Handout 8 MATLAB Code for the Projection Method and BC Details for the Lid-Driven Cavity Problem

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1 Handout 8 MATLAB Code for the Projection Method and BC Details for the Lid-Driven Cavity Problem Let s simplify the equations of the last page of Handout 7 for the case h = Δx = Δy because the code given in the next page is written for this case. Also drop the body force term for simplicity. u i,j v i,j Simplified equations used in the code for h = Δx = Δy and no body force = u n i,j + Δt [ 1 h (u e u u u e u u w u u w + u u n v u n u u s v u s ) + μ n ρh 2 (u i+1,j + u i 1,j + u i,j+1 + u i,j 1 4u i,j )] = v n i,j + Δt [ 1 h (u e v v v e u v w v v w + v v n v v n v v s v v s ) + μ n ρh 2 (v i+1,j + v i 1,j + v i,j+1 + v i,j 1 4v i,j )] p i+1,j + p i 1,j + p i,j+1 + p i,j 1 4p i,j = ρh Δt (u i,j u i 1,j + v i,j u n+1 i,j = u i,j Δt (p i+1,j p i,j ) ρ h (20a) (20b) v i,j 1 ) (18) (21a) v n+1 i,j = v i,j Δt (p i,j+1 p i,j ) ρ h (21b) Boundary Conditions with Ghost Cells for the Lid-Driven Carvity Problem The code given at the of this handout is written specifically for the lid-driven cavity problem. Let s study the BCs of this problem first. y u T = U lid v T = 0 u L = 0 v L = 0 L u B = 0 v B = 0 u R = 0 v R = 0 x 1

2 Consider the following mesh of (Nx = 3) x (Ny = 3) p cells. Thick lines are the domain boundaries. All around the domain we placed ghost cells. For structured grids ghost cells provide an easy way to implement the BCs. There are (Nx+2) x (Ny+2) = 25 pressure nodes. There are (Nx+1) x (Ny+2) = 20 u-velocity nodes. There are (Nx+2) x (Ny+1) = 20 v-velocity nodes. 2

3 BCs for u: We need to calculate only 6 u values, the ones for the inner u-cells, i.e. u(2:nx, 2:Ny+1). u on the left and right boundaries are given as u L = 0 and u R = 0. Do not solve for them. To calculate ghost node values we use linear interpolation. For example for u 1,1 Or for u 3,5 u 1,1 + u 1,2 2 u 3,5 + u 3,4 2 = u B u 1,1 = 2u B u 1,2, where u B is the u at the bottom boundary = u T u 3,5 = 2u T u 3,4, where u T is the u at the top boundary After the (or beginning) of each time step ghost node values are updated using equations similar to these. 3

4 BCs for v: We need to calculate only 6 v values, the ones for the inner v-cells, i.e. v(2:nx+1, 1:Ny). v on the bottom and top boundaries are given as v B = 0 and v T = 0. Do not solve for them. To calculate ghost node values we use linear interpolation. For example for v 1,2 Or for v 5,3 v 1,2 + v 2,2 2 v 5,3 + v 4,3 2 = v L v 1,2 = 2v L v 2,2, where v L is the v at the left boundary = v R v 5,3 = 2v R v 4,3, where v R is the v at the right boundary After the (or beginning) of each time step ghost node values are updated using equations similar to these. 4

5 BCs for the PPE: We need to calculate 9 p values, i.e. p(2:nx+1, 2:Ny+1) For the p-cells at the boundaries we need to derive a special Eqn. (18). Consider cell p 2,2. Eqn. (16) for it is Eqn (15) for this p-cell becomes h ( u n+1 2,2 u 1,2 n+1 u L + v 2,2 n+1 n+1 v 2,1 ) = 0 (16 ) v B u n+1 2,2 = u i,j Δt (p 3,2 p 2,2 ) ρ h p 3,2 p 2,2 = ρh Δt (u 2,2 u n+1 2,2 ) (17a ) This is not necesary (17b ) 5

