# channel flow

Hi,

I try to solve 2D channel flow. But I have some problems. Before explaining the problems, I mention my settings.

1. I use D2Q9 lattice. Also I use LBGK.

2. I use one way bounce back scheme. (on north and south walls)

For example south boundary
f(2,i,0)=f(4,i,0)
f(5,i,0)=f(7,i,0)
f(6,i,0)=f(8,i,0)

1. I give velocity BC at west domain with Zou-He scheme. Al so I give fully developed velocity profile.

uy(0,j)=1.5d0uo(1.d0-(4.d0/h**2)((jdy)-(h/2.d0))*2)
rhow=(f(0,0,j)+f(2,0,j)+f(4,0,j)+2.d0
(f(3,0,j)+f(6,0,j)+f(7,0,j)))/(1.d0-uy(0,j))
f(1,0,j)=f(3,0,j)+2.d0rhowuy(0,j)/3.d0
f(5,0,j)=f(7,0,j)+((f(4,0,j)-f(2,0,j))/2.d0)+rhowuy(0,j)/6.d0
f(8,0,j)=f(6,0,j)-((f(4,0,j)-f(2,0,j))/2.d0)+rhow
uy(0,j)/6.d0

1. As a outlet BC, I give zero gradient velocity (first order)

f(3,n,j)=f(3,n-1,j)
f(6,n,j)=f(6,n-1,j)
f(7,n,j)=f(7,n-1,j)

1. For calculating the equilibrium distribution function,

feq= wirho(1+3(eiu)+4.5(eiu)*2-1.5(uu))

1. My process sequence is;

As a initial calculate feq for initial condition, and f=feq,
And iterations begin. Collision step, streaming step, BC (velocity, bounce back and outflow), calculatong rho, u and v for new iteration.

My problem is, although flow is fully developed, veloctity is decreased from that I gave. Also density is decreased. So I lose mass.

Daralcan

Hello :

Could you explain how to deal with the 4 corners?

`````` ----------------------------------
``````

## –> flow in

D2Q9 lattice:
625
301
748

Take the inlet-south corner for example. f1 f5 f8 and f2 f5 f6 are unknown, and f6-f8 are inverse so the information here is insufficient.

In my opinion we can set f6=f8=0.5*(rho-(f1+f2+f3+f4+f5+f7+f0)), and do you also set like this?

If so, how do you get the rho above?

Do you just exrapolate the rho above the the inlet-south corner directly? In LB, rho(x=1,y=1)=rho(x=1,y=2)

and then substitute rho(x=1,y=1) into the equation f6=f8=0.5*(rho-(f1+f2+f3+f4+f5+f7+f0))?
----------------------------------

## . <- rho(x=1,y=2)

y
l
l
l_ _ _ _x

Thanks

Hi Peterson,

First of all, thank you for reply.

Yes, I used special corner treatment like this paper “On pressure and velocity ?ow boundary conditions and bounceback for the lattice Boltzmann BGK model ---- Qisu Zou and Xiaoyi He” . Also you mention the same corner special cornet treatment.

Also, in outlet BC, because of 1st order zero velocity gradient, I rearranged special corner treatment as (outlet-south);
f3=f1 / f4=f2 / f5=f7 / f6=f8 ,so I didn´t use f5=f7=0.5*(rho-(f1+f2+f3+f4+f6+f8+f0)). Is this correct or uncorrect?

But, I didn´t rho extrapolate like you. I am going to try this extrapolation and I will look my results.

Daralcan

I did the rho extrapolation but there is no changing!!!