A diffuse-interface method for the containerless freezing of three-phase flows in complex geometries

TL;DR

This work presents a unified diffuse-interface framework for three-phase freezing in complex geometries by coupling a gas–liquid phase field, a solid–liquid enthalpy-based energy equation, and Navier–Stokes flow, with an embedded wettability energy that avoids explicit boundary condition discretization on irregular boundaries. An LB solver triplet for flow, phase field, and temperature (enthalpy) dynamics is developed, incorporating a mass source term to capture volume changes due to density differences during solidification and a Wettability-enabled energy functional $\mathcal{F}_\phi$ to govern gas–liquid–solid interfaces. The method is validated on conduction freezing, three-phase Stefan problems, and droplet freezing on flat and curved substrates, showing good agreement with analytical, experimental, and theoretical results, and is extended to freezing in rough fractures and unsaturated porous media. The approach provides an efficient, robust tool for simulating containerless freezing in complex geometries with realistic boundary effects, with potential applications in energy, geoscience, and materials processing where phase-change in heterogeneous media is critical.

Abstract

In this work, we first propose a diffuse-interface model for the freezing processes of three-phase flows in complex geometries, and the core of the model to intergratge the Navier-Stokes equations for fluid flows, a modified phase-field equation for gas-liquid interfaces, and an enthalpy approach for solid-liquid phase-change processes in a unified diffuse-interface framework. The volume expansion or shrinkage of the liquid phase caused by the density change during the phase-change process is considered by introducing a mass source term into the continuity equation. The wettability effect in such a gas-liquid-solid multiphase system is also included in the phase-field free energy, thereby avoiding the direct discretization of wetting boundary condition on the complex fluid-solid boundary. Then, we develop a mesoscopic lattice Boltzmann (LB) method to solve the diffuse-interface model for the freezing processes in multiphase systems, and test the accuracy and efficiency of the LB method through some benchmark problems, including the conduction-induced freezing in a semi-infinite space, the three-phase Stefan problem, the droplet solidification on the flat and curved surfaces. It is found that the numerical results are in good agreement with the experimental data and theoretical solutions. Finally, the LB method is further extended to study the freezing dynamics of multiphase flows in a fracture and porous medium, and the numerical results show that the developed method is efficient in the study of freezing processes of multiphase flows in complex geometries.

Figures (14)

• Figure 1: Schematic of three-phase freezing process in a complex geometry. (a) The gas-liquid-solid system with complex interfaces: the regions denoted by $\Omega_g$ (white region, $\phi=-1 \cap f_s=0 \cap \phi_0=0$), $\Omega_l$ (yellow region, $\phi=1 \cap f_s=0 \cap \phi_0=0$), and $\Omega_s$ (blue region, $\phi=1 \cap f_s=1 \cap \phi_0=0$) are filled with gas, liquid and solid phases. The complex region occupied by another solid phase is denoted as $\Omega_0$ (grey region, $\phi=0 \cap f_s=1 \cap \phi_0=1$). $\Gamma_{sl}$ indicates the freezing front between the solid and liquid phases, and $\Gamma_{mg}$ represents the interface between the phase-change material and gas phase. (b) The moving contact lines in gas–liquid–solid system, where $\theta$ is the contact angle between the liquid phase and pre-existing solid phase, while $\psi$ is the contact angle between the liquid phase and solid phase formed by freezing.
• Figure 2: Distribution of bulk free energy density $f_b\left(\phi, \phi_0\right)=\frac{3 \sigma}{4 \varepsilon}(1-\phi)^2(1+\phi)^2+\frac{9 \sigma}{2 \varepsilon} \phi_0^2 \phi^2$ with $\sigma=0.02$ and $\varepsilon=3$.
• Figure 3: Schematic of the one-phase freezing problem in the semi-infinite space (a). Comparisons of the temperature distribution (b) and the solid-liquid interface evolution (c) between the numerical results and analytical solutions under different solid-liquid density ratios $\rho_s/ \rho_l$.
• Figure 4: Schematic diagram of the two-phase freezing by the conduction (a). Comparisons of the temperature distribution $T$ (b) and the solid-liquid interface evolution (c) between the numerical results and the analytical solutions under different values of thermal conductivity ratio $\lambda_s/ \lambda_l$.
• Figure 5: The schematic of the three-phase Stefan problem (a), from left to right, shows the initial stage before freezing, the intermediate stage where part of the liquid phase has been frozen into the solid phase, and the final moment when the freezing is complete. The evolutions of the freezing front position $h(t)$ under different values of solid-liquid density ratio $\rho_s/\rho_l$ (b).