Lattice Boltzmann Method

Our research line in the lattice Boltzmann methods (LBM) focuses the analysis of the collision operator.
The collision operator is of critical importance in the LBM method, as it is responsible of the proper modelization of the physics.

The Lattice Boltzmann method (LBM) is becoming one of the most interesting alternative because it is able to overcome some of the traditional CFD limitations.

However, the LBM is a relatively young method, where these limitations are still addressing. The limitation of the “standard” LBM (BGK collision operator) is the stability of the method at high Reynolds and Mach numbers. This limitation is important to address in the aeronautical industry because the flight regime is at high Reynolds number and there are many transonic and supersonic applications inside this industry.

Among others advanced LBM, the Multiple-Relaxation Time (MRT) collision operator was developed to overcome this limitation.

Using von Neumann stability analysis (Figure 6) to extract dispersion and diffusion errors, we compare the D2Q9 and D3Q19 lattice scheme with different advanced collision operators: the single relaxation time (BGK), the multiple-relaxation time with raw moments (MRT–RM) and with central moments (MRT–CM).


We propose an optimization, for the free parameters, based on the k-1% dispersion-error rule, that states that waves with dispersive errors above 1% should be dissipated since they pollute the solution and may cause instabilities.

To this aim we increase dissipation in the scheme for waves with dispersive errors above 1%.

In particular, we show that the optimized MRT–CM can cope with lower viscosities, higher velocities and coarser meshes that their predecessors. Finally, the optimized MRT–CM is tested for a shear layers flow (GIFs DSL_MRT_CM_Standard_64 y DSL_MRT_CM_Optimized_64) and Taylor Green Vortex (GIF TGV) to illustrate the enhanced stability and accuracy of the proposed technique.

Vorticity isocountours for N =64 and standard MRT-CM with Ma = 0.2 and Re = 3 · 104 at t/tc = 1.
Vorticity isocountours for N =64 and optimized MRT-CM with Ma = 0.2 and Re = 3 · 104 at t/tc = 1.
Isosurface of the Q-criterion colored by kinetic energy for Re = 1600 on a N = 256 grid compared with BGK collision operator.