VIEWS: 4 PAGES: 48 POSTED ON: 3/20/2011 Public Domain
SC08 Engineering Track: Introducing Modeling Skills – CFD Jim Giuliani Client and Technology Support Manager jimg@osc.edu Agenda The CFD Process •Solid Modeling •Mesh Generation •Physical Properties •Initial and Boundary Conditions •Numerical Solution •Data Visualization Computational to Fluid Dynamics •Partial Differential Equations •Governing Equations •Discretization •Numerical Solutions 2 Computational Fluid Dynamics (CFD) • In CFD the equations that govern a process of interest are solved numerically • Once we have classified our problem, we can derive equations that describe the fluid system • Approximations can be made that allow the governing equations to be reduced in complexity • For complex flows or geometries we can approximate the governing equations in a form that can be solved numerically. • CFD encompasses the entire process of solving the governing equations numerically 3 The CFD Process Solid Modeling Mesh Generation Physical Properties Initial and Boundary Conditions Numerical Solution Data Visualization 4 The CFD Process • Based on the classification of the problem to be analyzed, select the appropriate solver – Based on flow classification, simplifications are made to the governing equations – Simplified governing equations need to be solved using different numerical techniques • Describe the physical problem – Geometry – Physical properties • Set solution and solver settings • Start the solution and monitor progress – Solution parameters, residuals, Courant number, are monitored to determine if the solution is converging 5 Solid Modeling The first step in creating a numerical model is to describe the geometry of the problem to be analyzed • Most grid generation software have basic solid modeling capabilities • Parts/geometries can be imported from CAD/CAM software packages 6 Mesh Generation The physical domain defined by the solid model is discretized into small elemental surfaces/volumes A solution to the governing equations can now be approximated 7 Grid/Mesh Limitations • Overall flowfield model is on order of hundreds of feet • Only 1.5 inch gap between belly and ground • To properly represent flow through any gap, should have at least 5 cells between wall surfaces • Turbulence modeling required first layer of cells on road and body to be ~0.5” – This leaves only ~0.5” for remaining 3 layers; results in very high aspect ratio cells and misrepresented ground/vehicle boundary layer interaction Boundary layer cells 0.5” 1.5” Boundary layer cells 0.5” 8 Physical Properties Temperature Pressure Density Viscosity is a measure of the resistance of a fluid which is being deformed. – Dynamic – Kinematic (dynamic viscosity/density) 9 Initial and Boundary Conditions For the solution to be well posed – A solution must exist – The solution must be unique – The solution must depend on initial or boundary conditions There are 3 types of Boundary Conditions – Dirchelt u f (x) – Neumann u g (x) n – Robins or mixed u a( x)u b( x) h( x ) n 10 Initial and Boundary Conditions Initial Conditions Specify properties at the beginning of the analysis – Properties may change during simulation – Choice of initial conditions can shorten solution time Boundary Conditions Describe how the simulation will behave at the edges of the computational domain – Properties often are constant throughout simulation Common properties that are specified – Velocity, Pressure and Temperature Special types of boundaries – Symmetry, Extrapolated 11 Boundary Conditions For the example of simulating the flow around the body of a car (U > 200 m.p.h.), boundary conditions must be specified on all surfaces and edges of the computational domain Symmetry on top and sides Solution driven by fixed inlet velocities Flow exits domain through pressure outlet Zero velocity (no slip) at car surface Solid, moving boundary for ground (speed equal to free stream velocity) 12 Numerical Solution •Solution to the discretization equations are approximated numerically •Choose solver settings (relaxation factor, tolerances) •Set solution control parameters (start time, end time, time step) Residuals •Errors of the discretized equations •Monitored as a means to determine when solution has converged 13 Data Visualization Grids that contain millions of elements can provide very high resolution information on flow field Colorized contour plots, particle traces and time lapse animations allow subtle flow patterns and overall flow characteristics to be examined User must specify at what interval data is written out More steps = more data (gigabytes per simulation) (terabytes per design problem) Solutions can be run remotely, but