Semi-implicit-linearized multiple-relaxation-time formulation of lattice Boltzmann schemes for mixture modeling

A lattice Boltzmann model for mixture modeling is developed by applying the multiple-relaxation-time (MRT) approach to the Hamel model, which allows one to derive from a general framework different model equations independently proposed, like the Gross-Krook model and the Sirovich model. By imposing some physical constraints, the MRT lattice-Boltzmann Hamel model reduces to the generalized MRT lattice-Boltzmann Gross-Krook model (involving the local Maxwellian centered on the barycentric velocity), which allows one to tune independently the species diffusivity, the mixture kinematic viscosity, and the mixture bulk viscosity. Reducing the number of moving particles over the total is possible to deal effectively with mass particle ratios far from unity and, for this reason, to model the pressure-driven diffusion. A convenient numerical approach is proposed for solving the developed model, which essentially widens the stability range of conventional schemes in terms of dimensionless relaxation frequencies, by solving explicitly the advection operator together with the nonlinear terms of the collisional operator and solving implicitly the residual linear terms. In this way, the calculations are drastically reduced and the operative matrices can be computed once for all, at the beginning of the calculation (implying moderate additional computational demand). Following this approach, a semi-implicit-linearized backward Euler scheme, ideal for parallel implementations, is proposed. In order to achieve the previous results, the asymptotic analysis, recently suggested for analyzing the macroscopic equations corresponding to lattice-Boltzmann schemes in the low-Mach-number limit, proves to be an effective tool. Some numerical tests are reported for proving the consistency of the proposed method with both the Fick model and Maxwell-Stefan model in the macroscopic limit.