 # Temperature Dirichlet boundary condition

Dear all,

In order to simulate thermal lattice Boltzmann method (conduction), what is the right way to apply Dirichlet boundary condition on wall (for example, west wall is maintain at constant temperature T=Tw) for which distribution functions f1, f5 and f8 are unknowns? Similarly other walls are maintained at constant temperatures.

6 2 5
\ | /
\ | /
3-----0-----1
/ |
/ |
7 4 8

Type (1)

``````        f(5,1,1)=w(5)*tw+w(7)*tw-f(7,2,2)
do j=2,ny-1
f(1,1,j)=w(1)*tw+w(3)*tw-f(3,2,j)
f(5,1,j)=w(5)*tw+w(7)*tw-f(7,2,j+1)
f(8,1,j)=w(8)*tw+w(6)*tw-f(6,2,j-1)
end do
f(8,1,ny)=w(8)*tw+w(6)*tw-f(6,2,ny-1)
``````

Type (2)

``````     do j=1,ny
f(1,1,j)=w(1)*tw+w(3)*tw-f(3,1,j)
f(5,1,j)=w(5)*tw+w(7)*tw-f(7,1,j)
f(8,1,j)=w(8)*tw+w(6)*tw-f(6,1,j)
end do
``````

Results with Type (1) close with FVM results but it spoils the boundary temperature, on the other hand, results with type (2) maintain the correct boundary temperature but interior temperatures are not matches with FVM results.

Hi vipul.mech140
I did not understand type 1. but I think type 2 works correctly.

I’m working the TLBM , I used this configuration get goot results

For example in a flow whit hot wall

``````                           /      |      \
g7     g4     g8
__________________________________
``````

inlet flow >>>> >> outlet T = T*
T = tw
______________________________
hot wall
g6 g2 g5
\ | /

Inlet T = tw
g(y,x,1) = tw * w(1)+ w(3) - g(y,x,3)
g(y,x,5) = tw * w(5)+ w(7) - g(y,x,7)
g(y,x,8) = tw * w(8)+ w(6) - g(y,x,6)

Outlet I used Second order open boundary condition for Termal field and Works

g(y,x,1) = 2 * g(y,x-1,1) - g(y,x-2,1)
g(y,x,2) = 2 * g(y,x-1,2) - g(y,x-2,2)
g(y,x,3) = 2 * g(y,x-1,3) - g(y,x-2,3)
g(y,x,4) = 2 * g(y,x-1,4) - g(y,x-2,4)
g(y,x,5) = 2 * g(y,x-1,5) - g(y,x-2,5)
g(y,x,6) = 2 * g(y,x-1,6) - g(y,x-2,6)
g(y,x,7) = 2 * g(y,x-1,7) - g(y,x-2,7)
g(y,x,8) = 2 * g(y,x-1,8) - g(y,x-2,8)
g(y,x,0) = 2 * g(y,x-1,0) - g(y,x-2,0)

Bounce bakc for hot wall- adiabatic
Hot, I used only the g whit take part in the calculation

``````    g(y,x,2) = th * (t(2)+ t(4)-g(y,1,4))
g(y,x,5) = th * (t(5)+ t(7)-g(y,1,7))
g(y,x,6) = th * (t(6)+ t(8)-g(y,1,8))
``````

``````g(y,x,4) = g(y-1,x,2)
g(y,x,7) = g(y-1,x,5)
g(y,x,8) = g(y-1,x,6)``````
1 Like

Yes, Type (2) works very well.

As per literature,

q= k*(T1-T2)/deltax (suppose, heat transfer in poistive X direction (i.e. from west wall)

Hence, T1=T2+(q*deltax/k)

``````   =>gk(1)=gk(2)+(q*deltax/k)

where k= direction of unknown distribution function (in my case,for west wall 1,5,8)
1= nodes on west wall
``````

If anyone of you have used this, then please let me know.
Thank you all.

Hi, have u solved this problem? can u guide me about the implementation of constant heat flux BC??