6 v n+1 2,2 = v 2,2 Δt (p 2,3 p 2,2 ) ρ h METU, Dept. of Mechanical Eng. p 2,3 p 2,2 = ρh Δt (v 2,2 v n+1 2,2 ) (17c ) This is not necesary (17d ) Add Eqn (17a*) and Eqn (17c*) p 3,2 + p 2,3 2p 2,2 = ρh Δt ( u 2,2 + v 2,2 Compared to the original Eqn (18) the changes are u 2,2 n+1 v 2,2 n+1 = u L v B according to Eqn (16 )) (18 ) Number of unknowns decreased from 5 to 3 on the left hand side. If we solve the system using Gauss- Seidel, this can be taken care automatically by setting ghost cell pressure values to zero and always keep them like that. That way the only difference of the left hand compared to the original Eqn (18) becomes the change of the coefficient of the center unknown, i.e. p 2,2 from -4 to 22. On the right hand side there seem to be a change, but actually not. Because in the code u L and v B will be seen as u 1,2 and u 2,1, making the right hand side of Eqn (18*) the same as the original Eqn (18), so no change is required. If we repeat this for a non-corner boundary cell, such as p 4,3, we ll see that the only difference compared to Eqn (18) will be the change of the coefficient of the center unknown from -4 to -3. To summarize, boundaries will affect Eqn (18) as follows Set the center coefficient of the corner cells to -3. Set the center coefficient of the non-corner cells to -2. In the meantime keep the ghost cell pressures at zero. 6

7 % 2D flow solver using the projection method. % It is design ed to solve the lid driven cavity problem. % Uses forward time and central space differencing. % Uses staggered mesh % Uses ghost cells for boundary conditions clc; clear all; close all; % Clear the command window % Clear previously defined variables % Close previously opened figure windows refsoln = load ('Ghia_Re1000_Uvel'); % u velocity variation along the vertical centerline % from Ghia's reference solution. The data is available % at 4 different Reynolds numbers (100, 1000, 3200 and % 5000). refsoln has 2 columns, y and u. L = 1; % Cavity size. Preferred value is 1. Nx = 40; % Number of pressure cells in x and y directions Ny = 40; % (not including ghost cells). % Provide an even value to avoid a possible error in % the post-processing part. rho = 1.0; % Density [kg/m^3] visc = 1/1000; % Dynamic viscosity [Pa s] dt = 0.02; % Time step nstep = 10000; % Maximum number of time levels maxgsiter = 1; % Maximum iteration value for the Gauss-Seidel solver u_top = 1.0; % Tangential velocities at the boundaries [m/s] u_bottom = 0.0; v_left = 0.0; v_right = 0.0; Be careful!!! h = L/Nx; % h is the cell size (=dx=dy). % Allocate memory. Note that staggered grid with ghost cells is being used. u = zeros(nx+1,ny+2); % x velocity values v = zeros(nx+2,ny+1); % y velocity values p = zeros(nx+2,ny+2); % Pressure values ustar = zeros(nx+1,ny+2); vstar = zeros(nx+2,ny+1); % Temporary x velocities % Temporary y velocities The mesh is staggered. %% % Initialize the solution %% u(:,:) = 0.0; v(:,:) = 0.0; p(:,:) = 0.0; uold = u; vold = v; % uold and vold are used to check steady state convergence 7

8 %% % Set the coefficient of the center unknown of the PPE %% METU, Dept. of Mechanical Eng. coefppe = zeros(nx+1,ny+1); for i = 2:Nx+1 for j = 2:Ny+1 if (i == 2 && j == 2) (i == Nx+1 && j == 2)... (i == 2 && j == Ny+1) (i == Nx+1 && j == Ny+1) coefppe(i,j) = -2; elseif (i == 2) (i == Nx+1) (j == 2) (j == Ny+1) coefppe(i,j) = -3; else coefppe(i,j) = -4; Set the coefficient of the center unknown of the PPE = -4 for non-boundary cells = -2 for corner cells = -3 for non-corner % Print the header for the data that'll be written on the screen fprintf(' n u at (0.5,0.5) v at (0.5,0.5)\n'); fprintf('=========================================\n'); for n = 1:nStep % Time loop % Calculate ghost cell velocities using linear interpolation for i = 1:Nx+1 u(i,1) = 2*u_BOTTOM - u(i,2); % Bottom boundary u(i,ny+2) = 2*u_TOP - u(i,ny+1); % Top boundary Calculate ghost cell velocities. for j = 1:Ny+1 v(1,j) = 2*v_LEFT - v(2,j); % Left boundary v(nx+2,j) = 2*v_RIGHT - v(nx+1,j); % Right boundary % STEP 1a. Calculate the temporary u velocities. Eqn (20a) of Handout 8 for i = 2:Nx for j = 2:Ny+1 ue = 0.5 * (u(i,j) + u(i+1,j)); uw = 0.5 * (u(i,j) + u(i-1,j)); un = 0.5 * (u(i,j) + u(i,j+1)); us = 0.5 * (u(i,j) + u(i,j-1)); Use central differencing to get face velocities vn = 0.5 * (v(i,j) + v(i+1,j)); vs = 0.5 * (v(i,j-1) + v(i+1,j-1)); ustar(i,j) = u(i,j) - dt / h * (ue*ue + un*vn - uw*uw - us*vs)... + dt * visc / rho / (h^2) * (u(i+1,j) + u(i,j+1)... + u(i-1,j) + u(i,j-1) - 4 * u(i,j)); Eqn (20a) of Handout 8 8

