Hey everybody!
In order to boost the stability of my LBM algorithms I tried to incorporate Smagorinsky turbulence models into both my BGK and TRT collision operators that increase the viscosity locally. While the BGK Smagorinsky algorithm seems to work fine across a broad range of Reynolds numbers, my results with the TRT collision operator and the turbulence model are significantly different for very high Reynolds numbers.
As I could not find a detailed derivation of TRT in population space I tried to include it similarly to the BGK model. My reasoning is as follows (C is the Smagorinsky constant whereas c_s is the lattice speed of sound):
 As the symmetric relaxation time tau_p in TRT is linked to the viscosity nu = nu_0 + (C Delta)^2 = (tau_p  0.5)*c_s^2 = (tau  0.5)*c_s^2 = (tau_p0  0.5)*c_s^2 + tau_T*c_s^2 the relaxation can be split up into an additional turbulent contribution.
 The strain rate tensor S may be calculated from the approximating the first order contribution of the momentum flux by the nonequilibrium momentum flux (In BGK this can be seen from the ChapmanEnskog expansion but I have also seen papers that use this approach for MRT models. As TRT could be seen as a special form of MRT I suppose this should still hold for TRT as well).
 Thus the overall strain rate tensor is given by S = 1/(2 rho c_s^2 tau) sqrt( 2 Pi_ij^(1) Pi_ij^(1) ) with Pi_ij^(1) approx Pi_ij^(neq) = sum_alpha e_{alpha i} e_{alpha j} ( f_alpha^(eq)  f_alpha ).
 The turbulent relaxation time can be calculated from these two equations to tau_T = 1/2 { sqrt[tau_p0^2 + 2 sqrt(2) (C Delta)^2/(rho c_s^4) sqrt( Pi_ij^(neq) Pi_ij^(neq) ) ]  tau_p0 }, so just like with the BGK operator.
 Now i simply add the turbulent relaxation time to the positive relaxation time of TRT tau_p = tau_p0 + tau_T while calculating the negative relaxation parameter from the magic parameter lambda and the positive relaxation time tau_p0 without the turbulent part rearranging the equation lambda = (tau_p0  0.5) (tau_m  0.5).
I could not find a paper that features a detailed derivation (CE expansion) of TRT in population space (all I know are in momentum space which I have no experience yet) or a paper combining these two models. Does somebody know a paper featuring the two or has anyone an idea where my mistake is?
Thanks a lot!
