Multiresolution relaxation times lattice Boltzmann schemes with projection

Abstract

We propose to extend the multiresolution relaxation times lattice Boltzmann schemes with an additional projection step. For the explicit example of the D2Q9 scheme, we define this extended method. We prove that in general the projection step does not change the asymptotic partial differential equations at second order. We present four numerical test cases. One concerns linear stability with a Fourier analysis with a single-vertex scheme. Three bidimensional fluid flows with a coarse mesh have been tested: the Minion and Brown sheared flow, the Ghia, Ghia and Shin lid-driven cavity and an unsteady acoustic wave. Our results indicate that the bulk viscosity can be dramatically reduced with a better stability than the initial scheme.

Figures (19)

• Figure 1: The nine velocities of the D2Q9 scheme LL00.
• Figure 2: Comparison of linear stability zones for advection speed $u_0 = 0.35 \, c_0$ and $\, v_0 = 0$. Traditional D2Q9 scheme LL00 on the left and D2Q9 MRT with projection on the right. The stability zone is extended for higher values of the relaxation parameter $\, s_e$.
• Figure 3: Minion-Brown test case MB97 for Reynolds number $\, Re = 10^4$, 128 grid points and $\, N_T = 5541 \,$ discrete time iterations. BGK results: $\, s_e = s_q = s_h = s_\mu$ given by the relation (\ref{['minion-sx']}). Vorticity field; the results are not satisfying.
• Figure 4: Minion-Brown test case MB97 for Reynolds number $\, Re = 10^4$, 128 grid points and $\, N_T \,$ discrete time iterations (\ref{['minion-NT']}). MRT results on the left with $\, s_e = 1.715 _,$ and results for the MRT scheme with projection on the right with the same parameter $\, s_e$. These vorticity fields are qualitatively correct.
• Figure 5: Minion-Brown test case MB97 for Reynolds number $\, Re = 10^4$, 128 grid points and $\, N_T = 5541 \,$ discrete time iterations. Results for the MRT scheme with projection. The bulk viscosity is reduced by using the parameter $\, s_e = 1.9999$. Vorticity field.