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driven_cavity_2d_vorticity_nonuniform.m
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driven_cavity_2d_vorticity_nonuniform.m
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% ----------------------------------------------------------------------- %
% __ __ __ _ __ __ %
% |\/| _ |_ | _ |_ |__| / |_ | \ _ (_ |__) |_ %
% | | (_| |_ | (_| |_) | \__ | |__/ (_) | | \ | %
% %
% ----------------------------------------------------------------------- %
% %
% Author: Alberto Cuoci <alberto.cuoci@polimi.it> %
% CRECK Modeling Group <http://creckmodeling.chem.polimi.it> %
% Department of Chemistry, Materials and Chemical Engineering %
% Politecnico di Milano %
% P.zza Leonardo da Vinci 32, 20133 Milano %
% %
% ----------------------------------------------------------------------- %
% %
% This file is part of Matlab4CFDofRF framework. %
% %
% License %
% %
% Copyright(C) 2019 Alberto Cuoci %
% Matlab4CFDofRF is free software: you can redistribute it and/or %
% modify it under the terms of the GNU General Public License as %
% published by the Free Software Foundation, either version 3 of the %
% License, or (at your option) any later version. %
% %
% Matlab4CFDofRF is distributed in the hope that it will be useful, %
% but WITHOUT ANY WARRANTY; without even the implied warranty of %
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the %
% GNU General Public License for more details. %
% %
% You should have received a copy of the GNU General Public License %
% along with Matlab4CRE. If not, see <http://www.gnu.org/licenses/>. %
% %
%-------------------------------------------------------------------------%
% %
% Code: 2D driven-cavity problem in vorticity/streamline formulation %
% The code is adapted and extended from Tryggvason, Computational %
% Fluid Dynamics http://www.nd.edu/~gtryggva/CFD-Course/ %
% %
% ----------------------------------------------------------------------- %
close all;
clear variables;
% Basic setup
Ly_over_Lx = 1.; % ratio between y and x lengths
nx=25; % number of grid points along x
ny=25; % number of grid points along y
deltax=2; % stretching factor along x
deltay=2; % stretching factor along y
Re=100; % Reynolds number [-]
tau=20; % total time of simulation [-]
% Data for reconstructing the velocity field
L=1; % length along x [m] (used as reference length)
nu=1e-3; % kinematic viscosity [m2/s]
Uwall=nu*Re/L; % wall velocity [m/s]
% Parameters for SOR
max_iterations=10000; % maximum number of iterations
beta=1.5; % SOR coefficient
max_error=0.0001; % error for convergence
% Grid construction (dimensionless)
x = zeros(nx,1);
for i=1:nx
x(i) = 0.5*(1+tanh(deltax*((i-1)/(nx-1)-0.5))/tanh(deltax/2));
end
y = zeros(ny,1);
for i=1:ny
y(i) = 0.5*(1+tanh(deltay*((i-1)/(ny-1)-0.5))/tanh(deltay/2));
end
y = Ly_over_Lx*y;
% Time step
h2 = (x(2)-x(1))*(y(2)-y(1)); % minimum cell volume
sigma = 0.5; % safety factor for time step (stability)
dt_diff=h2*Re/4; % time step (diffusion stability)
dt_conv=4/Re; % time step (convection stability)
dt=sigma*min(dt_diff, dt_conv); % time step (stability)
nsteps=tau/dt; % number of steps
fprintf('Time step: %f\n', dt);
fprintf(' - Diffusion: %f\n', dt_diff);
fprintf(' - Convection: %f\n', dt_conv);
% Memory allocation
psi=zeros(nx,ny); % streamline function
omega=zeros(nx,ny); % vorticity
psio=zeros(nx,ny); % streamline function at previous time
omegao=zeros(nx,ny); % vorticity at previous time
u=zeros(nx,ny); % reconstructed dimensionless x-velocity
v=zeros(nx,ny); % reconstructed dimensionless y-velocity
U=zeros(nx,ny); % reconstructed x-velocity [m/s]
V=zeros(nx,ny); % reconstructed y-velocity [m/s]
% Mesh construction (only needed in graphical post-processing)
[X,Y] = meshgrid(x,y); % mesh
% Allocation of vectors for non-uniform grid (see the Poisson equation function)
% Along the x direction
ae = zeros(nx,1); ax = zeros(nx,1); aw = zeros(nx,1);
for i=2:nx-1
a = x(i)-x(i-1); b = x(i+1)-x(i-1); c = x(i+1)-x(i);
ae(i) = 2/(b*c);
ax(i) = 2/(a*c);
aw(i) = 2/(a*b);
end
% Along the y direction
an = zeros(ny,1); ay = zeros(ny,1); as = zeros(ny,1);
for j=2:ny-1
a = y(j)-y(j-1); b = y(j+1)-y(j-1); c = y(j+1)-y(j);
an(j) = 2/(b*c);
ay(j) = 2/(a*c);
as(j) = 2/(a*b);
end
% Time loop
t = 0;
for istep=1:nsteps
% ------------------------------------------------------------------- %
% Poisson equation (SOR)
% ------------------------------------------------------------------- %
[psi, iter] = Poisson2D( psi, x, y, omega, ...
beta, max_iterations, max_error, ...
ae, ax, aw, an, ay, as );
% ------------------------------------------------------------------- %
% Find vorticity on boundaries
% ------------------------------------------------------------------- %
omega(2:nx-1,1)=-2.0*psi(2:nx-1,2)/(y(2)-y(1))^2; % south
omega(2:nx-1,ny)=-2.0*psi(2:nx-1,ny-1)/(y(ny)-y(ny-1))^2 ...
-2.0/(y(ny)-y(ny-1))*1; % north
omega(1,2:ny-1)=-2.0*psi(2,2:ny-1)/(x(2)-x(1))^2; % east
omega(nx,2:ny-1)=-2.0*psi(nx-1,2:ny-1)/(x(nx)-x(nx-1))^2; % west
% ------------------------------------------------------------------- %
% Reconstruction of dimensionless velocity field
% ------------------------------------------------------------------- %
u(:,ny)=1;
for i=2:nx-1
for j=2:ny-1
u(i,j) = (psi(i,j+1)-psi(i,j-1))/(y(j+1)-y(j-1));
v(i,j) = -(psi(i+1,j)-psi(i-1,j))/(x(i+1)-x(i-1));
end
end
% ------------------------------------------------------------------- %
% Find new vorticity in interior points
% ------------------------------------------------------------------- %
[omega] = AdvectionDiffusion(omega, x, y, u, v, Re, dt);
% Message on video
if (mod(istep,50)==1)
fprintf('Step: %d - Time: %f - Poisson iterations: %d\n', istep, t, iter);
end
% Advance time
t=t+dt;
% ------------------------------------------------------------------- %
% Reconstruction of velocity field [m/s]
% ------------------------------------------------------------------- %
U = u*Uwall;
V = v*Uwall;
% ------------------------------------------------------------------- %
% Graphics only
% ------------------------------------------------------------------- %
plot_2x4 = false; % plotting the 2x4 plot
if (plot_2x4 == true)
subplot(241);
contour(x,y,omega');
axis('square'); title('omega'); xlabel('x'); ylabel('y');
subplot(245);
contour(x,y,psi');
axis('square'); title('psi'); xlabel('x'); ylabel('y');
subplot(242);
contour(x,y,u');
axis('square'); title('u'); xlabel('x'); ylabel('y');
subplot(246);
contour(x,y,v');
axis('square'); title('v'); xlabel('x'); ylabel('y');
subplot(243);
plot(x,u(:, round(ny/2)));
hold on;
plot(x,v(:, round(ny/2)));
axis('square'); legend('u', 'v');
title('velocities along HA'); xlabel('x'); ylabel('velocities');
hold off;
subplot(247);
plot(y,u(round(nx/2),:));
hold on;
plot(y,v(round(nx/2),:));
axis('square'); legend('u', 'v');
title('velocities along VA'); xlabel('y'); ylabel('velocities');
hold off;
subplot(244);
quiver(x,y,u',v');
axis('square', [0 1 0 Ly_over_Lx]);
title('velocity vectors'); xlabel('x'); ylabel('y');
pause(0.001);
end
end
% ------------------------------------------------------------------- %
% Write final maps
% ------------------------------------------------------------------- %
subplot(231);
surface(x,y,u');
axis('square'); title('u'); xlabel('x'); ylabel('y');
subplot(234);
surface(x,y,v');
axis('square'); title('v'); xlabel('x'); ylabel('y');
subplot(232);
surface(x,y,omega');
axis('square'); title('omega'); xlabel('x'); ylabel('y');
subplot(235);
surface(x,y,psi');
axis('square'); title('psi'); xlabel('x'); ylabel('y');
subplot(233);
contour(x,y,psi', 30, 'b');
axis('square');
title('stream lines'); xlabel('x'); ylabel('y');
subplot(236);
quiver(x,y,u',v');
axis([0 1 0 Ly_over_Lx], 'square');
title('stream lines'); xlabel('x'); ylabel('y');
% ------------------------------------------------------------------- %
