Forcing term in single-phase and Shan-Chen-type multiphase lattice Boltzmann models

Abstract

Numerous schemes have been proposed to incorporate a bulk forcing term into the lattice Boltzmann equation. In this paper we present a simple and straightforward comparative analysis of five popular schemes [Shan and Chen, Phys. Rev. E 47, 1815 (1993); Phys Rev Lett. 81, 1618 (1998); He et al., Phys. Rev. E 57, R13 (1998); Guo et al., Phys. Rev. E 65, 046308 (2002); Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)] in which their differences and similarities are identified. From the analysis we classify the schemes into two groups; the behaviors of the schemes in each group are proven to be identical up to second order. Numerical test simulating the two-dimensional unsteady Taylor-Green vortex flow problem demonstrate that all five schemes are of comparable accuracy for single-phase flow. However, for two-phase flow the situation is different, which is demonstrated by incorporating these schemes into different Shan-Chen-type multiphase models. The forcing scheme in the original Shan-Chen (SC) multiphase model turns out to be inaccurate in terms of the resulting surface tension for different density ratios and relaxation times. In the numerical tests, a typical equation of state and interparticle interactions including next-nearest neighbors were incorporated into the SC model. Our results confirm that the surface-tension values obtained from the original SC lattice Boltzmann method (LBM) simulation depend on the value of the relaxation time τ. For τ<0.7Δt, the surface tension agree well with the analytical solutions. However, when τ>0.7Δt, the surface tension turns out to be systematically larger than the analytical one, exceeding it by more than a factor of 2 for τ=2Δt. In contrast, with the application of the scheme proposed by He et al., the SC LBM produces very accurate surface tensions independent of the value of τ. We also found that the densities of the coexisting liquid and gas can be adjusted to match those at thermodynamic equilibrium if the particle interaction term includes next-nearest-neighbor contributions. The obtained results will be useful for further studies of two-phase flow with high density ratios using the SC LBM approach.

Lattice Boltzmann modeling of multiphase flows at large density ratio with an improved pseudopotential model

“…For some of the above-mentioned techniques, as reported by Huang et al [24], their achievable largest density ratio will drop rapidly when the viscosity decreases. Meanwhile, most previous studies of the pseudopotential LB model are focused on stationary or quasistationary multiphase problems.…”

“…(24). To achieve thermodynamic consistency, the parameter σ in the improved forcing scheme can be slightly changed with a.…”

“…The periodical boundary conditions are applied in the x and y directions. The density field is initialized as follows [24]: Fig. 1.…”

Multiphase lattice Boltzmann simulations for porous media applications

“…Recently, several methods have been developed to alleviate the limitations of the original inter-particle potential model and improve its performance. These techniques include incorporating a realistic equation of state into the model [111,112], increasing the isotropy order of the interactive force [113,114], improving the force scheme [115][116][117][118], and using the Multi-Relaxation Time (MRT) scheme [109,119] instead of the BGK approximation. These techniques have been demonstrated to be effective in reducing the magnitude of spurious velocities, eliminating the unphysical dependence of equilibrium density and interfacial tension on viscosity (relaxation time), and increasing the viscosity and density ratios in simple systems [120][121][122][123][124].…”

Modeling for Collapsing Cavitation Bubble near Rough Solid Wall by Mulit-Relaxation-Time Pseudopotential Lattice Boltzmann Model

“…A vapor bubble with radius of 80 R = is initially placed at the center of the domain. The density field is initialized as [33] ( ) ( ) ( ) …”