[Paper Review] Axisymmetric lattice Boltzmann model for multiphase flows with large density ratio

Hong Liang, Yang Li+2 more|arXiv (Cornell University)|Oct 22, 2018

Lattice Boltzmann Simulation Studies38 references4 citations

TL;DR

This paper presents a novel axisymmetric lattice Boltzmann model based on the Allen-Cahn phase-field method for simulating multiphase flows with large density ratios—previously unattainable in existing axisymmetric LB models. The model employs two separate lattice Boltzmann equations: one for interface tracking with a tailored source term and equilibrium distribution, and another for hydrodynamics using a forcing distribution function. It achieves high accuracy and low spurious velocities, with excellent agreement to analytical and experimental data across a wide range of density ratios.

ABSTRACT

In this paper, a novel lattice Boltzmann (LB) model based on the Allen-Cahn phase-field theory is proposed for simulating axisymmetric multiphase flows. The most striking feature of the model is that it enables to handle multiphase flows with large density ratio, which are unavailable in all previous axisymmetric LB models. The present model utilizes two LB evolution equations, one of which is used to solve fluid interface, and another is adopted to solve hydrodynamic properties. To simulate axisymmetric multiphase flows effectively, the appropriate source term and equilibrium distribution function are introduced into the LB equation for interface tracking, and simultaneously, a simple and efficient forcing distribution function is also delicately designed in the LB equation for hydrodynamic properties. Unlike many existing LB models, the source and forcing terms of the model arising from the axisymmetric effect include no additional gradients, and consequently, the present model contains only one non-local phase field variable, which in this regard is much simpler. We further conducted the Chapman-Enskog analysis to demonstrate the consistencies of our present MRT-LB model with the axisymmetric Allen-Cahn equation and hydrodynamic equations. A series of numerical examples, including static droplet, oscillation of a viscous droplet, breakup of a liquid thread, and bubble rising in a continuous phase, are used to test the performance of the proposed model. It is found that the present model can generate relatively small spurious velocities and can capture interfacial dynamics with higher accuracy than the previously improved axisymmetric LB model. Besides, it is also found that our present numerical results show excellent agreement with analytical solutions or available experimental data for a wide range of density ratios, which highlights the strengths of the proposed model.

Motivation & Objective

• To develop a lattice Boltzmann model capable of simulating axisymmetric multiphase flows with large density ratios, which remain challenging in existing models.
• To overcome the limitations of previous axisymmetric LB models that fail to handle large density contrasts due to numerical instabilities and inaccurate interfacial dynamics.
• To ensure numerical stability and accuracy by incorporating an advanced multiple-relaxation-time (MRT) collision model and minimizing non-local dependencies.
• To achieve consistent recovery of axisymmetric hydrodynamic and phase-field equations through Chapman-Enskog analysis.

Proposed method

• The model uses two distinct lattice Boltzmann equations: one for phase-field evolution (interface tracking) and another for hydrodynamic properties (fluid motion).
• A source term and equilibrium distribution function are specifically designed for the interface-tracking equation to account for axisymmetric effects without introducing additional gradients.
• A simple and efficient forcing distribution function is introduced into the hydrodynamic equation to model the axisymmetric effects, avoiding complex gradient terms.
• The multiple-relaxation-time (MRT) scheme is applied to the collision operator to enhance numerical stability and reduce viscosity-dependent artifacts.
• The model is derived via Chapman-Enskog analysis, demonstrating that it recovers the axisymmetric Cahn-Hilliard and Navier-Stokes equations under incompressible flow conditions.
• Hydrodynamic pressure is reconstructed from the zeroth moment of the distribution function using a derived expression involving relaxation parameters and velocity terms.

Experimental results

Research questions

• RQ1 Can a lattice Boltzmann model accurately simulate axisymmetric multiphase flows with large density ratios, which are outside the scope of previous axisymmetric LB models?
• RQ2 How can the axisymmetric effects be incorporated into the lattice Boltzmann framework without introducing additional gradients or non-local terms?
• RQ3 What is the impact of the MRT collision model on the numerical stability and accuracy of the interfacial dynamics in large-density-ratio flows?
• RQ4 To what extent does the model reduce spurious velocities compared to existing axisymmetric LB models?
• RQ5 How well does the model reproduce analytical solutions and experimental data across a wide range of density ratios?

Key findings

• The proposed model successfully simulates multiphase flows with large density ratios—previously unattainable in axisymmetric lattice Boltzmann models—demonstrating its novelty and capability.
• The model generates significantly smaller spurious velocities than the previously improved axisymmetric LB model, enhancing interfacial accuracy.
• Numerical results for static droplet, oscillating droplet, thread breakup, and bubble rising show excellent agreement with analytical solutions and experimental data across a wide range of density ratios.
• The Chapman-Enskog analysis confirms that the MRT-LB model recovers the correct axisymmetric hydrodynamic and phase-field equations under incompressible flow assumptions.
• The model’s design, which includes only one non-local phase field variable and avoids additional gradient terms, results in a simpler and more robust formulation compared to prior approaches.
• The pressure calculation method based on the zeroth moment of the distribution function provides consistent and accurate hydrodynamic pressure, validated through asymptotic analysis.