two criteria? As you can see, the curvature refinement has
              higher priority than the small solid features refinement if the curvature criterion is smaller
              than 2/3 π. Note that COSMOSFloWorks-specified values of the curvature criterion are
              always smaller than 2/3 π.

                          In other words, if you did not set the CRC greater than 2/3 π and if
                          the SSFRL and NCRL are smaller than the CRL, then the small solid
                          feature refinement would be idle.
              Nevertheless, the advantage of the small solid features refinement is that being sensitive to
              relatively small geometry features it does not “notice” the large-scale curvatures, thus
              avoiding refinements in the entire computational domain but resolving only the areas of
              small features. At the same time, the curvature refinement can be used to resolve the
              large-scale curvatures. So both the refinements have their own coverage providing a
              flexible tool for creating an optimal mesh.




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    Chapter Local Mesh Settings


    2.4 Tolerance Refinement
            Any surface is approximated by a set of polygons whose vertices are the points of
            intersection of this surface with the cells' edges. This approach accurately represents flat
            faces though curved surfaces are represented by some approximation (e.g. as a circle can
            be represented by a polygon). The tolerance refinement criterion controls the precision of
            this approximation. A cell will be split if the distance between the outermost point of the
            surface within the cell and the polygon approximating this surface is larger than the
            specified criterion value.
                 Small Solid Feature Refinement                            Tolerance Refinement
                                                                 Tolerance criterion = 0.1               Tolerance
                                                                                                      criterion = 0.08




                      Curvature Refinement                               Tolerance Refinement
                Refines cells taking into account the           Tolerance criterion = 0.1                Tolerance
                          curvature only.               Refines cells only if the solid part cut by   criterion = 0.03
                                                         the polygon is large enough (h > 0.1)




    3    Local Mesh Settings

            The local mesh settings influence only the initial mesh and do not affect the basic mesh in
            the local area, but are basic mesh sensitive in that all refinement levels are set with respect
            to the basic mesh cell.
            The local mesh settings are applied to the cells intersected with the local mesh region
            which can be represented by a component, face, edge or vertex.
            If a cell intersects with different local mesh setting regions, the refinement settings in this
            cell will be used to achieve the maximum refinement.
            Cell Type Refinement. The refinement level of cells of a specific type (all cells, fluid and
            partial cells, solid and partial cells, or only partial cells) denotes the minimum level to
            which the corresponding cells must be split if it doesn’t contradict the Cell Mating rule.

                         The minimum level means the lower bound to which it is obligatory
                         to split cells, though the cells can be split further if it is required to
                         satisfy the other criteria such as Small solid features refinement,
                         Curvature refinement, Narrow channels refinement or Solid
                         Boundary Refinement.

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              If different cell types are to be refined, the refinement level of partial cells is set as the
              maximum level among all selected levels.
              The local mesh settings have higher priority over the initial mesh settings. Therefore, the
              local mesh cells will be split to the specified local refinement levels regardless of the
              general SSFRL, CRL and NCRL (specified in the Initial Mesh dialog box). This, however,
              may cause refinement of cells located outside of the local region due to imposing the Cell
              Mating rule.

       4   Irregular Cells

              When analyzing the computational mesh from the results file obtained with the earlier
              versions of COSMOSFloWorks, you may notice the presence of irregular cells. An
              irregular cell is a computational mesh cell lying at the solid/fluid interface (or solid/solid
              interface in cases where two or more different solids are within the cell), partly in one
              substance and partly in another, and characterized by the impossibility to define the
              solid/fluid interface position within the cell, given the cell’s nodes positions relative to
              solid region and the intersections of the solid/fluid interface with the cell. Please note that
              there are no irregular cells in the newly generated meshes, because the solid/fliud interface
              is now always resolved properly.
              You can use use the Results Summary to find out whether irregular cells are present and
              use the Mesh Visualization tool to detect where they are located.

       5   The "Optimize thin walls resolution" option

              In the earlier versions of COSMOSFloWorks refinement of the mesh around model's walls
              was needed to resolve thin walls properly, but it could also lead to increase in number of
              cells in adjacent fluid regions, especially in narrow channels between walls. If this
              additional mesh refinement is critical for obtaining proper results and you want to perform
              calculation on the same mesh as in the earlier version of COSMOSFloWorks, clear the
              Optimize thin walls resolution check box. In this case the mesh will be almost the same
              as in the previous version, the main difference is the absence of irregular cells. (see
              Fig.5.1).




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    Chapter Postamble




                                                                     solid/fluid
                                                                     interfaces




                                  (a)                                                  (b)

                                                      Fig.5.1
         Mesh refinement around a thin wall: (a) the Optimize thin walls resolution option is switched
        off, i.e. the mesh cells are split as in the previous versions of COSMOSFloWorks; (b) the Optimize
            thin walls resolution option is selected (the default state), i.e. the mesh cells are not split.



    6      Postamble

               The problem of resolving a model with the computational mesh is always model-specific.
               In general, a denser mesh will provide better accuracy but you should tend to create an
               optimal mesh and to avoid redundant refinement.
               When performing a calculation, try different mesh settings and analyze the obtained
               results carefully in order to understand whether it is necessary to refine the mesh or a
               coarser resolution is acceptable for the desired accuracy.

    7      Glossary

               Nx, Ny, Nz – Number of basic mesh cells per X, Y and Z directions, respectively.

               SSFRL – Small solid features refinement level.

               CRL – Curvature refinement level.

               CRC – Curvature refinement criterion.

               TRL – Tolerance refinement level.

               TRC – Tolerance refinement criterion.

               NCRL – Narrow channel refinement level.

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              CNC – Characteristic number of cells across a narrow channel.

              NCHmin – The minimum height of narrow channels.

              NCHmax – The maximum height of narrow channels.

              SDRC – Square difference refinement criterion.




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    Chapter Glossary




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                                                                                                      4
                                                                 Validation Examples




       Introduction

              A series of calculation examples presented below validate the ability of
              COSMOSFloWorks to predict the essential features of various flows, as well as to solve
              conjugate heat transfer problems (i.e. flow problems with heat transfer in solids). In order
              to perform the validation accurately and to present clear results which the user can check
              independently, relatively simple examples have been selected. For each of the following
              examples, exact analytical expression or well-documented experimental results exist.
              Each of the examples focus on one or two particular physical phenomena such as: laminar
              flow with or without heat transfer, turbulent flows including vortex development,
              boundary layer separation and heat transfer, compressible gas flow with shock and
              expansion waves. Therefore, these examples validate the ability of COSMOSFloWorks to
              predict fundamental flow features accurately. The accuracy of predictions can be
              extrapolated to typical industrial examples (encountered every day by design engineers
              and solved using COSMOSFloWorks), which may include a combination of the
              above-mentioned physical phenomena and geometries of arbitrary complexity.




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    Chapter Introduction




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       1     Flow through a Cone Valve

                  Let us see how COSMOSFloWorks predicts incompressible turbulent 3D flows in a 3D
                  cone valve taken from Ref.14 (the same in Ref.2) and having a complex flow passage
                  geometry combining sudden 3D contractions and expansions at different turning angles ϕ
                  (Fig. 1.1.). Following the Refs.2 and 14 recommendations on determining a valve’s
                  hydraulic resistance correctly, i.e. to avoid any valve-generated flow disturbances at the
                  places of measuring the flow total pressures upstream and downstream of the valve, the
                  inlet and outlet straight pipes of the same diameter D and of enough length (we take 7D
                  and 17D) are connected to the valve, so constituting the experimental rig model (see Fig.
                  1.2.). As in Ref.14, a water flows through this model. Its temperature of 293.2 K and fully
                  developed turbulent inlet profile (see Ref.1) with mass-average velocity U ≈ 0.5 m/s (to
                  yield the turbulent flow’s Reynolds number based on the pipe diameter ReD = 105) are
                  specified at the model inlet, and static pressure of 1 atm is specified at the model outlet.




                             Fig. 1.1. The cone valve under consideration: D = 0.206 m, Dax = 1.515⋅D, α = 13°40′.

                  The corresponding model used for these predictions is shown in Fig. 1.2.. The valve’s
                  turning angle ϕ is varied in the range of 0…55° (the valve opening diminishes to zero at
                  ϕ = 82°30′).
           Inlet velocity
           profile




                                                                                                  Outlet static pressure

                                                                                                  P = 101325 Pa
                            Fig. 1.2. The model for calculating the 3D flow in the cone valve.

                  The flow predictions performed with COSMOSFloWorks are validated by comparing the
                  valve’s hydraulic resistance ζv, and the dimensionless coefficient of torque M (see Fig.
                  1.1.) acting on the valve, m, to the experimental data of Ref.14 (Ref.2).

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    Chapter Flow through a Cone Valve


            Since Ref.14 presents the valve’s hydraulic resistance (i.e. the resistance due to the flow
            obstacle, which is the valve) ζv, whereas the flow calculations in the model (as well as the
            experiments on the rig) yield the total hydraulic resistance including both ζv and the tubes’
            hydraulic resistance due to friction, ζf , i.e. ζ = ζv + ζf , then, to obtain ζv from the flow
            predictions (as well as from the experiments), ζf is calculated (measured in the
            experiments) separately, at the fully open valve (ϕ = 0); then ζv = ζ - ζf .
            In accordance with Ref.14, both ζ and ζf are defined as (Po inlet - Po outlet)/(ρU2/2), where
            Po inlet and Po outlet are the flow total pressures at the model’s inlet and outlet, accordingly,
            ρ is the fluid density. The torque coefficient is defined as m = M/[D3⋅(ρU2/2)⋅(1+ ζv)],
            where M is the torque trying to slew the valve around its axis (vertical in the left picture in
            Fig. 1.1.) due to a non-uniform pressure distribution over the valve’s inner passage
            (naturally, the valve’s outer surface pressure cannot contribute to this torque). M is
            measured directly in the experiments and is integrated by COSMOSFloWorks over the
            valve’s inner passage.
            The COSMOSFloWorks predictions have been performed at result resolution level of 5
            with manual setting of the minimum gap size to the valve’s minimum passage in the Y = 0
            plane and the minimum wall thickness to 3 mm (to resolve the valve’s sharp edges).
            COSMOSFloWorks has predicted ζf = 0.455, ζv shown in Fig. 1.3., and m shown in Fig.
            1.4. It is seen that the COSMOSFloWorks predictions well agree with the experimental
            data.
            This cone valve's 3D vortex flow pattern at ϕ = 45° is shown in Fig. 1.5. by flow
            trajectories colored by total pressure. The corresponding velocity contours and vectors at
            the Y = 0 plane are shown in Fig. 1.6..
            The predictions have been performed on a computational mesh consisting of about
            150000 cells and have required about 270 MB memory and about 7 hours to run on a
            1GHz PIII platform for each specified ϕ .




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                 v
                100
                                                                                                   Experimental data
                                                                                                   Calculation



                 10




                  1




                 0.1                                                                       ϕ(° )
                       15       20    25     30      35      40       45      50      55

             Fig. 1.3. Comparison of the COSMOSFloWorks predictions with the Ref.14 experimental data on the
             cone valve’s hydraulic resistance versus the cone valve turning angle.


                  m
                  0.16
                                                                                                   Experimental data

                  0.14                                                                             Calculation

                  0.12

                     0.1

                  0.08

                  0.06

                  0.04

                  0.02

                       0                                                                     (°)
                           15    20    25     30      35      40      45      50      55


              Fig. 1.4. Comparison of the COSMOSFloWorks predictions with the Ref.14 experimental data on the
              cone valve’s torque coefficient versus the cone valve turning angle.




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    Chapter Flow through a Cone Valve




                Fig. 1.5. Flow trajectories colored by total pressure at ϕ = 45°.




                 Fig. 1.6. The cone valve’s velocity contours and vectors at ϕ = 45°.




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       2   Laminar Flows Between Two Parallel Plates

              Let us consider two-dimensional (planar) steady-state laminar flows of Newtonian,
              non-Newtonian, and compressible liquids between two parallel stationary plates spaced at
              a distance of 2h (see Fig. 2.1.).
              In the case of Newtonian and non-Newtonian liquids the channel has a 2h = 0.01 m height
              and a 0.2 m length, the inlet for these liquids have standard ambient temperature (293.2 K)
              and a uniform inlet velocity profile of uaverage = 0.01 m/s (entrance disturbances are
              neglected). The inlet pressure is not known beforehand, since it will be obtained from the
              calculations in accordance with the specified channel exit pressure of 1 atm. (The fluids
              pass through the channel due to a pressure gradient.)




                uaverage or m




                                Fig. 2.1. Flow between two parallel plates.

              Since the Reynolds number based on the channel height is equal to about Re2h=100, the
              flow is laminar.
              As for the liquids, let us consider water as a Newtonian liquid and four non-Newtonian
              liquids having identical density of 1000 kg/m3, identical specific heat of 4200 J/(kgK) and
              identical thermal conductivity of 10 W/(mK), but obeying different non-Newtonian liquid
              laws available in COSMOSFloWorks.
              The considered non-Newtonian liquids' models and their governing characteristics are
              presented in Table 4.1. These models are featured by the function connecting the flow
              shear stress (τ) with the flow shear rate ( γ ), i.e. τ
                                                           &            = f (γ& ) , or, following Newtonian
              liquids, the liquid dynamic viscosity (η) with the flow shear rate ( γ ), i.e.
                                                                                    &          τ = η (γ& ) ⋅ γ& :
              1 the Herschel-Bulkley model:     τ = K ⋅ (γ& )n + τ o , where K is the consistency
                 coefficient, n is the power-law index, and τ o is the yield stress (a special case with
                 n = 1 gives the Bingham model);

                                              = K ⋅ (γ& ) , i.e., η = K ⋅ (γ& ) , which is a special case of
                                                                               n −1
              2 the power-law model: τ
                                                         n


                 Herschel-Bulkley model with        τ o = 0;


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    Chapter Laminar Flows Between Two Parallel Plates



            3 the Carreau model:                                               [
                                       τ = η ⋅ γ& , η = η ∞ + (η o − η∞ ) ⋅ 1 + (K1 ⋅ γ& )2   ](   n −1) / 2
                                                                                                               , where η ∞
                is the liquid dynamic viscosity at infinite shear rate, i.e. the minimum dynamic
                viscosity, η o is the liquid dynamic viscosity at zero shear rate, i.e. the maximum
                dynamic viscosity, K1 is the time constant, n is the power-law index (this model is a
                smooth version of the power-law model).
                Non-Ne w tonia n liquid No.                 1              2             3                         4
                Non-Newtonian liquid m odel          Herschel-Bulkley   B ingham     Power law                  Carreau

             Consistency coefficient, K (P a⋅s n )        0.001          0.001         0.001                        -

                     Power law index, n                    1.5             1            0.6                       0.4
                    Y ield stress, o (Pa)                 0.001          0.001           -                         -
              Minimum dynamic viscosity,       ∞
                                                            -              -             -                        10 -4
                           (Pa⋅ s)
              M axim um dynamic viscosity,      o
                                                            -              -             -                        10 -3
                           (Pa⋅ s)
                   Tim e constant, K 1 (s)                  -              -             -                         1


            In accordance with the well-known theory presented in Ref.1, after some entrance length,
            the flow profile u(y) becomes fully developed and invariable. It can be determined from
                                                                dP dτ          τ
            the Navier-Stokes x-momentum equation                 =   = const = w corresponding to
                                                                dx dy           h
            this case in the coordinate system shown in Fig. 2.1. (y = 0 at the channel's center plane,
             dP                                                              du
                is the longitudinal pressure gradient along the channel, γ =
                                                                          &     in the flow under
             dx                                                              dy
            consideration).
            As a result, the fully developed u(y) profile for a Newtonian fluid has the following form:

                        1 dP 2
            u(y) = -         (h − y 2 ) ,
                       2η dx
            where η is the fluid dynamic viscosity and η is the half height of the channel,

             dP    3η uaverage
                =−             ,
             dx       h2
            where uaverage is the flow's mass-average velocity defined as the flow's volume flow rate
            divided by the area of the flow passage cross section.
            For a non-Newtonian liquid described by the power-law model the fully developed u(y)
            profile and the corresponding pressure gradient can be determined from the following
            formulae:


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                                                  n +1
                                                                                                                        n
                                  2n + 1   y  n  dP                                               u          2n + 1 
               u ( y ) = uaverage
                                                               K
                                         1 −  h   , dx = − h
                                  n +1    
                                                                                                    ⋅  average ⋅
                                                                                                       h           n 
                                                                                                                          .
                                                                                                                      
              For a non-Newtonian liquid described by the Herschel-Bulkley model the fully developed
              u(y) profile can be determined from the following formulae:


                                                      (τ w − τ o ) n at y < τ o h ,
                                                                  n +1
              u ( y ) = umax =
                                          h       n
                                      K 1/ nτ w n + 1                       τw

                                                n +1
                                                      
                                       y     n 
                                    τ w −τ o  
              u ( y ) = umax ⋅ 1 −  h          at y > τ o h ,
                                 τ w −τ o  
                                                        τw
                                              
                                                     
              where the unknown wall shear stress τ                        is determined numerically by solving the
                                                                    w
              nonlinear equation

                                                                 n +1     n τ w −τ o 
                                                   ⋅ (τ w − τ o ) n ⋅ 1 −
                                  h             n
              uaverage =                  ⋅                            2n + 1 τ       ,
                                                                                      
                              K 1/ nτ w       n +1                              w    
              e.g. with the Newton method, as described in this validation. The corresponding pressure

                                                  dP τ w
              gradient is determined as              =   .
                                                  dx   h
              For a non-Newtonian liquid described by the Carreau model the fully developed u(y)
              profile can not be determined analytically in an explicit form, so in this validation example
              it is obtained by solving the following parametric equation:


               y=
                    h
                    τw
                          (                       (
                         p µ ∞ + ( µ 0 − µ ∞ ) 1 + λ2 p 2   )
                                                            ( n −1) / 2
                                                                          ),
                                  hµ∞ 2 ( µ0 − µ ∞ )h                                                       2 1  ( µ0 − µ∞ )h
                                                                   (                     )
                                                                                             ( n −1) / 2
               u = umax −              p −            1 + λ 2 p2                                            np − 2  −             ,
                                  2τ w     (n + 1)τ w                                                           λ  (n + 1)τ w λ 2
              where p is a free parameter varied within the ±p max range,


                                                     (
               τ w = pmax  µ ∞ + ( µ 0 − µ ∞ ) 1 + λ2 pmax 2
                              
                                                                    )( n −1) / 2
                                                                                   
                                                                                    ,
                                                                                   


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    Chapter Laminar Flows Between Two Parallel Plates



                      hµ ∞ 2     ( µ − µ∞ )h                                                  2       1  ( µ 0 − µ ∞ )h
                                                      (                    )
                                                                               ( n −1) / 2
             umax =        pmax + 0           1 + λ 2 pmax
                                                       2
                                                                                              npmax − 2  +
                      2τ w         (n + 1)τ w                                                        λ  (n + 1)τ wλ 2
                                                                      h                      pmax       dy
            The p value is varied to satisfy huaverage =          ∫
                                                                  0
                                                                          udy = ∫
                                                                                         0
                                                                                                    u
                                                                                                        dp
                                                                                                           dp .

                                                                           dP τ w
            The corresponding pressure gradient is equal to                   =   .
                                                                           dx   h
            The SolidWorks model for the 2D calculation is shown in Fig. 2.2.. The boundary
            conditions are specified as mentioned above and the initial conditions coincide with the
            inlet boundary conditions. The results of the calculations performed with
            COSMOSFloWorks at result resolution level 5 are presented in Figs.2.3 - 2.8. The channel
            exit u(y) profile and the channel P(x) profile were obtained along the sketches shown by
            green lines in Fig. 2.2..




             Fig. 2.2. The model for calculating 2D flow between two parallel plates with COSMOSFloWorks.



                U, m/s
                 0.016
                                                                                                            Water, theory
                 0.014
                                                                                                            Water, calculation
                 0.012                                                                                      Liquid #3, theory
                  0.01                                                                                      Liquid #3, calculation
                 0.008
                 0.006

                 0.004
                 0.002
                      0
                                                                                                           Y, m
                      -0.005     -0.003      -0.001       0.001            0.003               0.005

              Fig. 2.3. The water and liquid #3 velocity profiles u(y) at the channel outlet.




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                 P, Pa
                101325.3
                                                                                                Water, theory
               101325.25                                                                        Water, calculation

                101325.2                                                                        Liquid #3, theory

                                                                                                Liquid #3,
               101325.15                                                                        calculation

                101325.1

               101325.05

                  101325
                                                                                                X, m
                             0            0.05         0.1           0.15           0.2

                   Fig. 2.4. The water and liquid #3 longitudinal pressure change along the channel, P(x).



               U, m/s
                0.016
                                                                                                Liquid #1, theory
                0.014
                                                                                                Liquid #1,
                0.012                                                                           calculation
                 0.01                                                                           Liquid #2, theory

                0.008                                                                           Liquid #2,
                                                                                                calculation
                0.006

                0.004

                0.002

                    0
                    -0.005       -0.003      -0.001      0.001       0.003        0.005 Y, m

                Fig. 2.5. The liquids #1 and #2 velocity profiles u(y) at the channel outlet.


              From Figs.2.4, 2.6, and 2.8 you can see that for all the liquids under consideration, after
              some entrance length of about 0.03 m, the pressure gradient governing the channel
              pressure loss becomes constant and nearly similar to the theoretical predictions. From
              Figs.2.3, 2.5, and 2.7 you can see that the fluid velocity profiles at the channel exit
              obtained from the calculations are close to the theoretical profiles.




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    Chapter Laminar Flows Between Two Parallel Plates



              P, Pa
              101325.6                                                                           Liquid #1, theory

              101325.5                                                                           Liquid #1,
                                                                                                 calculation
              101325.4
                                                                                                 Liquid #2, theory

              101325.3
                                                                                                 Liquid #2,
                                                                                                 calculation
              101325.2

              101325.1

                101325                                                                    X, m
                          0            0.05           0.1            0.15           0.2

             Fig. 2.6. The liquids #1 and #2 longitudinal pressure change along the channel, P(x).


                 U, m/s
                  0.014                                                                          Liquid #4, theory
                  0.012
                                                                                                 Liquid #4,
                   0.01                                                                          calculation

                  0.008

                  0.006

                  0.004

                  0.002

                       0
                       -0.005      -0.003      -0.001       0.001       0.003       0.005 Y, m


             Fig. 2.7. The liquid #4 velocity profile u(y) at the channel outlet.



            In the case of compressible liquids the channel has the height of 2h = 0.001 m and the
            length of 0.5 m, the liquids at its inlet had standard ambient temperature (293.2 K) and a
            uniform inlet velocity profile corresponding to the specified mass flow rate of
             &
             m = 0.01 kg/s.
            The inlet pressure is not known beforehand, since it will be obtained from the calculations
            as providing the specified mass flow rate under the specified channel exit pressure of 1
            atm. (The fluids pass through the channel due to the pressure gradient).
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                P, Pa

                101325.14
                                                                                              Liquid #4, theory
                101325.12
                                                                                              Liquid #4,
                 101325.1                                                                     calculation

                101325.08

                101325.06

                101325.04

                101325.02

                   101325
                            0            0.05          0.1          0.15          0.2 X, m

              Fig. 2.8. The liquid #4 longitudinal pressure change along the channel, P(x).


              Let us consider two compressible liquids whose density obeys the following laws:
                 • the power law:
                            n              , where ρ , P , B and n are specified: ρ is the liquid's density
                     ρ     P + B under the 0reference pressure P , B and n are constants,
                                                  0                        0
                         =                                      0
                     ρ0    P0 + B
                 • the logarithmic law:
                                   ρ0             , where ρ0, P0, B and C are specified: ρ0 is the liquid's
                     ρ=                           density under the reference pressure P0, B and C are
                                        B+P
                          1 − C *ln               constants.
                                        B + P0
              In this validation example these law's parameters values have been specified as ρ0=103
              kg/m3, P0 = 1 atm, B = 107 Pa, n = 1.4, C = 1, and these liquids have the 1Pa·s dynamic
              viscosity.
              Since this channel is rather long, the pressure gradient along it can be determined as

               ∂P   3η m , where η is the liquids' dynamic viscosity, m is the liquid mass
                       &                                                  &
                  =− 2
               ∂x   h S ρ flow rate, S is the channel's width, ρ is the liquid density.

              Therefore, by substitution the compressible liquids' ρ(P) functions, we obtain the
              following equations for determining P(x) along the channel:
                 • for the power-law liquid:


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    Chapter Laminar Flows Between Two Parallel Plates



                         3µ m  P0 + B 
                                                           1/ n
                    ∂P       &
                       =− 2
                    ∂x   h S ρ0  P + B 
                                       
                    its solution is
                                              n    3µ m
                                                      &
                    (P + B)
                                 1+ 1                              1
                                        n
                                                 = 2     ( P0 + B ) n x + C1 ,
                                            n + 1 h S ρ0

                    where C1 is a constant determined from the boundary conditions;
                • for the logarithmic-law liquid:

                      ∂P   3µ m 
                               &             P+B 
                         =− 2      1 − C ln         , this equation is solved numerically.
                      ∂x   h S ρ0           P0 + B 
            Both the theoretical P(x) distributions and the corresponding distributions computed
            within COSMOSFloWorks on a 5*500 computational mesh are presented in Figs.2.9 and
            2.10. It is seen that the COSMOSFloWorks calculations agree with the theoretical
            distributions.
              P, Pa

              18000000
                                                                                            LN, theory
              16000000
                                                                                            LN, ca lculation
              14000000
              12000000
              10000000
                8000000
                6000000
                4000000
                2000000
                         0
                             0               0.1     0.2          0.3   0.4      0.5 X, m

            Fig. 2.9. The logarithmic-law compressible liquid's longitudinal pressure change along the channel,
            P(x).