% Write velocity profiles along the centerlines for exp comparison
% ------------------------------------------------------------------- %
u_profile = u(round(nx/2),:);
fileVertical = fopen('vertical.out','w');
for i=1:ny
fprintf(fileVertical,'%f %f\n',y(i), u_profile(i));
end
fclose(fileVertical);
v_profile = v(:,round(ny/2));
fileHorizontal = fopen('horizontal.out','w');
for i=1:nx
fprintf(fileHorizontal,'%f %f\n',x(i), v_profile(i));
end
fclose(fileHorizontal);
% ------------------------------------------------------------------- %
% Compare with exp data (available only for Re=100, 400, and 1000)
% ------------------------------------------------------------------- %
% Read experimental data from file
exp_u_along_y = dlmread('experimental_data/u_along_y.exp', '', 1, 0);
exp_v_along_x = dlmread('experimental_data/v_along_x.exp', '', 1, 0);
% Comparison with exp data
% Be careful: cols 1,2 for Re=100, 3,4 for Re=400, 5,6 for Re=1000
figure;
plot(exp_u_along_y(:,1), exp_u_along_y(:,2), 'o', y, u_profile, '-');
axis('square'); title('u along y (centerline)'); xlabel('y'); ylabel('u');
figure;
plot(exp_v_along_x(:,1), exp_v_along_x(:,2), 'o', x, v_profile, '-');
axis('square'); title('v along x (centerline)'); xlabel('x'); ylabel('v');
% --------------------------------------------------------------------------------------
% Poisson equation solver
% The second order derivative is discretized over a non uniform grid
% d2(psi)/dx2 = ae*psi(i+1)-ac*psi(i)+aw*psi(i-1)
% ae = 2/(b*c), ax = 2/(a*c), aw = 2/(a*b)
% a=x(i)-x(i-1), b=x(i+1)-x(i-1), c=x(i+1)-x(i)
% --------------------------------------------------------------------------------------
function [psi, iter] = Poisson2D( psi, x, y, omega, beta, max_iterations, max_error, ...
ae, ax, aw, an, ay, as)
nx = length(x);
ny = length(y);
for iter=1:max_iterations
% Update solution
for i=2:nx-1
for j=2:ny-1
psi(i,j)=( psi(i+1,j)*ae(i) + psi(i-1,j)*aw(i) + ...
psi(i,j+1)*an(j) + psi(i,j-1)*as(j) + ...
omega(i,j) ) / (ax(i)+ay(j))*beta + ...
(1.0-beta)*psi(i,j);
end
end
% Estimate the error
epsilon=0.0;
for i=2:nx-1
for j=2:ny-1
epsilon=epsilon+abs( psi(i+1,j)*ae(i) - psi(i,j)*ax(i) + psi(i-1,j)*aw(i) + ...
psi(i,j+1)*an(j) - psi(i,j)*ay(j) + psi(i,j-1)*as(j) + ...
omega(i,j) );
end
end
epsilon = epsilon/(nx-2)/(ny-2);
% Check the error
if (epsilon <= max_error) % stop if converged
break;
end
end
end
% --------------------------------------------------------------------------------------
% Advection-diffusion equation: forward Euler + centered discretization
% --------------------------------------------------------------------------------------
function [omega] = AdvectionDiffusion(omega, x, y, u, v, Re, dt)
nx = length(x);
ny = length(y);
omegao=omega;
for i=2:nx-1
ax = x(i)-x(i-1); bx = x(i+1)-x(i-1); cx = x(i+1)-x(i);
for j=2:ny-1
ay = y(j)-y(j-1); by = y(j+1)-y(j-1); cy = y(j+1)-y(j);
advection_x = -u(i,j)*(omegao(i+1,j)-omegao(i-1,j))/bx;
advection_y = -v(i,j)*(omegao(i,j+1)-omegao(i,j-1))/by;
diffusion_x = 1/Re*( ax*omegao(i+1,j)-bx*omegao(i,j)+...
cx*omegao(i-1,j))/(0.5*ax*bx*cx);
diffusion_y = 1/Re*( ay*omegao(i,j+1)-by*omegao(i,j)+...
cy*omegao(i,j-1))/(0.5*ay*by*cy);
omega(i,j)=omegao(i,j) + ...
dt*( advection_x + advection_y + ...
diffusion_x + diffusion_y );
end
end
end