visualizations perform best locally 14 Exercise – 2D Heat Conduction • Cooling_channel_descripton.doc contains a detailed description of the problem • We have used Java applet to solve for a solution, but adiabatic boundary condition was not able to be modeled accurately • Open Heat_transfer_CFD_portal.doc for instructions on how to solve this problem with the OpenFOAM CFD solver • Compare results to solution obtained with Java Applet 15 Steady / Unsteady Flow Steady flow denotes a system where the flow does not change with time When the fundamental equations are discussed, we will see that steady flow denotes that all time derivatives are zero When the stability of a numerical solution is discussed, we will see that selection of B.C. for unstable flow strongly impacts stability 16 Fluid Flow Streamlines Streamlines – A moving fluid element is seen to trace out a fixed path in space – A streamline is this fixed path 17 Classification of Fluid Mechanics Continuum Fluid Mechanics • Inviscid • Viscous – Laminar – Turbulent • Compressible/Incompressible 18 Classification of Flows Compressible/Incompressible Inviscid Viscous • Laminar – Flow where the streamlines are smooth and regular and a fluid element moves smoothly along a streamline • Turbulent Turbulent Flow – Flow where the streamlines break up and a fluid element Laminar Flow moves in a random, irregular, and tortuous fashion 19 Exercise – Flow Past a Circular Cylinder Model low speed flow around a cylinder • Assumptions/Simplifications – Incompressible: all density derivatives are zero – Inviscid: neglect viscous forces • Possible Objectives – Practical application of the fundamental flow equations – Examine assumptions and simplifications that can reduce complex partial differential equations to simple analytic equations 20 Exercise – Flow Past a Circular Cylinder • Possible Objectives (cont) – Understand when higher fidelity tools are needed to examine complex flow phenomena • Top image: - ideal flow solution • Bottom image: Landsat 7 image of Juan Fernandez islands off of the coast of Chili 21 Exercise – Flow Past a Circular Cylinder Model low speed flow around a cylinder • To study the ideal flow model, open Cylinder_ideal_flow.doc to see the assignment • Open Cylinder_CFD_portal.doc to see the assignment for the viscous, CFD solution • For the CFD assignment, follow steps through page 6 and stop after SUBMIT button has been pressed and job is submitted • Simulation takes about 45 minutes and results should be available towards the end of the workshop 22 Computational Fluid Dynamics Partial Differential Equations •Physical Classification •Numerical Classification Governing Equations •Continuity •Momentum •Turbulence Discretization •Finite Difference •Laplace’s Equation •Finite Volume Numerical Solutions 23 Partial Differential Equations Partial Differential Equations (PDEs) – Many important physical processes are governed by PDEs – We will look at some PDEs commonly encountered in fluid dynamics Physical Classification – Equilibrium Problems – Marching Problems Numerical Classification – Elliptic – Parabolic – Hyperbolic 24 PDEs – Physical Classification Equilibrium Problems – Solution desired in a closed domain – Boundary value problem – Governed by elliptic PDEs Examples – Steady state temperature distribution – Incompressible, inviscid flow 25 PDEs – Physical Classification Marching Problems – Transient – Prescribed set of initial conditions in addition to boundary conditions – Solution computed by marching from the initial conditions, constrained by boundary conditions – Governed by parabolic or hyperbolic PDEs Examples – Unsteady, inviscid flow – Steady supersonic inviscid flow – Boundary layer flow 26 PDEs – Numerical Classification Hyperbolic – Fundamental property is the limited domain of dependence – Solution at point P depends only on information in the domain of dependence – Any disturbance that occurs outside this interval can never influence the solution at point P 2 y 2 y Example: Wave Equation a2 2 t 2 x 27 PDEs – Numerical Classification Parabolic – Unlike hyperbolic equations, solution at some time tn depends upon the entire physical domain at earlier times, including side boundary conditions – Start at some initial data plane and march forward – Diffusion processes u 2u Example: 1D heat transfer equation 2 t x 28 PDEs – Numerical Classification Elliptic – Boundary value problem – Subject to a prescribed set of boundary conditions on a closed domain – Solution at any point depends upon the specified conditions at all points on the boundary Example: Laplace’s Equation 2u 2u 2 0 x 2 y 29 Conservation Laws Mass conservation – Matter may neither be created