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               P, Pa

               35000000
                                                                                         Power, theory
               30000000                                                                  Power, calculation

               25000000

               20000000

               15000000

               10000000

                 5000000

                        0
                            0        0.1        0.2        0.3        0.4        0.5 X, m
             Fig. 2.10. The power-law compressible liquid's longitudinal pressure change along the channel, P(x).




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    Chapter Laminar Flows Between Two Parallel Plates




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       3   Laminar and Turbulent Flows in Pipes

                Having been encouraged by the 2D results presented in the previous example, let us now
                see how the 3D flow through a straight pipe is predicted. Let us consider water (at
                standard 293.2 K temperature) flowing through a long straight pipe with circular cross
                section of d = 0.1 m (see Fig. 3.1.). At the pipe inlet the velocity is uniform and equal to
                uinlet. At the pipe outlet the static pressure is equal to 1 atm.
                The SolidWorks model used for all the 3D pipe flow calculations is shown in Fig. 3.2. The
                initial conditions have been specified to coincide with the inlet boundary conditions. The
                computational domain is reduced to domain (Z ≥ 0, Y ≥ 0) with specifying the flow
                symmetry planes at Z = 0 and Y = 0.




           uinlet




                                             Fig. 3.1. Flow in a pipe.




                    Fig. 3.2. The SolidWorks model for calculating 3D flow in a pipe with COSMOSFloWorks.


                According to theory (Ref.1), the pipe flow velocity profile changes along the pipe until it
                becomes a constant, fully developed profile at a distance of Linlet from the pipe inlet.
                According to Ref.1, Linlet is estimated as:




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    Chapter Laminar and Turbulent Flows in Pipes



                                       0.03 ⋅ d ⋅ Red ≥ 3 ⋅ d , Red = 0.1...2500
                                       
                           Linlet =        100 ⋅ d ,     Re d = 2500...6000
                                              40 ⋅ d ,    Re d = 6000...106
                                       
            where Red = ρ⋅U⋅d/µ is the Reynolds number based on the pipe diameter d, U is the
            mass-average flow velocity, ρ is the fluid density, and µ is the fluid dynamic viscosity.
            Therefore, to provide a fully developed flow in the pipe at Red under consideration, we
            will study the cases listed in Table 1:. Here, Lpipe is the overall pipe length. All the
            COSMOSFloWorks predictions concerning the fully developed pipe flow characteristics
            are referred to the pipe section downstream of the inlet section.
                              Table 1: Pipe inlet velocities and lengths.
                   Red                uinlet, m/s            Linlet, m             Lpipe, m
                         0.1                  10-6                  0.3                  0.45
                         100                 0.001                  0.3                  0.45
                         1000                 0.01                   3                   4.5
                              4
                         10                   0.1                  4 (5)*               6 (10)*
                              5
                         10                    1                   4 (5)*               6 (10)*
                         106                   10                 4 (5)*6               6 (10)*
                                  *) the lengths in brackets are for the rough pipes.
            The flow regime in a pipe can be laminar, turbulent, or transitional, depending on Red.
            According to Ref.1, Red = 4000 is approximately the boundary between laminar pipe flow
            and turbulent one (here, the transitional region is not considered).
            Theory (Refs. 1 and 4) states that for laminar fully developed pipe flows
            (Hagen-Poiseuille flow) the velocity profile u(y) is invariable and given by:
                                                     1 dP 2
                                      u( y ) = −           (R − y2 ) ,
                                                    4 µ dx
            where R is the pipe radius, and dP/dx is the longitudinal pressure gradient along the pipe,
            which is also invariable and equal to:
                                            dP    8µ uinlet .
                                               =−
                                            dx      R2
            The COSMOSFloWorks predictions of dP/dx and u(y) of the laminar fully developed pipe
            flow at Red = 100 performed at result resolution level 6 are presented in Fig. 3.3. and Fig.
            3.4. The presented predictions relate to the smooth pipe, and similar ones not presented
            here have been obtained for the case of the rough tube with relative sand roughness of
            k/d=0.2…0.4 %, that agrees with the theory (Ref.1).


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                   Pressure (Pa)
                  101325.0020
                                                                                                    Theory
                  101325.0018
                                                                                                    Calculation
                  101325.0016
                  101325.0014
                  101325.0012
                  101325.0010
                  101325.0008
                  101325.0006
                  101325.0004
                  101325.0002
                  101325.0000
                                 0            0.1            0.2        0.3            0.4   0.5       X (m)

              Fig. 3.3. The longitudinal pressure change (pressure gradient) along the pipe at Red ≈ 100.



                Velocity (m/s)
                 0.0025
                                                                                                    Theory
                                                                                                    Calculation
                   0.002

                 0.0015


                   0.001

                 0.0005


                       0
                           0           0.01           0.02           0.03             0.04   0.05   Y (m)



               Fig. 3.4. The fluid velocity profile at the pipe exit for Red ≈ 100.

              From Fig. 3.3. one can see that after an entrance length of about 0.15 m the pressure
              gradient predicted by COSMOSFloWorks coincides with the one predicted by theory.
              Therefore, the prediction of pipe pressure loss is excellent. As for local flow features, from
              Fig. 3.4. one can see that the fluid velocity profiles predicted at the pipe exit are rather
              close to the theoretical profile.




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    Chapter Laminar and Turbulent Flows in Pipes


            The velocity profile and longitudinal pressure distribution in a smooth pipe at Red = 105,
            i.e., in a turbulent pipe flow regime, predicted by COSMOSFloWorks at result resolution
            level 6 are presented in Figs.3.5 and 3.6 and compared to theory (Ref.1, the Blasius law of
            pressure loss, the 1/7-power velocity profile).

              Pressure (Pa)
               102400
                                                                                                 Theory
               102200                                                                            Calculation

               102000


               101800

               101600

               101400

               101200
                        0           2            4            6             8           10   X (m)


            Fig. 3.5. The longitudinal pressure change (pressure gradient) along the pipe at Red = 105.

            Then, to stand closer to engineering practice, let us consider the COSMOSFloWorks
            predictions of the pipe friction factor used commonly and defined as:
                                                       ∆P      d
                                              f =       2
                                                             ⋅   ,
                                                       uinlet L
                                                     ρ
                                                         2
            where L is length of the pipe section with the fully developed flow, along which pressure
            loss ∆P is measured.




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                 Velocity (m/s)
                 1.4
                                                                                                Theory
                 1.2                                                                            Calculation
                   1

                 0.8

                 0.6

                 0.4

                 0.2

                   0
                                                                                              Y (m)
                       0           0.01           0.02           0.03           0.04   0.05


              Fig. 3.6. The fluid velocity profile at the pipe exit at Red = 105.

              In Figs. 3.7 and 3.8 (scaled up) you can see the COSMOSFloWorks predictions performed
              at result resolution level 5 for the smooth pipes in the entire Red range (both laminar and
              turbulent), and compared with the theoretical and empirical values determined from the
              following formulae which are valid for fully-developed flows in smooth pipes (Refs.1, 2,
              and 4):

                            64
                            Re ,           Re d ≤ 2300 − laminar flows,
                               d
                           
                       f = 0.316 ⋅ Re −1 4 , 4000 < Re d < 10 5 − turbulent flows,
                                        d
                                              −2
                            1.8 ⋅ log Red  , Re ≥ 105 − turbulent flows
                           
                                      6.9 
                                                      d



              It can be seen that the friction factor values predicted for smooth pipes, especially in the
              laminar region, are fairly close to the theoretical and empirical curve.
              As for the friction factor in rough pipes, the COSMOSFloWorks predictions for the pipes
              having relative wall roughness of k/d=0.4% (k is the sand roughness) are presented and
              compared with the empirical curve for such pipes (Refs.1, 2, and 4) in Fig. 3.8.. The
              underprediction error does not exceed 13%.
              Additionally, in the full accordance with theory and experimental data the
              COSMOSFloWorks predictions show that the wall roughness does not affect the friction
              factor in laminar pipe flows.




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    Chapter Laminar and Turbulent Flows in Pipes




             Friction factor
             1.E+03
                                                                                                   Smooth pipes,
                                                                                                   theoretical and
             1.E+02
                                                                                                   empirical data
                                                                                                   Calculation
             1.E+01


             1.E+00


              1.E-01


              1.E-02


              1.E-03
                   1.E-01   1.E+00   1.E+01   1.E+02   1.E+03    1.E+04   1.E+05   1.E+06   Re d


            Fig. 3.7. The friction factor predicted by COSMOSFloWorks for smooth pipes in comparison with the
            theoretical and empirical data (Refs.1, 2, and 4).


              Friction Factor
              0.100                                                                           Smooth wall,
                                                                                              theoretical and
                                                                                              empirical data
                                                                                              Smooth wall,
                                                                                              calculation

                                                                                              Rough wall,
                                                                                              k/d=0.4%, empirical
                                                                                              data
                                                                                              Rough wall,
                                                                                              k/d=0.4%,
                                                                                              calculation


              0.010
                                                                                                   Re d
                 1.E+03                 1.E+04               1.E+05                1.E+06

             Fig. 3.8. The friction factor predicted by COSMOSFloWorks for smooth and rough pipes in
             comparison with the theoretical and empirical data (Refs.1, 2, and 4).




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       4   Flows Over Smooth and Rough Flat Plates

               In the previous example we have presented a validation for laminar and turbulent flows in
               smooth and rough pipes for a wide range of Reynolds numbers. Now let us consider
               uniform flows over smooth and rough flat plates with laminar and turbulent boundary
               layers, so that COSMOSFloWorks predictions of a flat plate drag coefficient are validated.
               We consider the boundary layer development of incompressible uniform 2D water flow
               over a flat plate of length L (see Fig. 4.1.). The boundary layer develops from the plate
               leading edge lying at the upstream computational domain boundary. The boundary layer at
               the leading edge is considered laminar. Then, at some distance from the plate leading edge
               the boundary layer automatically becomes turbulent (if this distance does not exceed L).
               The SolidWorks model is shown in Fig. 4.2.. The problem is solved as internal in order to
               avoid a conflict situation in the corner mesh cell where the external flow boundary and the
               model wall intersect. In the internal flow problem statement, to avoid any influence of the
               upper model boundary or wall on the flow near the flat plate, the ideal wall boundary
               condition has been specified on the upper wall. The plate length is equal to 10 m, the
               channel height is equal to 2 m, the walls’ thickness is equal to 0.5 m.

                        Water



                                                           L

                                       Fig. 4.1. Flow over a flat plate.


             Inlet                                                             Ideal wall




                                                                                                        Outlet




                      Rough or smooth wall



                Fig. 4.2. The model for calculating the flow over the flat plate with COSMOSFloWorks.




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    Chapter Flows Over Smooth and Rough Flat Plates


            To solve the problem, an incoming uniform water flow of a certain velocity (see below),
            temperature of 293.2 K, turbulence intensity of 1%, and turbulence length of 0.01 m is
            specified at the channel inlet, whereas the water static pressure of 1 atm is specified at the
            channel outlet.
            The flow computation is aimed at predicting the flat plate drag coefficient, defined as (see
            Refs. 1 and 4):
                                                       F
                                             Cd =
                                                     ρV 2 ,
                                                         A
                                                      2
            where F is the plate drag force, A is the plate surface area, ρ is fluid density, and V is the
            fluid velocity.
            According to Refs.1 and 4, the plate drag coefficient value is governed by the Reynolds
            number, based on the distance L from the plate leading edge (ReL = ρVL/µ, where ρ is the
            fluid density, V is the incoming uniform flow velocity, and µ is fluid dynamic viscosity),
            as well as by the relative wall roughness L/k, where k is the sand roughness. As a result,
            Refs.1 and 4 give us the semi-empirical flat plate CD (ReL) curves obtained for different
            L/k from the generalized tubular friction factor curves and presented in Fig. 4.3. (here, ε ≡
            k). If the boundary layer is laminar at the plate leading edge, then the wall roughness does
            not affect CD until the transition from the laminar boundary layer to the turbulent one, i.e.,
            the CD (ReL) curve is the same as for a hydraulically smooth flat plate. The transition
            region’s boundaries depend on various factors, the wall roughness among them. Here is
            shown the theoretical transition region for a hydraulically smooth flat plate. The transition
            region's boundary corresponding to fully turbulent flows (i.e., at the higher ReL) is marked
            in Fig. 4.3. by a dashed line. At the higher ReL, the semi-empirical theoretical curves have
            flat parts along which ReL does not affect C D at a fixed wall roughness. These flat parts of
            the semi-empirical theoretical curves have been obtained by a theoretical scaling of the
            generalized tubular friction factor curves to the flat plate conditions under the assumption
            of a turbulent boundary layer beginning from the flat plate leading edge.
            To validate the COSMOSFloWorks flat plate CD predictions within a wide ReL range, we
            have varied the incoming uniform flow velocity at the model inlet to obtain the ReL values
            of 105, 3⋅105, 106, 3⋅106, 107, 3⋅107, 108, 3⋅108, 109.To validate the wall roughness
            influence on CD, the wall roughness k values of 0, 50, 200, 103, 5⋅103, 104 µm have been
            considered. The COSMOSFloWorks calculation results obtained at result resolution level
            5 and compared with the semi-empirical curves are presented in Fig. 4.3..
            As you can see from Fig. 4.3., CD (ReL) of rough plates is somewhat underpredicted by
            COSMOSFloWorks in the turbulent region, at L/k ³ 1000 the CD (ReL) prediction error
            does not exceed about 12%.




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                   CD
                0.014                                                                                L/k=1e 3
                                                                                                     L/k=2e 3
                0.012                                                                                L/k=1e 4
                                                                                                     L/k=5e 4
                 0.01                                                                                L/k=2e 5
                                                                                                     Sm ooth

                0.008


                0.006


                0.004


                0.002


                    0
                  1.00E+05          1.00E+06         1.00E+07          1.00E+08         1.00E+09     Re


              Fig. 4.3. The flat plate drag coefficient predicted with COSMOSFloWorks for rough and hydraulically
              smooth flat plates in comparison with the semi-empirical curves (Refs.1 and 4).




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    Chapter Flows Over Smooth and Rough Flat Plates




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       5   Flow in a 90-degree Bend Square Duct

              In the previous examples we have considered laminar and turbulent flows over flat plates
              and in straight pipes. Let us now see how COSMOSFloWorks predicts 3-dimensional
              incompressible flow in a 90o-bend square duct.
              Following Ref.8, we will consider a steady-state flow of water (at 293.2 K inlet
              temperature and Uinlet = 0.0198 m/s inlet uniform velocity) in a 40×40 mm square
              cross-sectional duct having a 90°- angle bend with ri = 72 mm inner radius (ro = 112 mm
              outer radius accordingly) and attached straight sections of 1.8 m upstream and 1.2 m
              downstream (see Fig. 5.1.). Since the flow's Reynolds number, based on the duct's
              hydraulic diameter (D=40 mm), is equal to ReD = 790, the flow is laminar.




                Fig. 5.1. The 90°-bend square duct's configuration indicating the velocity measuring
                stations and the dimensionless coordinates used for presenting the velocity profiles.


              The COSMOSFloWorks prediction was performed at result resolution level 7.
              The predicted dimensionless (divided by Uinlet) velocity profiles are compared in Figs.5.2,
              5.3 with the ones measured with a laser-Doppler anemometry at the following duct cross
              sections: XH = -5⋅D, -2.5⋅D, 0 (or θ=0°) and at the θ=30°, 60°, 90° bend sections. The z
                                                                   r − ro       z
              and r directions are represented by coordinates              and       , where z1/2 = 20 mm.
                                                                   ri − ro     z1/ 2
              The dimensionless velocity isolines (with the 0.1 step) at the duct's θ= 60° and 90°
              sections, both measured in Ref.8 and predicted with COSMOSFloWorks, are shown in
              Figs.5.4 and 5.5.
              It is seen that the COSMOSFloWorks predictions are close to the Ref.8 experimental data.


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    Chapter Flow in a 90-degree Bend Square Duct




              Fig. 5.2. The duct's velocity profiles predicted by COSMOSFloWorks (red lines) in comparison with
              the Ref.8 experimental data (circles).



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                                z/z1/2=0.5                                     z/z1/2=0




                            z/z1/2=0.5                                         z/z1/2=0
                Fig. 5.3. The duct's velocity profiles predicted by COSMOSFloWorks (red lines) in comparison
                with the Ref.8 experimental data (circles).




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    Chapter Flow in a 90-degree Bend Square Duct




                 Fig. 5.4. The duct's velocity isolines at the θ = 60° section predicted by COSMOSFloWorks (left)
                 in comparison with the Ref.8 experimental data (right).




                                                             θ
                 Fig. 5.5. The duct's velocity isolines at the =90° section predicted by COSMOSFloWorks (left)
                 in comparison with the Ref.8 experimental data (right).




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       6   Flows in 2D Channels with Bilateral and Unilateral Sudden Expansions

              In this example we will consider both turbulent and laminar incompressible steady-state
              flows through 2D (plane) channels with bilateral and unilateral sudden expansions and
              parallel walls, as shown in Figs.6.1 and 6.2. At the 10 cm inlet height of the
              bilateral-sudden-expansion channels a uniform water stream at 293.2 K and 1 m/s is
              specified. The Reynolds number is based on the inlet height and is equal to Re = 105,
              therefore (since Re > 104) the flow is turbulent. At the 30 mm height inlet of the
              unilateral-sudden-expansion channel an experimentally measured water stream at 293.2 K
              and 8.25 mm/s mean velocity is specified, so the Reynolds number based on the inlet
              height is equal to Re = 250, therefore the flow is laminar. In both channels, the sudden
              expansion generates a vortex, which is considered in this validation from the viewpoint of
              hydraulic loss in the bilateral-expansion channel (compared to Ref.2) and from the
              viewpoint of the flow velocity field in the unilateral-expansion channel (compared to
              Ref.13).

                                                   Y


                                     inlet                                                 outlet
                   Water

                   1 m/s                                                X


                Fig. 6.1. Flow in a 2D (plane) channel with a bilateral sudden expansion.


                                   Inlet experimental velocity profile           Outlet static pressure
                                          h = 15
                           30 mm




                                     Y
                                                       recirculation
                                      0                            Lr                               X
                    20 mm                                                   400 mm

                Fig. 6.2. Flow in a 2D (plane) channel with a unilateral sudden expansion.

              In accordance with Ref.2, the local hydraulic loss coefficient of a bilateral sudden
              expansion (the so-called total pressure loss due to flow) for a turbulent (Re > 104) flow
              with a uniform inlet velocity profile depends only on the expansion area ratio and is
              determined from the following formula:
                                                                                     2
                                                              P −P        A 
                                                         ζ s = 0 2 1 = 1 − 0  ,
                                                                       
                                                               ρu0        A1 
                                                                              
                                                                2
              where A0 and A1 are the inlet and outlet cross sectional areas respectively, P0 and P1 are
              the inlet and outlet total pressures, and ρu02/2 is the inlet dynamic head.

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    Chapter Flows in 2D Channels with Bilateral and Unilateral Sudden Expansions


            In a real sudden expansion the flow hydraulic loss coefficient is equal to ζ = ζf + ζs, where
            ζf is the friction loss coefficient. In order to exclude ζf from our comparative analysis,
            we have imposed the ideal wall boundary condition on all of the channel walls.
            In this validation example the channel expansion area ratios under consideration are: 1.5,
            2.0, 3.0, and 6.0. To avoid disturbances at the outlet due to the sudden expansion, the
            channel length is 10 times longer than its height. The 1 atm static pressure is specified at
            the channel outlet.
            The ζ s values predicted by COSMOSFloWorks at result resolution level 8 for different
            channel expansion area ratios A0/A1 are compared to theory in Fig. 6.3.
            From Fig. 6.3., one can see that COSMOSFloWorks overpredicts ζ s by about
            4.5...7.9 %.

                s


               1
                                                                                                    Theory
              0.9
              0.8                                                                                   Calculation
              0.7
              0.6
              0.5
              0.4
              0.3
              0.2
              0.1
               0
                                                                                            A0/A1
                    0         0.2          0.4          0.6          0.8           1


            Fig. 6.3. Comparison of COSMOSFloWorks calculations to the theoretical values (Ref.2) for the sudden
            expansion hydraulic loss coefficient versus the channel expansion area ratio.

            The model used for the unilateral-sudden-expansion channel's flow calculation is shown
            in Fig. 6.4. The channel's inlet section has a 30 mm height and a 20 mm length. The
            channel's expanded section (downstream of the 15 mm height back step) has a 45 mm
            height and a 400 mm length (to avoid disturbances of the velocity field compared to the
            experimental data from the channel's outlet boundary condition). The velocity profile
            measured in the Ref.13 at the corresponding Reh = 125 (the Reynolds number based on
            the step height) is specified as a boundary condition at the channel inlet. The 105 Pa static
            pressure is specified at the channel outlet.




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                                     Inlet velocity profile




                                                   Outlet static pressure
                       Fig. 6.4. The SolidWorks model for calculating the 2D flow in the
                       unilateral-sudden-expansion channel with COSMOSFloWorks.



              The flow velocity field predicted by COSMOSFloWorks at result resolution level 8 is
              compared in Figs.6.5, 6.6, and 6.7 to the values measured in Ref.13 with a laser
              anemometer. The flow X-velocity (u/U, where U = 8.25 mm/s) profiles at several X =
              const (-20 mm, 0, 12 mm, … 150 mm) cross sections are shown in Fig. 6.5. It is seen that
              the predicted flow velocity profiles are very close to the experimental values both in the
              main stream and in the recirculation zone. The recirculation zone's characteristics, i.e. its
              length LR along the channel's wall, (plotted versus the Reynolds number Reh based on the
              channel's step height h, where Reh =125 for the case under consideration), the separation
              streamline, and the vortex center are shown in Figs.6.6 and 6.7. It is seen that they are very
              close to the experimental data.




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    Chapter Flows in 2D Channels with Bilateral and Unilateral Sudden Expansions




         Fig. 6.5. The unilateral-sudden-expansion channel's velocity profiles predicted by COSMOSFloWorks
         (red lines) in comparison with the Ref.13 experimental data (black lines with dark circles).
                                                                         12


                                                                         10
                                      L /h - recirculation zone length




                                                                         8


                                                                         6


                                                                         4


                                                                         2


                                                                         0
                                                                              0   50   100    150   200     250 Reh


          Fig. 6.6. The unilateral-sudden-expansion channel's recirculation zone length predicted by
          COSMOSFloWorks (red square) in comparison with the Ref.13 experimental data (black signs).


             15



                                                                                                                          separation streamlines,
             10                                                                                                           calculation
                                                                                                                          vortex center, calculation



              5




              0
                  0         0.2                                           0.4           0.6           0.8             1

          Fig. 6.7. The unilateral-sudden-expansion channel recirculation zone's separation streamlines and
          vortex center, both predicted by COSMOSFloWorks (red lines and square) in comparison with the
          Ref.13 experimental data (black signs).


             As one can see, both the integral characteristics (hydraulic loss coefficient) and local
             values (velocity profiles and recirculation zone geometry) of the turbulent and laminar
             flow in a 2D sudden expansion channel under consideration are adequately predicted by
             COSMOSFloWorks.

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       7   Flow over a Circular Cylinder

              Until now, we have considered only internal flows. Let us now consider an external
              incompressible flow example. In this example, water at a temperature of 293.2 K and a
              pressure of 1 atm flows over a cylinder of 0.01 m or 1 m diameter. The flow pattern of this
              example substantially depends on the Reynolds number which is based on the cylinder
              diameter. At low Reynolds numbers (4 < Re < 60) two steady vortices are formed on the
              rear side of the cylinder and remain attached to the cylinder, as it is shown schematically
              in Fig. 7.1. (see Refs.3).




                 Fig. 7.1. Flow past a cylinder at low Reynolds numbers (4 < Re < 60).


              At higher Reynolds numbers the flow becomes unstable and a von Karman vortex street
              appears in the wake past the cylinder. Moreover, at Re > 60…100 the eddies attached to
              the cylinder begin to oscillate and shed from the cylinder (Ref.3). The flow pattern is
              shown schematically in Fig. 7.2..




                 Fig. 7.2. Flow past a cylinder at Reynolds numbers Re > 60…100.

              To calculate the 2D flow (in the X-Y plane) with COSMOSFloWorks, the model shown in
              Fig. 7.3. has been created. The cylinder diameter is equal to 0.01 m at Re ≤104 and 1 m at
              Re>104. The incoming stream turbulence intensity has been specified as 1%. To take the
              flow’s physical instability into account, the flow has been calculated by
              COSMOSFloWorks using the time-dependent option. All the calculations have been
              performed at result resolution level 7.




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    Chapter Flow over a Circular Cylinder




                 Fig. 7.3. The SolidWorks model used to calculate 2D flow over a cylinder.

             In accordance with the theory, steady flow patterns have been obtained in these
             calculations in the low Re region. An example of such calculation at Re=41 is shown in
             Fig. 7.4. as flow trajectories over and past the cylinder in comparison with a photo of such
             flow from Ref.9. It is seen that the steady vortex past the cylinder is predicted correctly.




             Fig. 7.4. Flow trajectories over and past a cylinder at Re=41 predicted with COSMOSFloWorks
             (above) in comparison with a photo of such flow from Ref.9 (below).




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              The unsteady vortex shedding from a cylinder at Re > 60..100, yields oscillations of both
              drag and lateral forces acting on the cylinder and a von Karman vortex street is formed
              past the cylinder. An X-velocity field over and past the cylinder is shown in Fig. 7.5. The
              COSMOSFloWorks prediction of the cylinder drag and lateral force oscillations'
              frequency in a form of Strouhal number (Sh = D/(tU), where D is the cylinder diameter, t
              is the period of oscillations, and U is the incoming stream velocity) in comparison with
              experimental data for Re≥103 is shown in Fig. 7.6..




