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

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.

💡 Research Summary

This paper addresses a long‑standing difficulty in phase‑field simulations of droplet‑laden turbulence: the excessive loss of dispersed‑phase volume that occurs when droplets undergo strong deformation and repeated breakup, especially at high Weber numbers. Although several volume‑correction strategies have been proposed—global mass‑conservation terms, localized diffusion‑suppressing corrections, and mobility‑modulation techniques—most have only been tested in laminar or weakly turbulent flows. The authors first benchmark a representative set of these models (standard Cahn‑Hilliard, conservative Allen‑Cahn, CH‑PC, CH‑FC, CH‑IC, singular‑mobility variants CH‑B and CH‑Y, and a global correction CH‑G) in direct numerical simulations (DNS) of a single large droplet immersed in fully developed homogeneous isotropic turbulence (HIT) with Taylor‑microscale Reynolds number Reλ≈42. Simulations are performed with a lattice‑Boltzmann solver on a cubic domain (L=256Δx) using a droplet radius r0=30Δx and Weber numbers ranging from 10 to 50. The results show that, for Weber numbers above ≈30, all existing models suffer noticeable volume loss (0.5 %–3 %) despite preserving global mass of the order parameter. The global correction, in particular, induces artificial coarsening because it redistributes dissolved mass uniformly, which is physically unjustified for droplets of different sizes.

To overcome these shortcomings, the authors propose a simple modification of the conservative Allen‑Cahn (CAC) equation. They introduce a curvature‑dependent counter‑diffusion term that acts locally on the interface: a coefficient proportional to the magnitude of the curvature κ=∇·n is multiplied by the diffusion operator ∇·(·∇ϕ). This term counteracts the spurious diffusion that drives small, high‑curvature droplets to dissolve, effectively acting as a “precipitation‑like” mechanism that restores volume in a statistical sense while leaving the global mass balance untouched. The modified equation retains the original CAC structure, so no additional source terms are required, and the extra computational load is modest (<5 % of total runtime).

Extensive DNS confirm the efficacy of the new model. Across the tested Weber numbers, the curvature‑dependent correction maintains droplet volume within ±0.1 % of the initial value, essentially eliminating the systematic loss observed with other methods. Spurious velocities are reduced by 15–20 % relative to the baseline CAC, and the interface thickness remains close to the prescribed diffuse‑interface width (≈4Δx). Importantly, the method does not cause numerical instability, does not violate global mass conservation, and does not trigger artificial coarsening. The authors also demonstrate that the approach is computationally inexpensive, requiring only a simple additional divergence term.

The paper discusses the physical interpretation of the correction, its implementation details, and the parameter α that scales the curvature dependence. Sensitivity tests indicate that a modest α yields the best trade‑off between volume preservation and stability. Limitations are acknowledged: the study assumes constant density and viscosity ratios and a fixed interface thickness; extension to multiphysics problems (heat transfer, chemical reactions) will likely require retuning of α. Future work is outlined, including validation against experimental data, application to flows with varying material properties, and integration with higher‑order phase‑field formulations.

In summary, by embedding a curvature‑dependent counter‑diffusion term into the conservative Allen‑Cahn framework, the authors provide a robust, low‑cost solution to droplet‑volume loss in high‑Weber‑number turbulent multiphase simulations. This advancement paves the way for more reliable DNS of complex droplet‑laden turbulence in engineering and natural‑science applications.