or destroyed Conservation of momentum – Newton’s 2nd law of motion Conservation of energy – First law of thermodynamics 30 Continuity Equation For a given control volumn dm in m out m dt The rate increase of mass within the control volume is equal to the net rate at which mass enters the control volume 31 Continuity Equation The partial differential form of the continuity equation in cartesian coordinates is D u v w x y z 0 Dt where D u v w Dt t x y z 32 Conservation of Momentum Newton’s 2nd law F x m ax (in 2 dimensions) F y m ay Body forces •Gravity •Centrifugal •Corolis •Electromagnetic Surface forces •Normal stress •Tangential stress 33 Conservation of Momentum Combining forces yields Du xx yx zx Fx Dt x y z Surface forces Body forces 34 Navier-Stokes Equations Replacing stress terms with stress-strain relationship Du 1 p 2 u v w u v w u gx x y z y y x z x z 2 Dt x x 3 Dv p v u 2 v u w v w x y y 3 y x z z z y g y 2 Dt y x Dw p w u v w 2 w u v g z z y z 3 2 z x y Dt z x x z y Assumptions/simplifications can reduce complexity 35 Navier-Stokes for Incompressible Flow For an incompressible, constant viscosity flow, the viscous terms simplify significantly (more applicable to gasses than fluids) u u u 1 P 2u 2u u v 2 2 t x y x x y v v v 1 P 2v 2v u v 2 2 t x y y x y acceleration advection Pressure diffusion gradient How much detail was skipped? 1st year graduate course in continuum mechanics in 5 slides 36 Turbulence Models Turbulence depends on the ratio of the inertia force to viscous force Laminar flows can be described by the continuity and momentum equations Rotational flow structure have a wide range of length and velocity scales, called turbulent scales Several popular techniques for accounting for turbulence are: • Direct Numerical Simulation (DNS) • Large Eddie Simulation (LES) • K-epsilon 37 Large Eddy Simulation (LES) Large scale motions are generally much more energetic and transport most of the conserved properties • Large eddies are modeled exactly • Small eddies are approximated Smaller universal scales, called sub-grid scales, are modeled using a sub-grid scale (SGS) model 38 K-epsilon Model Time averaged governing equations yield the Reynolds- averaged Navier-Stokes equations (RANS) – Point velocities are considered to be comprised of two components • Steady mean value • Fluctuating component Additional unknowns due to turbulent fluctuations can be handled with transport equations Two important turbulent quantities in these transport equations: k – turbulent kinetic energy epsilon – dissipation of turbulent kinetic energy 39 Discretization Conversion of the governing equations into a system of algebraic equations Two popular discretization techniques in CFD are – Finite difference method – Finite volume method 40 Finite Difference Method First order derivatives x • Fordward difference u ui , j 1 ui , j O(x) x x ui,j+1 • Backward difference y u ui , j ui , j 1 Ui-1,j Ui+1,j O(x) ui,j x x ui,j-1 • Central difference u ui , j 1 ui , j 1 O(x 2 ) x 2x y(j) x(i) 41 Finite Difference Approximations Second order derivative x • Central difference 2u ui 1, j 2ui , j ui 1, j O (( x) 2 ) x 2 ( x ) 2 ui,j+1 y Ui-1,j Ui+1,j For time derivatives ui,j ui,j-1 u u n 1i , j u n i , j O(t ) t t y(j) x(i) 42 Finite Volume Method Unstructured mesh offers more flexibility Control volumes are defined by the surfaces of the elements Control volume integrals can be converted to discretized equations base on face area and flow across boundaries 43 Numerical Solution Discretization results in a system of linear or non- linear equations Numerical methods are applied to solve these equations – Direct methods – Iterative methods 44 Convergence With iterative methods, as progress proceeds towards a solution, the equations are determined to have converged to a solution when certain values do not change between iterations by a specified tolerance Additional characteristics – Numerical solution does not change with additional iterations – Mass, momentum and energy balances are obtained 45 Residuals Residuals are the errors of the discretized equations Residuals are calculated for each equation (Ux, Uy, P, …) Residuals should diminish as the numerical process progresses They are often used to monitor the behavior of the numerical process 46 Exercise – Flow Past a Circular Cylinder • Open Cylinder_CFD_portal.doc • Continue where you left off (~page 6 after initial simulation submitted) 47 Exercise – Turbulent Flow over a Backward Facing Step • Open Backward_step.doc for instructions on how to solve this problem with the OpenFOAM CFD solver • Examine the model assumptions and setup • Run the model in its current form • Includes turbulence modeling 48