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Lecture Objectives:

Lecture Objectives:. Advance discretization methods. Advection diffusion equation 1-D, steady-state. D x. D x. P. E. W. D x. e. w. y. Q. Example: Equation for temperature of water flowing through hot pipe

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Lecture Objectives:

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  1. Lecture Objectives: Advance discretization methods

  2. Advection diffusion equation 1-D, steady-state Dx Dx P E W Dx e w y Q Example: Equation for temperature of water flowing through hot pipe Assume that diffusion in y direction ins negligible: This is incorrect assumption introduced jut to simplify example to 1-D problem! Temperature is changing along x x Q model Vx T1 T2 T3 T4 … Tn

  3. Advection diffusion equation 1-D, steady-state Dx Dx N N+1 N-1 Different notation: Dx General equation

  4. Advection equation 1-D, steady-state Dx Dx P E W Dx Vx>0 1) Upwind scheme: Vx<0 2) Central differencing scheme: 3) Hybrid of upwind and central differencing scheme Higher order differencing scheme: Quadratic upwind differencing Scheme (QUICK) N+1 N+2 N-1 N N-2 P WW E EE W We need to find coefficients aP, aW, aE, aWW, aEE,

  5. Central Difference and Upwind Differencing Schemes Notation of coefficients in book Stability of the Central Difference Schemes Pe = Diffusion / Advection = (Vx)/(/∆x)<2 It defined by the requirement that coefficient aE>0 Requirement for “boundedness” D=/∆x F=Vx

  6. Hybrid Differencing Schemes Notation of coefficients in the book T N W E P S B

  7. Higher order differencing scheme: Quadratic upwind differencing Scheme (QUICK) N+1 N+2 N-1 N N-2 P WW E EE W We need to find coefficients aP, aW, aE, aWW, aEE,

  8. Quadratic upwind differencing Scheme (QUICK) Coefficients: Advection coefficient: Source: Diffusion coefficients : For advection only:

  9. HW3 questions

  10. Relaxation Relaxation with iterative solvers: When the equations are nonlinear it can happen that you get divergency in iterative procedure for solving considered time step divergence variable solution convergence Solution is Under-Relaxation: Y*=f·Y(n)+(1-f)·Y(n-1) Y – considered parameter , n –iteration , f – relaxation factor For our example Y*in iteration101=f·Y(100)+(1-f) ·Y(99) f = [0-1] – under-relaxation -stabilize the iteration f = [1-2] – over-relaxation - speed-up the convergence iteration Value which is should be used for the next iteration Under-Relaxation is often required when you have nonlinear equations!

  11. Example of relaxation Example: Advection diffusion equation, 1-D, steady-state, 4 nodes 1) Explicit format: 4 3 1 2 2) Guess initial values: 3) Substitute and calculate: Substitute and calculate: 4) Substitute and calculate: ………………………….

  12. Residual calculation for CFD • Residual for the cell RFijk=Fkijk-Fk-1ijk • Total residual for the simulation domain RFtotal=S|RFijk| • Scaled (normalized) residual RF=S|RFijk|/FF iteration cell position Variable: p,V,T,… For all cells Flux of variable F used for normalization Vary for different CFD software

  13. Algorithms

  14. Navier Stokes EquationsCFD Specific Solver Continuity equation This velocities that constitute advection coefficients: F=rV Momentum x Momentum y Momentum z Pressure is in momentum equations which already has one unknown In order to use linear equation solver we need to solve two problems: • find velocities that constitute in advection coefficients 2) link pressure field with continuity equation

  15. Pressure and velocities in NS equations How to find velocities that constitute advection coefficients? For the first step use Initial guess And for next iterative steps use the values from previous iteration

  16. Pressure and velocities in NS equations How to link pressure field with continuity equation? SIMPLE (Semi-Implicit Method for Pressure-Linked Equations ) algorithm Dx Dx P E W Dx Ae Aw Aw=Ae=Aside We have two additional equations for y and x directions The momentum equations can be solved only when the pressure field is given or is somehow estimated. Use * for estimated pressure and the corresponding velocities

  17. SIMPLE algorithm Guess pressure field: P*W, P*P, P*E, P*N , P*S, P*H, P*L 1) For this pressure field solve system of equations: x: ……………….. y: ……………….. z: Solution is: 2)The pressure and velocity correction P = P* + P’ P’ – pressure correction For all nodes E,W,N,S,… V = V* + V’ V’ – velocity correction Substitute P=P* + P’ into momentum equations (simplify equation) and obtain V’=f(P’) V = V* + f(P’) 3) Substitute V = V* + f(P’) into continuity equation solve P’ and then V 4) Solve T , k , e equations

  18. SIMPLE algorithm start Guess p* p=p* Step1: solve V* from momentum equations Step2: introduce correction P’ and express V = V* + f(P’) Step3: substitute V into continuity equation solve P’ and then V Step4: Solve T , k , e equations no Converged (residual check) yes end

  19. Other methods SIMPLER SIMPLEC variation of SIMPLE PISO COUPLED - use Jacobeans of nonlinear velocity functions to form linear matrix ( and avoid iteration )

  20. Newton-Raphson method(example of Jacobean solver) • Faster convergence • Used in many professional tools (MathCAD, EES, MatLab, Mathematica, etc) More complex for programming • Requires linear solver Based on Taylor-Series Expansion • You need first derivative for each function to create the Jacobean matrix • Equations in the form where all side are on one side of equality sign Our simple example: X-Y/2=-1 → X-Y/2+1=0 X2-Y=-3 → X2-Y+3=0

  21. Function values for guessed variables Jacobean matrix Unknowns (correction Dxi) Newton-Raphson method Section 6.11 of handouts Our simple example: f1 = X-Y/2+1=0 f2 = X2-Y+3=0 Steps: 0) Find derivatives d(f1)/dX =1 , d(f1)/dY =-1/2 d(f2)/dX =2X , d(f2)/dY =-1 1) Initial guess: Y(0)=2, X(0)=2 2) Find f1(Y(0),X(0))=2-2/2+1=2 f2(Y(0),X(0))=22-2+3=5 3) Using derivatives and guess values find the Jacobean matrix 4) Solve the matrix using linear solver and find DX and DY 5) Find Y(1)=Y(0)+ DY, X(1)=X(0)+ DX, Repeat step (2) with Y(1) and X(1) ….. Follow the procedure till convergence

  22. Course Review (so far)

  23. Conservation Equations Navier Stokes Equations

  24. Reynolds Averaged Navier Stokes equations Continuity: 1) Momentum: 2) 3) 4) Similar is for STy and STx 4 equations 5 unknowns → We need to model

  25. Modeling of Turbulent Viscosity Fluid property – often called laminar viscosity Flow property – turbulent viscosity MVM: Mean velocity models TKEM: Turbulent kinetic energy equation models Additional models: LES: Large Eddy simulation models RSM: Reynolds stress models

  26. Discretization and equation solver SIMPLE algorithm Discretization of RANS Guess p* p=p* Step1: solve V* from momentum equations Step2: introduce correction P’ and express V = V* + f(P’) Step3: substitute V into continuity equation solve P’ and then V Step4: Solve T , k , e equations no Converged (residual check) yes end

  27. Write down this • 10.157.6.25 • CAEE-LicSrv01.austin.utexas.edu

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