An effective correction method for droplet volume conservation in direct numerical simulation of droplet-laden turbulence

TL;DR

This work addresses persistent droplet-volume loss in phase-field simulations of droplet-laden turbulence by first evaluating representative volume-correction models, which fail at high $We_d$. It then introduces a curvature-dependent counter-diffusion term within the conservative Allen-Cahn framework, yielding statistical droplet-volume conservation without inducing artificial coarsening or numerical instability. Validation in direct numerical simulations of droplet-laden homogeneous isotropic turbulence demonstrates robust volume preservation across a broad range of Weber numbers and density ratios, with preserved interfacial area and Hinze-scale droplet-size distributions and only weak spurious currents. By tying the correction to Hinze’s law, the method remains inactive in laminar cases and scalable to turbulent regimes, providing a practical and efficient path to long-time, high-$We_d$ phase-field DNS of multiphase turbulence.

Abstract

Accurately preserving the volume of the dispersed droplets remains a significant challenge in phase-field simulations of droplet-laden turbulence, especially under conditions that feature strong interfacial deformation and breakup. While modified phase-field equations have been developed to mitigate volume loss, their effectiveness has not been systematically assessed in the context of fully developed turbulent flows. In this work, we first evaluate the performance of several representative volume-corrected phase-field models in direct numerical simulations of droplet-laden homogeneous isotropic turbulence. Our results reveal that, at sufficiently high Weber numbers, none of the existing models provides satisfactory droplet-volume preservation. To address this limitation, we then propose a simple yet effective modification of the conservataive Allen-Cahn equation by incorporating a curvature-dependent counter-diffusion correction. Direct numerical simulations in turbulent regimes demonstrate that the proposed model achieves conservation of droplet volume in a statistical sense, while avoiding common adverse effects, such as numerical instability, violation of global mass conservation, increased computational cost, artificial coarsening, or enhanced spurious velocities.

Figures (14)

• Figure 1: Schematic of a droplet in homogeneous isotropic turbulence.
• Figure 2: The evolution of droplet volume for CAC, CH, CH-PC, CH-FC, CH-IC, and our modified CAC models
• Figure 3: Visualizations of the two-phase interface at time step $=50,000$ at ${\rm We}_{\rm d} = 20$. The colorbar in each subplot shows the velocity magnitude in the lattice unit.
• Figure 4: Comparison of droplet volume evolution under different tunable parameters for the CH-PC, CH-FC, and CH-IC models.
• Figure 5: The evolution of droplet volume for the two modified CH models with singular mobility, CH-B and CH-Y.