Hi, everyone

I want to simulate the flow around a cylinder. The Inlet adopts velocity boundary condition, Outlet adopts pressure boundary condition, Wall boundary adopts bounce-back condition. I found a sample Matlab code about this problem on the internet. But after checking the total mass and total momentum in x and y direction, I found that they are not conserved. So I think there may be something wrong with the code. Could anyone help me check this? Thank you very much.

```
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clear
% GENERAL FLOW CONSTANTS
lx = 400; % number of cells in x-direction
ly = 100; % number of cells in y-direction
obst_x = lx/5+1; % position of the cylinder; (exact
obst_y = ly/2+3; % y-symmetry is avoided)
obst_r = ly/10+1; % radius of the cylinder
uMax = 0.1; % maximum velocity of Poiseuille inflow
Re = 100; % Reynolds number
nu = uMax * 2.*obst_r / Re; % kinematic viscosity
omega = 1. / (3*nu+1./2.); % relaxation parameter
maxT = 1001; % total number of iterations
tPlot = 50; % cycles
total_mass = zeros(maxT,2);
total_momentum_x = zeros(maxT,2);
total_momentum_y = zeros(maxT,2);
% 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];
col = [2:(ly-1)];
in = 1; % position of inlet
out = lx; % position of outlet
[y,x] = meshgrid(1:ly,1:lx); % get coordinate of matrix indices
obst = ... % Location of cylinder
(x-obst_x).^2 + (y-obst_y).^2 <= obst_r.^2;
obst(:,[1,ly]) = 1; % Location of top/bottom boundary
bbRegion = find(obst); % Boolean mask for bounce-back cells
% INITIAL CONDITION: Poiseuille profile at equilibrium
L = ly-2; y_phys = y-1.5;
ux = 4 * uMax / (L*L) * (y_phys.*L-y_phys.*y_phys);
% for i=2:lx
% for j=1:ly
% ux(i,j) = 0;
% end
% end
uy = zeros(lx,ly);
rho = 1;
for i=1:9
cu = 3*(cx(i)*ux+cy(i)*uy);
fIn(i,:,:) = rho .* t(i) .* ...
( 1 + cu + 1/2*(cu.*cu) - 3/2*(ux.^2+uy.^2) );
end
% 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;
total_mass(cycle,1) = cycle;
total_mass(cycle,2) = sum(sum(rho));
total_momentum_x(cycle,1) = cycle;
total_momentum_x(cycle,2) = sum(sum(rho.*ux));
total_momentum_y(cycle,1) = cycle;
total_momentum_y(cycle,2) = sum(sum(rho.*uy));
%if (total_mass(cycle,2)-total_mass(1,2))>200, break, end
% MACROSCOPIC (DIRICHLET) BOUNDARY CONDITIONS
% Inlet: Poiseuille profile
y_phys = col-1.5;
ux(:,in,col) = 4 * uMax / (L*L) * (y_phys.*L-y_phys.*y_phys);
uy(:,in,col) = 0;
rho(:,in,col) = 1 ./ (1-ux(:,in,col)) .* ( ...
sum(fIn([1,3,5],in,col)) + 2*sum(fIn([4,7,8],in,col)) );
% Outlet: Constant pressure
rho(:,out,col) = 1;
ux(:,out,col) = -1 + 1 ./ (rho(:,out,col)) .* ( ...
sum(fIn([1,3,5],out,col)) + 2*sum(fIn([2,6,9],out,col)) );
uy(:,out,col) = 0;
% MICROSCOPIC BOUNDARY CONDITIONS: INLET (Zou/He BC)
fIn(2,in,col) = fIn(4,in,col) + 2/3*rho(:,in,col).*ux(:,in,col);
fIn(6,in,col) = fIn(8,in,col) + 1/2*(fIn(5,in,col)-fIn(3,in,col)) ...
+ 1/2*rho(:,in,col).*uy(:,in,col) ...
+ 1/6*rho(:,in,col).*ux(:,in,col);
fIn(9,in,col) = fIn(7,in,col) + 1/2*(fIn(3,in,col)-fIn(5,in,col)) ...
- 1/2*rho(:,in,col).*uy(:,in,col) ...
+ 1/6*rho(:,in,col).*ux(:,in,col);
% MICROSCOPIC BOUNDARY CONDITIONS: OUTLET (Zou/He BC)
fIn(4,out,col) = fIn(2,out,col) - 2/3*rho(:,out,col).*ux(:,out,col);
fIn(8,out,col) = fIn(6,out,col) + 1/2*(fIn(3,out,col)-fIn(5,out,col)) ...
- 1/2*rho(:,out,col).*uy(:,out,col) ...
- 1/6*rho(:,out,col).*ux(:,out,col);
fIn(7,out,col) = fIn(9,out,col) + 1/2*(fIn(5,out,col)-fIn(3,out,col)) ...
+ 1/2*rho(:,out,col).*uy(:,out,col) ...
- 1/6*rho(:,out,col).*ux(:,out,col);
% 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
% OBSTACLE (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)==1)
u = reshape(sqrt(ux.^2+uy.^2),lx,ly);
u(bbRegion) = nan;
imagesc(u');
colorbar
title(num2str(cycle));
axis equal off; drawnow
end
end
```