D2Q9 TRT Lid driven cavity flow

[Alex]

Hello!
I finished the solution of D2Q9 BGK Lid driven cavity flow and now start the TRT collision operator approximation. I used several papers and book of Mohamad about LBM. If i understand correctly, I need to modify only collision step. I split distribution function into symmetric and asymmetric parts. tau_symmetric is calculated as ( 3*nu_lb + 0.5 ), tau_asymmetric = 0.5*tau_s/(tau_s-0.5) [magic paramter = 0.25]. With this parameters the code is unstable. Notably, when tau_asymmetric = tau_symmetric i have BGK solution, as it should be. So i think the problem is in calculation of tau_asymmetric. The relevant part of the code is as follows:

opposite=[1 4 5 2 3 8 9 6 7];
    t1 = u.*u + v.*v;
    for k = 1:9
 t2 = cx(k)*u + cy(k)*v;
        feq(:,:,k) = w(k)*rho.*(1 + 3*t2 + 4.5*t2.^2 - 1.5*t1);
              
               f_s(:,:,k)=0.5*(f(:,:,k)+f(:,:,opposite(k)));
               f_a(:,:,k)=0.5*(f(:,:,k)-f(:,:,opposite(k)));
               feq_s(:,:,k)=0.5*(feq(:,:,k)+feq(:,:,opposite(k)));
               feq_a(:,:,k)=0.5*(feq(:,:,k)-feq(:,:,opposite(k)));
f(:,:,k)=f(:,:,k)-omega_s*(f_s(:,:,k)-feq_s(:,:,k))-omega_a*(f_a(:,:,k)-feq_a(:,:,k));

Can someone suggest any guides or help me with this problem?
Thanks a lot!

[stephan]

Hi Alex

Your implementation seems to be correct. Also the fact that setting \tau_s = \tau_a yields the BGK solution is a good sign.

The specific values for the magic parameter \Lambda = (\tau_s – 0.5)*(\tau_a – 0.5) mentioned in the literature are a typical result for certain flow types and Reynolds number regimes.

If your code is supposed to run at high Reynolds numbers, I suggest you try different (smaller or much smaller) values for \Lambda, since it is connected to the Reynolds number via \tau_s. For high Reynolds numbers, the magnitude of \tau_a is way too large with a fixed \Lambda=0.25. Maybe try a few decimal digits below that, e.g. \Lambda=0.0025 and work your way up.

With the calibration of \tau_a through a smaller \Lambda, you should obtain stable results. In case this does not work, fix \tau_a=1 and try again.

If you encounter any special results (highly increased stability, or none at all), please let me know.

Anyway, good luck with your tests.

BR,
Stephan

PS: A similar question was posted and answered about two month ago:
https://www.openlb.net/forum/topic/smagorinsky-turbulence-model-and-trt-collision-operator/

[Alex]

Hello Stephan,

Thank you! I will try it

kind regards,
Alex

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