A decoupled and stabilized lattice Boltzmann method for multiphase flow with large density ratio at high Reynolds and Weber numbers

Highlights

• Decoupled MRT is developed to remove dependency between relaxation times and viscosity.

• Second-order $u^3$ numerical error is eliminated in Decoupled MRT (DMRT).

• Stabilization scheme is proposed to achieve large-density-ratio multiphase flow at high Re and We.

• All collision regimes are reproduced by DMRT for droplet collision, splashing and impact.

Abstract

A decoupled lattice Boltzmann method (LBM) with improved numerical stability is developed from the conventional multiple relaxation time (MRT) model by eliminating the second-order mutual effects of viscosity modes. The dependency between relaxation times and viscosity is decoupled to remove the O($u^3$) errors in the original model, and thus a stabilization strategy is proposed by adjusting relaxation times without changing the local viscosity under the mass and momentum conservations. This new model still stays in the conventional MRT framework and keeps the advantages of simplicity and efficiency of LBM implementation. The decoupled MRT is demonstrated to be valid for high-speed flow with Galilean invariance. Moreover, the pseudopotential model is employed in this paper to present the improvement of numerical stability in simulating the large-density-ratio multiphase flows at high Reynolds (Re) and Weber (We) numbers. Droplet collision is achieved at density ratio 1153, Re = 34474 and We = 28212. Droplet splashing is achieved at density ratio 6637, Re = 34474 (and 71928) and We = 30538 (and 259). Droplet impacting on dry wall is achieved at density ratio 5720, Re = 11462 (and 22982) and We = 13573 (and 166). All collision regimes are simulated by the decoupled MRT from low Re (We) to high Re (We).

Introduction

Lattice Boltzmann method (LBM) has been developed for decades due to its kinetic nature and mesoscopic approach that provides a promising method to construct underlying mesoscopic and microscopic physics. It has been applied to a variety of problems of porous media, multiphase flow, turbulence, micro-flow, electromagnetic hydrodynamics, droplet interaction, and bubble nucleation [1], [2]. However, numerical stability at high Reynolds and Weber numbers still restricts its application in scientific research and engineering design. Starting from lattice gas automata method [3], [4], several variants of LBM are proposed to handle the more accurate and stable simulations, including single-relaxation-time Bhatnagar–Gross–Krook (SBGK) method [5], [6], multiple-relaxation-time (MRT) method [7], [8], entropic LBM [9], [10], lattice Boltzmann flux solver (LBFS) [11], and cascaded LBM [12], [13] etc. It should be noted that these methods are also developed for large-density-ratio multiphase flows at high Reynolds and Weber numbers.

In the scope of LBM multiphase flows, a variety of multiphase models are proposed to achieve phase segregation, including the color-gradient LBM [14], the free-energy LBM [15], the pseudopotential LBM [16], [17], and phase-field LBM [18], [19] etc. The multiphase LBM has been applied to study fuel cells, micro-channel flow, droplet collision and adhesion [20]. Generally, low spurious current, large density ratio, high Reynolds number, and high Weber number are the major issues when LBM is used to research the dynamic multiphase flow. Much progress has been made to improve the numerical stability in simulating dynamic multiphase flows in the aforementioned situations, which will reveal a more complex interfacial mechanism and thermal process.

In the latest models, the achievable density ratio in dynamic flow is about 1000 for both of the immiscible fluid and miscible fluid. Reynolds number and Weber number is achieved up to about 6000 respectively in droplet collision and splashing application, but not for both parameters Re and We [13], [19], [21], [22], [23]. Chen et al. [24] proposed a simplified LBM with the phase-field method to simulation a droplet splashing of Re =8000 and We =8000 successfully, although no droplet break-up is observed like other simulations and experiments [11], [23], [25], [26]. Compared with other interfacial capturing and tracking methods used in conventional Navier-Stokes equation, the multiphase LBM still need to improve its capability of enhancing these parameters, and extend LBM to more realistic scientific and technical applications further [28], [29].

Since the MRT-LBM was proposed, it has been used in many multiphase frameworks to give better numerical stability than SBGK method. A variety of researches realized the MRT can have a better performance by adjusting its relaxation times of energy mode, energy flux mode, and shear viscosity mode [8], [27]. In addition, Krüger et al. [30], [31] proved the second-order convergence of stress tensor in SBGK and gave the expression of viscosity stress tensor. Wang et al. [32] developed it in MRT to evaluate the shear rate term ($\mathrm{\nabla }\mathbf{u}+{\left(\mathrm{\nabla }\mathbf{u}\right)}^{\mathrm{T}}$) for the LBFS solution, and thus it provided the decoupling scheme of relaxation time and fluid viscosity for non-Newtonian power-law fluid. Although MRT has the satisfactory Galilean invariance at low speed and second-order accuracy, the MRT still remain two instability-sources. Firstly, it introduces some numerical-error terms like ${\partial }_x\left(u_x{\partial }_y\left(\rho u_x{u}_y\right)\right)$, which are inherited from the single-relaxation BGK method, for recovering the viscosity stress tensor in second-order expansion [33]. This error may cause numerical instability when it encounters the multiphase flow with a large density ratio. Secondly, the relaxation times of viscosity modes are used to control the bulk and shear viscosities, which means the MRT must use the extremely unstable over-relaxation of viscosity modes to produce low viscosity in most computation domain. If the viscosity in MRT can be introduced independently, the further optimization solution of numerical stability is possibly achieved by adjusting the local relaxation times of viscosity modes. Therefore, this article seeks to show a decoupled MRT LBM and the optimization scheme for better numerical stability in processing dynamic multiphase flow at high Reynolds and Weber numbers. A simpler force scheme is also proposed for the corresponding decoupled MRT framework. This new model still stays in conventional MRT framework and keeps the advantages of simplicity and efficiency of LBM. Moreover, the dynamic droplet collision, splashing and impacting are conducted to verify its stability performance to handle with the multiphase flow at low viscosity and surface tension. The pseudopotential model is employed for phase separation.

The basic MRT framework is summarized in Sec. 2. The decoupled scheme and the generic stabilization strategies are developed in Sec. 3. Sec. 4 shows several benchmarks to validate the accuracy, Galilean invariance, force scheme, mass and momentum conservations, and surface tension. In Sec. 4.1, the coexistence curve is shown to confirm the simple force term is still consistent with the previous research. In Sec. 4.2, Taylor-Green vortex flow is conducted to illustrate its second-order numerical accuracy, and it also proves the decoupled viscosity won't change with relaxation times. Galilean invariance and isotropy are also studied. Sec. 4.3 simulates the shear wave flow on a moving frame to research the Galilean invariance and numerical error at high Mach number. Sec. 4.4 shows the numerical accuracy of the decoupled MRT by the steady Poiseuille flow with body force. The surface tension is measured and compared by the Laplace's law in Sec. 4.5. Sec. 5 presents its capability to simulate dynamic complex-interface multiphase flows of droplet collision, droplet splashing, and droplet impacting on dry wall. Sec. 6 is the conclusion.