# 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.

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: