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homework4_nonuniform.m
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homework4_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) 2020 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/>. %
% %
% ----------------------------------------------------------------------- %
close all;
clear variables;
%% Input data
%--------------------------------------------------------------------------
% Basic setup
nx=25; % number of grid points along x
ny=nx; % number of grid points along y
Lx=1; % length along x [m]
Ly=Lx; % length along y [m]
deltax=2; % stretching factor along x
deltay=2; % stretching factor along y
Uwall=0.1; % wall velocity [m/s]
nu=1e-3; % kinematic viscosity [m2/s]
ttot=200; % total time of simulation [s]
% Parameters for SOR
max_iterations=10000; % maximum number of iterations
beta=1.5; % SOR coefficient
max_error=0.0001; % error for convergence
%% Pre-processing operations
%--------------------------------------------------------------------------
L=max(Lx,Ly); % reference length [m]
V=Uwall; % reference velocity [m/s]
Re=V*L/nu; % Reynolds' number [-]
tref=L/V; % reference time [s]
lengthx=Lx/L; % dimensionless length along x [-]
lengthy=Ly/L; % dimensionless length along y [-]
tautot=ttot/tref; % total time of simulation (dimensionless) [-]
uwall = Uwall/V; % dimensionless wall velocity [-]
% Grid
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
x = lengthx*x;
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 = lengthy*y;
% Time step
hx_min = min( x(2:end)-x(1:end-1) ); % minimum grid step along x [-]
hy_min = min( y(2:end)-y(1:end-1) ); % minimum grid step along y [-]
h_min = min( hx_min, hy_min); % minimum grid step [-]
sigma = 0.5; % safety factor for time step (stability)
dtau_diff=h_min^2*Re/4; % time step (diffusion stability) [-]
dtau_conv=4/Re; % time step (convection stability) [-]
dtau=sigma*min(dtau_diff, dtau_conv); % time step (stability) [-]
nsteps=tautot/dtau; % number of steps
fprintf('Time step: %f\n', dtau);
fprintf(' - Diffusion: %f\n', dtau_diff);
fprintf(' - Convection: %f\n', dtau_conv);
% Memory allocation
psi=zeros(nx,ny); % streamline function [-]
omega=zeros(nx,ny); % vorticity [-]
u=zeros(nx,ny); % reconstructed dimensionless x-velocity [-]
v=zeros(nx,ny); % reconstructed dimensionless y-velocity [-]
% 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
%% Numerical solution
%--------------------------------------------------------------------------
tau = 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 );
% ------------------------------------------------------------------- %
% Reconstruction of dimensionless velocity field
% ------------------------------------------------------------------- %
[u,v] = ReconstructDimensionlessVelocity(u,v,psi,x,y,uwall);
% ------------------------------------------------------------------- %
% 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
% ------------------------------------------------------------------- %
% Advection-diffusion equation (new vorticity in interior points)
% ------------------------------------------------------------------- %
[omega] = AdvectionDiffusion2D(omega, x,y, u,v, Re, dtau);
% ------------------------------------------------------------------- %
% Advancing time
% ------------------------------------------------------------------- %
if (mod(istep,25)==1)
fprintf('Step: %d - Time: %f - Iterations: %d\n', ...
istep, tau, iter);
end
tau=tau+dtau;
% ------------------------------------------------------------------- %
% On-the-fly graphical post-processing
% ------------------------------------------------------------------- %
if (mod(istep,25)==0)
contour(x,y,psi', 30, 'b');
axis('square');
pause(0.01);
end
end
%% Final post-processing operations
% ------------------------------------------------------------------- %
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 lengthx 0 lengthy], 'square');
title('stream lines'); xlabel('x'); ylabel('y');
%% ------------------------------------------------------------------------
% 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 [f,iter] = Poisson2D( f, x,y, S, beta,max_iterations,max_error, ...
ae,ax,aw,an,ay,as)
nx = length(x);
ny = length(y);
for iter=1:max_iterations
for i=2:nx-1
for j=2:ny-1
f(i,j) = beta*( ae(i)*f(i+1,j) + aw(i)*f(i-1,j) + ...
an(j)*f(i,j+1) + as(j)*f(i,j-1) + ...
-S(i,j) ) / (ax(i)+ay(j)) + ...
(1-beta)*f(i,j);
end
end
res = 0;
for i=2:nx-1
for j=2:ny-1
res = res+abs( f(i+1,j)*ae(i) - f(i,j)*ax(i) + f(i-1,j)*aw(i) + ...
f(i,j+1)*an(j) - f(i,j)*ay(j) + f(i,j-1)*as(j) + ...
-S(i,j) );
end
end
res = res/(nx-2)/(ny-2);
if (res <= max_error)
break;
end
end
end
%% ------------------------------------------------------------------------
% Reconstruction of velocity field (dimensionless)
% ------------------------------------------------------------------------
function [u,v] = ReconstructDimensionlessVelocity(u,v,psi,x,y,uwall)
nx = length(x);
ny = length(y);
u(:,ny) = uwall;
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
end
%% ------------------------------------------------------------------------
% Advection-diffusion equation: forward Euler + centered discretization
% ------------------------------------------------------------------------
function [f] = AdvectionDiffusion2D(f, x,y, u,v, Re, dtau)
nx = length(x);
ny = length(y);
fo = f;
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)*(fo(i+1,j)-fo(i-1,j))/bx;
advection_y = -v(i,j)*(fo(i,j+1)-fo(i,j-1))/by;
diffusion_x = 1/Re*( ax*fo(i+1,j)-bx*fo(i,j)+...
cx*fo(i-1,j))/(0.5*ax*bx*cx);
diffusion_y = 1/Re*( ay*fo(i,j+1)-by*fo(i,j)+...
cy*fo(i,j-1))/(0.5*ay*by*cy);
f(i,j) = fo(i,j) + ...
dtau*( advection_x + advection_y + ...
diffusion_x + diffusion_y );
end
end
end