              Fig. 7.5. Velocity contours of flow over and past the cylinder at Re=140.

                   Sh

                   0.4
                  0.35
                   0.3
                  0.25
                   0.2
                  0.15
                   0.1
                  0.05
                     0
                    1.E+01      1.E+02       1.E+03       1.E+04      1.E+05       1.E+06    1.E+07     Re



             Fig. 7.6. The cylinder flow's Strouhal number predicted with COSMOSFloWorks (red triangles) in
             comparison with the experimental data (blue line with dashes, Ref.4).




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    Chapter Flow over a Circular Cylinder


             The time-averaged cylinder drag coefficient is defined as
                                                           FD
                                               CD =
                                                       1
                                                         ρU 2 DL
                                                       2
             where FD is the drag force acting on the cylinder, ρU2/2 is the incoming stream dynamic
             head, D is the cylinder diameter, and L is the cylinder length. The cylinder drag
             coefficient, predicted by COSMOSFloWorks is compared to the well-known CD(Re)
             experimental data in Fig. 7.7..

                 CD
                100




                 10




                  1




                0.1                                                                                     Re
                  1.E-01    1.E+00    1.E+01    1.E+02    1.E+03   1.E+04   1.E+05    1.E+06   1.E+07

             Fig. 7.7. The cylinder drag coefficient predicted by COSMOSFloWorks (red diamonds) in comparison
             with the experimental data (black marks, Ref.3)




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       8   Supersonic Flow in a 2D Convergent-Divergent Channel

              Until now we have only considered incompressible flows, so now we will study a
              compressible, supersonic flow.
              The first example is a supersonic flow of air in a 2D (plane) convergent-divergent channel
              whose scheme is shown on Fig. 8.1.




              Fig. 8.1. Supersonic flow in a 2D convergent-divergent channel.

              A uniform supersonic stream of air, having a Mach number M=3, static temperature of
              293.2 K, and static pressure of 1 atm, is specified at the channel inlet between two parallel
              walls. In the next convergent section (see Fig. 8.2.) the stream decelerates through two
              oblique shocks shown schematically in Fig. 8.1. as lines separating regions 1, 2, and 3.
              Since the convergent section has a special shape adjusted to the inlet Mach number, so the
              shock reflected from the upper plane wall and separating regions 2 and 3 comes to the
              section 3 lower wall edge, a uniform supersonic flow occurs in the next section 3 between
              two parallel walls. In the following divergent section the supersonic flow accelerates thus
              forming an expansion waves fan 4. Finally, the stream decelerates in the exit channel
              section between two parallel walls when passing through another oblique shock.




              Fig. 8.2. Dimensions (in m) of the 2D convergent-divergent channel including a reference line for
              comparing the Mach number.


              The SolidWorks model of this 2D channel is shown in Fig. 8.3.




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    Chapter Supersonic Flow in a 2D Convergent-Divergent Channel


            Since the channel was designed for the inviscid flow of an ideal gas, the ideal wall
            boundary condition has been specified and the laminar only flow has been considered
            instead of turbulent. The computed Mach number along the reference line and at the
            reference points (1-5) are compared with the theoretical values in Fig. 8.4..




              Fig. 8.3. The model for calculating the 2D supersonic flow in the 2D convergent-divergent
              channel with COSMOSFloWorks.


            To obtain the most accurate results possible with COSMOSFloWorks, the calculations
            have been performed at result resolution level 8. The predicted Mach number at the
            selected channel points (1-5) and along the reference line (see Fig. 8.2.), are presented in
            Table.8.1 and Fig. 8.4. respectively.

             Table.8.1 Mach number values predicted with COSMOSFloWorks with comparison
                              to the theoretical values at the reference points.
                      P oint                    1             2             3            4             5
             X coordinate of point, m        0.0042         0.047        0.1094        0.155        0.1648
             Y coordinate of point, m        0.0175        0.0157         0.026        0.026        0.0157
                  Theoretical M               3.000         2.427         1.957        2.089         2.365
              COSMO SFloWorks
                 prediction of M              3.000         2.429        1.965         2.106         2.380
               P rediction error,%             0.0           0.1          0.4           0.8           0.6
            From Table.8.1 and Fig. 8.4. it can be seen that the COSMOSFloWorks predictions are
            very close to the theoretical values. In Fig. 8.4. one can see that COSMOSFloWorks
            properly predicts the abrupt parameter changes when the stream passes through the shock
            and a fast parameter change in the expansion fan.




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               M
               3.1       1                                                                                 Theory
                 3
               2.9                                                                                         Calculation
               2.8
               2.7
               2.6
               2.5                         2
                                                                                            5
               2.4
               2.3
               2.2
                                                                                     4
               2.1
                                                                   3
                 2
               1.9
                     0       0.02   0.04       0.06   0.08   0.1       0.12   0.14       0.16   0.18x, m

               Fig. 8.4. Mach number values predicted with COSMOSFloWorks along the reference line (the
               reference points on it are marked by square boxes with numbers) in comparison with the theoretical

              To show the full flow pattern, the predicted Mach number contours of the channel flow are
              shown in Fig. 8.5..




                                Fig. 8.5. Mach number contours predicted by COSMOSFloWorks.

              This example illustrates that COSMOSFloWorks is capable of capturing shock waves with
              a high degree of accuracy. This high accuracy is possible due to the COSMOSFloWorks
              solution adaptive meshing capability. Solution adaptive meshing automatically refines the
              mesh in regions with high flow gradients such as shocks and expansion fans.




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    Chapter Supersonic Flow in a 2D Convergent-Divergent Channel




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       9   Supersonic Flow over a Segmental Conic Body

              Now let us consider an external supersonic flow of air over a segmental conic body shown
              in Fig. 9.1. The general case is that the body is tilted at an angle of α with respect to the
              incoming flow direction. The dimensions of the body whose longitudinal (in direction t,
              see Fig. 9.1.) and lateral (in direction n) aerodynamic drag coefficients, as well as
              longitudinal (with respect to Z axis) torque coefficient, were investigated in Ref.5 are
              presented in Fig. 9.2. They were determined from the dimensionless body sizes and the
              Reynolds number stated in Ref.5.

                                                                      n




              External air flow                                          y
              M∞ = 1.7


                                                             Center of                   x
                                                             gravity                 a


                                                                                              t



                            Fig. 9.1. Supersonic flow over a segmental conic body.




                          Fig. 9.2. Model sketch dimensioned in centimeters.

              The model of this body is shown in Fig. 9.3..


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    Chapter Supersonic Flow over a Segmental Conic Body


            To compare the COSMOSFloWorks predictions with the experimental data of Ref.5, the
            calculations have been performed for the case of incoming flow velocity of Mach number
            1.7. The undisturbed turbulent incoming flow has a static pressure of 1 atm, static
            temperature of 660.2 K, and turbulence intensity of 1%. The flow Reynolds number of
            1.7×106 (defined with respect to the body frontal diameter) corresponds to these
            conditions, satisfying the Ref.5 experimental conditions.




             Fig. 9.3. The SolidWorks model for calculating the 3D flow over the 3D segmental conic
             body with COSMOSFloWorks.

            To compare the flow prediction with the experimental data of Ref.5, the calculations have
            been performed for the body tilted at α = 0°, 30°, 60°, 90°, 120°, 150° and 180° angles. To
            reduce the computational resources, the Z = 0 flow symmetry plane has been specified in
            all of the calculations. Additionally, the Y = 0 flow symmetry plane has been specified at
            α = 0° and 180°.
            The calculations have been performed at result resolution level 6.
            The comparison is performed on the following parameters:
               • longitudinal aerodynamic drag coefficient,

                                                       Ft
                                             Ct =            ,
                                                    1
                                                      ρU 2 S
                                                    2
            where Ft is the aerodynamic drag force acting on the body in the t direction (see Fig. 9.1.),
            ρU2/2 is the incoming stream dynamic head, S is the body frontal cross section (being
            perpendicular to the body axis) area;



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                 • lateral aerodynamic drag coefficient,
                                                            Fn
                                                Cn =             ,
                                                        1
                                                          ρU 2 S
                                                        2
              where Fn is the aerodynamic drag force acting on the body in the n direction (see Fig.
              9.1.), ρU2/2 is the incoming stream dynamic head, S is the body frontal cross section
              (being perpendicular to the body axis) area;
                 • on the longitudinal (with respect to Z axis) aerodynamic torque coefficient,
                                                           Mz
                                                mz =
                                                        1         ,
                                                          ρU 2 SL
                                                        2
              where Mz is the aerodynamic torque acting on the body with respect to the Z axis (see Fig.
              9.1.), ρU2/2 is the incoming stream dynamic head, S is the body frontal cross section
              (being perpendicular to the body axis) area, L is the reference length.
              The calculation results are presented in Figs.9.4 and 9.5.
               Ct , Cn
                1.6
                                                                                              Ct experiment
                1.4
                1.2                                                                           Ct calculation
                  1                                                                           Cn experiment
                0.8
                                                                                              Cn calculation
                0.6
                0.4
                0.2
                  0
               -0.2
               -0.4
               -0.6
               -0.8
                 -1                                                                      Attack angle
                      0       30         60        90         120        150       180     (degree)


              Fig. 9.4. The longitudinal and lateral aerodynamic drag coefficients predicted with COSMOSFloWorks
              and measured in the experiments of Ref.5 versus the body tilting angle.

              From Fig. 9.4., it is seen that the COSMOSFloWorks predictions of both Cn and Ct are
              excellent.




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      COSMOSFloWorks                                                             4-45
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    Chapter Supersonic Flow over a Segmental Conic Body


            As for the longitudinal aerodynamic torque coefficient (mz) prediction, it is also close to
            the experimental data of Ref.5, especially if we take into account the measurements error.
              mz
              0.08
                                                                                            Experiment
              0.06                                                                          Calculation

              0.04

              0.02

                 0

             -0.02

             -0.04                                                                      Attack angle
                     0       30        60         90        120          150      180     (degree)

            Fig. 9.5. The longitudinal aerodynamic torque coefficient predicted with COSMOSFloWorks and
            measured in the experiments (Ref.5) versus the body tilting angle.


            To illustrate the quantitative predictions with the corresponding flow patterns, the Mach
            number contours are presented in Figs. 9.6, 9.7, and 9.8. All of the flow patterns presented
            on the figures include both supersonic and subsonic flow regions. The bow shock consists
            of normal and oblique shock parts with the subsonic region downstream of the normal
            shock. In the head subsonic region the flow gradually accelerates up to a supersonic velocity
            and then further accelerates in the expansion fan of rarefaction waves. The subsonic wake
            region past the body can also be seen.




                           Fig. 9.6. Mach number contours at   α = 0o.




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                            Fig. 9.7. Mach number contours at α = 60°.




                           Fig. 9.8. Mach number contours at α = 90°.

              As the forward part becomes sharper, the normal part of the bow shock and the
              corresponding subsonic region downstream of it become smaller. In the presented
              pictures, the smallest nose shock (especially its subsonic region) is observed at a = 60o.


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    Chapter Supersonic Flow over a Segmental Conic Body




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       10 Flow over a Heated Plate

              Until now we have only considered flow in or around bodies with adiabatic walls. We will
              now consider flows with other thermal boundary conditions.

              The first example, is a uniform 2D flows with a laminar boundary layer on a heated flat
              plate, see Fig. 10.1.. The incoming uniform air stream has a velocity of 1.5 m/s, a
              temperature of 293.2 K, and a static pressure of 1 atm. Thus, the flow Reynolds number
              defined on the incoming flow characteristics and on the plate length of 0.31 m is equal to

              3.1⋅104, therefore the boundary layer beginning from the plate’s leading edge is laminar
              (see Ref.6).

              Then, let us consider the following three cases:

              Case #1: the plate over its whole length (within the computational domain) is 10°C
              warmer than the incoming air (303.2 K), both the hydrodynamic and the thermal boundary
              layer begin at the plate's leading edge coinciding with the computational domain
              boundary;

              Case #2: the upstream half of the plate (i.e. at x ≤ 0.15 m) has a fluid temperature of 293.2
              K, and the downstream half of the plate is 10°C warmer than the incoming air (303.2 K),
              the hydrodynamic boundary layer begins at the plate's leading edge coinciding with the
              computational boundary;

              Case #3: plate temperature is the same as in case #1, the thermal boundary layer begins at
              the inlet computational domain boundary, whereas the hydrodynamic boundary layer at
              the inlet computational domain boundary has a non-zero thickness which is equal to that
              in case #2 at the thermal boundary layer starting.




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      COSMOSFloWorks                                                             4-49
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    Chapter Flow over a Heated Plate


            The calculation goal is to predict the local coefficient of heat transfer from the wall to the
            fluid, as well as the local skin-friction coefficient.
                                        Computational domain




                      Air flow
                                                                            Heated plate
                      V = 1.5 m/s
                                                                            T = 303.2 K
                      T = 293.2 K



                        Fig. 10.1. Laminar flow over a heated flat plate.


            The SolidWorks model used for calculating the 2D flow over the heated flat plate with
            COSMOSFloWorks is shown in Fig. 10.2.. The problem is solved as internal in order to
            avoid the conflict situation when the external flow boundary with ambient temperature
            conditions intersects the wall with a thermal boundary layer.

            To avoid any influence of the upper wall on the flow near the heated lower wall, the ideal
            wall boundary condition has been specified on the upper wall. To solve the internal
            problem, the incoming fluid velocity is specified at the channel inlet, whereas the fluid
            static pressure is specified at the channel exit. To specify the external flow features, the
            incoming stream's turbulent intensity is set to 1% and the turbulent length is set to 0.01 m,
            i.e., these turbulent values are similar to the default values for external flow problems.




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                                                                    Ideal wall


         Inlet velocity Ux                                                                  Static pressure opening




                             Heated wall



                Fig. 10.2. The SolidWorks model used for calculating the 2D flow over heated flat plate with
                COSMOSFloWorks.


                The heat transfer coefficient h and the skin-friction coefficient Cf are COSMOSFloWorks

                output flow parameters. The theoretical values for laminar flow boundary layer over a flat
                plate, in accordance with Ref.6 can be determined from the following equations:

                                                                  kNu x
                                                             h=            ,
                                                                    x

                where

                        k is the thermal conductivity of the fluid,
                        x is the distance along the wall from the start of the hydrodynamic boundary layer,
                        Nux is the Nusselt number defined on a heated wall as follows:
                         Nux = 0.332 ⋅ Pr1 / 3 Re1 / 2
                                                 x

                        for a laminar boundary layer if it’s starting point coincides with the thermal
                        boundary layer starting point, and

                                 0.332 ⋅ Pr 1 / 3 Re 1 / 2
                      Nu x =                         x
                                  3   1 − ( x0 / x )3 / 4


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    Chapter Flow over a Heated Plate


                for a laminar boundary layer if the thermal boundary layer begins at point x0 lying
                downstream of the hydrodynamic boundary layer starting point, in this case Nux is
                defined at x>x 0 only;

                               µ Cp
            where   Pr    =    ---------- is the Prandtl number, µ is the fluid dynamic viscosity, Cp is the
                                   k
                                                                    ρVx      -
                                                                    ---------- is the Reynolds number
            fluid specific heat at constant pressure,   R ex   =
                                                                        µ
            defined on x, ρ is the fluid density, and V is the fluid velocity;


                      0, 664 at Re ≤ 5 ⋅105 , i.e., with a laminar boundary layer.
             C fx =               x
                         Re

            As for the hydrodynamic boundary layer thickness δ needed for specification at the
            computational domain boundary in case #3, in accordance with Ref.6, it has been

            determined from the following equation: δ = 4.64 ⋅ x / Re0.5 , so δ = 0.00575 m in this
                                                                     x

            case. For these calculations all fluid parameters are determined at the outer boundary of
            the boundary layer.

            The COSMOSFloWorks predictions of h and Cf performed at result resolution level 7, and

            the theoretical curves calculated with the formulae presented above are shown in Figs.10.3
            and 10.4. It is seen that the COSMOSFloWorks predictions of the heat transfer coefficient
            and the skin-friction coefficient are in excellent agreement with the theoretical curves.




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               h (W/m^2/K)
               25
                                                                                            Case #1, theory
                                                                                            Case #1, calculation
               20
                                                                                            Case #2, theory
                                                                                            Case #2, calculation
               15
                                                                                            Case #3, theory

               10                                                                           Case #3, calculation


                5


                0
                                                                                            X (m)
                    0       0.05       0.1        0.15       0.2        0.25       0.3

              Fig. 10.3. Heat transfer coefficient change along a heated plate in a laminar boundary layer:
              COSMOSFloWorks predictions compared to theory.




               Cf
                0.02
                                                                                            Cases #1 and #2,
               0.018                                                                        theory
                                                                                            Case #1, calculation
               0.016
               0.014                                                                        Case #2, calculation
               0.012
                                                                                            Case #3, theory
                0.01
                                                                                            Case#3, calculation
               0.008
               0.006
               0.004
               0.002
                    0
                        0     0.05       0.1       0.15       0.2       0.25       0.3       X (m)

              Fig. 10.4. Skin-friction coefficient change along a heated plate in a laminar boundary layer:
              COSMOSFloWorks predictions compared to theory.




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    Chapter Flow over a Heated Plate




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       11 Convection and Radiation in an Annular Tube

               We will now consider incompressible laminar flow in a portion of an annular tube, whose
               outer shell is a heat source having constant heat generation rate Q1 with a heat-insulated
               outer surface, and whose central body fully absorbs the heat generated by the tube’s outer
               shell (i.e. the negative heat generation rate Q2 is specified in the central body); see Fig.
               11.1.. (The tube model is shown in Fig. 11.2.). We will assume that this tube is rather long,
               so the tube's L=1 m portion under consideration has fully developed fluid velocity and
               temperature profiles at the inlet, and, since the fluid properties are not
               temperature-dependent, the velocity profile also will not be temperature-dependent.
                     Y                                                          Y
                                                     P = 1 atm                          Q1 or T1

             U, T
                                                                       Q2 or T2
                                                      X                                        X




                                   1m                            ∅0.4m

                                                                         ∅1.2m
                                                                         ∅1.4 m


                            Fig. 11.1. Laminar flow in a heated annular tube.




             Fig. 11.2. A model created for calculating 3D flow within a heated annular tube using
             COSMOSFloWorks.




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    Chapter Convection and Radiation in an Annular Tube


            To validate the COSMOSFloWorks capability for solving conjugate heat transfer
            problems both with and without radiation, let us solve the following three problems: 1) a
            conjugate heat transfer problem with convection only, 2) radiation heat transfer only
            problem, and 3) a conjugate heat transfer problem with both convection and radiation.
            In the first problem we specify Q2 = -Q1, so the convective heat fluxes at the tube inner
            and outer walls are constant along the tube. The corresponding laminar annular pipe flow's
            fully developed velocity and temperature profiles, according to Ref.6, are expressed
            analytically as follows:

                        r  2   r  2  ln( r / r ) 
                                 1
            u(r) = ϕ ⋅   −   − 1
                                                     2
                                                         − 1 ,
                         r   r        ln( r1 / r2 ) 
                        2    2      
                                                           

                         q2      r
            T(r)= T2 −      r2 ln  ,
                                 r 
                         k        2
                                                  2u
            where ϕ =                     
                                   2                                    2
                          r1                                  r 
                          
                           r        − 1  / ln( r1 / r 2 ) −  1 
                                                                 r        − 1   ,
                          2                                  2 
                                          

            u is the fluid velocity,
            T is the fluid temperature,
            r is the radial coordinate,
            r1 and r2 are the tube outer shell’s inner radius and tube’s central body radius,
            respectively,

            u is the volume-average velocity, defined as the volume flow rate divided by the tube
            cross-section area,
            q2 is the the heat flux from the fluid to the tube’s central body,
            k is the fluid thermal conductivity,
            T2 is the surface temperature of the central body.
            The heat flux from the fluid to the tube's central body (negative, since the heat comes from
            the fluid to the solid) is equal to

                                             q2 = k 
                                                      ∂T            Q2
                                                               =
                                                     ∂r  r = r2 2π ⋅ r2 ⋅ L




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              Let Q1 = - Q2 = 107.235 W and u ≈13.59 m/s (ϕ = -10 m/s), the fluid has the following
              properties: k = 0.5 W/(m⋅K), Cp = 500 J/(kg⋅K), µ = 0.002 Pa⋅s, ρ = 0.1 kg/m3. Since the
              corresponding (defined on the equivalent tube diameter) Reynolds number Red ≈ 815 is
              rather low, the flow has to be laminar. We specify the corresponding velocity and
              temperature profiles as boundary conditions at the model inlet and as initial conditions,
              and Pout = 1 atm as the tube outlet boundary condition.
              To reduce the computational domain, let us set Y=0 and X=0 flow symmetry planes
              (correspondingly, the specified Q1 and Q2 values are referred to the tube section's quarter
              lying in the computational domain). The calculation have been performed at result
              resolution level 7.
              The fluid temperature profile predicted at 0.75 m from the tube model inlet is shown in
              Fig. 11.3. together with the theoretical curve.
                T
                500
                                                                                                    Theory
                475                                                                                 Calculation
                450
                425
                400
                375
                350
                325
                300
                275
                250
                      0      0.1       0.2      0.3       0.4      0.5      0.6       0.7    Y, m


              Fig. 11.3. Fluid temperature profiles across the tube in the case of convection only, predicted with
              COSMOSFloWorks and compared to the theoretical curve.

              It is seen that this prediction practically coincides with the theoretical curve.
              Before solving the third problem coupling convection and radiation, let us determine the
              radiation heat fluxes between the tube's outer and inner walls under the previous problem's
              wall temperatures. In addition to holding the outer shell's temperature at 450 K and the
              central body's temperature at 300 K as the volume sources, let us specify the emissivity of
              ε1 = 0.95 for the outer shell and ε2 = 0.25 for the central body. To exclude any convection,
              let us specify the liquid velocity of 0.001 m/s and thermal conductivity of 10-20 W/(m⋅K).




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    Chapter Convection and Radiation in an Annular Tube


            Let J2 denote the radiation rate leaving the central body, and G2 denotes the radiation rate
            coming to the central body, therefore Q2r = J2 - G2 (the net radiation rate from the central
            body). In the same manner, let J1 denote the radiation rate leaving the outer shell's inner
            surface, and G1 denote the radiation rate coming to the outer shell's inner surface,
            therefore Q1r = J1 - G1 (the net radiation rate from the outer shell's inner surface). These
            radiation rates can be determined by solving the following equations:
                  J2 = A2σε2T24 + G2(1-ε2),
                  G2 = J1F1-2 ,
                  J1 = A1σε1T14 + G1(1-ε1),
                  G1 = J2F2-1 + J1F1-1 ,

            where σ=5.669⋅10-8 W/m2⋅K4 is the Stefan-Boltzmann constant, F1-2 , F2-1, F1-1 are these
            surfaces' radiation shape factors, under the assumption that the leaving and incident
            radiation fluxes are uniform over these surfaces, Ref.6 gives the following formulas:

            F1-2=(1/X) - (1/π/X){arccos(B/A) - (1/2/Y)[(A2 + 4A - 4X2 + 4)1/2arccos(B/X/A) +
            + Barcsin(1/X)-πA/2]},

            F1-1=1-(1/X)+(2/ π/X)arctan[2(X2-1)1/2/Y]-(Y/2/π/X){[(4X2+Y2)1/2 /Y]arcsin{[4(X2-1)+
             + (Y/X)2(X2-2)]/[Y2+4(X2-1)]}-arcsin[(X2-2)/X2]+(π/2)[(4X2+Y2)1/2/Y-1]}

            F2-1 = F1-2⋅A1/A2, where X=r1/r2, Y=L/r2, A=X2+Y2-1, B=Y2-X2+1.
            These net and leaving radiation rates (over the full tube section surface), both calculated
            by solving the equations analytically and predicted by COSMOSFloWorks at result
            resolution level 7, are presented in Table 2:.
    Table 2: Radiation rates predicted with COSMOSFloWorks with comparison
                               to the theoretical values.
                                                        COSMOSFloWorks predictions
             Parameter     Theory (Ref.6), W            Value, W    Prediction error,%
                Q2 r           -383.77                   -388.30            1.2%
                J2 r           1728.35                   1744.47            0.9%
                Q1 r           4003.68                   3931.87           -1.8%
                J1 r           8552.98                   8596.04            0.5%

            It is seen that the prediction errors are quite small. To validate the COSMOSFloWorks
            capabilities on the third problem, which couples convection and radiation, let us add the
            theoretical net radiation rates, Q1 r and Q2 r scaled to the reduced computational domain,
            i.e., divided by 4, to the Q1 and Q2 values specified in the first problem. Let us specify Q1
            = 1108.15 W and Q2 = -203.18 W, so theoretically we must obtain the same fluid
            temperature profile as in the first considered problem.


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              The fluid temperature profile predicted at 0.75 m from the tube model inlet at the result
              resolution level 7 is shown in Fig. 11.4. together with the theoretical curve. It is seen that
              once again this prediction virtually coincides with the theoretical curve.
               T, K
               475
                                                                                                 Theory
               450                                                                               Calculation

               425

               400

               375

               350

               325

               300

               275
                      0     0.1       0.2      0.3      0.4      0.5      0.6       0.7     Y, m

              Fig. 11.4. Fluid temperature profiles across the tube in the case coupling convection and radiation,
              predicted with COSMOSFloWorks and compared to the theoretical curve.




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    Chapter Convection and Radiation in an Annular Tube




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       12 Heat Transfer from a Pin-fin Heat Sink

                Heat sinks play an important role in electronics cooling. Following the experimental work
                presented in Ref.19 and numerical study presented in Ref.21, let us consider heat transfer
                from an electrically heated thermofoil which is mounted flush on a plexiglass substrate,
                coated by an aluminum pin-fin heat sink with a 9×9 pin fin array, and placed in a closed
                plexiglass box. In order to create more uniform ambient conditions for this box, it is
                placed into another, bigger, plexiglass box and attached to the heat-insulated thick wall,
                see Figs. 12.1, 12.2. Following Ref.19, let us consider the vertical position of these boxes,
                as it is shown in Fig. 12.1. (c) (here, the gravity acts along the Y axis).




         Fig. 12.1. The pin-fin heat sink nestled within two plexiglass boxes: lcp=Ls=25.4 mm, hcp=0.861 mm, Hp=5.5
         mm, Hb=1.75 mm, Sp=1.5 mm, Sps = Ls/8, L=127 mm, H=41.3 mm, Hw=6.35 mm (from Ref.19).




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    Chapter Heat Transfer from a Pin-fin Heat Sink


             The corresponding model used in the calculations is shown in Fig. 12.2.. In this model's
             coordinate system the gravitational acceleration vector is directed along the X axis. The
             computational domain envelopes the outer surface of the external box, and the Z=0
             symmetry plane is used to reduce the required computer resources.




                          a                                                             b
          Fig. 12.2. A model created for calculating the heat transfer from the pin-fin heat sink through the two
          nested boxes into the environment: (a) the internal (smaller) box with the heat sink; (b) the whole model.


             According to Ref.19, both the heat sink and the substrate are coated with a special black
             paint to provide a surface emissivity of 0.95 (the other plexiglass surfaces are also opaque,
             diffuse and gray, but have an emissivity of 0.83).
             The maximum steady-state temperature Tmax of the thermofoil releasing the heat of
             known power Q was measured. The constant ambient temperature Ta was measured at the
             upper corner of the external box. As a result, the value of
                   Rja = (Tmax - Ta)/Q (12.1)
             was determined at various Q (in the 0.1...1 W range).
             The ambient temperature is not presented in Ref.19, so, proceeding from the suggestion
             that the external box in the experiment was placed in a room, we have varied the ambient
             temperature in the relevant range of 15...22ºC. Since Rja is governed by the temperature
             difference Tmax - Ta, (i.e. presents the two boxes’ thermal resistance), the ambient
             temperature range only effects the resistance calculations by 0.6ºC/W at Q = 1W, (i.e. by
             1.4% of the experimentally determined Rja value that is 43ºC/W). As for the boundary
             conditions on the external box’s outer surface, we have specified a heat transfer coefficient
             of 5.6 W/m2 K estimated from Ref. 20 for the relevant wind-free conditions and an
             ambient temperature lying in the range of 15...22ºC (additional calculations have shown
             that the variation of the constant ambient temperature on this boundary yield nearly
             identical results). As a result, at Q = 1W (the results obtained at the other Q values are
             shown in Ref.21) and Ta =20ºC we have obtained Rja = 41ºC/W, i.e. only 5% lower than
             the experimental value.

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              The flow streamlines visualized in Ref.19 using smoke and obtained in the calculations
              are shown in Fig. 12.3..




                 Fig. 12.3. Flow streamlines visualized by smoke in the Ref.19 experiments (left) and
                 obtained in the calculations (colored in accordance with the flow velocity values) (right).




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    Chapter Heat Transfer from a Pin-fin Heat Sink




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       13 Unsteady Heat Conduction in a Solid

              Until now, we have studied various flow problems including those with heat flow from the
              model walls to the fluid, but we have not considered heat conduction in solids (i.e., a
              conjugate heat transfer). To validate this capability, let us consider unsteady heat
              conduction in a solid. To compare the COSMOSFloWorks predictions with the analytical
              solution (Ref.6), we will solve a one-dimensional problem.
              A warm solid rod having the specified initial temperature and the heat-insulated side
              surface suddenly becomes and stays cold (at a constant temperature of T=300 K) at both
              ends (see Fig. 13.1.). The rod inner temperature evolution is studied. The constant initial
              temperature distribution along the rod is considered: Tinitial (x)=350K.


                                     Tinitial = 350 K
                                                                                          T = 300 K
            T = 300 K


                                                                                   X
                                                    L



              Fig. 13.1. A warm solid rod cooling down from an initial temperature to the temperature at
              the ends of the rod.


              The problem is described by the following differential equation:
                                                  ∂ 2T ρC ∂T ,
                                                       =
                                                  ∂x 2   k ∂τ
              where ρ, C, and k are the solid material density, specific heat, and thermal conductivity,
              respectively, and τ is the time, with the following boundary condition: T=T0 at x = 0 and
              at x = L.
              In the general case, i.e., at an arbitrary initial condition, the problem has the following
              solution:
                                            ∞
                                                                               nπ x ,
                              T = T0 + Cne − ( nπ / L) kτ /( ρC ) sin
                                           ∑
                                                               2


                                           n =1
                                                                                 L
              where coefficients Cn are determined from the initial conditions (see Ref.6).
              With the uniform initial temperature profile, according to the initial and boundary
              conditions, the problem has the following solution:

                                          4 ∞ 1 −[nπ / L ]2 kτ /( ρC )      nπx (K).
                        T = 300 + 50        ∑n e
                                          π n =1
                                                                       sin(
                                                                             L
                                                                               )

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    Chapter Unsteady Heat Conduction in a Solid


            To perform the time-dependent analysis with COSMOSFloWorks, a SolidWorks model
            representing a solid parallelepiped with dimensions 1×0.2×0.1 m has been created (see
            Fig. 13.2.).




                    Fig. 13.2. The SolidWorks model used for calculating heat conduction in a solid rod with
                    COSMOSFloWorks (the computational domain envelopes the rod).



            The evolution of maximum rod temperature, predicted with COSMOSFloWorks and
            compared with theory, is presented in Fig. 13.3.. The COSMOSFloWorks prediction has
            been performed at result resolution level 5. One can see that it coincide with the
            theoretical curve.
                                355
                                                                                                   Theory

                                345                                                                Calculation


                                335
              Temperature (K)




                                325



                                315



                                305
                                                                                             Physical time (s)
                                      0   2000   4000        6000        8000        10000


            Fig. 13.3. Evolution of the maximum rod temperature, predicted with COSMOSFloWorks and
            compared to theory.


            The temperature profiles along the rod at different time moments, predicted by
            COSMOSFloWorks, are compared to theory and presented in Fig. 13.4.. One can see that
            the COSMOSFloWorks predictions are very close to the theoretical profiles. The
            maximum prediction error not exceeding 2K occurs at the ends of the rod and is likely
            caused by calculation error in the theoretical profile due to the truncation of Fourier series.


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                              325
                                                                                        t=5000s/theory
                                                                                        t=5000s/calculation
                              320
                                                                                        t=10000s/theory
                Temperature (K)                                                         t=10000s/calculation
                              315


                              310


                              305


                              300
                                    0   0.2   0.4        0.6         0.8         1 X (m)

              Fig. 13.4. Evolution of the temperature distribution along the rod, predicted with COSMOSFloWorks
              and compared to theory.




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    Chapter Unsteady Heat Conduction in a Solid




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       14 Tube with Hot Laminar Flow and Outer Heat Transfer

              Let us now consider an incompressible laminar flow of hot fluid through an externally
              cooled circular tube (Fig. 14.1.). The fluid flow has fully developed velocity and
              temperature profiles at the tube inlet, whereas the heat transfer conditions specified at the
              tube outer surface surrounded by a cooling medium sustain the self-consistent fluid
              temperature profile throughout the tube.
                                                               r
               Polystyrene                                                     Te(z)
                                                                               αe = const



               Liquid


               Laminar flow




                  Fig. 14.1. Laminar flow in a tube cooled externally.


              In accordance with Ref.6, a laminar tube flow with a fully developed velocity profile has a
              self-consistent fully developed temperature profile if the following two conditions are
              satisfied: the fluid's properties are temperature-independent and the heat flux from the
              tube inner surface to the fluid (or vise versa) is constant along the tube. These conditions
              provide the following fully developed tube flow temperature profile:

                                          q R        r 2 1  r  4  4q ⋅ ( z − z )
              T(r, z) = T(r=0, z=zinlet) - w i        −    + w                inlet
                                                                                          ,
                                           k         Ri  4  Ri      ρ C pumax Ri
                                                                     
              where
              T is the fluid temperature,
              r is a radial coordinate (r = 0 corresponds to the tube axis, r = Ri corresponds to the tube
              inner surface, i.e., Ri is the tube inner radius),
              z is an axial coordinate (z = zinlet corresponds to the tube inlet),

              qw is a constant heat flux from the fluid to the tube inner surface,
              k is the fluid thermal conductivity,
              ρ   is the fluid density,
              Cp is the fluid specific heat under constant pressure,


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    Chapter Tube with Hot Laminar Flow and Outer Heat Transfer


             umax is the maximum fluid velocity of the fully developed velocity profile

                                              2
                                        r  .
                       u ( r ) = umax 1 −   
                                        Ri  
                                            
                                               
            Since the tube under consideration has no heat sinks and is cooled by surrounding fluid
            medium, let us assume that the fluid medium surrounding the tube has certain fixed
            temperature Te, and the heat transfer between this medium and the tube outer surface is
            determined by a specified constant heat transfer coefficient αe.
            By assuming a constant thermal conductivity of the tube material, ks, specifying an
            arbitrary αe, and omitting intermediate expressions, we can obtain the following
            expression for Te:

                                                                3         1    1   R  4 q w ⋅  z − zinlet 
                                                                                                
                                                                                                
                                                                                                             
                                                                                                             
             T e (z ) = T ( r = 0 , z = z           ) − q w Ri     +         −   ln i  +                       ,
                                            inlet               4k
                                                                       α e Ro k s R o 
                                                                                          ρC p u max Ri

            where Ro is the tube outer radius.

            In the validation example under consideration (Fig. 14.2.) the following tube and fluid
            characteristics have been specified: Ri = 0.05 m, Ro = 0.07 m, z - zinlet = 0.1 m, the tube
            material is polystyrene with thermal conductivity ks = 0.082 W/(m⋅K), umax = 0.002 m/s,
            T(r=0, z=zi) = 363 K, qw = 147.56 W/m2, k = 0.3 W/(m⋅K), Cp = 1000 J/(kg⋅K), fluid
            dynamic viscosity µ = 0.001 Pa⋅s, ρ = 1000 kg/m3 (these fluid properties provide a
            laminar flow condition since the tube flow Reynolds number based on the tube diameter is
            equal to Red = 100). The T(r,zinlet) and u(r) profiles at the tube inlet, the Te(z) distribution
            along the tube, αe = 5 W/(m2⋅K), and tube outlet static pressure Pout = 1 atm have been
            specified as the boundary conditions.
            The inlet flow velocity and temperature profiles have been specified as the initial
            conditions along the tube.
            To reduce the computational domain, the calculations have been performed with the Y=0
            and X=0 flow symmetry planes. The calculations have been performed at result resolution
            level 7.
            The fluid and solid temperature profiles predicted at z = 0 are shown in Fig. 14.3. together
            with the theoretical curve. It is seen that the prediction practically coincides with the
            theoretical curve (the prediction error does not exceed 0.4%).




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                  Sketch line for                                                                Outlet static
                  temperature profile                                                            pressure opening
                  determination




                                                                                                 Computational
               Inlet velocity                                                                    domain
               opening




              Fig. 14.2. The model used for calculating the 3D flow and the conjugate heat transfer in the tube
              with COSMOSFloWorks.




                   T, K
                   370
                                                                                               Theory liquid
                   360
                                                                                               Theory solid
                   350                                                                         Calculation
                   340
                   330
                   320
                   310
                   300
                   290
                          0     0.01    0.02   0.03      0.04   0.05    0.06     0.07   R, m


              Fig. 14.3. Fluid and solid temperature profiles across the tube, predicted with COSMOSFloWorks
              and compared with the theoretical curve.




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    Chapter Tube with Hot Laminar Flow and Outer Heat Transfer




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       15 Flow over a Heated Cylinder

              Let us now return to the earlier validation example of incompressible flow over a cylinder
              and modify it by specifying a heat generation source inside the cylinder (see Fig. 15.1.).
              The cylinder is placed in an incoming air stream and will acquire certain temperature
              depending on the heat source power and the air stream velocity and temperature.

                                                                  Y

              External air flow

                                                                                     X

                                                         Heat source q




                                        Fig. 15.1. 2D flow over a heated cylinder.



              Based on experimental data for the average coefficient of heat transfer from a heated
              circular cylinder to air flowing over it (see Ref.6), the corresponding Nusselt number can
              be determined from the following formula:

                                              NuD = C ⋅ (Re D )n ⋅ Pr1 3 ,

              where constants C and n are taken from the following table:

                                            ReD                   C                  n

                                  0.4 - 4                    0.989           0.330
                                  4 - 40                     0.911           0.385
                                  40 - 4000                  0.683           0.466
                                  4000 - 40000               0.193           0.618

                                  40000 - 400000             0.0266          0.805

              Here, the Nusselt number, NuD = (h×D)/k (where h is the heat transfer coefficient
              averaged over the cylinder, and k is fluid thermal conductivity), the Reynolds number,
              ReD = (U×D)/µ (where U is the incoming stream velocity, and µ is fluid dynamic
              viscosity), and the Prandtl number, Pr=µ×Cp/k (where µ is fluid dynamic viscosity, Cp is
              fluid specific heat at constant pressure, and k is fluid thermal conductivity) are based on
              the cylinder diameter D and on the fluid properties taken at the near-wall flow layer.
              According to Ref.6, Pr = 0.72 for the entire range of ReD.

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    Chapter Flow over a Heated Cylinder


            To validate the COSMOSFloWorks predictions, the air properties have been specified to
            provide Pr = 0.72: k = 0.0251375 W/(m×K), µ = 1.8×10-5 Pa×s, specific heat at constant
            pressure Cp = 1005.5 J/(kg×K). Then, the incoming stream velocity, U, has been specified
            to obtain ReD = 1, 10, 100, 10 3, 104, 5×104, 105, 2×105, and 3×105 for a cylinder
            diameter of D = 0.1 m (see Table 3:).
            This validation approach consists of specifying the heat generation source inside the
            cylinder with a power determined from the desired steady-state cylinder temperature and
            the average heat transfer coefficient, h = (NuD×k)/D. NuD is determined from the
            specified ReD using the empirical formula presented above. The final cylinder surface
            temperature, that is also required for specifying the heat source power Q (see Table 3:) is
            assumed to be 10°C higher than the incoming air temperature. The initial cylinder
            temperature and the incoming air temperature are equal to 293.15 K. The cylinder material
            is aluminum. Here, the heat conduction in the solid is calculated simultaneously with the
            flow calculation, i.e., the conjugate heat transfer problem is solved.
            As a result of the calculation, the cylinder surface has acquired a steady-state temperature
            differing from the theoretical one corresponding to the heat generation source specified
            inside the cylinder. Multiplying the theoretical value of the Nusselt number by the ratio of
            the obtained temperature difference (between the incoming air temperature and the
            cylinder surface temperature) to the specified temperature difference, we have determined
            the predicted Nusselt number versus the specified Reynolds number. The values obtained
            by solving the steady-state and time dependent problems at result resolution level 5 are
            presented in Fig. 15.2. together with the experimental data taken from Ref.6.
         Table 3: The COSMOSFloWorks specifications of U and Q for the problem under
                                     consideration.


                                             ReD       U , m /s       Q ,W
                                                                 -4
                                               1       1.5× 10        0.007
                                                                 -3
                                              10       1.5× 10        0.016
                                                   2
                                             10         0.0 15        0.041
                                                   3
                                             10         0.1 5         0.121
                                                   4
                                             10          1. 5         0.405
                                                   5
                                             10           15          1.994

            From Fig. 15.2., it is seen that the predictions made with COSMOSFloWorks, both in the
            time-dependent approach and in the steady-state one, are excellent within the whole ReD
            range under consideration.




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                 Nu D
                1000
                                                                                             Ca lcula tion,
                                                                                             ste a dy-sta te
                                                                                             Ca lcula tion,
                 100                                                                         tim e -de pe nde nt



                   10



                    1



                  0.1
                   1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 Re D

              Fig. 15.2. Nusselt number for air flow over a heated cylinder: COSMOSFloWorks predictions and the
              experimental data taken from Ref.6.




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    Chapter Flow over a Heated Cylinder




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       16 Natural Convection in a Square Cavity

              In the previous validation examples we have considered forced convection, i.e., heat
              transfer between a wall and a fluid while the fluid flow is caused by some driving force
              other than temperature gradient, so that its characteristics are specified via the flow
              parameters at model openings (internal problems) or at far-field boundaries (external
              problems). Let us now consider a heat transfer due to a heat-induced natural convection
              within a closed cavity.
              Here we will consider a 2D square cavity with a steady-state natural convection, for which
              a highly-accurate numerical solution has been proposed in Ref.10 and used as a
              benchmark for about 40 computer codes in Ref.11, besides it well agrees with the
              semi-empirical formula proposed in Ref.12 for rectangular cavities. This cavity's
              configuration and imposed boundary conditions, as well as the used coordinate system,
              are presented in Fig. 16.1.. Here, the left and right vertical walls are held at the constant
              temperatures of T1 = 305 K and T2 = 295 K, accordingly, whereas the upper and bottom
              walls are adiabatic. The cavity is filled with air.
                                              Y


                                                  305 K              g

                                                  Adiabatic walls


                                                          295 K
                                                                            X

                                                          L

                           Fig. 16.1. An enclosed 2D square cavity with natural convection.

              The square cavity's side dimension, L, is varied within the range of 0.0111...0.111 m in
              order to vary the cavity's Rayleigh number within the range of 103…106. Rayleigh
              number descibes the characteristics of the natural convection inside the cavity and is
              defined as follows:

                    βgρ 2C p L3∆T
               Ra =                       ,
                         kµ

              where   β = 1T      is the volume expansion coefficient of air,
                      g is the gravitational acceleration,
                      Cp is the air's specific heat at constant pressure,
                      ∆T=T1 - T2 = 10 K is the temperature difference between the walls,

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    Chapter Natural Convection in a Square Cavity


                   k is the thermal conductivity of air,
                   µ is the dynamic viscosity of air.
            The cavity's model is shown in Fig. 16.2..




                     Fig. 16.2. The model created for calculating the 2D natural convection
                     flow in the 2D square cavity using COSMOSFloWorks.

            Due to gravity and different temperatures of the cavity's vertical walls, a steady-state
            natural convection flow (vortex) with a vertical temperature stratification forms inside the
            cavity. The Ra = 105 flow's prediction performed with COSMOSFloWorks is shown in
            Fig. 16.3..




              Fig. 16.3. The temperature, X-velocity, Y-velocity, the velocity vectors, and the streamlines,
              predicted by COSMOSFloWorks in the square cavity at Ra = 105.


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              A quantitative comparison of the COSMOSFloWorks predictions performed at result
              resolution level 8 with References.10, 11 (computational benchmark) and 12
              (semi-empirical formula) for different Ra values is presented in Figs.16.4 - 16.6. The
              Nusselt number averaged over the cavity's hot vertical wall (evidently, the same value
              must be obtained over the cavity's cold vertical wall)   Nuav = qwav ⋅ L /(∆T ⋅ k ) , where
              qw av is the heat flux from the wall to the fluid, averaged over the wall, is considered in
              Fig. 16.4..
              Here, the dash line presents the Ref.12 semi-empirical formula

               Nuav = 0.28 ⋅ Ra1/ 4 ( L / D) −1/ 4 ,
              where D is the distance between the vertical walls and L is the cavity height (D=L in the
              case under consideration). One can see that the COSMOSFloWorks predictions practically
              coincide with the benchmark at Ra ≤ 105 and are close to the semi-empirical data.
                Nuav
               10
                                                                                             Refs.10, 11
                 9                                                                           Ref.12
                 8                                                                           Calculation
                 7
                 6
                 5
                 4
                 3
                 2
                 1
                 0
                                                                                        Ra
                1.E+03              1.E+04             1.E+05              1.E+06

              Fig. 16.4. The average sidewall Nusselt number vs. the Rayleigh number.




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    Chapter Natural Convection in a Square Cavity


            The dimensionless velocities of the natural convection flow in the X and Y directions,
                 U ⋅ L ⋅ C p ⋅ ρ and        V ⋅ L ⋅Cp ⋅ ρ
             U=                        V =                  (which are maximum along the cavity's
                        k                          k
            mid-planes, i.e., U max along the vertical mid-plane and Vmax along the horizontal

                                                                                                x and
            mid-plane) are considered in Fig. 16.5.. The dimensionless coordinates,
                                                                                           x=
                                                                                                L
                                  y , of these maximums' locations (i.e., y for U and x for V ) are
             y=                                                                  max         max
                                  L
            presented in Fig. 16.6.. One can see that the COSMOSFloWorks predictions of the natural
            convection flow's local parameters are fairly close to the benchmark data at Ra ≤ 105.
                         1000
                                                                                        Vmax, Refs.10, 11
               Vmax




                                                                                        Vmax, calculation
                                                                                        Umax, Refs.10, 11
               max,




                                 100
                                                                                        Umax, calculation
               Dimensionless U




                                 10




                                   1
                                                                                      Ra
                                  1.E+03        1.E+04         1.E+05        1.E+06

            Fig. 16.5. Dimensionless maximum velocities vs. Rayleigh number.




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                                               1
                                                                                           Y Umax, Refs.10, 11
                                              0.9

                Dimensionless X Vmax, YUmax
                                                                                           Y Umax, calculation
                                              0.8
                                                                                           X Vmax, Refs.10, 11
                                              0.7
                                              0.6                                          X V max,

                                              0.5
                                              0.4
                                              0.3
                                              0.2
                                              0.1
                                                0
                                               1.E+03   1.E+04   1.E+05      1.E+06        Ra

              Fig. 16.6. Dimensionless coordinates of the maximum velocities' locations.




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    Chapter Natural Convection in a Square Cavity




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       17 Particles Trajectories in Uniform Flows

              Let us now consider the COSMOSFloWorks capability to predict particles trajectories in a
              gas flow (i.e. two-phase flow of fluid + liquid droplets or solid particles).
              In accordance with the particles motion model accepted in COSMOSFloWorks, particle
              trajectories are calculated after completing a fluid flow calculation (which can be either
              steady or time-dependent). That is, the particles mass and volume flow rates are assumed
              substantially lower than those of the fluid stream, so that the influence of particles’
              motions and temperatures on the fluid flow parameters is negligible, and motion of the
              particles obeys the following equation:

                                  dV p    ρ f (V f − Vp ) ⋅ V f − V p           ,
                              m        =−                             Cd A + Fg
                                   dt                  2
              where m is the particle mass, t is time, Vp and Vf are the particle and fluid velocities
              (vectors), accordingly, ρf is the fluid density, Cd is the particle drag coefficient, A is the
              particle frontal surface area, and Fg is the gravitational force.
              Particles are treated as non-rotating spheres of constant mass and specified (solid or
              liquid) material, whose drag coefficient is determined from Henderson’s semi-empirical
              formula (Ref.7). At very low velocity of particles with respect to carrier fluid (i.e., at the
              relative velocity’s Mach number M → 0) this formula becomes
                                    24             4.12
                           Cd =        +                         + 0.38
                                    Re 1 + 0 .03 ⋅ Re + 0 .48 Re
              where Reynolds number is defined as
                                                       ρ f V f − Vp d ,
                                               Re =
                                                              µ
              d is the diameter of particles, and µ is the fluid dynamic viscosity.
              To validate COSMOSFloWorks, let us consider three cases of injecting a particle
              perpendicularly into an incoming uniform flow, Fig. 17.1.. Since both the fluid flow and
              the particle motion in these cases are 2D (planar), we will solve a 2D (i.e. in the XY-plane)
              flow problem.
              Uniform
              fluid flow



                                                     Particle injection

                       Fig. 17.1. Injection of a particle into a uniform fluid flow.



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    Chapter Particles Trajectories in Uniform Flows


             Due to the same reason as in the previous validation examples with flow over flat plates,
             we will solve this validation as an internal problem. The corresponding SolidWorks model
             is shown in Fig. 17.2.. Both of the walls are ideal, the channel has length of 0.233 m and
             height of 0.12 m, all the walls have thickness of 0.01 m. We specify the uniform fluid
             velocity Vinlet, the fluid temperature of 293.2 K, and the default values of turbulent flow
             parameters with the laminar boundary layer at the channel inlet, and the static pressure of
             1 atm at the channel outlet. All the fluid flow calculations are performed at a result
             resolution level of 5.
                  Inlet                                                                  Ideal Wall




         Origin and particle
                                                                                                          Outlet
         injection point


                                                   Fig. 17.2. The model.


             To validate calculations of particles trajectories by comparing them with available
             analytical solutions of the particle motion equation, we consider the following three cases:
                 a) the low maximum Reynolds number of Re max = 0.1 (air flow with Vinlet = 0.002
                    m/s, gold particles of d = 0.5 mm, injected at the velocity of 0.002 m/s
                    perpendicularly to the wall),
                 b) the high maximum Reynolds number of Re max = 105 (water flow with Vinlet = 10
                    m/s, iron particles of d = 1 cm, injected at the velocities of 1, 2, 3 m/s
                    perpendicularly to the wall),
                 c) a particle trajectory in the Y-directed gravitational field (gravitational acceleration
                    gy = -9.8 m/s2, air flow with Vinlet = 0.6 m/s, an iron particle of d = 1 cm, injected
                    at the 1.34 m/s velocity at the angle of 63.44o with the wall).
             In the first case, due to small Re values, the particle drag coefficient is close to Cd=24/Re
             (i.e., obeys the Stokes law). Then, neglecting gravity, we obtain the following analytical
             solution for the particle trajectory:

                                                         d 2ρp                                       18µ
                      X (t ) = X          + V fx ⋅ t +           (V px           − V fx ) ⋅ exp( −         t) ,
                                     t =0
                                                         18µ              t =0                       d 2ρp

                                                       d 2ρ p                                      18µ
                      Y (t ) = Y        + V fy ⋅ t +            (V py          − V fy ) ⋅ exp( −          t) ,
                                   t =0
                                                       18µ              t =0                       d 2ρ p




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              where Vfx, Vpx, Vfy, Vpy are the X- and Y-components of the fluid and particle velocities,
              accordingly, ρp is the particle material density. The COSMOSFloWorks calculation and
              the analytical solution are shown in Fig. 17.3.. It is seen that they are very close to one
              another. Special calculations have shown that the difference is due to the CD assumptions
              only.

