Objective Numerical methods SIMPLE CFD Algorithm SIMPLE SemiImplicit

Objective • Numerical methods – SIMPLE CFD Algorithm SIMPLE Semi-Implicit Method for Pressure-Linked Equations – Define Relaxation

General Transport Equation unsteady-state H N W P E Equation in the algebraic format: S L We have to solve the system matrix for each time step ! Transient term: Are these values for step or + ? Unsteady-state 1 -D If: - - explicit method - + - implicit method

General Transport Equation unsteady-state Fully explicit method: Or different notation: Implicit method For Vx>0 For Vx<0

Steady state vs. Unsteady state Steady state Unsteady state We use iterative solver to get solution and We iterate for each time step Make the difference between - Calculation for different time step - Calculation in iteration step

1 D example multiple (N) volumes N unknowns 1 2 3 i Equation for volume 1 N equations Equation for volume 2 ……………… Equation matrix: For 1 D problem 3 -diagonal matrix N-1 N

3 D problem Equation in the general format: H N W P E S L Wright this equation for each discretization volume of your discretization domain A F 60, 000 cells (nodes) N=60, 000 elements x = 7 -diagonal matrix This is the system for only one variable ( ) When we need to solve p, u, v, w, T, k, e, C system of equation is larger

Iteration method Alternative to use matrix solver tool is to use iterations You can use excel if you are not familiar with matrix solver tools General Iteration Procedure: 1) Express equation in explicit form 2) Guess initial values 3) Substitute initial values and calculate new values 4) Substitute new values and calculate newer values 6) Repeat step 4) until convergence is achieved example Iterations -residual Difference of value between two iteration Value: T 1 Residual initial guess 22 iteration 1 23 1. 00000 iteration 2 23. 25 0. 25000 iteration 3 23. 390625 0. 14063 iteration 4 23. 483459 0. 09283 iteration 5 …… --- 23. 96441 Iteration 98 23. 96444 2 2 2 0. 00003

Numerical instability divergency variable divergence solution convergence iteration

Navier Stokes Equations Continuity equation This velocities that constitute advection coefficients: F=r. 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 in 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 x W Aw P x x 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 )

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 from homework assignment) 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