Lattice Boltzmann models for the hydrodynamic equations in multiphase flow with high density ratio

TL;DR

This paper tackles the challenge of simulating multiphase flows with high density ratios using lattice Boltzmann methods by developing a P–u LB scheme derived from a lattice kinetic scheme (LKS) mapped to LB with a filter collision operator. The approach directly solves velocity and pressure fields in a compatible discrete space, mitigating high-order truncation errors that plague traditional LB formulations and reduce dependence on absolute pressure, while preserving momentum transfer across interfaces. Validation across static and dynamic droplets, Poiseuille flow, droplet collisions, bubble rising, and dam-break scenarios demonstrates improved Galilean invariance and robustness, with notably more accurate airflow fields induced by water motion. These improvements enhance the practicality of LB-based multiphase simulations at high density ratios and offer a path toward more reliable engineering predictions of interfacial flows.

Abstract

Multiphase flows with high density ratios, such as water and air flows, have recently been simulated using the lattice Boltzmann (LB) method. This approach corresponds to solving the phase field equations, such as the Cahn-Hilliard and Allen-Cahn equations, and the hydrodynamic equations, typically the Navier-Stokes and pressure equations for pseudo-incompressible fluids. Due to the high density ratio, the higher-order numerical truncation errors associated with spatial density gradients can become significant. These errors can lead to problems such as inaccuracies in shear stress, violations of Galilean invariance, and undesirable dependencies on absolute pressure for the pseudo-incompressible solutions. To overcome such problems, the moments of the distribution function and the equilibrium state must be carefully designed while ensuring robustness. In this work, we propose a new scheme based on the lattice kinetic scheme (LKS), which directly solves the velocity and pressure fields in the similar discrete space as the LB method. When mapping the LKS-based models to the LB models, the original LKS models are simplified for computational efficiency and the filter collision operator is implemented. Benchmark test cases confirm that the proposed scheme effectively addresses these issues, achieving high accuracy and robustness while eliminating the iterative steps typically required in the LKS. One of the most significant improvements is the accuracy of the airflow field induced by water motion, likely due to improved momentum transfer across the interface.

Figures (18)

• Figure 1: Velocity profiles of the two-layer Poiseuille flow using the original $P \hbox{-} \rho u$ scheme fakhari2016mass and the $P \hbox{-} \rho u$ scheme with higher order corrections otomo2019improved are plotted with the square and cross symbols, respectively. The analytical solution is represented by a line.
• Figure 2: The order parameter $\varphi$ profiles at 3.20$\times 10^4$ (left) and 3.36$\times 10^4$ (right) timesteps using the $P \hbox{-} \rho u$ scheme fakhari2017improved for a case where velocity of 0.025 in the $x$-direction is initially applied everywhere. Similarly, the pressure and velocity in the $x$-direction are shown at the center and bottom.
• Figure 3: The order parameter $\varphi$ profiles using the $P/\rho \hbox{-} u$ model fakhari2017improved where the initial pressure is set to 0.01 (left) or 1.0(right) everywhere.
• Figure 4: Pressure difference across the interface, $dP$, vs. inverted droplet radius, $1/R$, compared to solid lines following Laplace's law when $\sigma=1.0\times 10^{-2}$ and $\sigma=1.0\times 10^{-3}$. Two sets of viscosities are applied, $\nu_\mathrm{set1} = \left\{ \nu_\mathrm{air}, \nu_\mathrm{water} \right\}= \left\{ 1.67\times10^{-1}, 1.10\times10^{-2} \right\}$, $\nu_\mathrm{set2} = \left\{ \nu_\mathrm{air}, \nu_\mathrm{water} \right\}= \left\{ 5.56\times10^{-3}, 3.67\times10^{-4} \right\}$.
• Figure 5: Profiles of $\varphi$ (top), $P$ (middle), and $u_x$ (bottom) at 3.20$\times 10^4$ (left) and 3.36$\times 10^4$ (right) timesteps using the LB model in Sec. \ref{['sec_formulation']} where the initial homogeneous velocity in the $x$-direction of 0.025 is applied.