               Y (m)
               0.035
                                                                                                               Analytical
               0.030                                                                                           solution
                                                                                                               Calculation
               0.025

               0.020

               0.015

               0.010

               0.005

               0.000
                    0.00               0.05           0.10         0.15               0.20             0.25 X (m)

              Fig. 17.3. Particle trajectories in a uniform fluid flow at Re max = 0.1, predicted by COSMOSFloWorks
              and obtained from the analytical solution.

              In the second case, due to high Re values, the particle drag coefficient is close to Cd=0.38.
              Then, neglecting the gravity, we obtain the following analytical solution for the particle
              trajectory:

                                                      ρ pd                                       0.285ρ
                  Y (t ) = Y          + V fy ⋅ t +            (V py          − V fy ) ⋅ ln(1 +          t)      ,
                               t =0
                                                     0.285ρ           t =0                         ρ pd

                                                       ρ pd                                        0.285 ρ
                  X (t ) = X           + V fx ⋅ t +             (V px          − V fx ) ⋅ ln(1 +           t) .
                                t =0
                                                      0.285 ρ           t =0                         ρ pd

              The COSMOSFloWorks calculations and the analytical solutions for three particle
              injection velocities, Vpy(t=0) = 1, 2, 3 m/s, are shown in Fig. 17.4.. It is seen that the
              COSMOSFloWorks calculations coincide with the analytical solutions. Special
              calculations have shown that the difference is due to the CD assumptions only.
              In the third case, the particle trajectory is governed by the action of the gravitational force
              only, the particle drag coefficient is very close to zero, so the analytical solution is:
                                                                                                   2
                                                        1  X − X t =0  .
                               Y = Y t =0 + V py t + g ⋅              
                                                t =0  y 2  V px       
                                                                      

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    Chapter Particles Trajectories in Uniform Flows


             The COSMOSFloWorks calculation and the analytical solution for this case are presented
             in Fig. 17.5.. It is seen that the COSMOSFloWorks calculation coincides with the
             analytical solution.

             Y (m)
             0.08
                                                                                                Vp = 1 m/s,
             0.07                                                                               analytical solution
                                                                                                Vp = 1 m/s,
             0.06
                                                                                                Calculation
             0.05                                                                               Vp = 2 m/s,
                                                                                                analytical solution
             0.04
                                                                                                Vp = 2 m/s,
             0.03                                                                               Calculation
                                                                                                Vp = 3 m/s,
             0.02
                                                                                                analytical solution
             0.01                                                                               Vp = 3 m/s,
                                                                                                Calculation
             0.00
                 0.00             0.05            0.10             0.15          0.20 X (m)

            Fig. 17.4. Particle trajectories in a uniform fluid flow at Re max = 105, predicted by COSMOSFloWorks
            and obtained from the analytical solution.

    I




              Y (m)
              0.08
                                                                                                     Calculation
              0.07

              0.06                                                                                  Theory

              0.05

              0.04

              0.03

              0.02

              0.01

                0
                 0.00           0.03           0.06         0.09          0.12           0.15
                                                                                                X (m)


             Fig. 17.5. Particle trajectories in the Y-directed gravity, predicted by COSMOSFloWorks and
             obtained from the analytical solution.




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       18 Porous Screen in a Non-uniform Stream

              Let us now validate the COSMOSFloWorks capability to calculate fluid flows through
              porous media.
              Here, following Ref.2, we consider a plane cold air flow between two parallel plates,
              through a porous screen installed between them, see Fig. 18.1.. At the channel inlet the air
              stream velocity profile is step-shaped (specified). The porous screen (gauze) levels this
              profile to a more uniform profile. This effect depends on the screen drag, see Ref.2.
                                                 Y

                Air
                                                       Porous screen
                                                                                                      X

                                                       L


               Fig. 18.1. Leveling effect of a porous screen (gauze) on a non-uniform stream.

              The SolidWorks model used for calculating the 2D (in XY-plane) flow is shown in Fig.
              18.2.. The channel has height of 0.15 m, the inlet (upstream of the porous screen) part of
              the 0.3 m length, the porous screen of the 0.01 m thickness, and the outlet (downstream of
              the porous screen) part of the 0.35 m length. All the walls have thickness of 0.01 m.




               Fig. 18.2. The SolidWorks model used for calculating the 2D flow between two parallel plates and
               through the porous screen with COSMOSFloWorks.




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    Chapter Porous Screen in a Non-uniform Stream


            Following Ref.2, we consider porous screens (gauzes) of different drag, ζ:
            ζ = 0.95, 1.2, 2.8, and 4.1, defined as:
                                                       2 ∆P
                                                 ζ =           ,
                                                       ρV 2
            where ∆P is the pressure difference between the screen sides, ρV2/2 is the dynamic
            pressure (head) of the incoming stream.
            Since in COSMOSFloWorks a porous medium’s resistance to flow is characterized by
            parameter k = - gradP/ρV, then for the porous screens k = V⋅ζ /(2⋅L), where V is the fluid
            velocity, L is the porous screen thickness. In COSMOSFloWorks, this form of a porous
            medium’s resistance to flow is specified as k = (A⋅V+B)/ρ, so A = ρ⋅ζ /(2⋅L), B = 0 for the
            porous screens under consideration. Therefore, taking L = 0.01 m and ρ = 1.2 kg/m3 into
            account, we specify A = 57, 72, 168, and 246 kg/m-4 for the porous screens under
            consideration. In accordance with the screens’ nature, their permeability is specified as
            isotropic.
            According to the experiments presented in Ref.2, the step-shaped velocity profiles V(Y)
            presented in Fig. 18.3. have been specified at the model inlet. The static pressure of 1 atm
            has been specified at the model outlet.
             V (m/s)
             25
                                                                                                ζ=0, 0.95
                                                                                                ζ=1.2
             20
                                                                                                ζ=2.8
                                                                                                ζ=4.1
             15


             10


              5


              0
                  0     0.02     0.04     0.06         0.08     0.1       0.12   0.14   Y (m)

                                         Fig. 18.3. Inlet velocity profiles.

            The air flow dynamic pressure profiles at the 0.3 m distance downstream from the porous
            screens, both predicted by COSMOSFloWorks at result resolution level 5 and measured in
            the Ref.2 experiments, are presented in Fig. 18.4. for the ζ = 0 case (i.e., without screen)
            and Figs.18.5-18.8 for the porous screens of different ζ .




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              It is seen that the COSMOSFloWorks predictions agree well, both qualitatively and
              quantitatively, with the experimental data both in absence of a screen and for all the
              porous screens (gauzes) under consideration, demonstrating the leveling effect of the
              gauze screens on the step-shaped incoming streams. The prediction error in the dynamic
              pressure maximum does not exceed 30%.
                                        350
                                                                                                                  Calculation
                                        300                                                                       Experiment
                Dynamic Pressure (Pa)




                                        250


                                        200


                                        150


                                        100


                                         50


                                          0
                                                                                                          Y (m)
                                              0   0.02    0.04    0.06    0.08    0.1    0.12     0.14

              Fig. 18.4. The dynamic pressure profiles at ζ = 0, predicted by COSMOSFloWorks and compared to
              the Ref.2 experiments.



                                        250
                                                                                                                 Calculation
                                                                                                                 Experiment
                                        200
                Dynamic Pressure (Pa)




                                        150



                                        100



                                        50



                                         0
                                              0   0.02   0.04    0.06    0.08    0.1    0.12    0.14     Y (m)


              Fig. 18.5. The dynamic pressure profiles at ζ = 0.95, predicted by COSMOSFloWorks and compared
              to the Ref.2 experiments.




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    Chapter Porous Screen in a Non-uniform Stream




                                        200
                                                                                                                            Calculation
                                                                                                                            Experiment

                                        150
                Dynamic Pressure (Pa)




                                        100




                                         50




                                             0
                                                 0    0.02    0.04     0.06     0.08     0.1     0.12     0.14      Y (m)


            Fig. 18.6. The dynamic pressure profiles at ζ = 1.2, predicted by COSMOSFloWorks and
            compared to the Ref.2 experiments.



                            120
                                                                                                                         Calculation
              Dynamic Pressure (Pa)




                                                                                                                         Experiment
                            100


                                        80

                                        60


                                        40


                                        20

                                        0
                                             0       0.02    0.04    0.06     0.08     0.1     0.12     0.14     Y (m)


            Fig. 18.7. The dynamic pressure profiles at ζ = 2.8, predicted by COSMOSFloWorks and compared
            to the Ref.2 experiments.




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                                        90
                                                                                                         Calculation
                                        80
                                                                                                         Experiment
                Dynamic Pressure (Pa)   70

                                        60

                                        50

                                        40

                                        30

                                        20

                                        10

                                         0
                                             0   0.02   0.04   0.06   0.08   0.1   0.12   0.14   Y (m)

              Fig. 18.8. The dynamic pressure profiles at ζ = 4.1, predicted by COSMOSFloWorks and compared
              to the Ref.2 experiments.




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    Chapter Porous Screen in a Non-uniform Stream




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       19 Lid-driven Flows in Triangular and Trapezoidal Cavities

              In the previous validation examples we have considered COSMOSFloWorks predictions
              of pressure-driven incompressible flows in various channels with stationary (motionless)
              walls. Since COSMOSFloWorks allows the motion of walls, let us now see how
              COSMOSFloWorks predicts lid-driven (i.e., shear-driven) 2D recirculating flows in
              closed 2D triangular and trapezoidal cavities with one or two moving walls (lids) in
              comparison with the calculations performed in Refs.15 and 16.
              These two cavities are shown in Fig. 19.1.. The triangular cavity has a moving top wall,
              the trapezoidal cavity has a moving top wall also, whereas its bottom wall is considered in
              two versions: as motionless and as moving at the top wall velocity. The no-slip conditions
              are specified on all the walls.
                           2                                                     1

                                                                                       U
                               U

                                                                        1
                                             h
                                                                                           Uwa

                                                                                           U

                                                                                  2

              Fig. 19.1. The 2D triangular (left) and trapezoidal (right) cavities with the moving walls (the
              motionless walls are shown with dashes).


              Shown in Refs. 15 and 16, the shear-driven recirculating flows in these cavities are fully
              governed by their Reynolds numbers Re = ρ·Uwall·h/µ, where ρ is the fluid density, µ is
              the fluid dynamic viscosity, Uwall is the moving wall velocity, h is the cavity height. So,
              we can specify the height of the triangular cavity h = 4 m, the height of the trapezoidal
              cavity h = 1 m, Uwall = 1 m/s for all cases under consideration, the fluid density ρ = 1
              kg/m3, the fluid dynamic viscosity µ=0.005 Pa·s in the triangular cavity produces a Re =
              800, and µ= 0.01, 0.0025, 0.001 Pa·s in the trapezoidal cavity produces a Re = 100, 400,
              1000, respectively.




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    Chapter Lid-driven Flows in Triangular and Trapezoidal Cavities


             The cavities’ models are shown in Fig. 19.2.. The COSMOSFloWorks calculation of flow
             in the triangular cavity has been performed on the 48×96 computational mesh. The results
             in comparison with those from Ref.15 are presented in Fig. 19.3. (streamlines) and in Fig.
             19.4. (the fluid velocity X-component along the central vertical bisector shown by a green
             line in Fig. 19.2.). A good agreement of these calculations is clearly seen.




            Fig. 19.2. The models for calculating the lid-driven 2D flows in the triangular (left) and trapezoidal
            (right) cavities with COSMOSFloWorks.




            Fig. 19.3. The flow trajectories in the triangular cavity, calculated by COSMOSFloWorks (right) and
            compared to the Ref.15 calculation (left).




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               Vx /Uwa ll
                1.2
                                                                                                          Calculation
                  1

                0.8

                0.6

                0.4

                0.2

                  0
                      0     0.5        1       1.5        2       2.5       3        3.5       4
               -0.2
                                                                                                   Y, m
               -0.4

              Fig. 19.4. The triangular cavity’s flow velocity X-component along the central vertical bisector,
              calculated by COSMOSFloWorks (red line) and compared to the Ref.15 calculation (black line with
              circlets).

              The COSMOSFloWorks calculations of flows in the trapezoidal cavity with one and two
              moving walls at different Re values have been performed with the 100×50 computational
              mesh. Their results in comparison with those from Ref.16 are presented in Fig.
              19.5.-19.10 (streamlines) and in Fig. 19.11. (the fluid velocity X-component along the
              central vertical bisector shown by a green line in Fig. 19.2.). A good agreement of these
              calculations is seen.




               Fig. 19.5. The flow streamlines in the trapezoidal cavity with a top only moving wall at Re = 100,
               calculated by COSMOSFloWorks (right) and compared to the Ref.16 calculation (left).




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    Chapter Lid-driven Flows in Triangular and Trapezoidal Cavities




              Fig. 19.6. The flow streamlines in the trapezoidal cavity with a top only moving wall at Re = 400,
              calculated by COSMOSFloWorks (right) and compared to the Ref.16 calculation (left).




              Fig. 19.7. The flow streamlines in the trapezoidal cavity with a top only moving wall at Re = 1000,
              calculated by COSMOSFloWorks (right) and compared to the Ref.16 calculation (left).




              Fig. 19.8. The flow streamlines in the trapezoidal cavity with two moving walls at Re = 100,
              calculated by COSMOSFloWorks (right) and compared to the Ref.16 calculation (left).




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               Fig. 19.9. The flow streamlines in the trapezoidal cavity with two moving walls at Re = 400,
               calculated by COSMOSFloWorks (right) and compared to the Ref.16 calculation (left).




               Fig. 19.10. The flow streamlines in the trapezoidal cavity with two moving walls at Re = 1000,
               calculated by COSMOSFloWorks (right) and compared to the Ref.16 calculation (left).

               Vx /Uwall
                 1
                                                                                                         Calculation
               0.8
               0.6
               0.4
               0.2
                 0
               -0.2 0            0.2            0.4            0.6            0.8             1

               -0.4
               -0.6
               -0.8
                                                                                                  Y, m
                 -1

              Fig. 19.11. The flow velocity X-component along the central vertical bisector in the trapezoidal
              cavity with two moving walls at Re = 400, calculated by COSMOSFloWorks (red line) and compared
              to the Ref.16 calculation (black line).




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    Chapter Lid-driven Flows in Triangular and Trapezoidal Cavities




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       20 Flow in a Cylindrical Vessel with a Rotating Cover

              In the previous COSMOSFloWorks validation example we considered a lid-driven 2D
              recirculating flow in a 2D cavity. Since COSMOSFloWorks allows to consider rotating
              walls, let us now see how COSMOSFloWorks predicts a 3D recirculating flow in a
              cylindrical vessel closed by a rotating cover (see Fig. 20.1.) in comparison with the
              experimental data presented in Ref.17 (also in Ref.18). This vessel of R = h = 0.144 m
              dimensions is filled with a glycerol/water mixture. The upper cover rotates at the angular
              velocity of Ω. The other walls of this cavity are motionless. The default no-slip boundary
              condition is specified for all walls.

                                  Ω                                                   rotating cover




                                                          h


                                                                       R




                            Fig. 20.1. The cylindrical vessel with the a rotating cover.


              Due to the cover rotation, a shear-driven recirculating flow forms in this vessel. Such
              flows are governed by the Reynolds number Re = ρ·Ω·R2/µ, where ρ is the fluid density, µ
              is the fluid dynamic viscosity, Ω is the angular velocity of the rotating cover, R is the
              radius of the rotating cover. In the case under consideration the 70/30% glycerol/water
              mixture has ρ = 1180 kg/m3, µ = 0.02208 Pa·s, the cover rotates at Ω = 15.51 rpm, so Re =
              1800.
              The COSMOSFloWorks calculation has been performed on the 82×41×82 computational
              mesh. The formed flow pattern (toroidal vortex) obtained in this calculation is shown in
              Fig. 20.2. using the flow velocity vectors projected onto the XY-plane. The tangential and
              radial components of the calculated flow velocity along four vertical lines arranged in the
              XY-plane at different distances from the vessel axis in comparison with the Ref.17
              experimental data are presented in Figs.20.3-7 in the dimensionless form (the
              Y-coordinate is divided by R, the velocity components are divided by Ω·R). There is good
              agreement with the calculation results and the experimental data shown.



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    Chapter Flow in a Cylindrical Vessel with a Rotating Cover




                       Fig. 20.2. The vessel's flow velocity vectors projected on the




          Fig. 20.3. The vessel's flow tangential and radial velocity components along the X = 0.6 vertical,
          calculated by COSMOSFloWorks (red) and compared to the Ref.17 experimental data.




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           Fig. 20.4. The vessel's flow tangential and radial velocity components along the X = 0.7 vertical,
           calculated by COSMOSFloWorks (red) and compared to the Ref.17 experimental data.




           Fig. 20.5. The vessel's flow tangential and radial velocity components along the X = 0.8 vertical,
           calculated by COSMOSFloWorks (red) and compared to the Ref.17 experimental data.




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    Chapter Flow in a Cylindrical Vessel with a Rotating Cover




         Fig. 20.6. The vessel's flow tangential and radial velocity components along the X = 0.9 vertical,
         calculated by COSMOSFloWorks (red) and compared to the Ref.17 experimental data.




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       21 Flow in an Impeller

              In the previous validation example we had considered COSMOSFloWorks predictions of
              a recirculating flow in a cylindrical vessel with a rotating cover. The calculation had been
              performed in the coordinate system related to the stationary object (vessel), and a part
              (cover) rotated in this coordinate system. Let us now validate the COSMOSFloWorks
              ability to perform calculations in a rotating coordinate system related to a rotating solid.
              Following Ref.22, we will consider the flow of water in a 9-bladed centrifugal impeller
              having blades tilted at a constant 60º angle with respect to the intersecting radii and
              extending out from the 320 mm inner diameter to the 800 mm outer diameter (see Fig.
              21.1.). The water in this impeller flows from its center to its periphery. To compare the
              calculation with the experimental data presented in Ref.22, the impeller's angular velocity
              of 32 rpm and volume flow rate of 0.00926 m3/s are specified.




                                 Fig. 21.1. The impeller's blades geometry.

              Since the impeller's inlet geometry and disk extension serving as the impeller's vaneless
              diffuser have no exact descriptions in Ref.22, to perform the validating calculation we
              arbitrarily specified the annular inlet as 80 mm in diameter with an uniform inlet velocity
              profile perpendicular to the surface in the stationary coordinate system.The impeller's
              disks external end was specified as 1.2 m diameter, as shown in Fig. 21.2..




              Fig. 21.2. The model used for calculating the 3D flow in the impeller.

              The above-mentioned volume flow rate at the annular inlet and the potential pressure of 1
              atm at the annular outlet are specified as the problem's flow boundary conditions.


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    Chapter Flow in an Impeller


            The COSMOSFloWorks 3D flow calculation is performed on the computational mesh
            using the result resolution level of 5 and the minimum wall thickness of 2 mm (since the
            blades have constant thickness). To further capture the curvature of the blades a local
            initial mesh was also used in the area from the annular inlet to the blades' periphery. As a
            result, the computational mesh has a total number of about 1,000,000 cells.
            Following Ref.22, let us compare the passage-wise flow velocities (ws, see their definition
            in Fig. 21.3., β = 60°) along several radial lines passing through the channels between the
            blades (lines g, j, m, p in Fig. 21.4.) at the mid-height between the impeller's disks.




                          Fig. 21.3. Definition of the passage-wise flow velocity.




              Fig. 21.4. Definition of the reference radial lines along which the passage-wise flow
              velocity was measured in Ref.22 (from a to s in the alphabetical order).


            The passage-wise flow velocities divided by Ω⋅r2, where Ω is the impeller's angular
            velocity and r2 = 400 mm is the impeller's outer radius, which were measured in Ref.22
            and obtained in the performed COSMOSFloWorks calculations, are shown in Fig. 21.5., 6,
            7, and 8. In these figures, the distance along the radial lines is divided by the line's length.
            The COSMOSFloWorks results are presented in each of these figures by the curve
            obtained by averaging the corresponding nine curves in all the nine flow passages between
            the impeller blades. The calculated passage-wise flow velocity's cut plot covering the
            whole computational domain at the mid-height between the impeller's disks is shown in
            Fig. 21.9.. Here, the g, j, m, p radial lines in each of the impeller's flow passages are
            shown. A good agreement of these calculation results with the experimental data is seen.


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                                              0 .7 5


              relative passagewise velocity
                                                                                                                    ca lc u la tio n
                                                                                                                    ex p erim en t

                                                 0 .5
                        (averaged)


                                              0 .2 5



                                                   0
                                                        0          0 .2       0 .4        0 .6       0 .8       1
                                                                re lative d is tanc e alo ng the rad ial line

              Fig. 21.5. The impeller's passage-wise flow velocity along the g (see Fig. 21.4.) radial line,
              calculated by COSMOSFloWorks and compared to the experimental data.
                 relative passagewise velocity




                                                 0.75
                                                                                                                    c alc u latio n

                                                                                                                    e xp e rime n t

                                                   0 .5
                           (averaged)




                                                 0.25



                                                        0
                                                            0        0.2        0.4        0.6       0.8        1
                                                                relative dis tanc e along the rad ial line

              Fig. 21.6. The impeller's passage-wise flow velocity along the j (see Fig. 21.4.) radial line, calculated
              by COSMOSFloWorks and compared to the experimental data.




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    Chapter Flow in an Impeller



      relative passagewise velocity                          0 .7 5
                                                                                                                                                   c a lc u la tio n
                                                                                                                                                   e x p e rim e n t
                                      (averaged)

                                                                         0 .5



                                                             0 .2 5



                                                                            0
                                                                                  0                             0 .5                          1
                                                                                      re lative d is ta n c e a lo n g th e ra d ia l lin e

                                              Fig. 21.7. The impeller's passage-wise flow velocity along the m (see Fig. 21.4.) radial line,
                                              calculated by COSMOSFloWorks and compared to the experimental data.



                                                                         0 .7 5
                                         relative passagewise velocity




                                                                                                                                                  c a lc u latio n

                                                                                                                                                  e x p erim e n t

                                                                          0 .5
                                                   (averaged)




                                                                         0 .2 5



                                                                             0
                                                                                  0           0 .2       0 .4          0 .6     0 .8          1
                                                                                          re lative d is tanc e alo ng the rad ial line

                                              Fig. 21.8. The impeller's passage-wise flow velocity along the p (see Fig. 21.4.) radial line,
                                              calculated by COSMOSFloWorks and compared to the experimental data.




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            Fig. 21.9. A cut plot of the impeller's passage-wise flow velocity calculated by
            COSMOSFloWorks.




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    Chapter Flow in an Impeller




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       22 Cavitation on a hydrofoil

              When the local pressure at some point in the liquid drops below the liquid's vapour
              pressure at the local temperature, the liquid undergoes phase transition and form cavities
              filled with the liquid's vapor with an addition of gas that has been dissolved in the liquid.
              This phenomenon is called cavitation.
              In this validation example we consider COSMOSFloWorks abilities to model cavitation
              on the example of water flow around a symmetric hydrofoil in a water-filled tunnel. The
              calculated results were compared with the experimental data from Ref. 23.
              The problem is solved in the 2D setting. A symmetric hydrofoil with the chord c of
              0.305 m is placed in a water-filled tunnel with the angle of attack of 3.5°. The part of the
              tunnel being modelled has the following dimensions: length l = 2 m and height
              h = 0.508 m. The calculation is performed four times with different values of the
              cavitation number σ defined as follows:
                                                     P ∞ – Pv
                                                 σ = ------------------
                                                                      -
                                                       1
                                                       -- ρ U ∞
                                                        -          2
                                                       2

              where P ∞ is the inlet pressure, Pv is the saturated water vapor pressure equal to 2340 Pa at
              given temperature (293.2 K), ρ is the water density at inlet, and U ∞ is the water velocity
              at inlet (see Fig. 22.1.).
              The inlet boundary condition is set up as Inlet Velocity of 8 m/s. On the tunnel outlet
              an Environment Pressure is specified so that by varying it one may tune the cavitation
              number to the needed value. The project fluid is water with the cavitation option switched
              on, while the other parameters are default. A local initial mesh was created in order to
              resolve the cavitation area better. The resulting mesh contains about 30000 cells.




                                         Fig. 22.1. The model geometry.




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    Chapter Cavitation on a hydrofoil


             The qualitative comparison in a form of cut plots with Vapor Volume Fraction as the
             visualization parameter are shown on Fig. 22.2.




                          σ=1.1                                                σ=0.97




                        σ=0.9                                                σ=0.88
       Fig. 22.2. A comparison of calculated and experimentally observed cavitation areas for different σ




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              The calculated length of the cavitation area was derived from the distribution of the
              Vapor Volume Fraction parameter over the hydrofoil’s surface as the distribution’s
              width at half-height. The results are presented on Fig. 22.3.




