Double-D2Q9 lattice Boltzmann models with extended equilibrium for two-dimensional magnetohydrodynamic flows

Abstract

The vast majority of the existing lattice Boltzmann methods (LBMs) suggest to relax relevant quantities to a second-order truncated equilibrium state. Despite its simplicity and popularity, this choice does not fully exploit the potential of any lattice discretization. In this paper, an extended equilibrium state is adopted to evaluate the suitability of different LBMs (i.e., the Bhatnagar–Gross–Krook, the multiple-relaxation-time in terms of raw and central moments, and the simplified one) to simulate two-dimensional magnetohydrodynamic flows by means of the D2Q9 velocity space. Two sets of particle distribution functions are employed: one for the flow field and the other for the magnetic one. Even if the minimal five-velocities discretization is sufficient to represent the evolution of the latter, a nine-velocities model enhances the capability to enforce the divergence-free condition of the magnetic field, as shown. Therefore, a double-D2Q9 approach is herein devised. Eventually, the computational cost involved by all the schemes is discussed both in terms of virtual memory and run time. Interestingly, the simplified LBM for magnetohydrodynamic flows is herein presented for the first time.

One-stage simplified lattice Boltzmann method for two- and three-dimensional magnetohydrodynamic flows

“…In this section, first the simplified LBM by De Rosis et al 49 is recalled. Secondly, our one-stage simplified LBM for two-dimensional magnetohydrodynamic flows is devised.…”

“…A. Two-dimensional two-stages simplified LBM by De Rosis et al 49 Let us consider a two-dimensional Cartesian space x = [x, y] and the D2Q9 velocity discretization, where lattice directions are c i = [c ix , c iy ] with…”

“…In a very recent paper, De Rosis et al 49 have performed a comprehensive comparison of four different LBMs for MHD: (i) the BGK LBM, the multiple-relaxation-time (MRT) LBMs based on (ii) raw and (iii) central moments, and (iv) the simplified lattice Boltzmann method. From the reported extensive analysis, two main results have been found.…”

“…Secondly, the divergence-free condition of the magnetic field (which should be ensured by any scheme for MHD flows [50][51][52][53][54] ) is enforced in an excellent manner only by the SLBM, which shows results very close to the round-off error. However, the SLBM shows the lowest accuracy, and this is particularly emphasised when coarse grid resolutions are adopted 49 . This should be addressed to its level of numerical dissipation that is generally higher than the one obtained by the BGK or MRT LBMs 55 .…”

Improved color-gradient method for lattice Boltzmann modeling of two-phase flows

“…LBM received a growing interest motivated by its scalability and its low dissipation properties, making it particularly well-suited for acoustic long distance dispersion problems 20 . It has been applied to a wide range of applications such as, reservoir permeability [21][22][23] , combustion flows 24 , blood flow 25 , boiling flow 26 , thermocapillarity flows 27 , non-Newtonian flows 28 , magneto-hydrodynamic flows 29 , cryogenic flows 30 , etc.…”