Modeling realistic multiphase flows using a non-orthogonal multiple-relaxation-time lattice Boltzmann method

Abstract

In this paper, we develop a three-dimensional multiple-relaxation-time lattice Boltzmann method (MRT-LBM) based on a set of nonorthogonal basis vectors. Compared with the classical MRT-LBM based on a set of orthogonal basis vectors, the present non-orthogonal MRT-LBM simplifies the transformation between the discrete velocity space and the moment space and exhibits better portability across different lattices. The proposed method is then extended to multiphase flows at large density ratio with tunable surface tension, and its numerical stability and accuracy are well demonstrated by some benchmark cases. Using the proposed method, a practical case of a fuel droplet impacting on a dry surface at high Reynolds and Weber numbers is simulated and the evolution of the spreading film diameter agrees well with the experimental data. Furthermore, another realistic case of a droplet impacting on a super-hydrophobic wall with a cylindrical obstacle is reproduced, which confirms the experimental finding of Liu et al. ["Symmetry breaking in drop bouncing on curved surfaces," Nat. Commun. 6, 10034 (2015)] that the contact time is minimized when the cylinder radius is comparable with the droplet radius.

Lattice Boltzmann simulation of water droplet impacting a hydrophobic plate with a cylindrical pore

“…An efficient nonorthogonal multiple-relaxation-time (MRT) lattice Boltzmann model (LBM) [25] is employed in this study. In the study of Hoang et al [28], it was shown that the 2D simulation delayed the droplet breakup due to ignoring the Laplace pressure caused by the out-droplet plane.…”

“…Multiplication of the explicit form of Eq. (1) by the transformation matrix M leads to the collision equation in the moment space as follows [25,31]:…”

“…Recently, Li et al [35] further showed that a nonorthogonal MRT-LBM can retain the numerical accuracy and simplify the implementation of its orthogonal counterpart. Based on a generalized multiple-relaxation-time scheme [36,37], Fei et al proposed another nonorthogonal MRT-LBM [25]. Compared with the classical MRT-LBM, the proposed nonorthogonal MRT-LBM simplifies the transformation between the discrete velocity space and the moment space, and exhibits better portability across different discrete velocity models.…”

“…On the other hand, the basic LB method is incapable of handling a large density ratio (between the liquid phase and vapor phase), and there are severe spurious currents near the phase interface which may reduce the numerical stability and accuracy. In this study, a 3D nonorthogonal multiple-relaxation-time LB model proposed by Fei et al [25] is applied to enhance the numerical stability. The improved multiphase pseudopotential model proposed by Li et al [26,27] is employed to achieve a large density ratio, leading to LB simulation of a realistic water droplet impact.…”

Multiple-relaxation-time discrete Boltzmann modeling of multicomponent mixture with nonequilibrium effects

“…In fact, the idea of using a finite set of discrete speeds appeared early in the seminal work [40], where the discrete velocity model was constructed for the Boltzmann equation, and the collision integral was expressed as a nonlinear quadratic term. In the last three decades, the lattice Boltzmann method (LBM), developed from the lattice gas automaton [41,42] and originally based on the discrete simplified Boltzmann equations [25,[43][44][45][46][47][48], has been successfully used as an alternative tool of various partial differential equations for complex systems with a multiphase [49][50][51][52][53][54], multicomponent [55][56][57], mass diffusion [58,59], external force [60], and/or chemical reactions [61,62], etc. In the evolution of the discrete Boltzmann equation, the particle velocity space is dicretized besides the discretization in physical space.…”