Dear all,
I got a trouble when adding the smagorinsky turbulence model?SGS?to the D2Q9 incompressible flow with Re=14000. The result seemed incorrect as there was no irregular vortex in the flow field. The key problem in incorporating SGS into LBM is shown in the figure below. I implemented the second formula by " dsqrt(2*( Q11Q11 + Q11Q12 + Q11Q21 + Q11Q22 + Q12Q12 + Q12Q21 + Q12Q22 + Q21Q21 + Q21Q22 + Q22Q22 ) ) " ,since Qij is a 2x2 matrix. Does it correct? Or simply( Q11Q11 + Q12Q12 + Q21Q21 + Q22Q22). If anyone did the same same work(LBM+SGS), please tell me the specifc solutions about the two eqs. shown below. Appreciate for any help.

the formula involves the “magnitude” of the tensor Q which is in 2D defined as
Q[sub]ij[/sub]Q[sub]ij[/sub] = Q[sub]11[/sub][sup]2[/sup] + Q[sub]12[/sub][sup]2[/sup] + Q[sub]21[/sub][sup]2[/sup] + Q[sub]22[/sub][sup]2[/sup]
The first definition you mentioned is wrong.

Thank you very much for your answer, I have got the result by the latter one in the simulation of flow (Re=7000) past a cylinder (2D), while it seemed not good. There existed numerical noise in some areas. I am now finding solutions.
In addition, I found that it would quickly (Tstep=700) suffer unstableness when the smagorinsky constant Cs was 0.1 and Delta = h (Lattice Space). When Cs was 0.1 and Delta = 4h( Chow and Moin found that, for a second order finite difference scheme, a filter-grid ratio of four should be used for the discretization errors to be smaller than the subgrid terms.), it was still unstable (Tstep=47000). Could you give me some advices, if you did similar work before?