                        Fig. 22.3. A comparison of calculated and measured cavitation lengths

              According to Ref. 23, the "clear appearance" of the cavity becomes worse for larger cavity
              lengths. The experimental data also confirm that the amount of uncertainty increases with
              increasing cavity extent. Taking these factors into account together with the comparison
              performed above, we can see that the calculated length of the cavitation area agrees well
              with the experiment for a wide range of cavitation numbers.
              Pressure measurements were performed on the hydrofoil surface at x/c = 0.05 in order to
              calculate the pressure coefficient defined as follows:
                                                   P∞ – P x ⁄ c = 0.05
                                                                                       -
                                            – Cp = -------------------------------------
                                                              1
                                                              -- ρ U ∞
                                                               -          2
                                                              2

              A comparison of the calculated and experimental values of this parameter is presented on
              Fig. 22.4. and also shows a good agreement.




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    Chapter Cavitation on a hydrofoil




                   Fig. 22.4. A comparison of calculated and measured pressure coefficient




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       23 Thermoelectric Cooling

              COSMOSFloWorks has the ability to model the work of a thermoelectric cooler (TEC),
              also known as Peltier element. The device used in this example has been developed for
              active cooling of an infrared focal plane array detector used during the Mars space mission
              (see Ref. 24).
              According to the hardware requirements, the cooler (see Fig. 23.1.) has the following
              dimensions: thickness of 4.8 mm, cold side of 8X8 mm2 and hot side of 12X12 mm2. It
              was built up of three layers of semiconductor pellets made of (Bi,Sb)2(Se,Te)3-based
              material. The cooler was designed to work at temperatures of hot surface in the range of
              120-180 K and to provide the temperature drop of more than 30 K between its surfaces.




              Fig. 23.1. Structure of the thermoelectric cooler.      Fig. 23.2. The thermoelectric module test
                                                                      setup. (Image from Ref.   24)

              To solve the engineering problem using COSMOSFloWorks, the cooler has been modelled
              by a truncated pyramidal body with fixed temperature (Temperature boundary
              condition) on the hot surface and given heat flow (Heat flow boundary condition) on the
              cold surface (see Fig. 23.3.).




                                            Fig. 23.3. The model geometry.


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    Chapter Thermoelectric Cooling


                       The TEC characteristics necessary for the modelling, i.e. temperature dependencies of the
                       maximum pumped heat, maximum temperature drop, maximum current strength and
                       maximum voltage, were represented in the COSMOSFloWorks Engineering Database as a
                       linear interpolation between the values taken from Ref. 24 (see Fig. 23.4.).




                                 Fig. 23.4. The TEC’s characteristics in the Engineering Database.

                       As it can be seen on Fig. 23.5., the temperature drop between the cooler’s hot and cold
                       surfaces in dependence of current agrees well with the experimental data.
                        50

                        45

                        40

                        35

                        30
          Delta T, K




                        25
                                                                                   Th=160 K - Ex perimental
                        20
                                                                                   Th=160 K - Simulated
                        15
                                                                                   Th=180 K - Ex perimental
                        10
                                                                                   Th=180 K - Simulated
                         5

                         0
                             0             0.2            0.4                0.6              0.8             1
                                                                      I, A


                                 Fig. 23.5. ∆T as a function of current under various Th.




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               The dependency of ∆T against heat flow under various Th (see Fig. 23.6.) is also in a good
               agreement with the performance data, as well as the coefficient of performance COP (see
               Fig. 23.7.) defined as follows:
                                                                    Qc             Qc
                                                             COP = ------- = ------------------
                                                                                              -
                                                                   Pin       Q h – Qc


               where Pin is the cooler’s power consumption, and Qc and Qh are the heat flows on the cold
               and hot faces, respectively.



                           50

                           45                                                                 Th=160 K - Perf ormanc e Curv e

                                                                                              Th=160 K - Simulation
                           40

                           35                                                                 Th=180 K - Perf ormanc e Curv e

                                                                                              Th=180 K - Simulation
                           30
              Delta T, K




                           25

                           20

                           15

                           10

                            5

                            0
                                0.0             0.1           0.2                0.3              0.4           0.5             0.6
                                                                               Qc , W



                                      Fig. 23.6. ∆T as a function of heat flow under various Th.




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    Chapter Thermoelectric Cooling




                0.16

                                                                            Th=160 K - Performance Curve
                0.14
                                                                            Th=160 K - Simulation
                0.12
                                                                            Th=180 K - Performance Curve
                0.10
                                                                            Th=180 K - Simulation
          COP




                0.08

                0.06

                0.04

                0.02

                0.00
                       0.0            10.0             20.0                30.0            40.0            50.0
                                                              Delta T, K


                         Fig. 23.7. COP as a function of ∆T under various Th.

                Finally, we may conclude that COSMOSFloWorks reproduces thermal characteristics of
                the thermoelectric coolers at various currents and temperatures with good precision.




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       References

              1 Schlichting, H., Boundary Layer Theory. 7th ed., McGraw –Hill, New York, 1979.
              2 Idelchik, I.E., Handbook of Hydraulic Resistance. 2nd ed., Hemisphere, New York,
                 1986.
              3 Panton, R.L., Incompressible Flow. 2nd ed., John Wiley & Sons, Inc., 1996.
              4 White, F.M., Fluid Mechanics. 3rd ed., McGraw-Hill, New York, 1994.
              5 Artonkin, V.G., Petrov, K.P., Investigations of aerodynamic characteristics of
                 segmental conic bodies. TsAGI Proceedings, No. 1361, Moscow, 1971 (in Russian).
              6 Holman, J.P., Heat Transfer. 8th ed., McGraw-Hill, New York, 1997.
              7 Henderson, C.B. Drag Coefficients of Spheres in Continuum and Rarefied Flows.
                 AIAA Journal, v.14, No.6, 1976.
              8 Humphrey, J.A.C., Taylor, A.M.K., and Whitelaw, J.H., Laminar Flow in a Square
                 Duct of Strong Curvature. J. Fluid Mech., v.83, part 3, pp.509-527, 1977.
              9 Van Dyke, Milton, An Album of Fluid Motion. The Parabolic press, Stanford,
                 California, 1982.
              10 Davis, G. De Vahl: Natural Convection of Air in a Square Cavity: a Bench Mark
                 Numerical Solution. Int. J. for Num. Meth. in Fluids, v.3, p.p. 249-264 (1983).
              11 Davis, G. De Vahl, and Jones, I.P.: Natural Convection in a Square Cavity: a
                 Comparison Exercise. Int. J. for Num. Meth. in Fluids, v.3, p.p. 227-248 (1983).
              12 Emery, A., and Chu, T.Y.: Heat Transfer across Vertical Layers. J. Heat Transfer, v.
                 87, p. 110 (1965).
              13 Denham, M.K., and Patrick, M.A.: Laminar Flow over a Downstream-Facing Step in a
                 Two-Dimensional Flow Channel. Trans. Instn. Chem. Engrs., v.52, p.p. 361-367
                 (1974).
              14 Yanshin, B.I.: Hydrodynamic Characteristics of Pipeline Valves and Elements.
                 Convergent Sections, Divergent Sections, and Valves. “Mashinostroenie”, Moscow,
                 1965.
              15 Jyotsna, R., and Vanka, S.P.: Multigrid Calculation of Steady, Viscous Flow in a
                 Triangular Cavity. J. Comput. Phys., v.122, No.1, p.p. 107-117 (1995).
              16 Darr, J.H., and Vanka, S.P.: Separated Flow in a Driven Trapezoidal Cavity. J. Phys.
                 Fluids A, v.3, No.3, p.p. 385-392 (1991).
              17 Michelsen, J. A., Modeling of Laminar Incompressible Rotating Fluid Flow, AFM
                 86-05, Ph.D. Dissertation, Department of Fluid Mechanics, Technical University of
                 Denmark, 1986.
              18 Sorensen, J.N., and Ta Phuoc Loc: Higher-Order Axisymmetric Navier-Stokes Code:
                 Description and Evaluation of Boundary Conditions. Int. J. Numerical Methods in
                 Fluids, v.9, p.p. 1517-1537 (1989).



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    Chapter References


            19 Enchao Yu, Yogendra Joshi: Heat Transfer Enhancement from Enclosed Discrete
               Components Using Pin-Fin Heat Sinks. Int. J. of Heat and Mass Transfer, v.45, p.p.
               4957-4966 (2002).
            20 Kuchling, H., Physik, VEB FachbuchVerlag, Leipzig, 1980.
            21 Balakin, V., Churbanov, A., Gavriliouk, V., Makarov, M., and Pavlov, A.: Verification
               and Validation of EFD.Lab Code for Predicting Heat and Fluid Flow, In: CD-ROM
               Proc. Int. Symp. on Advances in Computational Heat Transfer “CHT-04”, April 19-24,
               2004, Norway, 21 p.
            22 Visser, F.C., Brouwers, J.J.H., Jonker, J.B.: Fluid flow in a rotating low-specific-speed
               centrifugal impeller passage. J. Fluid Dynamics Research, 24, pp. 275-292 (1999).
            23 Wesley, H. B., and Spyros, A. K.: Experimental and computational investigation of
               sheet cavitation on a hydrofoil. Presented at the 2nd Joint ASME/JSME Fluid
               Engineering Conference & ASME/EALA 6th International Conference on Laser
               Anemometry. The Westin Resort, Hilton Head Island, SC, USA August 13 - 18, 1995.
            24 Yershova, L., Volodin, V., Gromov, T., Kondratiev, D., Gromov, G., Lamartinie, S.,
               Bibring, J-P., and Soufflot, A.: Thermoelectric Cooling for Low Temperature Space
               Environment. Proceedings of 7th European Workshop on Thermoelectrics, Pamplona,
               Spain, 2002.




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                                                                                                      5
                                                                Technical Reference




       1   Physical Capabilities of COSMOSFloWorks

              With COSMOSFloWorks it is possible to study a wide range of fluid flow and heat
              transfer phenomena that include the following:
                 • External and internal fluid flows
                 • Steady-state and time-dependent fluid flows
                 • Compressible gas and incompressible fluid flows (either in different projects or
                   simultaneously in different regions not in contact with each other)
                 • Free, forced, and mixed convection
                 • Fluid flows with boundary layers, including wall roughness effects
                 • Laminar and turbulent fluid flows
                 • Multi-species fluids and multi-component solids
                 • Heat conduction in fluid, solid and porous media with/without conjugate heat
                   transfer and/or contact heat resistance between solids and/or radiation heat transfer
                   between opaque solids (some solids can be considered transparent for radiation),
                   and/or volume (or surface) heat sources, e.g. due to Peltier effect, etc.
                 • Various types of thermal conductivity in solid medium, i.e. isotropic, unidirectional,
                   biaxial/axisymmetrical, and orthotropic
                 • Fluid flows and heat transfer in porous media
                 • Flows of non-Newtonian liquids
                 • Flows of compressible liquids
                 • Real gases
                 • Two-phase (fluid + particles) flows
                 • Equilibrium volume condensation of water from steam and its influence on fluid
                   flow and heat transfer

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    Chapter Governing Equations


               • Relative humidity in gases and mixtures of gases
               • Fluid flows in models with moving/rotating surfaces and/or parts
               • Cavitation in incompressible water flows
               • Periodic boundary conditions.

    2   Governing Equations

    2.1 The Navier-Stokes Equations for Laminar and Turbulent Fluid Flows
            COSMOSFloWorks solves the Navier-Stokes equations, which are formulations of mass,
            momentum and energy conservation laws for fluid flows. The equations are supplemented
            by fluid state equations defining the nature of the fluid, and by empirical dependencies of
            fluid density, viscosity and thermal conductivity on temperature. Inelastic non-Newtonian
            fluids are considered by introducing a dependency of their dynamic viscosity on flow
            shear rate and temperature, and compressible liquids are considered by introducing a
            dependency of their density on pressure. A particular problem is finally specified by the
            definition of its geometry, boundary and initial conditions.
            COSMOSFloWorks is capable of predicting both laminar and turbulent flows. Laminar
            flows occur at low values of the Reynolds number, which is defined as the product of
            representative scales of velocity and length divided by the kinematic viscosity. When the
            Reynolds number exceeds a certain critical value, the flow becomes turbulent, i.e. flow
            parameters start to fluctuate randomly.
            Most of the fluid flows we encounter in engineering practice are turbulent, so
            COSMOSFloWorks was mainly developed to simulate and study turbulent flows. To
            predict turbulent flows, we use the Favre-averaged Navier-Stokes equations, where
            time-averaged effects of the flow turbulence on the flow parameters are considered,
            whereas the other, i.e. large-scale, time-dependent phenomena are taken into account
            directly. Through this procedure, extra terms known as the Reynolds stresses appear in the
            equations for which additional information must be provided. To close this system of
            equations, COSMOSFloWorks employs transport equations for the turbulent kinetic
            energy and its dissipation rate, the so-called k- ε model.
            COSMOSFloWorks employs one system of equations to describe both laminar and
            turbulent flows. Moreover, transition from a laminar to turbulent state and/or vice versa is
            possible.
            Flows in models with moving walls (without changing the model geometry) are computed
            by specifying the corresponding boundary conditions. Flows in models with rotating parts
            are computed in coordinate systems attached to the models rotating parts, i.e. rotating with
            them, so the models' stationary parts must be axisymmetric with respect to the rotation
            axis.




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              The conservation laws for mass, angular momentum and energy in the Cartesian
              coordinate system rotating with angular velocity Ω about an axis passing through the
              coordinate system's origin can be written in the conservation form as follows:
               ∂ρ    ∂
                  +      (ρui ) = 0                                                                   (5.1)
               ∂t   ∂ xi
               ∂ ρui    ∂                ∂p    ∂
                     +      (ρui u j ) +     =      (τ ij + τ ij ) + S i i = 1,2,3
                                                               R
                                                                                                      (5.2)
                ∂t     ∂x j              ∂ xi ∂ x j
               ∂ρ H ∂ρui H ∂                             ∂p R ∂u
                ∂t
                   +
                     ∂xi
                          =
                            ∂xi
                                         (
                                u j (τ ij + τ ij ) + qi + − τ ij i + ρε + Si ui + QH , (5.3)
                                               R

                                                         ∂t
                                                             )  ∂x j

                           u2
               H = h+         ,
                           2
              where u is the fluid velocity, ρ is the fluid density,     S i is a mass-distributed external force
              per unit mass due to a porous media resistance (Si porous), a buoyancy (Sigravity = - ρgi,
              where gi is the gravitational acceleration component along the i-th coordinate direction),
              and the coordinate system’s rotation (Sirotation), i.e., Si = Siporous + Sigravity + Sirotation, h is
              the thermal enthalpy,    Q H is a heat source or sink per unit volume, τ ik is the viscous
              shear stress tensor,   q i is the diffusive heat flux. The subscripts are used to denote
              summation over the three coordinate directions.
              For calculations with the High Mach number flow option enabled, the following energy
              equation is used:

                                p
                     ∂ρ u i  E + 
                            
              ∂ρ E               ρ ∂                                  R ∂u i

               ∂t
                   +        
                            ∂ xi
                                  =
                                     ∂xi
                                                (       R
                                                                     )
                                         u j (τ ij + τ ij ) + q i − τ ij
                                                                         ∂x j
                                                                              + ρε + S i u i + Q H , (5.4)


                    u2
              E = e+ ,
                    2
              where e is the internal energy.
              For Newtonian fluids the viscous shear stress tensor is defined as:

                        ∂ui ∂u j 2 ∂uk 
              τ ij = µ     +    − δ ij                                                              (5.5)
                        ∂x   ∂xi 3 ∂xk 
                        j              


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    Chapter Governing Equations


            Following Boussinesq assumption, the Reynolds-stress tensor has the following form:

                         ∂ui        ∂u j2 ∂u  2
            τ ijR = µt 
                       
                                 +      − δ ij k  − ρkδ ij                                    (5.6)
                         ∂x j       ∂xi 3 ∂xk  3
                                                 

            Here   δ i j is the Kronecker delta function (it is equal to unity when i = j, and zero
            otherwise),    µ is the dynamic viscosity coefficient, µ t is the turbulent eddy viscosity
            coefficient and k is the turbulent kinetic energy. Note that   µ t and k are zero for laminar
            flows. In the frame of the k-ε turbulence model,  µ t is defined using two basic turbulence
            properties, namely, the turbulent kinetic energy k and the turbulent dissipation ε,

                        C µ ρk 2
            µt = f µ                                                                            (5.7)
                            ε

            Here f µ is a turbulent viscosity factor. It is defined by the expression


                                        2       20,5  ,
                    [                        ]
             f µ = 1 − exp (− 0.025R y ) ⋅  1 +
                                                 RT 
                                                                                               (5.8)
                                                     

            where R = ρk , R = ρ k y
                        2


                                     µε           µ
                   T        y



            and y is the distance from the wall. This function allows us to take into account
            laminar-turbulent transition.
            Two additional transport equations are used to describe the turbulent kinetic energy and
            dissipation,

            ∂ρk                                     
                       (ρu i k ) = ∂   µ + µ t  ∂k  + S k ,
                   ∂                             
                +                               ∂x 
                                                                                                (5.9)
             ∂t   ∂ xi            ∂ xi     σk  i 

            ∂ρε                                        ∂ε   
                +
                   ∂
                       ( ρuiε ) = ∂   µ + µ t                + Sε ,                         (5.10)
             ∂t   ∂x i           ∂ xi  
                                          σε           ∂x
                                                         i
                                                               
                                                               




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              where the source terms    S k and S ε are defined as

                           ∂ui                                                                  (5.11)
               Sk = τ ij
                       R
                                − ρε + µt PB
                           ∂x j

                           ε  R ∂ui
                              f1τ ij
                                                                 ρε 2 .                        (5.12)
               Sε = Cε1                    + µtCB PB  − Cε 2 f 2
                           k        ∂x j           
                                                                  k

              Here   P B represents the turbulent generation due to buoyancy forces and can be written as

                        gi 1 ∂ρ
               PB = −                                                                           (5.13)
                        σ B ρ ∂xi

              where   g i is the component of gravitational acceleration in direction x i , the constant
              σB = 0.9, and constant C B    is defined as: CB = 1 when   P B > 0 , and 0 otherwise;
                                   3
                         0.05  ,
               f1 = 1 + 
                         fµ 
                                  f 2 = 1 − exp − RT
                                                    2
                                                       (     )                                  (5.14)
                              

              The constants    C µ , C ε 1 , C ε 2 , σ k , σ ε are defined empirically. In COSMOSFloWorks
              the following typical values are used:
              Cµ = 0.09, Cε1 = 1.44, Cε2 = 1.92, σ ε = 1.3,
              σk = 1                                                                            (5.15)

              Where Lewis number Le=1 the diffusive heat flux is defined as:

                    µ    µ t  ∂ h , i = 1, 2, 3.
                    Pr + σ  ∂ x
              qi =           
                                                                                                (5.16)
                           c     i


              Here the constant σ c = 0.9, Pr is the Prandtl number, and h is the thermal enthalpy.

              These equations describe both laminar and turbulent flows. Moreover, transitions from
              one case to another and back are possible. The parameters k and     µ t are zero for purely
              laminar flows.




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    Chapter Governing Equations


        Laminar/turbulent Boundary Layer Model
            A laminar/turbulent boundary layer model is used to describe flows in near-wall regions.
            The model is based on the so-called Modified Wall Functions approach. This model is
            employed to characterize laminar and turbulent flows near the walls, and to describe
            transitions from laminar to turbulent flow and vice versa. The modified wall function uses
            a Van Driest's profile instead of a logarithmic profile. If the size of the mesh cell near the
            wall is more than the boundary layer thickness the integral boundary layer technology is
            used. The model provides accurate velocity and temperature boundary conditions for the
            above mentioned conservation equations.

        Constitutive Laws and Thermophysical Properties
            The system of Navier-Stokes equations is supplemented by definitions of thermophysical
            properties and state equations for the fluids. COSMOSFloWorks provides simulations of
            gas and liquid flows with density, viscosity, thermal conductivity, specific heats, and
            species diffusivities as functions of pressure, temperature and species concentrations in
            fluid mixtures, as well as equilibrium volume condensation of water from steam can be
            taken into account when simulating steam flows.
            Generally, the state equation of a fluid has the following form:
            ρ = f ( p,T, y),
                          ,                                                                   (5.17)


            where y =(y1, ... yM) is the concentration vector of the fluid mixture components.
            Excluding special cases (see below subsections concerning Real Gases, Equilibrium
            volume condensation of water from steam), gases are considered ideal, i.e. having the
            state equation of the form
                   P ,
             ρ=                                                                               (5.18)
                   RT
            where R is the gas constant which is equal to the universal gas constant Runiv divided by
            the fluid molecular mass M, or, for the mixtures of ideal gases,
                         y
              R = Runiv∑ m ,                                                                  (5.19)
                       m Mm


            where y m , m=1, 2, ...,M, are the concentrations of mixture components, and     M m is the
            molecular mass of the m-th component.
            Specific heat at constant pressure, as well as the thermophysical properties of the gases,
            i.e. viscosity and thermal conductivity, are specified as functions of temperature. In
            addition, proceeding from Eq. (5.18), each of such gases has constant specific heat ratio
            Cp/Cv.



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              Excluding special cases (see below subsections Compressible Liquids, Non-Newtonian
              Liquids), liquds are considered incompressible, i.e. the density of an individual liquid
              depends only on temperature:
                  ρ = f (T ) ,                                                                   (5.20)

              and the state equation for a mixture of liquids is defined as
                              −1
                     y 
              ρ = ∑ m 
                   ρ 
                                                                                                 (5.21)
                   m m
              The specific heat and the thermophysical properties of the liquid (i.e. viscosity and
              thermal conductivity), are specified as functions of temperature.

           Real Gases
              The state equation of ideal gas (5.18) become inaccurate at high pressures or in close
              vicinity of the gas-liquid phase transition curve. Taking this into account, a real gas state
              equation together with the related equations for thermodynamical and thermophysical
              properties should be employed in such conditions. At present, this option may be used
              only for a single gas, probably mixed with ideal gases.
              In case of user-defined real gas, COSMOSFloWorks uses a custom modification of the
              Redlich-Kwong equation that may be expressed in dimensionless form as follows:

                        1           a·F    
                        Φ − b − Φ ·(Φ + c) 
              pr = Tr ·                    
                                                                                                 (5.22)
                        r        r    r    

              where pr = p/pc, Tr = T/Tc, Φr = Vr·Zc, Vr=V/Vc, F=Tr-1.5, pc, Tc, and Vc are the
              user-specified critical parameters of the gas, i.e. pressure, temperature, and specific
              volume at the critical point, and Zc is the gas compressibility factor that also defines the a,
              b, and c constants. A particular case of equation (5.22) with Zc=1/3 (which in turn means
              that b=c) is the original Riedlich-Kwong equation as given in Ref. 11.
              Alternatively, one of the modifications (Ref. 11) taking into account the dependence of F
              on temperature and the Pitzer acentricity factor ω may be used: the Wilson modification,
              the Barnes-King modification, or the Soave modification.
              The specific heat of real gas at constant pressure (C p) is determined as the combination of
              the user-specified temperature-dependent "ideal gas" specific heat (Cpideal) and the
              automatically calculated correction. The former is a polynomial with user-specified order
              and coefficients. The specific heat at constant volume (Cv) is calculated from Cp by means
              of the state equation.




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    Chapter Governing Equations


            Likewise, the thermophysical properties are defined as a sum of user-specified "basic"
            temperature dependency (which describes the corresponding property in extreme case of
            low pressure) and the pressure-dependent correction which is calculated automatically.
            The basic dependency for dynamic viscosity η of the gas is specified in a power-law form:
            η = a·Tn. The same property for liquid is specified either in a similar power-law form
            η = a·Tn or in an exponential form: η = 10a(1/T-1/n). As for the correction, it is given
            according to the Jossi-Stiel-Thodos equation for non-polar gases or the Stiel-Thodos
            equations for polar gases (see Ref. 11), judging by the user-specified attribute of polarity.
            The basic dependencies for thermal conductivities λ of the substance in gaseous and liquid
            states are specified by the user either in linear λ = a+n·T or in power-law λ = a·Tn forms,
            and the correction is determined from the Stiel-Thodos equations (see Ref. 11).
            All user-specified coefficients must be entered in SI unit system, except those for the
            exponential form of dynamic viscosity of the liquid, which should be taken exclusively
            from Ref. 11.
            In case of pre-defined real gas, the custom modification of the Riedlich-Kwong equation
            of the same form as Eq. (5.22) is used, with the distinction that the coefficients a, b, and c
            are specified explicitly as dependencies on Tr in order to reproduce the gas-liquid phase
            transition curve at P < Pc and the critical isochore at P > Pc with higher precision.




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              When the calculated (p, T) point drops out of the region bounded by the temperature and
              pressure limits (zones 1 - 8 on Fig.2.1) or gets beyond the gas-liquid phase transition
              curve (zone 9 on Fig.2.1), the corresponding warnings are issued and properties of the real
              gas are extrapolated linearly.




                                                   Fig.2.1

              If a real gas mixes with ideal gases (at present, mixtures of multiple real gases are not
              considered), properties of this mixture are determined as an average weighted with mass
              or volume fractions:
                      N

               ν =   ∑ Yi νi                                                                  (5.23)
                     i=1

              where ν is the mixture property (i.e., Cp, µ, or λ), N is the total number of the mixture
              gases (one of which is real and others are ideal), Yi is the mass fraction (when calculating
              Cp) or the volume fraction (when calculating µ and λ) of the i-th gas in the mixture.
              The real gas model has the following limitations and assumptions:
                 • The precision of calculation of thermodynamic properties at nearly-critical
                   temperatures and supercritical pressures may be lowered to some extent in
                   comparison to other temperature ranges. Generally speaking, the calculations
                   involving user-defined real gases at supercritical pressures are not recommended.
                 • The user-defined dependencies describing the specific heat and transport properties
                   of the user-defined real gases should be applicable in the whole Tmin...Tmax range
                   (or, speaking about liquid, in the whole temperature range where the liquid exists).