9 % STEP 1b. Calculate the temporary v velocities. Eqn (20b) of Handout 8 METU, Dept. of Mechanical Eng. for i = 2:Nx+1 for j = 2:Ny ve = 0.5 * (v(i,j) + v(i+1,j)); vw = 0.5 * (v(i,j) + v(i-1,j)); vn = 0.5 * (v(i,j) + v(i,j+1)); vs = 0.5 * (v(i,j) + v(i,j-1)); Use central differencing to get face velocities ue = 0.5 * (u(i,j) + u(i,j+1)); uw = 0.5 * (u(i-1,j) + u(i-1,j+1)); vstar(i,j) = v(i,j) - dt / h * (ve*ue + vn*vn - vw*uw - vs*vs)... + dt * visc / rho / (h^2) * (v(i+1,j) + v(i,j+1)... + v(i-1,j) + v(i,j-1) - 4 * v(i,j)); Eqn (20b) of Handout 8 % STEP 2. Solve the pressure Poisson equation (PPE) using Gauss-Seidel % Eqn (18) of Handout 8. % Calculate the RHS of the PPE system. for i = 2:Nx+1 for j = 2:Ny+1 rhsppe(i,j) = (rho*h/dt) * (ustar(i,j) - ustar(i-1,j) + vstar(i,j) - vstar(i,j-1)); % Gauss-Seidel loop for iter = 1:maxGSiter pold = p; for i = 2:Nx+1 for j = 2:Ny+1 p(i,j) = (rhsppe(i,j) - p(i+1,j) - p(i-1,j) - p(i,j+1) - p(i,j-1)) / coefppe(i,j); % Here we typically check the convergence of Gauss-Seidel. But for this steady % problem it is not very critical to solve the PPE accurately so we skipped % this check. Solve the PPE, i.e. Eqn (18) of Handout 8 using Gauss-Seidel. % STEP 3. Correct the temporary velocities. Eqn (21a-b) of Handout 8 % Correct u velocities for i = 2:Nx for j = 2:Ny+1 u(i,j) = ustar(i,j) - dt / (rho * h) * (p(i+1,j) - p(i,j)); Eqn (21a) of Handout 8 9

10 % Correct v velocities for i = 2:Nx+1 for j = 2:Ny v(i,j) = vstar(i,j) - dt / (rho * h) * (p(i,j+1) - p(i,j)); Eqn (21b) of Handout 8 % Write some results on the screen % Calculate u and v at (0.5,0.5) ucenter = u(nx/2+1, Ny/2+1); vcenter = u(nx/2+1, Ny/2+1); fprintf('%5d %8.3e %8.3e\n', n, ucenter, vcenter); uold = u; vold = v; % Plot the u velocity variation along the vertical centerline of the % cavity Postprocessing if mod(n,10) == 0 ycenter = h/2:h:1-h/2; ucenter = u(nx/2+1:nx/2+1, 2:Ny+1); plot(refsoln(:,2), refsoln(:,1), ucenter, ycenter, 'ko', 'MarkerFaceColor', 'k') axis([ ]); grid on xlabel('u [m/s]'); ylabel('y [m]'); title('u velocity component along the vertical centerline'); leg('current', 'Ghia', 'Location', 'NorthWest') drawnow; % End of time loop 10