# LES-LBM for turbulent simulation

Dear All:

I am new here. I want to simulate turbulent water duct flow, Channel height about 1 cm and Re=10000. 2D simulation would be tried first and if it works I will extend it to 3D simulation.

For the 2D case, the Zou/He B.C.'s were applied ( inlet/outlet/wall nodes entirely follow the paper ‘‘On pressure and velocity flow boundary conditions and bounceback for the lattice Boltzmann BGK model’’), inlet involve velocity condition with uniform inlet velocity u=0.02 while the pressure condition on the outlet with rho_out=1.0 in the LB coding. On the 4 corner nodes since the information may be insufficient for the evaluation of the density, the dnesity from the neighboring cells were extrapolated (take inlet-bottom node for example, rho = (f(x,y+1,9)+f(x,y+1,2)+f(x,y+1,4)+2.*(f(x,y+1,3)+f(x,y+1,6)+f(x,y+1,7)))/(1.-ux), here ux=0.02 to get rho=rho_in).

The LES-LBM approach was also involved to solve turulence problem, following the paper (LARGE-EDDY SIMULATIONS WITH A MULTIPLE-RELAXATION-TIME LBE MODEL, International Journal of Modern Physics B, Vol. 17, Nos. 1-2 (2003) 33-39), that’s multiple-relaxation-time approach, relaxation time tau calculated locally through the relation of strain and stress tensor.
But I am not sure how to get stress tensor in the LB.
In my procedure the stress tensor is computed as follows:

For 2D, stress tensor
Pi=
/ pi11 pi12 /
/ pi21 pi22 /

pi11=f(X,Y,1)-fq(1)+f(X,Y,3)-fq(3)+f(X,Y,5)-fq(5)+f(X,Y,6)-fq(6)+f(X,Y,7)-fq(7)+f(X,Y,8)-fq(8)
pi22=f(X,Y,2)-fq(2)+f(X,Y,4)-fq(4)+f(X,Y,5)-fq(5)+f(X,Y,6)-fq(6)+f(X,Y,7)-fq(7)+f(X,Y,8)-fq(8)
pi12=f(X,Y,5)-fq(5)-(f(X,Y,6)-fq(6))+f(X,Y,7)-fq(7)-(f(X,Y,8)-fq(8))
pi21=f(X,Y,5)-fq(5)-(f(X,Y,6)-fq(6))+f(X,Y,7)-fq(7)-(f(X,Y,8)-fq(8))
f()-fq() refers to the non-equilibrium part of distribution function

And the magnitude of the stress tensor pi = (pi11(x,y)**2+pi22(x,y)**2+pi12(x,y)**2+pi21(x,y)**2))**0.5

Are the processes above correct or not?

One more question here, does the collision take place at the wall? I mean, should the collision step (fnew(x,y,z,d) = (1.0-OMEGA) * f(x,y,z,d) + OMEGA *f_equilibrium(d)) be applied on the wall also?

In my opinion, the relaxation time tau should approach 0.5 for high Re flow, so I would try to tune it from 0.500001 to 0.6, and grids number (60300, 90600, … ) and inlet velocity (u=0.02, 0.01, 0.005, …) will be tried also, but seems no reasonable results, the program still diverging. It is OK for tau>0.8 without LES, but diverges for tau<0.65 with or without LES.

I appriciate anyone of you read this, since there are so many words here.

Please give me some advices, comments and suggestions, thanks in advance.

Regards

Peterson

hi. you should noitce some points:
1-if Re increase, tau will close to 0.5, and this tau is divegence your simulation.
2-but by turbulent modeling you can prevent this unstablity
3-use LES, and SRT
4-use smagorinsky model
5-your strain tenssor S, can obtain directly from non-equilbrian part of f,(see DNS and LES of decaying isotropic turbulence with
and without frame rotation using lattice Boltzmann method
Huidan Yu a, Sharath S. Girimaji a, Li-Shi Luo b,*)
6-im sure u can solve it.

Hello, Peterson

In the MRTLBM, we calculate the non-equlibrium moment mneq_i=(m_i-meq_i), not the (f_i-feq_i).
To get N-S eq. from the MRTLBE follow the Chapman-Enskog approach, the stress tensor can be got by the non-equlibrium moments.
s11=0.25d0*s(1)mneq_1(i,j)+0.75d0s(7)mneq_7(i,j)
s12=s21=1.5d0
s(8)mneq_8(i,j)
s22=0.25d0
s(1)mneq_1(i,j)-0.75d0s(7)*mneq_7(i,j)
where the s11,s12,s21,s22 is the stress tensor, s(1),s(7), and s(8) is the relaxation time rates for the second, 8th, 9th moments, respectively.

Dear Amir and worn222:

I am so glad to read your replies, and thanks a lot for your suggestions and encouragements.

I will follow your suggestions and keep trying.

^___^

Warmest regards

Peterson