Lecture Objectives Advance discretization methods Advection diffusion equation

Lecture Objectives: Advance discretization methods

Advection diffusion equation 1 -D, steady-state Dx W w Example: P Dx Dx E e y Equation for temperature of water flowing through hot pipe Q x Temperature is changing along x Q Assume that diffusion in y direction ins negligible: This is incorrect assumption introduced jut to simplify example to 1 -D problem! model Vx T 1 T 3 T 2 T 4 … Tn

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

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

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

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

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

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

HW 3 questions

Relaxation with iterative solvers: variable When the equations are nonlinear it can happen that you get divergency in iterative procedure for solving considered time step divergence solution convergence Solution is Under-Relaxation: iteration Y*=f·Y(n)+(1 -f)·Y(n-1) Y – considered parameter , n –iteration , f – relaxation factor Value which is should be used for the next iteration For our example Y*in iteration 101=f·Y(100)+(1 -f) ·Y(99) f = [0 -1] – under-relaxation -stabilize the iteration f = [1 -2] – over-relaxation - speed-up the convergence Under-Relaxation is often required when you have nonlinear equations!

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

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

Algorithms

Navier Stokes Equations CFD Specific Solver Continuity equation This velocities that constitute advection coefficients: F= V 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: 1) find velocities that constitute in advection coefficients 2) link pressure field with continuity equation

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

Pressure and velocities in NS equations How to link pressure field with continuity equation? SIMPLE (Semi-Implicit Method for Pressure-Linked Equations ) algorithm Dx W Aw P Dx Dx Ae 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 E

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’ V = V* + V’ P’ – pressure correction V’ – velocity correction For all nodes E, W, N, S, … 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

SIMPLE algorithm start p=p* Guess p* Step 1: solve V* from momentum equations Step 2: introduce correction P’ and express V = V* + f(P’) Step 3: substitute V into continuity equation solve P’ and then V Step 4: Solve T , k , e equations no Converged (residual check) yes end

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

Newton-Raphson method (example of Jacobean solver) - Faster convergence - Used in many professional tools (Math. CAD, EES, Mat. Lab, 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 2 -Y=-3 → → X-Y/2+1=0 X 2 -Y+3=0

Newton-Raphson method Our simple example: f 1 = X-Y/2+1=0 f 2 = X 2 -Y+3=0 Section 6. 11 of handouts Function values for guessed variables Jacobean matrix Steps: 0) Find derivatives d(f 1)/d. X =1 d(f 2)/d. X =2 X , d(f 1)/d. Y =-1/2 , d(f 2)/d. Y =-1 Unknowns (correction Dxi) 1) Initial guess: Y(0)=2, X(0)=2 2) Find f 1(Y(0), X(0))=2 -2/2+1=2 f 2(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

Course Review (so far)

Conservation Equations Navier Stokes Equations

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

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: RSM: Large Eddy simulation models Reynolds stress models

Discretization and equation solver Discretization of RANS SIMPLE algorithm Guess p* p=p* Step 1: solve V* from momentum equations Step 2: introduce correction P’ and express V = V* + f(P’) Step 3: substitute V into continuity equation solve P’ and then V Step 4: Solve T , k , e equations Converged (residual check) yes end no

Write down this • 10. 157. 6. 25 • CAEE-Lic. Srv 01. austin. utexas. edu