A diffuse-interface model for N-phase flows with liquid-solid phase change

Abstract

In this work, we first propose a diffuse interface model for simulating N phase flows with solid liquid phase change. In this model, a phase field approach is adopted to capture multiphase fluid interfaces, and an enthalpy based formulation is used to describe the phase change. The volume changes resulting from density differences during phase change are incorporated by introducing a source term into the continuity equation. The method also satisfies the reduction consistent property, allowing it to rigorously degenerate to both the conservative phase field method for N phase flows and the classical enthalpy method for solid liquid phase change. Then a coupled lattice Boltzmann (LB) method is developed to solve this diffuse interface model. Some numerical tests, including film freezing, single droplet freezing, and compound droplet freezing are performed, and the results are in good agreement with the analytical solutions and data reported in the previous works. Furthermore, the proposed method is applied to study freezing dynamics of complex systems with insoluble impurities, capturing the interaction between the advancing freezing front and embedded impurities. It is found that the proposed diffuse interface method is accurate and efficient for studying N phase systems with phase change.

Figures (13)

• Figure 1: Schematic of the compound droplet freezing problem. (a) Initial state of a compound droplet on a solid substrate. (b) Schematic of compound droplet freezing on a cold substrate. The phases involved are as follows: the ambient fluid is denoted by $\Omega_3$ ($\phi_1=0$, $\phi_2=0$, $\phi_3=1$, $f_s=0$), the fluid 1 is represented by $\Omega_{1,l}$ ($\phi_1=1$, $\phi_2=0$, $\phi_3=0$, $f_s=0$), the frozen fluid 1 is denoted by $\Omega_{1,s}$ ($\phi_1=1$, $\phi_2=0$, $\phi_3=0$, $f_s=1$), the fluid 2 is represented by $\Omega_{2,l}$ ($\phi_1=0$, $\phi_2=1$, $\phi_3=0$, $f_s=0$), the frozen fluid 2 is denoted by $\Omega_{2,s}$ ($\phi_1=0$, $\phi_2=1$, $\phi_3=0$, $f_s=1$). $\Gamma_{sl}$ indicates the freezing front between the solid and liquid phases, and $\Gamma_{mg}$ represents the external interface between the compound droplet system and the surrounding ambient fluid.
• Figure 2: (a) Schematic diagram of the single-component liquid film freezing process. The phase-change fluid $\Omega_l$, the ambient fluid $\Omega_g$, and the solid phase $\Omega_s$ formed by freezing are represented by blue, white, and yellow regions. The freezing front $\Gamma_{sl}$ is indicated by red solid line, while the interface $\Gamma_{gl}$ between the phase-change fluid and the ambient fluid is labeled by blue solid line. The positions of these interfaces are denoted by $s(t)$ and $h(t)$, respectively. (b) Comparisons of solid-liquid and gas-liquid interface positions among the present method, reference numerical results HuangPRE2024HuangIJHMT2025 and analytical solutions LyuJCP2021. (c) Comparisons of the interfacial velocities predicted by the present method and reported in the previous work HuangPRE2024.
• Figure 3: (a) Schematic of freezing of a liquid pool composed of immiscible liquid fluids. Fluid 1 $(\Omega_{1,l})$, fluid 2 $(\Omega_{2,l})$, and the ambient fluid $(\Omega_{g})$ are represented by gray, blue, and white, respectively, while the solid phase $(\Omega_{s})$ formed by freezing is indicated in yellow. The functions $s(t)$, $h_2(t)$, and $h(t)$ denote the positions of the freezing front, the interface between fluid 1 and fluid 2, and the interface between the ambient fluid and fluid 1, respectively. Temporal evolutions of the solidification front $\Gamma_{sl}$, fluid-fluid interface $\Gamma_{12}$, and gas-liquid interface $\Gamma_{gl}$ under different density ratios (b) $\rho_1^s/ \rho_1^l=1$, $\rho_2^s/ \rho_2^l=1.0$ and (c) $\rho_1^s/ \rho_1^l=1$ and $\rho_2^s/ \rho_2^l=0.95$.
• Figure 4: (a) Schematic of a liquid pool freezing process. The liquid pool is composed of two immiscible liquid fluids ($\Omega_{1,l}$, $\Omega_{2,l}$) distributed on the left and right sides, where fluid 1 is marked in gray and fluid 2 in blue. The solid phase $\Omega_{s}$ formed by freezing is denoted by yellow. Comparisons of phase interfaces before and after freezing for (b) $\rho_1^s/ \rho_1^l=1$, $\rho_2^s/ \rho_2^l=1$, (c) $\rho_1^s/ \rho_1^l=1.05$, $\rho_2^s/ \rho_2^l=0.95$, (d) $\rho_1^s/ \rho_1^l=1.1$, $\rho_2^s/ \rho_2^l=1.05$ and (e) $\rho_1^s/ \rho_1^l=1.1$, $\rho_2^s/ \rho_2^l=1$, where the gray line is the initial phase interface contour, and the red line and the black dashed line are the freezing front and the two phase interfaces after freezing.
• Figure 5: The evolutions of normalized phase volumes during the freezing processes under different density ratios (a) $\rho_1^s/ \rho_1^l=1$, $\rho_2^s/ \rho_2^l=1$, (b) $\rho_1^s/ \rho_1^l=1.05$, $\rho_2^s/ \rho_2^l=0.95$, (c) $\rho_1^s/ \rho_1^l=1.1$, $\rho_2^s/ \rho_2^l=1.05$ and (d) $\rho_1^s/ \rho_1^l=1.1$, $\rho_2^s/ \rho_2^l=1$. $V_s^p$ is the volume of the solid phase formed by the freezing of fluid $p$, and $V_{sl}^p$ is the total volume of the solid and liquid phases of fluid $p$, which are both normalized by the initial volume.