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               • Tmin for user-defined real gas should be set at least 5...10 K higher than the triple
                 point of the substance in question.

        Compressible Liquids
            Compressible liquids whose density depends on pressure and temperature can be
            considered within the following approximations:
               • obeying the logarithmic law:

                                             B+P 
                    ρ = ρ 0 / 1 − C ⋅ ln
                                                     ,
                                             B + P0 
                                                     
                    where ρo is the liquid's density under the reference pressure Po, C, B are
                    coefficients, here ρo, C, and B can depend on temperature, P is the calculated
                    pressure;
               • obeying the power law:
                                            1/ n
                              P+B 
                    ρ = ρ0 ⋅ 
                              P + B
                                                  ,
                              0    
                    where, in addition to the above-mentioned designations, n is a power index which
                    can be temperature dependent.

        Non-Newtonian Liquids
            COSMOSFloWorks is capable of computing laminar flows of inelastic non-Newtonian
            liquids. In this case the viscous shear stress tensor is defined, instead of Eq. (5.5), as
                              ∂u i       ∂u j 
             τ ij = µ (γ& ) ⋅ 
                              
                                      +        ,                                              (5.24)
                              ∂x j       ∂x i 
                                               
            where shear rate,
                                                       ∂ ui ∂ u j
            γ& = d ij 2 − d ii ⋅ d jj , d ij =             +
                                                       ∂x j ∂xi

            and for specifying a viscosity function µ (γ ) the following three models of inelastic
                                                        &
            non-Newtonian viscous liquids are available in COSMOSFloWorks:
                The Herschel-Bulkley model:
                                                                        τo
                                             µ (γ& ) = K ⋅ (γ& )n−1 +        ,
                                                                        γ&
               where K is the liquid's consistency coefficient, n is the liquid's power law index, and
               τo    is the liquid's yield stress. This model includes the following special cases:

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                 • n = 1, τ o = 0 describes Newtonian liquids, in this case K is the liquid's dynamic
                   viscosity;
                 • n = 1, τ o > 0 describes the Bingham model of non-Newtonian liquids, featured by a
                   non-zero threshold yield stress ( τ o ) below of which the liquid behaves as a solid, so
                   to achieve a flow this threshold shear stress must be exceeded. (In
                   COSMOSFloWorksthis threshold is modeled by automatically equating K, named
                   plastic viscosity in this case, to a substantially high value at τ < τ o );
                 • 0 < n < 1, τ o = 0 describes the power law model of shear-thinning non-Newtonian
                   liquids (see also below).
                 • n > 1, τ o = 0 describes the power law model of shear-thickening non-Newtonian
                   liquids (see also below).
                 The power law model: ,
                                               µ (γ& ) = K ⋅ (γ& )n−1 ,
                 in contrast to the Herschel-Bulkley model's special case, the µ values are restricted:
                 µmin ≤ µ ≤ µmax;
                 The Carreau model:
                                                            [
                                 µ = µ ∞ + (µo − µ ∞ ) ⋅ 1 + (K t ⋅ γ& )2   ](   n−1) / 2   ,

                 where:

                     µ∞ is the liquid's dynamic viscosity at infinite shear rate, i.e., the minimum
                    dynamic viscosity,

                     µo   is the liquid's dynamic viscosity at zero shear rate, i.e., the maximum dynamic
                    viscosity,
                    Kt is the time constant,
                    n is the power law index.
                 This model is a smooth version of the power law model with the µ restrictions. In these
                 models, all parameters with the exception of the dimensionless power law index are
                 temperature-dependent in a general case.

           Equilibrium volume condensation of water from steam
              If the gas whose flow is computed includes steam, COSMOSFloWorks can predict an
              equilibrium volume condensation of water from this steam (without any surface
              condensation) taking into account the corresponding changes of the steam temperature,
              density, enthalpy, specific heat, and sonic velocity. In accordance with the equilibrium
              approach, local mass fraction of the condensed water in the local total mass of the steam
              and the condensed water is determined from the local temperature of the fluid, pressure,
              and, if a multi-component fluid is considered, the local mass fraction of the steam. Since
              this model implies an equilibrium conditions, the condensation has no history, i.e. it is a
              local fluid property only.

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            In addition, it is assumed that
               • the volume of the condensed water is neglected, i.e. considered zero, so this
                 prediction works properly only if the volume fraction of the condensed water does
                 not exceed 5%,
               • the steam temperature falls into the range of 283...610 K and the pressure does not
                 exceed 10 MPa.

    2.2 Conjugate Heat Transfer
            COSMOSFloWorks allows to predict simultaneous heat transfer in solid and fluid media
            with energy exchange between them. Heat transfer in fluids is described by the energy
            conservation equation (5.3) where the heat flux is defined by (5.16). The phenomenon of
            anisotropic heat conductivity in solid media is described by the following equation:
              ∂ρe   ∂  ∂T 
                  =    λi   + QH ,
               ∂t ∂xi  ∂xi 
                                                                                             (5.25)
                           
            where e is the specific internal energy, e = cT, c is specific heat, QH is specific heat
            release (or absorption) per unit volume, and λi are the eigenvalues of the thermal
            conductivity tensor. It is supposed that the heat conductivity tensor is diagonal in the
            considered coordinate system. For isotropic medium λ1 = λ2 = λ3 = λ.
            If a solid consists of several solids attached to each other, then the thermal contact
            resistances between them (on their contact surfaces), specified in the Engineering database
            in the form of contact conductance (as m2·K/W ), can be taken into account when
            calculating the heat conduction in solids. As a result, a solid temperature step appears on
            the contact surfaces. In the same manner, i.e. as a thermal contact resistance, a very thin
            layer of another material between solids or on a solid in contact with fluid can be taken
            into account when calculating the heat conduction in solids, but it is specified by the
            material of this layer (its thermal conductivity taken from the Engineering database) and
            thickness. The surface heat source (sink) due to Peltier effect may also be considered (see
            "2.3. Thermoelectric Coolers" on page 5-13).
            The energy exchange between the fluid and solid media is calculated via the heat flux in
            the direction normal to the solid/fluid interface taking into account the solid surface
            temperature and the fluid boundary layer characteristics, if necessary.




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              If a solid medium is porous with a fluid flowing through it, then a conjugate heat transfer
              problem in this porous-solid/fluid medium can be solved also in the manner described
              below. The equations (5.3) and (5.25) are solved in a usual way, but with addition of
              energy exchange between the fluid and the porous solid matrix, defined via the volumetric
              heat exchange in the Eq. (5.25) in a form of   Q H p o r o s i t y = γ ⋅ ( T p – T ) , where
              γ is the user-defined volumetric coefficient of heat transfer between fluid and the porous
              matrix, Tp is the temperature of the porous matrix, T is the fluid temperature, and the same
              QH with the opposite sign is employed in Eq. (5.25) for the porous matrix. Note that the γ
              and c of the porous matrix used in Eq. (5.25) can differ from those of the corresponding
              bulk solid material. Naturally, both the fluid flow equations and the porous matrix heat
              transfer equation take into account the fluid and solid densities multiplied by the
              corresponding fluid and solid volume fractions in the porous matrix.

       2.3 Thermoelectric Coolers
              Thermoelectric cooler (TEC) is a flat sandwich consisting of two plates covering a circuit
              of p-n semiconductor junctions inside. When a direct electric current (DC) i runs through
              this circuit, in accordance with the Peltier effect the a·i·Tc heat, where a is the Seebeck
              coefficient, Tc is the TEC's "cold" surface temperature, is pumped from the TEC's "cold"
              surface to its "hot" surface (the "cold" and "hot" sides are determined from the DC
              direction). This heat pumping is naturally accompanied by the Joule (ohmic) heat release
              at both the TEC surfaces and the heat transfer from the hotter side to the colder (reverse to
              the Peltier effect). The ohmic heat release is defined as R·i2/2, where R is the TEC's
              electric resistance, while the heat transfer is defined as k·∆T , where k is the TEC's
              thermal conductivity, ∆T = Th - Tc , Th is the TEC's "hot" surface temperature. The net
              heat transferred from the TEC's "cold" surface to its "hot" surface, Qc, is equal to

               Qc = a ⋅ i ⋅ Tc − R ⋅ i 2 / 2 − k ⋅ ∆T ,
              Correspondingly, the net heat released at the TEC's "hot" surface, Qh, is equal to

               Qh = a ⋅ i ⋅ Th + R ⋅ i 2 / 2 − k ⋅ ∆T .
              In COSMOSFloWorks a TEC is specified by selecting a flat plate (box) in the model,
              assigning its "hot" face, and applying one of the TECs already defined by user in the
              Engineering Database. The following characteristics of TEC are specified in the
              Engineering Database:
                 • the maximum DC current, imax
                 • the maximum heat Qcmax transferred at this imax at ∆T = 0
                 • the maximum temperature difference ∆Tmax, attained at Qc = 0
                 • the voltage Vmax corresponding to imax.

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    Chapter Governing Equations


            All of these characteristics are specified for two Th values, in accordance with the
            information usually provided by the TEC suppliers. Proceeding from these characteristics,
            the a(T), R(T), and k(T) linear functions are determined. The functional boundary
            conditions are specified automatically on the TEC's "cold" and "hot" surfaces, which must
            be free from other boundary conditions.
            The temperature solution inside the TEC and on its surfaces is obtained using a special
            procedure differing from the standard COSMOSFloWorks calculation procedure for heat
            conduction in solids.
            The TEC's "hot" face must be in contact with other solids, i.e it must not be in contact with
            any fluid. In addition, it is required that the obtained TEC solution, i.e. Th and ∆T, lie
            within the TEC's operating range specified by its manufacturer.



    2.4 Radiation Heat Transfer Between Solids
            In addition to heat transfer in solids, COSMOSFloWorks is capable to calculate radiation
            heat transfer between solids whose surface emissivity is specified. If necessary, a heat
            radiation from the computational domain's far-field boundaries or the model's openings to
            the model surfaces can be defined and considered either as from solid surfaces, i.e. by
            specifying these boundaries' emissivity and temperature, or as a solar radiation defined by
            the specified location (on the surface of the Earth) and time (including date) or by constant
            or time-dependent direction and intensity.

        General Assumptions
            The radiation heat transfer is analyzed under the following assumptions:
               The heat radiation from the solid surfaces, both the emitted and reflected, is assumed
               diffuse (except for the symmetry radiative surface type), i.e. obeying the Lambert law,
               according to which the radiation intensity per unit area and per unit solid angle is the
               same in all directions.
               The propagating heat radiation passes through a body specified as radiation transparent
               without any refraction and/or absorption.
               The project fluids neither emit nor absorb heat radiation (i.e., they are transparent to
               the heat radiation), so the heat radiation concerns solid surfaces only.
               The radiative solid surfaces which are not specified as a blackbody or whitebody are
               assumed an ideal graybody, i.e. having a continuous emissive power spectrum similar
               to that of blackbody, so their monochromatic emissivity is independent of the emission
               wavelength. For certain materials with certain surface conditions, the graybody
               emissivity can depend on the surface temperature.




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           Ray Tracing
              In a general case, the surfaces participating in the heat radiation (hereinafter radiative
              surfaces) can emit, absorb, and reflect a heat radiation. Therefore, both the heat radiation
              L leaving a radiative surface and the net radiation N being the difference between the
              radiation heat leaving this surface and the radiation heat arriving at it, are calculated for
              each of these surfaces.

               L = ε ⋅σ ⋅ T 4 + ρ ⋅ I ,

               N = L − I = ε ⋅σ ⋅ T 4 + ( ρ − 1 ) ⋅ I ,
              where ε is this surface emissivity, σ is the Stefan-Boltzmann constant, T is the
              temperature of the surface (ε·σ·T4 is the heat radiated by this surface in accordance with
              the Stefan-Boltzman law), I is the radiation arriving at this surface, ρ is a reflection
              coefficient (ρ = 1 - ε for graybody walls and ρ = 0 for openings).
              In order to reduce the of memory requirements, the problem of determining the leaving
              and net heat radiation fluxes is solved using a discrete ray Monte-Carlo approach
              consisting of the following main elements:
                 To reduce the number of radiation rays and, therefore, the required calculation time and
                 resources, the computational mesh cells containing faces approximating the radiative
                 surfaces are joined in clusters by a special procedure that takes into account the face
                 area and angle between normal and face in each partial cell. The cells intersected by
                 boundaries between radiative surfaces of different emissivity are considered as
                 belonging to one of these surfaces and cannot be combined in one cluster. This
                 procedure is executed after constructing the computational mesh before the calculation
                 and after each solution-adaptive mesh refinement, if any.
                 From each cluster, a number of rays are emitted, equally distributed over the enclosing
                 unit hemisphere. Each ray is traced through the fluid and transparent solid bodies until
                 it intercepts the computational domain’s boundary or a cluster belonging to another
                 radiative surface, thus defining a ‘target’ cluster. Since the radiation heat is transferred
                 along these rays only, their number and arrangement govern the accuracy of
                 calculating the radiation heat coming from one radiative surface to another (naturally,
                 the net heat radiated by a radiative surface does not depend on number of these rays).
                 So, for each of the clusters, the hemisphere governed by the ray’s origin and the normal
                 to the face at this origin is uniformly divided into several nearly equal solid angles
                 generated by several zenith angles (at least 3 within the 0...90º range, including the




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               zero zenith angle of the normal to the face) and several azimuth angles (at least 12
               within the 0...360º range).




                              Fig.2.2Definition of rays emitted from cluster.

            The total number of emitted rays is

             N = (m − 1) ⋅ n + 1,
            where m is the number of different latitude values for the rays (including the polar ray),
            n is the number of different longitude values (n = 2 for 2D case),
            Θ and Φ are the zenith (latitudinal) and azimuth (longitudinal) angles, respectively.
            The value of m is defined directly by the View factor resolution level which can be
            changed by the user via the Calculation Control Options dialog box. The value of n
            depends on m as follows: n = m ⋅ 4 .
            The higher the View factor resolution level, the better the accuracy of the radiation heat
            transfer calculation, but the calculation time and required computer resources increase
            significantly when high values of View factor resolution level are specified.
            Periodically during the calculation, a radiation ray is emitted in each of the solid angles in
            a direction that is defined randomly within this solid angle. These radiation rays are traced
            until intersection with either another radiative surface or the boundary of the
            computational domain. To increase the accuracy of heat radiation calculation, the number
            of radiation rays emitted from each cluster can be increased automatically during the
            calculation, depending on the surface temperature and emissivity, to equalize the radiation
            heat emitting through the solid angles.
               When a radiation ray intercepts a cluster of other radiative surfaces, the radiation heat
               carried by this ray is uniformly distributed over the area of this cluster. The same
               procedure is performed if several radiation rays hit the same cluster. To smooth a
               possible non-uniformity of the incident radiation heat distribution over a radiative
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                  surface, a fraction of the radiation heat arriving with rays at a cluster can be transferred
                  to the neighboring clusters also. In addition, small fluctuations are smoothed by the
                  heat conduction in solid regions.

           View Factor Calculation
              The view factor between two clusters is the fraction of the total radiation energy emitted
              from one of the clusters that is intercepted by other clusters. The following relations are
              used in the code to define the view factor.

           3D case
              View factor for each ray (except for the polar ray) are defined as follows:
                        εk                                   
                                                                    2

                          , ε k = (2k − 1) ⋅ 
               Fk =                                  n
                                                               , k = 1, 2,..., m − 1.
                        n                     (m − 1) ⋅ n + 1
              View factor for the Polar ray is:

                              ( m − 1) ⋅ n 
                                                     2

               Fpolar   = 1−                    .
                              ( m − 1) ⋅ n + 1 
                                               
           2D case

                        εk                         π                      π              
               Fk =              ε k = 2 ⋅ sin               ⋅ sin  2 ⋅ (2m − 1) (2k − 1) , k = 1, 2,..., m − 1.
                                               2 ⋅ (2m − 1) 
                             ,
                        2                                                                  

                           π ⋅ (m − 1) 
               Fpolar = cos
                            2m − 1 
                                          .
                                        
           Set of Equations

               I i = ∑ F ji L j is the Incident radiation flux;
                        j


               Li − ρ i ∑ ( Fji L j ) = ε iσTi 4
                             j

              where:
              ε is the emissivity coefficient, ρ is the reflection coefficient (ρ = 1-ε for walls, ρ = 0 for
              openings), and         σ = 5.672 × 10 −8 W                is the Stefan-Boltzmann constant.
                                                             m2K 4



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    Chapter Governing Equations


        Environment and Solar Radiation
            Environmental and solar radiation can be applied to external and internal problems. In
            fact, an environment radiation is the energy flux generated by the walls of an imaginary
            huge «room» that surrounds the body. This flux has predefined radiation parameters. In
            contrast to the environment radiation, the solar radiation is modeled by the directional
            energy flux. Therefore, we define solar radiation via its power flow (intensity) and its
            directional vector. Directed energy fluxes can be emitted by the surfaces that have the
            «Solar opening» boundary condition.

            The external radiation view factor can be calculated as F =   ∑ F ∗ S , where Fi are the
                                                                           i
                                                                               i

            view factors for the rays that have reached the boundaries of the computational domain,
            and S is the cluster area. Each «Solar opening» boundary condition produces one ray that
            follows the directional vector. After it reaches the outer boundary or the surface having
            appropriate radiation boundary condition, the view factor can be estimated as

                 (             )
            F = nsolar , nclust ∗ S .

        Radiative Surface Types


             Radiative surface type        Prescribed values              Dependent values

             Wall                          ε, Tr                          ρ = 1-ε, α = ε
             Opening/Outer boundary        ε, Tr                          ρ = 0, α = 1
             Solar opening                 n, W                           ε=1
             Symmetry                      No parameters

             Absorbent wall                No parameters

             Wall to ambient               ε, Tr                          ρ = 1-ε, α = ε
             Non-radiative                 No parameters

            For the «Wall» and «Wall to ambient» boundary condition, the program gets Tr from the
            current results set.
            For the «Opening/Outer Boundary» boundary condition, Tr is taken from the
            Engineering Database.
            The rays are emitted only from surfaces and boundaries on which the «Wall» or
            «Opening» boundary conditions are applied.
            Surfaces with the specified «Absorbent wall» boundary condition are taken into account
            during the calculation but they can act as absorbents only. This wall type takes all heat
            from the radiation that reaches it and does not emit any heat.

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              The «Symmetry» boundary condition forces the walls to which it is applied to reflect rays
              as an ideal mirror.
              The «Solar opening» boundary condition requires the wall to emit radiation like the outer
              solar radiation. It is specified by the direction vector and intensity. The solar radiation at
              the computational domain boundaries can be specified not only by the direction vector and
              intensity, but also by the location (on the surface of the Earth) and time.
              «Wall to ambient» reproduces the most elementary phenomenon among the radiation
              effects. The walls with this condition does not interact with any other surfaces. They can
              only exhaust energy into the space that surrounds the computational domain. Heat flux
              from the surface could be calculated as:

                        (
               O = σ ⋅ Tr 4 − ε out ⋅ Trout ,
                                         4
                                             )
              where   Trout is the temperature of the environmental radiation.
              «Non-radiative» boundary condition removes specific surfaces from the radiation heat
              transfer analysis, so they do not affect the results.
              After rays reach the surfaces for which «Opening», «Solar opening» or «Wall to
              ambient» radiative surface types are specified, they disappear. All energy that is carried
              out by these rays also dies away.

           Viewing Results
              The main result of the heat radiation calculation is the solids’ surface or internal
              temperatures. But these temperatures are influenced by calculations of heat transfer in
              solids and solid/fluid heat transfer also. To see the radiation calculation’s results only, the
              User may view the Leaving radiant flux and the distributions of the Net radiant flux over
              the selected radiative surfaces as Surface Plots, and their maximum, minimum, and
              average values over these surfaces in the Surface Parameters dialog boxes, as well as the
              Leaving radiation rate and Net radiation rate as an integral over these surfaces in the
              Surface Parameters dialog boxes.

       2.5 Global Rotating Reference Frame
              The rotation of the coordinate system is taken into account via the following
              mass-distributed force:

              S i rotation = −2 eijk Ω j ρuk + ρ Ω 2 ri ,

              where eijk is the Levy-Civita symbols (function), Ω is the angular velocity of the rotation,
              r is the vector coming to the point under consideration from the nearest point lying on the
              rotation axis.




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    2.6 Local rotating regions
            This option is employed for calculating time-dependent (transient) or steady-state flows in
            regions surrounding rotating non-axisymmetrical solids (e.g. impellers, mixers, propellers,
            etc), when a single global rotating reference cannot be employed. For example, local
            rotating regions can be used in analysis of the fluid flow in the model including several
            components rotating over different axes and/or at different speeds or if the computational
            domain has a non-axisymmetrical (with respect to a rotating component) outer solid/fluid
            interface. In accordance with the employed approach, each rotating solid component is
            surrounded by an axisymmetrical (with respect to the component's rotation axis) Rotating
            region, which has its own coordinate system rotating together with the component. If the
            model includes several rotating solid components having different rotation axes, the
            rotating regions surrounding these components must not intersect with each other. The
            fluid flow equations in the stationary (non-rotating) regions of the computational domain
            are solved in the inertial (non-rotating) Cartesian Global Coordinate System. The
            influence of the rotation's effect on the flow is taken into account in the equations written
            in each of the rotating coordinate systems.
            To connect solutions obtained within the rotating regions and in non-rotating part of the
            computational domain, special internal boundary conditions are set automatically at the
            fluid boundaries of the rotating regions. Since the coordinate system of the rotating region
            rotates, the rotating region’s boundaries are sliced into rings of equal width as shown on
            the Fig.2.3. Then the values of flow parameters transferred as boundary conditions from
            the adjacent fluid regions are averaged circumferentially over each of these rings.
                      Computational domain or fluid subdomain
                      Flow parameters are calculated in the inertial Global Coordinate System




                                         Local rotating region
                                         Flow parameters are calculated
                                         in the local rotating coordinate
                                         system


                                                                                 Rotation axis
                                             Flow parameters are
                                             averaged over these rings




                                                   Fig.2.3


            To solve the problem, an iterative procedure of adjusting the flow solutions in the rotating
            regions and in the adjacent non-rotating regions, therefore in the entire computational
            domain, is performed with relaxations.

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              Please note that even in case of time-dependent (transient) analysis the flow parameters
              within the rotating regions are calculated using a steady-state approach and averaged on
              the rotating regions' boundaries as described above.

       2.7 Mass Transfer in Fluid Mixtures
              The mass transfer in fluid mixtures is governed by species conservation equations. The
              equations that describe concentrations of mixture components can be written as

               ∂ρ y m    ∂                                           
                      +      (ρu i ym ) = ∂  Dmn + Dmn ∂yn
                                                 (  t
                                                                )      + S m , m = 1,2,..., M
                                                                      
                                                                                                    (5.26)
                ∂t      ∂ xi              ∂x i         ∂ xi          

                               t
              Here D mn,    D m n are the molecular and turbulent matrices of diffusion, Sm is the rate of
              production or consumption of the m-th component.
              In case of Fick's diffusion law:

                                                      µt
               Dmn = D ⋅ δ mn , Dmn = δ mn ⋅
                                 t                                                              (5.27)
                                                      σ
              The following obvious algebraic relation between species concentrations takes place:

               ∑ ym = 1 .                                                                       (5.28)
                m


       2.8 Flows in Porous Media
              Porous media are treated in COSMOSFloWorks as distributed resistances to fluid flow, so
              they can not occupy the whole fluid region or fill the dead-end holes. In addition, if the
              Heat conduction in solids option is switched on, the heat transfer between the porous
              solid matrix and the fluid flowing through it is also considered. Therefore, the porous
              matrix act on the fluid flowing through it via the Si, Siui, and (if heat conduction in solids
              is considered) QH terms in Eqs. (5.2) and (5.3), whose components related to porosity are
              defined as:

              S i porous = − k δ ij ρu j ,                                                        (5.29)


              QHporosit y = γ ⋅ ( Tp – T ) ,                                                      (5.30)




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    Chapter Governing Equations


            where k is the resistance vector of the porous medium (see below), γ is the user-defined
            volumetric porous matrix/fluid heat transfer coefficient, Tp is the temperature of the
            porous matrix, T is temperature of the fluid flowing through the matrix, and the other
            designations are given in Section 1. In addition, the fluid density in Eqs. (5.1)-(5.3) is
            multiplied by the porosity n of the porous medium, which is the volume fraction of the
            interconnected pores with respect to the total medium volume.
            In the employed porous medium model turbulence disappears within a porous medium
            and the flow becomes laminar.
            If the heat conduction in porous matrix is considered, then, in addition to solving
            Eqs. (5.1)-(5.3) describing fluid flow in porous medium, an Eq. (5.25) describing the heat
            conduction in solids is also considered within the porous medium. In this equation the
            source QH due to heat transfer between the porous matrix and the fluid is defined in the
            same manner as in Eq. (5.30), but with the opposite sign. The values of γ and c for the
            porous matrix may differ from those of the corresponding bulk solid material and hence
            must be specified independently. Density of the solid material is multiplied by the solid
            volume fraction in the porous matrix, i.e. by (1-n).
            Thermal conductivity of the porous matrix can be specified as anisotropic in the same
            manner as for the solid material.
            The conjugate heat transfer problem in a porous medium is solved under the following
            restrictions:
               • heat conduction in a porous medium not filled with a fluid is not considered,
               • porous media are considered transparent for radiation heat transfer,
               • heat sources in the porous matrix can be specified in the forms of heat generation
                 rate or volumetric heat generation rate only; heat sources in a form of constant or
                 time-dependent temperature can not be specified.
            To perform a calculation in COSMOSFloWorks, you have to specify the following porous
            medium properties: the effective porosity of the porous medium, defined as the volume
            fraction of the interconnected pores with respect to the total medium volume. Later on, the
            permeability type of the porous medium must be chosen among the following:
               • isotropic (i.e., the medium permeability is independent of direction),
               • unidirectional (i.e., the medium is permeable in one direction only),
               • axisymmetrical (i.e., the medium permeability is fully governed by its axial and
                 transversal components with respect to a specified direction),
               • orthotropic (i.e., the general case, when the medium permeability varies with
                 direction and is fully governed by its three components determined along three
                 principal directions).
            Then you have to specify some constants needed to determine the porous medium
            resistance to fluid flow, i.e., vector k defined as k = - grad(P)/(ρ⋅V), where P, ρ, and V are
            fluid pressure, density, and velocity, respectively. It is calculated according to one of the
            following formulae:
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                 • k = ∆P⋅S/(m⋅L), where ∆P is the pressure difference between the opposite sides of a
                   sample parallelepiped porous body, m is the mass flow rate through the body, S and
                   L are the body cross-sectional area and length in the selected direction, respectively.
                   You can specify ∆P as a function of m, whereas S and L are constants. Instead of
                   mass flow rate you can specify volume flow rate, v. In this case COSMOSFloWorks
                   calculates m = v⋅ρ. All these values do not specify the porous body for the
                   calculation, but its resistance k only.
                 • k = (A⋅V+B)/ρ, where V is the fluid velocity, A and B are constants, ρ is the fluid
                   density. Here, only A and B are specified, since V and ρ are calculated.
                 • k= µ/(ρ⋅ D2), where µ and ρ are the fluid dynamic viscosity and density, D is the
                   reference pore size determined experimentally. Here, only D is specified, since µ
                   and ρ are calculated.
                 • k= µ/( ρ⋅D2)⋅f(Re), differing from the previous formula by the f(Re) factor, yielding a
                   more general formula. Here, in addition to D, f(Re) as a formula dependency is
                   specified.
              To define a certain porous body, you specify both the body position in the model and, if
              the porous medium has a unidirectional or axisymmetrical permeability, the reference
              directions in the porous body.

