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Introduction to Computational Fluid Dynamics Course Notes (CFD 4) Karthik Duraisamy Department of Aerospace Engineering University of Glasgow Course organization • Homeworks – 25 % • Report -- 15 % • Final exam -- 60 % Interaction is extremely important Learning Objectives • Make the student understand the role of C in FD, its applicability, potential and limitations • Give a basic foundation in numerical analysis, by teaching the relevance of accuracy and stability • Give a working idea of the various choices of numerical methods and discretization schemes by applying them to simple model equations. In doing this, always remind them of the connection with the big picture. • Make the student knowledgeable about the various terminologies in practical CFD (Grids, BCs, Approximations, Schemes etc) • Ingrain the basics of good CFD practice (be aware of the applicability/feasibility of a particular model, its limitations, choose the right boundary conditions, ascertain grid/time independence, verification/validation) • By the end of the class, the student should be in a position to set up simple aerodynamic problems and analyze them Contents • Introduction (1.5) • Classification of PDE, Model equations (1.5) • Finite difference methods: Spatial discretization (2.5) Temporal discretization (1.5) Convergence, Consistency, Stability (1) • Grids/Boundary conditions (1) • Euler equations (0.5) • DNS/LES (1) • RANS Equations and Turbulence modeling (1) • Case studies & Best practices in CFD (1.5) • Hands-on CFD/Lab sessions (8) (.) – Approximate number of lectures Introduction What is CFD/FD ? • CFD is a branch of Fluid dynamics • So what really is Engineering Fluid Dynamics in the first place? Lets look at some examples: We are interested in the forces (pressure , viscous stress etc.) acting on surfaces (Example: In an airplane, we are interested in the lift, drag, power, pressure distribution etc) We would like to determine the velocity field (Example: In a race car, we are interested in the local flow streamlines, so that we can design for less drag) We are interested in knowing the temperature distribution (Example: Heat transfer in the vicinity of a computer chip) • Roughly put, in Engineering fluid dynamics, we would like to determine certain flow properties in a certain region of interest, so that the information can be used to predict the behaviour of systems, to design more efficient systems etc.. • Theoretical Fluid Dynamics Most important branch of fluid dynamics. Crucial in understanding concepts (Example: L = ρUΓ), Usually good in predicting trends (Example: δ ~ Re-1/2) Can obtain a lot of information using simplifying assumptions, sometimes enough for detailed design (Example: the SR-71 Blackbird was designed completely using theoretical ideas) However, doesn’t always provide sufficient information • Experimental Only way to obtain reliable data in many situations. However, costly, difficult to achieve exact conditions, difficult to isolate effects, sometimes difficult to assess error, sometimes not repeatable • Computational (CFD) Becoming important as computers are getting faster and cheaper. Potential to provide tremendous amount of data at a fraction of the cost of experiments. But sometimes unreliable because of numerical/modeling/human errors. Sometimes more expensive than experiments Very important to validate with theory/experiments Words of wisdom (To be taken with a huge helping of salt :) • Theoretical Fluid dynamics: Most important. Everyone HAS to learn it. • Experimental Fluid dynamics: Important. Usually, everyone believes it except the person that conducted the experiment. • Computational Fluid dynamics: Also important. Usually, no one believes it except the person that performed the calculations. • A good engineer understands the pro’s and con’s of all three methods, and should be in a position to assess which one is best under the circumstances • More importantly, should not be prejudiced against any of the three approaches Courtesy: CFD society of Canada Courtesy: CFD society of Canada Sample Application – 1 [Simulation to understand physics] Flow over F-16 at 45o angle of attack Surface Pressure contours and streamtraces Courtesy: Kyle Squires, ASU Sample Application -2 [Validation with Experiment] Experiment Computation Flow over fixed wing – Expt. vs CFD of velocity contours Sample Application -3 [Simulation to aid theoretical understanding] Merger of co-rotating vortices due to Elliptical instability (Movie) Courtesy: CERFACS Procedures in CFD • Identification of right approximation (Viscous/Inviscid, Laminar/Turbulent, Incompressible / compressible, Single-phase/multi- phase) • Identification of right solution method (Finite Element / Difference/Volume, Structured/Unstructured mesh, Order of accuracy) • Pre-processing (Generate computational grid, assign boundary conditions, set initial conditions, compile code, prepare input parameters) • Solution (Run the code, monitor the solution) • Post-processing (Collect and organize data, analyze results) • Verification (Do the results make sense? Are the trends right? Does it agree with previous calculations on similar configurations?) • Validation (Does the result (or an aspect of the result)) agree with theory/experiment?) • At every step, good understanding of theoretical fluid dynamics is essential!!! Example: Flow over a pitching airfoil • Problem: Predict the loads acting on an airfoil pitching in a wind tunnel under the following conditions: α =10o + 10o sin(w t), Re = 3.8x106, M = 0.3, w = 0.06 • Identification of right approximation : Viscous, Turbulent, compressible, Single-phase • Identification of right solution method (Finite Volume, Structured mesh, second order accurate) Example: Flow over a pitching airfoil • Preprocessing: Example: Flow over a pitching airfoil • Solution: Example: Flow over a pitching airfoil • Post processing: Flow visualization (movie) Example: Flow over a pitching airfoil • Post processing: Loads comparison Governing Equations of fluid dynamics • Assumptions: Continuum flow, Newtonian fluid • Lets restrict ourselves to single phase, single species, perfect gases (this way, incompressible flow is a special case) • Ignore body forces • Unknowns: Density (ρ), Velocity (u,v,w), Pressure (p) • Dynamics of fluids is then given by Conservation of Mass (Continuity equation) [Law of common sense] Conservation of Momentum (Navier-Stokes equations) [Newton’s second law] Conservation of Energy (Energy equation) [First law of thermodynamics] • 5 equations to determine 5 unknowns. • All of fluid dynamics is contained in these equations Governing equations • How to derive these equations? Integral form Differential form • Reynolds transport theorem: Rate of change of “stuff” inside a control volume = Net flux of “stuff” entering/leaving the boundaries + generation of “stuff” – destruction of “stuff” • In addition, need some more info (such as stress-strain relation, temperature-heat flux relation etc.) The “stuff” U is nothing but mass, momentum and energy Governing equations for compressible perfect gas Governing equations for compressible perfect gas Example: One dimensional steady inviscid flow Sound familiar?
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