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% cavity2d.m: 2D cavity flow, simulated by a LB method
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% Lattice Boltzmann sample, Matlab script
% Copyright (C) 2006-2008 Jonas Latt
% Address: Rue General Dufour 24, 1211 Geneva 4, Switzerland
% E-mail: Jonas.Latt@cui.unige.ch
%
% Implementation of 2d cavity geometry and Zou/He boundary
% condition by Adriano Sciacovelli
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% This program 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 2
% of the License, or (at your option) any later version.
% This program 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 this program; if not, write to the Free
% Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
% Boston, MA 02110-1301, USA.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear
% GENERAL FLOW CONSTANTS
lx = 128;
ly = 128;
uLid = 0.05; % horizontal lid velocity
vLid = 0; % vertical lid velocity
Re = 100; % Reynolds number
nu = uLid *lx / Re; % kinematic viscosity
omega = 1. / (3*nu+1./2.); % relaxation parameter
maxT = 40000; % total number of iterations
tPlot = 10; % cycles for graphical output
% D2Q9 LATTICE CONSTANTS
t = [4/9, 1/9,1/9,1/9,1/9, 1/36,1/36,1/36,1/36];
cx = [ 0, 1, 0, -1, 0, 1, -1, -1, 1];
cy = [ 0, 0, 1, 0, -1, 1, 1, -1, -1];
opp = [ 1, 4, 5, 2, 3, 8, 9, 6, 7];
lid = [2: (lx-1)];
[y,x] = meshgrid(1:ly,1:lx);
obst = ones(lx,ly);
obst(lid,2:ly) = 0;
bbRegion = find(obst);
% INITIAL CONDITION: (rho=0, u=0) ==> fIn(i) = t(i)
fIn = reshape( t' * ones(1,lx*ly), 9, lx, ly);
% MAIN LOOP (TIME CYCLES)
for cycle = 1:maxT
% MACROSCOPIC VARIABLES
rho = sum(fIn);
ux = reshape ( (cx * reshape(fIn,9,lx*ly)), 1,lx,ly ) ./rho;
uy = reshape ( (cy * reshape(fIn,9,lx*ly)), 1,lx,ly ) ./rho;
% MACROSCOPIC (DIRICHLET) BOUNDARY CONDITIONS
ux(:,lid,ly) = uLid; %lid x - velocity
uy(:,lid,ly) = vLid; %lid y - velocity
rho(:,lid,ly) = 1 ./ (1+uy(:,lid,ly)) .* ( ...
sum(fIn([1,2,4],lid,ly)) + 2*sum(fIn([3,6,7],lid,ly)) );
% MICROSCOPIC BOUNDARY CONDITIONS: LID (Zou/He BC)
fIn(5,lid,ly) = fIn(3,lid,ly) - 2/3*rho(:,lid,ly).*uy(:,lid,ly);
fIn(9,lid,ly) = fIn(7,lid,ly) + 1/2*(fIn(4,lid,ly)-fIn(2,lid,ly))+ ...
1/2*rho(:,lid,ly).*ux(:,lid,ly) - 1/6*rho(:,lid,ly).*uy(:,lid,ly);
fIn(8,lid,ly) = fIn(6,lid,ly) + 1/2*(fIn(2,lid,ly)-fIn(4,lid,ly))- ...
1/2*rho(:,lid,ly).*ux(:,lid,ly) - 1/6*rho(:,lid,ly).*uy(:,lid,ly);
% COLLISION STEP
for i=1:9
cu = 3*(cx(i)*ux+cy(i)*uy);
fEq(i,:,:) = rho .* t(i) .* ...
( 1 + cu + 1/2*(cu.*cu) - 3/2*(ux.^2+uy.^2) );
fOut(i,:,:) = fIn(i,:,:) - omega .* (fIn(i,:,:)-fEq(i,:,:));
end
% MICROSCOPIC BOUNDARY CONDITIONS: NO-SLIP WALLS (bounce-back)
for i=1:9
fOut(i,bbRegion) = fIn(opp(i),bbRegion);
end
% STREAMING STEP
for i=1:9
fIn(i,:,: ) = circshift(fOut(i,:,: ), [0,cx(i),cy(i)]);
end
% VISUALIZATION
if (mod(cycle,tPlot)==0)
u = reshape(sqrt(ux.^2+uy.^2),lx,ly);
u(bbRegion) = nan;
imagesc(u(:,ly:-1:1)'./uLid);
colorbar
axis equal off; drawnow
end
end