# Laminar flow in channel

Hi everyone

i am new to LBM method and trying to simulate Laminar flow in channel with half-way bounce back method but i do not get 1.5*inlet velocity profile for my velocity profile. I post my code below I would be grateful if someone help me out thanks in advance.

clc
clear
close all
% GENERAL FLOW CONSTANTS
lx = 351; % number of cells in x-direction
ly = 52; % number of cells in y-direction
uMax = 0.166666667; % maximum velocity of Poiseuille inflow
Re = 100; % Reynolds number
nu = uMax * 50 / Re; % kinematic viscosity
omega = 1. / (3*nu+1./2.); % relaxation parameter
maxT = 40000; % total number of iterations
tPlot = 1000; % cycles

% 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=zeros(lx,ly);

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(1:lx,1:ly) = uMax;
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;
``````

ux(1,bbRegion) = 0 ;
uy(1,bbRegion) = 0 ;

``````% MACROSCOPIC (DIRICHLET) BOUNDARY CONDITIONS
% Inlet: Poiseuille profile
y_phys = col-1.5;
ux(:,in,col) =uMax;
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
``````

% % (BOUNCE-BACK _halfway)

``````fOut(5,:,ly)=fOut(3,:,ly);
fOut(8,:,ly)=fOut(6,:,ly);
fOut(9,:,ly)=fOut(7,:,ly);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fOut(3,:,1)=fOut(5,:,1);
fOut(6,:,1)=fOut(8,:,1);
fOut(7,:,1)=fOut(9,:,1);

% STREAMING STEP
for i=1:9
fIn(i,:,:) = circshift(fOut(i,:,:), [0,cx(i),cy(i)]);
end
``````

% fIn(4,out,:)=2fIn(4,out-1,:)-fIn(4,out-2,:);
% fIn(8,out,:)=2
fIn(8,out-1,:)-fIn(8,out-2,:);
% fIn(7,out,:)=2*fIn(7,out-1,:)-fIn(7,out-2,:);
% %

``````% VISUALIZATION
if (mod(cycle,tPlot)==0)
disp(cycle)
u = reshape(ux,lx,ly);
``````

% % u(bbRegion) = nan;
% imagesc(u’);
% axis equal off; drawnow

a=u(250,:)./uMax;
b=(0:ly-1)./(ly-1);
plot(b,a)
grid on
drawnow

``````end
``````

end