[Paper Review] A consistent and conservative Phase-Field model for thermo-gas-liquid-solid flows including liquid-solid phase change

Ziyang Huang, Guang Lin+1 more|arXiv (Cornell University)|Feb 13, 2021

Solidification and crystal growth phenomena103 references44 citations

TL;DR

This paper proposes a consistent and conservative phase-field model for thermo-gas-liquid-solid flows with liquid-solid phase change, uniquely capturing volume change during phase transition through unambiguous phase volume fractions. The model enforces mass, momentum, and energy conservation, introduces a localized surface tension force and a modified Carman-Kozeny drag to suppress solid-phase velocity, and demonstrates accurate agreement with analytical and experimental data across complex, high-property-ratio flows.

ABSTRACT

In the present study, a consistent and conservative Phase-Field model is developed to study thermo-gas-liquid-solid flows with liquid-solid phase change. The proposed model is derived with the help of the consistency conditions and exactly reduces to the consistent and conservative Phase-Field method for incompressible two-phase flows, the fictitious domain Brinkman penalization (FD/BP) method for fluid-structure interactions, and the Phase-Field model of solidification of pure material. It honors the mass conservation, defines the volume fractions of individual phases unambiguously, and therefore captures the volume change due to phase change. The momentum is conserved when the solid phase is absent, but it changes when the solid phase appears due to the no-slip condition at the solid boundary. The proposed model also conserves the energy, preserves the temperature equilibrium, and is Galilean invariant. A novel continuous surface tension force to confine its contribution at the gas-liquid interface and a drag force modified from the Carman-Kozeny equation to reduce solid velocity to zero are proposed. The issue of initiating phase change in the original Phase-Field model of solidification is addressed by physically modifying the interpolation function. The corresponding consistent scheme is developed to solve the model, and the numerical results agree well with the analytical solutions and the existing experimental and numerical data. Two challenging problems having a wide range of material properties and complex dynamics are conducted to demonstrate the capability of the proposed model.

Motivation & Objective

• To develop a consistent and conservative phase-field model for thermo-gas-liquid-solid flows including liquid-solid phase change.
• To resolve key issues in existing models, such as ambiguous phase volume fractions and spurious surface tension/drag forces.
• To ensure mass conservation during phase change by defining unambiguous volume fractions for each phase.
• To enforce zero velocity in the solid phase through a physically motivated drag force, avoiding reliance on unrealistically high viscosities.
• To validate the model against analytical solutions and experimental data, demonstrating accuracy in complex, high-property-ratio multiphase flows.

Proposed method

• Derives the model using consistency conditions, ensuring reduction to established methods: incompressible two-phase flow, FD/BP for fluid-structure interaction, and pure solidification phase-field models.
• Introduces a novel continuous surface tension force confined to the gas-liquid interface via a phase-field formulation.
• Proposes a modified Carman-Kozeny-based drag force to effectively suppress solid-phase velocity, avoiding numerical instability from high viscosities.
• Defines phase volume fractions unambiguously to conserve mass and capture volume changes during phase transition.
• Develops a consistent numerical scheme that preserves Galilean invariance, energy conservation, and temperature equilibrium.
• Uses a physically modified interpolation function to resolve the issue of phase change initiation in the original phase-field solidification model.

Experimental results

Research questions

• RQ1 Can a phase-field model consistently and conservatively model thermo-gas-liquid-solid flows with liquid-solid phase change while capturing volume changes?
• RQ2 How can surface tension forces be localized to the gas-liquid interface without affecting the gas-solid or liquid-solid interfaces?
• RQ3 What is an effective and stable method to enforce zero velocity in the solid phase without relying on unrealistically high solid viscosities?
• RQ4 How can phase change initiation be physically and consistently modeled in the phase-field framework?
• RQ5 To what extent does the proposed model reproduce analytical and experimental results in complex, high-property-ratio multiphase flows?

Key findings

• The proposed model accurately captures volume change during phase transition, with numerical mass conservation error remaining very small.
• The novel surface tension force successfully confines its effect to the gas-liquid interface, preventing unphysical forces at the gas-solid interface and enabling stable simulations.
• Increasing solid viscosity alone is insufficient to stop solid motion; the proposed drag force effectively enforces zero velocity in the solid phase.
• The modified Carman-Kozeny drag force produces stable and accurate results, with an alternative formulation showing similar performance but a broader effective region.
• The model achieves excellent agreement with analytical solutions for the Stefan problem and large-density-ratio advection, validating its consistency and conservation properties.
• Quantitative agreement with experimental and numerical data is achieved when the Gibbs-Thomson and linear kinetic coefficients (Γφ and µφ) are carefully calibrated, with the suggestion that they take the same value for optimal results.