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

Cheng Peng, Xuming Li+8 more|arXiv (Cornell University)|Feb 3, 2026

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TL;DR

The paper reviews existing phase-field models for droplet-volume conservation in droplet-laden turbulence, finds limitations at high Weber numbers, and introduces a curvature-dependent counter-diffusion correction to the conservative Allen–Cahn equation that achieves statistical droplet-volume conservation without major drawbacks.

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.

Motivation & Objective

• Evaluate how representative volume-correction phase-field models perform in fully developed droplet-laden turbulence.
• Identify limitations of current approaches in preserving droplet volume at high Weber numbers.
• Develop a simple, efficient correction to conserve droplet volume in a statistical sense while maintaining stability and mass balance.
• Demonstrate the effectiveness of the proposed model through 3D DNS of droplet-laden homogeneous isotropic turbulence across a range of Weber numbers.

Proposed method

• Compare several phase-field models (CAC, CH, CH-PC, CH-FC, CH-IC, CH-B, CH-Y, CH-G) in direct numerical simulations of droplet-laden HIT.
• Use a lattice Boltzmann solver with a diffuse-interface formulation and fixed interface thickness to assess droplet-volume preservation.
• Introduce a curvature-dependent counter-diffusion correction term into the CAC equation: -a ∇·n, with a linked to turbulence via Hinze’s law.
• Propose a practical, parameterized relation a = α M_AC (φ_D − φ_C) / d_crit and embed it into the CAC correction, preserving mass globally and computational efficiency.
• Validate the modified CAC model in 3D HIT DNS over a range of Weber numbers and compare to existing models.

Experimental results

Research questions

• RQ1 Do existing volume-correction phase-field models preserve droplet volume in fully developed turbulent flows at high Weber numbers?
• RQ2 Can a curvature-dependent counter-diffusion term in the CAC equation achieve statistical droplet-volume conservation without compromising stability or causing artificial coarsening?
• RQ3 How should the correction coefficient a be chosen or adapted to turbulence (e.g., via Hinze’s law) to maintain effectiveness across different We and density ratios?
• RQ4 What are the comparative impacts of the proposed method on droplets of varying sizes within turbulent multiphase flows?

Key findings

• Most existing models lose significant droplet volume in turbulent conditions with high Weber numbers.
• Among tested CH-based models, CH-IC retains more volume than others, but most methods still suffer substantial dissolution or coarsening.
• A simple modified CAC equation with a curvature-dependent counter-diffusion term can conserve droplet volume statistically without inducing instability or artificial mass gain.
• The counter-diffusion strength a, scaled with Hinze-like criteria, activates for small, highly curved droplets and vanishes in laminar limits, maintaining stability.
• Numerical tests indicate the proposed method preserves overall droplet volume while avoiding excessive numerical diffusion, coarsening, or global mass imbalance.