       2.9 Two-phase (fluid + particles) Flows
              COSMOSFloWorks calculates two-phase flows as a motion of spherical liquid particles
              (droplets) or spherical solid particles in a steady-state flow field. COSMOSFloWorks can
              simulate dilute two-phase flows only, where the particle’s influence on the fluid flow
              (including its temperature) is negligible (e.g. flows of gases or liquids contaminated with
              particles). Generally, in this case the particles mass flow rate should be lower than about
              30% of the fluid mass flow rate.




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    Chapter Governing Equations


            The particles of a specified (liquid or solid) material and constant mass are assumed to be
            spherical. Their drag coefficient is calculated with Henderson’s formula (Ref. 1), derived
            for continuum and rarefied, subsonic and supersonic, laminar, transient, and turbulent
            flows over the particles, and taking into account the temperature difference between the
            fluid and the particle. The particle/fluid heat transfer coefficient is calculated with the
            formula proposed in Ref. 2. If necessary, the gravity is taken into account. Since the
            particle mass is assumed constant, the particles cooled or heated by the surrounding fluid
            change their size. The interaction of particles with the model surfaces is taken into account
            by specifying either full absorption of the particles (that is typical for liquid droplets
            impinging on surfaces at low or moderate velocities) or ideal or non-ideal reflection (that
            is typical for solid particles). The ideal reflection denotes that in the impinging plane
            defined by the particle velocity vector and the surface normal at the impingement point,
            the particle velocity component tangent to surface is conserved, whereas the particle
            velocity component normal to surface changes its sign. A non-ideal reflection is specified
            by the two particle velocity restitution (reflection) coefficients, en and eτ, determining
            values of these particle velocity components after reflection, V2,n and V2,τ, as their ratio to
            the ones before the impingement, V1,n and V1,τ:
                                                        V2 ,n                  V2 ,τ
                                                en =                   eτ =
                                                        V1,n                   V1,τ

            As a result of particles impingement on a solid surface, the total erosion mass rate,
            RΣerosion , and the total accretion mass rate, RΣaccretion, are determined as follows:

                          N
            R∑ erosion = ∑        ∫ K i ⋅ V p i ⋅ f1 i ( α p i ) ⋅ f 2 i ( d p i )dm p i,
                                            b                                      &
                         i =1 M p i


                              N
            R∑ accretion = ∑ M p i ,
                           i =1

            where:
            N is the number of fractions of particles specified by user as injections in
            COSMOSFloWorks (the user may specify several fractions of particles, also called
            injections, so that the particle properties at inlet, i.e. temperature, velocity, diameter, mass
            flow rate, and material, are constant within one fraction),
            i is the fraction number,
            Mp i is the mass impinging on the model walls in unit time for the i-th particle fraction,
            Ki is the impingement erosion coefficient specified by user for the i-th particle fraction,
            Vp i is the impingement velocity for the i-th particle fraction,
            b is the user-specified velocity exponent (b = 2 is recommended),
             f1 i (αp i) is the user-specified dimensionless function of particle impingement angle αp i,
            f2 i (dp i) is the user-specified dimensionless function of particle diameter dp i.



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       2.10 Cavitation
              A liquid subjected to a low pressure above a threshold ruptures and forms vaporous
              cavities. More specifically, when the local pressure at a point in the liquid falls below the
              liquid's vapour pressure at the local temperature, the liquid undergoes phase transition and
              form cavities filled with the liquid's vapor with an addition of gas that has been dissolved
              in the liquid. This phenomenon is called cavitation.
              A homogeneous equilibrium model of cavitation in water is employed. The computational
              simplicity is the main advantage of the homogeneous model. The stationary homogeneous
              flow approach provides a simple technique for analyzing two-phase flows. It is accurate
              enough to handle a variety of practically important processes, including localized boiling
              of water due to intense heating.
              The fluid is assumed to be a homogenous gas-liquid mixture with the volume-averaged
              parameters and the gaseous phase comprising the liquid vapour and non-condensable
              (dissolved) gas. The liquid vapour to gas ratio is defined at the local equilibrium
              thermodynamic conditions. By default, the mass fraction of non-condensable air is set to
              10-5. This is a typical value under normal conditions and appropriate in most cases but it
              can be modified by the user in the range of 10-4…10-8.
              The homogeneous equilibrium cavitation model does not describe the detailed structure of
              the cavitation area, and the migration of individual vapour bubbles in the counter-gradient
              direction is not considered. The velocities and temperatures of the gaseous (including
              vapour and non-condensable gas) and liquid phases are assumed to be the same.
              The density of the gas-liquid mixture is calculated as:

                    1          RunivT                                R Tz (T , P)
               ρ=     , v = yg        + (1 − yg − yv )vl (T , P) + yv univ v
                    v          Pµg                                        Pµv     ,

              where v is the specific volume of the gas-liquid mixture, vl is the specific volume of
              liquid, zv (T,P) is the vapour compressibility ratio, P is the local static pressure, T is the
              local temperature, yv is the mass fraction of vapour,     yg is the mass fraction of the
              non-condensable gas; µg is the molar mass of the non-condensable gas, µv is the molar
              mass of vapour, Runiv is the universal gas constant.

              The mass fraction of vapour yv is computed numerically from the following non-linear
              equation for the full enthalpy gas-liquid mixture:
                                                                                     2
                                                                               I v
               H = y g hg (T , P) + (1 − y g − yv )hl (T , P) + yv hv (T , P) + C ,
                                                                                2



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    Chapter Governing Equations


            where temperature of the mixture T is a function of pressure P and yv. Here hg, hl, hv are
            the enthalpies of non-condensable gas, liquid and vapour, respectively,
             I C = (ρu x ) 2 + ( ρu y ) 2 + ( ρu z ) 2 is the squared impulse.
            The model has the following limitations and assumptions:
               • Cavitation is currently available only for incompressible water (when defining the
                 project fluids you should select Water SP from the list of Pre-Defined liquids);
                 cavitation in mixtures of different liquids cannot be calculated.
               • The properties of the non-condensable gas are set to be equal to those of air.
               • The temperature and pressure in the phase transition areas should be within the
                 following ranges:
                  T = 277.15 - 583.15 K, P = 800 - 107 Pa.
               • If the calculation has finished or has been stopped and the Cavitation option has
                 been enabled or disabled, the calculation cannot be resumed or continued and must
                 be restarted from the beginning.
               • The Cavitation option should not be selected if you calculate a water flow in the
                 model without flow openings (inlet and outlet).
               • The model does not describe the detailed structure of the cavitation area, i.e
                 parameters of individual vapour bubbles.
               • The fluid region where cavitation occurs should be well resolved by the
                 computational mesh.
               • The parameters of the flow at the inlet boundary conditions must be such that the
                 volume fraction of liquid water in the inlet flow would be at least 0.1.

    2.11 Boundary Conditions

        Internal Flow Boundary Conditions
            For internal flows, i.e., flows inside models, COSMOSFloWorks offers the following two
            options of specifying the flow boundary conditions: manually at the model inlets and
            outlets (i.e. model openings), or to specify them by transferring the results obtained in
            another COSMOSFloWorks calculation in the same coordinate system (if necessary, the
            calculation can be performed with another model, the only requirement is the flow regions
            at the boundaries must coincide).
            With the first option, all the model openings are classified into "pressure" openings,
            "flow" openings, and "fans", depending on the flow boundary conditions which you
            intend to specify on them.




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              A "pressure" opening boundary condition, which can be static pressure, or total pressure,
              or environment pressure is imposed in the general case when the flow direction and/or
              magnitude at the model opening are not known a priori, so they are to be calculated as part
              of the solution. Which of these parameters is specified depends on which one of them is
              known. In most cases the static pressure is not known, whereas if the opening connects the
              computational domain to an external space with known pressure, the total pressure at the
              opening is known. The Environment pressure condition is interpreted by
              COSMOSFloWorks as a total pressure for incoming flows and as a static pressure for
              outgoing flows. If, during calculation, a vortex crosses an opening with the Environment
              pressure condition specified at it, this pressure considered as the total pressure at the part
              of opening through which the flow enters the model and as the static pressure at the part of
              opening through which the flow leaves the model.
              Note that when inlet flow occurs at the "pressure" opening, the temperature, fluid mixture
              composition and turbulence parameters have to be specified also.
              A "flow" opening boundary condition is imposed when dynamic flow properties (i.e., the
              flow direction and mass/volume flow rate or velocity/ Mach number) are known at the
              opening. If the flow enters the model, then the inlet temperature, fluid mixture
              composition and turbulence parameters must be specified also. The pressure at the
              opening will be determined as part of the solution. For supersonic flows the inlet pressure
              must be specified also.
              A "fan" condition simulates a fan installed at a model opening. In this case the
              dependency of volume flow rate on pressure drop over the fan is prescribed at the opening.
              These dependencies are commonly provided in the technical documentation for the fans
              being simulated.
              With the second option, you specify the boundary conditions by transferring the results
              obtained in another COSMOSFloWorks calculation in the same coordinate system. If
              necessary, the calculation can be performed with another model, the only requirement is
              the flow regions at the boundaries must coincide. At that, you select the created boundary
              conditions’ type: either as for external flows (so-called "ambient" conditions, see the next
              Section), or as for "pressure" or "flow" openings, see above. If a conjugate heat transfer
              problem is solved, the temperature at the part of the boundary lying in a solid body is
              transferred from the other calculation.
              Naturally, the flow boundary conditions specified for an internal flow problem with the
              first and/or second options must be physically consistent with each other, so it is expedient
              to specify at least one "pressure"-type boundary condition and at least one "flow"-type
              boundary condition, if not only "ambient" boundary conditions are specified.

           External Flow Boundary Conditions

              For external problems such as flow over an aircraft or building, the parameters of the
              external incoming flow (so-called "ambient" conditions) must be defined. Namely the
              velocity, pressure, temperature, fluid mixture composition and turbulence parameters must
              be specified. Evidently, during the calculation they can be partly violated at the flow
              boundary lying downstream of the model.
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    Chapter Governing Equations


        Wall Boundary Conditions
            In COSMOSFloWorks the default velocity boundary condition at solid walls corresponds
            to the well-known no-slip condition. The solid walls are also considered to be
            impermeable. In addition, the wall surface's translation and/or rotation (without changing
            the model's geometry) can be specified. If a calculation is performed in a rotating
            coordinate system, then some of the wall surfaces can be specified as stationary, i.e. a
            backward rotation in this coordinate system (without changing the model geometry).
            COSMOSFloWorks also provides the "Ideal Wall" condition that corresponds to the
            well-known slip condition. For example, Ideal Walls can be used to model planes of flow
            symmetry.
            If the flow of non-Newtonian liquids is considered, then the following slip condition at
            solid walls can be specified: if the shear stress τ exceeds the yield stress value τ0,slip, then

            a slip velocity vslip determined from     vslip = C1 (τ − τ0,slip )C 2 , where C 1 and C2, as well
            as τ0,slip, can be specified by user, if they are not specified in the model of non-Newtonian
            liquid. If conjugate heat transfer in fluid and solid media is not considered, one of the
            following boundary conditions can be imposed at solid walls: either the wall temperature

             T = Tw ,                                                                             (5.31)


            or the heat flux,

             q = qw                                                                               (5.32)


            being positive for heat flows from fluid to solid, equal to zero for adiabatic
            (heat-insulated) walls, and negative for heat flows from solid to fluid.
            When considering conjugate heat transfer in fluid and solid media, the heat exchange
            between fluid and solid is calculated by COSMOSFloWorks, so heat wall boundary
            conditions are not specified at the walls.

        Internal Fluid Boundary Conditions
            If one or several non-intersecting axisymmetric rotating regions (local rotating reference
            frames) are specified, the flow parameters are transferred from the adjacent fluid regions
            and circumferentially averaged over rotating regions’ boundaries as boundary conditions.




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           Periodic Boundary Conditions
              The "periodicity" condition may be applied if the model consist of identical geometrical
              features arranged in periodic linear order. Periodic boundary conditions are specified at
              the pair of computational domain boundaries for the selected direction in which a
              geometrical feature or a group of features repeats regularly over distance. Periodic
              boundary conditions allows to reduce the analysis time by calculating the fluid flow only
              for a small group of identical geometrical features or even just for one feature, but taking
              into account influence of other identical features in the pattern. Please note that the
              number of basic mesh cells along the direction in which the "periodicity" condition is
              applied must be no less than five.

       3   Numerical Solution Technique

              The numerical solution technique employed in COSMOSFloWorks is robust and reliable,
              so it does not require any user knowledge about the computational mesh and the numerical
              methods employed. But sometimes, if the model and/or the problem being solved are too
              complicated, so that the COSMOSFloWorks standard numerical solution technique
              requires extremely high computer resources (memory and/or CPU time) which are not
              available, it is expedient to employ COSMOSFloWorks options which allow the
              adjustment of the automatically specified values of parameters governing the numerical
              solution technique. To employ these options properly and successfully, take into account
              the information presented below about COSMOSFloWorks’ numerical solution technique.
              Briefly, COSMOSFloWorks solves the governing equations with the finite volume (FV)
              method on a spatially rectangular computational mesh designed in the Cartesian
              coordinate system with the planes orthogonal to its axes and refined locally at the
              solid/fluid interface and, if necessary, additionally in specified fluid regions, at the
              solid/solid surfaces, and in the fluid region during calculation. Values of all the physical
              variables are stored at the mesh cell centers. Due to the FV method, the governing
              equations are discretized in a conservative form. The spatial derivatives are approximated
              with implicit difference operators of second-order accuracy. The time derivatives are
              approximated with an implicit first-order Euler scheme. The viscosity of the numerical
              scheme is negligible with respect to the fluid viscosity.

       3.1 Computational Mesh
              COSMOSFloWorks computational mesh is rectangular everywhere in the computational
              domain, so the mesh cells’ sides are orthogonal to the specified axes of the Cartesian
              coordinate system and are not fitted to the solid/fluid interface. As a result, the solid/fluid
              interface cuts the near-wall mesh cells. Nevertheless, due to special measures, the mass
              and heat fluxes are treated properly in these cells named partial.
              The rectangular computational domain is automatically constructed (may be changed
              manually), so it encloses the solid body and has the boundary planes orthogonal to the
              specified axes of the Cartesian coordinate system. Then, the computational mesh is
              constructed in the following several stages.

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    Chapter Numerical Solution Technique


            First of all, a basic mesh is constructed. For that, the computational domain is divided into
            slices by the basic mesh planes, which are evidently orthogonal to the axes of the
            Cartesian coordinate system. The user can specify the number and spacing of these planes
            along each of the axes. The so-called control planes whose position is specified by user
            can be among these planes also. The basic mesh is determined solely by the computational
            domain and does not depend on the solid/fluid interfaces.
            Then, the basic mesh cells intersecting with the solid/fluid interface are split uniformly
            into smaller cells in order to capture the solid/fluid interface with mesh cells of the
            specified size (with respect to the basic mesh cells). The following procedure is employed:
            each of the basic mesh cells intersecting with the solid/fluid interface is split uniformly
            into 8 child cells; each of the child cells intersecting with the interface is in turn split into 8
            cells of next level, and so on, until the specified cell size is attained.
            At the next stage of meshing, the mesh obtained at the solid/fluid interface with the
            previous procedure is refined (i.e. the cells are split further or probably merged) in
            accordance with the solid/fluid interface curvature. The criterion to be satisfied is
            established as follows: the maximum angle between the normals to the surface inside one
            cell should not exceeds certain threshold, otherwise the cell is split into 8 cells.
            Finally, the mesh obtained with these procedures is refined in the computational domain to
            satisfy the so-called narrow channel criterion: for each cell lying at the solid/fluid
            interface, the number of the mesh cells (including the partial cells) lying in the fluid region
            along the line normal to the solid/fluid interface and starting from the center of this cell
            must not be less than the criterion value. Otherwise each of the mesh cells on this line is
            split into 8 child cells.
            As a result of all these meshing procedures, a locally refined rectangular computational
            mesh is obtained and used then for solving the governing equations on it.
            Since all the above-mentioned meshing procedures are performed before the calculation,
            the obtained mesh is unable to resolve all the solution features well. To overcome this
            disadvantage, the computational mesh can be refined further at the specified moments
            during the calculation in accordance with the solution spatial gradients (both in fluid and
            in solid, see User’s Guide for details). As a result, in the low-gradient regions the cells are
            merged, whereas in the high-gradient regions the cells are split. The moments of the
            computational mesh refinement during the calculation are prescribed either automatically
            or manually.




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       3.2 Spatial Approximations
               The cell-centered finite volume (FV) method is used to obtain conservative
               approximations of the governing equations on the locally refined rectangular mesh. The
               governing equations are integrated over a control volume which is a grid cell, and then
               approximated with the cell-centered values of the basic variables. The integral
               conservation laws may be represented in the form of the cell volume and surface integral
               equation:

               ∂
               ∂t ∫
                    Udv + ∫ F ⋅ds = ∫ Qdv                                                         (5.33)



               are replaced by the discrete form

                ∂
                   (Uv ) +        ∑ F ⋅ S = Qv                                                    (5.34)
                ∂t             cell faces


               The second-order upwind approximations of fluxes F are based on the implicitly treated
               modified Leonard's QUICK approximations (Ref. 3) and the Total Variation Diminishing
               (TVD) method (Ref. 4).
               In COSMOSFloWorks, especially consistent approximations for the convective terms, div
               and grad operators are employed in order to derive a discrete problem that maintains the
               fundamental properties of the parent differential problem in addition to the usual
               properties of mass, momentum and energy conservation.

           Spatial Approximations at the Solid/fluid Interface
               Considering equation (5.34) for partial mesh cells (i.e., for the mesh cells cut by the
               solid/fluid interface), we introduce the additional boundary faces and the corresponding
               boundary fluxes taking the boundary conditions and geometry into account (see Fig.3.1),
               as well as use a special calculation procedure for them. As a result, the solid/fluid interface
               influence on the problem solution both in the fluid and in the solid is calculated very
               accurately.




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    Chapter Numerical Solution Technique




              Fig.3.1 Computational mesh cells at the solid/fluid interface.


    3.3 Temporal Approximations
            Time-implicit approximations of the continuity and convection/diffusion equations (for
            momentum, temperature, etc.) are used together with an operator-splitting technique
            (Ref. 5, Ref. 6, and Ref. 7). This technique is used to efficiently resolve the problem of
            pressure-velocity decoupling. Following the SIMPLE-like approach (Ref. 8), an elliptic
            type discrete pressure equation is derived by algebraic transformations of the originally
            derived discrete equations for mass and momentum, and taking into account the boundary
            conditions for velocity.




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       3.4 Form of the Numerical Algorithm
              Let index 'n' denotes the time-level, and '*' denotes intermediate values of the flow
              parameters. The following numerical algorithm is employed to calculate flow parameters
              on time-level (n+1) using known values on time-level (n):

               U* - U n
                 ∆t
                                (        )
                        + Ah U n , p n U* = S n ,
                                                                                                (5.35)




               Lhδp =
                              ( )
                        divh ρu*
                                 +
                                   1 ρ* − ρ n
                                              ,
                                                                                                (5.36)
                           ∆t      ∆t ∆t
               ρ* = ρ(pn+δp,T*,y*),

               ρu n +1 = ρu* − ∆t ⋅ grad h δp ,
                                                                                                (5.37)


               pn +1 = pn + δp ,                                                                (5.38)


               ρT n +1 = ρT * , ρκ n +1 = ρκ * , ρε n +1 = ρε * , ρy n +1 = ρy* ,
                                                                                                (5.39)


                          (                    )
               ρ n+1 = ρ p n+1 , T n +1 , y n +1 .                                              (5.40)


              Here U = (ρu, ρT, ρκ, ρε, ρy)Tis the full set of basic variables excluding pressure p,
              u=(u1,u 2,u3)T is the velocity vector, y = (y1, y 2, ..., yM)T is the vector of component
              concentrations in fluid mixtures, and δp = pn+1 - p n is an auxiliary variable that is called
              a pressure correction. These parameters are discrete functions stored at cell centers. They
              are to be calculated using the discrete equations (5.35)-(5.40) that approximate the
              governing differential equations. In equations (5.35)-(5.40) Ah, divh, gradh and Lh =
              divhgradh are discrete operators that approximate the corresponding differential operators
              with second order accuracy.
              Equation (5.35) corresponds to the first step of the algorithm when fully implicit discrete
              convection/diffusion equations are solved to obtain the intermediate values of momentum
              and the final values of turbulent parameters, temperature, and species concentrations.
              The elliptic type equation (5.36) is used to calculate the pressure correction δp. This
              equation is defined in such a way that the final momentum field ρun+1 calculated from
              (5.35) satisfies the discrete fully implicit continuity equation. Final values of the flow
              parameters are defined by equations (5.37)-(5.40).




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    Chapter Numerical Solution Technique


    3.5 Methods to Resolve Linear Algebraic Systems

        Iterative Methods for Nonsymmetrical Problems
            To solve the asymmetric systems of linear equations that arise from approximations of
            momentum, temperature and species equations (5.35), a preconditioned generalized
            conjugate gradient method (Ref. 9) is used. Incomplete LU factorization is used for
            preconditioning.

        Iterative Methods for Symmetric Problems
            To solve symmetric algebraic problem for pressure-correction (5.36), an original
            double-preconditioned iterative procedure is used. It is based on a specially developed
            multigrid method (Ref. 10).

        Multigrid Method
            The multigrid method is a convenient acceleration technique which can greatly decrease
            the solution time. Basic features of the multigrid algorithm are as follows. Based on the
            given mesh, a sequence of grids (grid levels) are constructed, with a decreasing number of
            nodes. On every such grid, the residual of the associated system of algebraic equations is
            restricted onto the coarser grid level, forming the right hand side of the system on that
            grid. When the solution on the coarse grid is computed, it is interpolated to the finer grid
            and used there as a correction to the result of the previous iteration. After that, several
            smoothing iterations are performed. This procedure is applied repeatedly on every grid
            level until the corresponding iteration meets the stopping criteria.
            The coefficients of the linear algebraic systems associated with the grid are computed
            once and stored.




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       References

              1 Henderson, C.B. Drag Coefficients of Spheres in Continuum and Rarefied Flows.
                 AIAA Journal, v.14, No.6, 1976.
              2 Carlson, D.J. and Hoglund, R.F. Particle Drag and Heat Transfer in Rocket Nozzles.
                 AIAA Journal, v.2, No.11, 1964.
              3 Roache, P.J., (1998) Fundamentals of Computational Fluid Dynamics, Hermosa
                 Publishers, Albuquerque, New Mexico, USA.
              4 Hirsch, C., (1988). Numerical computation of internal and external flows. John Wiley
                 and Sons, Chichester.
              5 Glowinski, R. and P. Le Tallec, (1989). Augmented Lagrangian Methods and
                 Operator-Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia.
              6 Marchuk, G.I., (1982). Methods of Numerical Mathematics, Springer-Verlag, Berlin.
              7 Samarskii, A.A., (1989). Theory of Difference Schemes, Nauka, Moscow (in Russian).
              8 Patankar, S.V., (1980). Numerical Heat Transfer and Fluid Flow, Hemisphere,
                 Washington, D.C.
              9 Saad, Y. (1996). Iterative methods for sparse linear systems, PWS Publishing
                 Company, Boston.
              10 Hackbusch, W. (1985). Multi-grid Methods and Applications, Springer-Verlag, NY,
                 USA.
              11 Reid R.C., Prausnitz J.M., Poling B.E. (1987). The properties of gases and liquids, 4th
                 edition, McGraw-Hill Inc., NY, USA.
              12 Idelchik, I.E. (1986). Handbook of Hydraulic Resistance, 2nd edition, Hemisphere,
                 New York, USA.




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    Chapter References




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