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Author Topic: Smagorinsky Turbulence Model and TRT collision operator
Posts: 4
Post Smagorinsky Turbulence Model and TRT collision operator
on: January 26, 2019, 19:47

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 non-equilibrium momentum flux (In BGK this can be seen from the Chapman-Enskog 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!

Posts: 26
Post Re: Smagorinsky Turbulence Model and TRT collision operator
on: January 29, 2019, 10:15

Hey tobit,

this is a good question! I think you are familar with the TRT papers and you know the common TRT magic parameters (lambda=0,25 or lambda=0,375). These values are proposed as universal values. They are calibrated for poiseuille flow or laminar shear flows. In the case of turbulence you have to go magnitudes lower to reach the same stability as BGK and lamda gets a function of your Reynolds number. What you have to do is a lambda recalibration for your cannonical flow type. Your derivation for Smagorinsky TRT seems to be correct, but don't expect an advantage in comparison for Smagorinsky BGK. If you get other results or see an increased stability, please let me know. I am interested in this topic.

Best Marc

Posts: 4
Post Re: Smagorinsky Turbulence Model and TRT collision operator
on: January 31, 2019, 19:47

Thanks Marc for your detailed reply!
I am familiar with the TRT papers and the derivation of the optimal parameters but I did not consider this. Your argument makes perfectly sense: Setting the anti-symmetric relaxation parameter using the magic parameter lambda (with 1/4 as I did) and a very high Reynolds number and thus a very low positive relaxation time (tau+ very close to 0.5) will result in a very high value for tau- (a very small omega-) and therefore the influence of the anti-symmetric part will be (almost) neglectable. Fixing tau+ and tau- to similar values makes TRT though lose its advantages over BGK as you have pointed out.
I think turbulence modelling in LBM in general is quite appealing indeed due to its simplicity and computational efficiency. Some time ago i benchmarked it and I think my performance dropped by around 1/3 (compared to the simple BGK operator) to 75 Mlups with a D3Q19 lattice on an octa-core processor.
I will let you know if I stumble across something. In the meanwhile thanks again for your kind reply!

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