Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A simple, fully-discrete, unconditionally energy-stable method for the two-phase Navier-Stokes Cahn-Hilliard model with arbitrary density ratios

Aaron Brunk, Marco F. P. ten Eikelder

TL;DR

The paper tackles energy stability in NSCH models with non-matching densities by introducing a simple, fully discrete, monolithic scheme that remains energy-stable for large density contrasts. It achieves this through a provable energy-dissipation formulation based on a positive-density extension and an equivalent reformulation using mass-averaged velocity and a volume-fraction order parameter. The authors prove mass conservation and unconditional energy stability at the discrete level and validate the approach with phase separation, convergence, and rising-bubble tests, demonstrating robustness and accuracy. This framework enhances reliability for simulations of diffuse-interface two-phase flows with high density ratios and provides a foundation for future extensions to multi-phase systems and alternative free-energy models.

Abstract

The two-phase Navier-Stokes Cahn-Hilliard (NSCH) mixture model is a key framework for simulating multiphase flows with non-matching densities. Developing fully discrete, energy-stable schemes for this model remains challenging, due to the possible presence of negative densities. While various methods have been proposed, ensuring provable energy stability under phase-field modifications, like positive extensions of the density, remains an open problem. We propose a simple, fully discrete, energy-stable method for the NSCH mixture model that ensures stability with respect to the energy functional, where the density in the kinetic energy is positively extended. The method is based on an alternative but equivalent formulation using mass-averaged velocity and volume-fraction-based order parameters, simplifying implementation while preserving theoretical consistency. Numerical results demonstrate that the proposed scheme is robust, accurate, and stable for large density ratios, addressing key challenges in the discretization of NSCH models.

A simple, fully-discrete, unconditionally energy-stable method for the two-phase Navier-Stokes Cahn-Hilliard model with arbitrary density ratios

TL;DR

The paper tackles energy stability in NSCH models with non-matching densities by introducing a simple, fully discrete, monolithic scheme that remains energy-stable for large density contrasts. It achieves this through a provable energy-dissipation formulation based on a positive-density extension and an equivalent reformulation using mass-averaged velocity and a volume-fraction order parameter. The authors prove mass conservation and unconditional energy stability at the discrete level and validate the approach with phase separation, convergence, and rising-bubble tests, demonstrating robustness and accuracy. This framework enhances reliability for simulations of diffuse-interface two-phase flows with high density ratios and provides a foundation for future extensions to multi-phase systems and alternative free-energy models.

Abstract

The two-phase Navier-Stokes Cahn-Hilliard (NSCH) mixture model is a key framework for simulating multiphase flows with non-matching densities. Developing fully discrete, energy-stable schemes for this model remains challenging, due to the possible presence of negative densities. While various methods have been proposed, ensuring provable energy stability under phase-field modifications, like positive extensions of the density, remains an open problem. We propose a simple, fully discrete, energy-stable method for the NSCH mixture model that ensures stability with respect to the energy functional, where the density in the kinetic energy is positively extended. The method is based on an alternative but equivalent formulation using mass-averaged velocity and volume-fraction-based order parameters, simplifying implementation while preserving theoretical consistency. Numerical results demonstrate that the proposed scheme is robust, accurate, and stable for large density ratios, addressing key challenges in the discretization of NSCH models.

Paper Structure

This paper contains 13 sections, 3 theorems, 59 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Navier-Stokes Cahn-Hilliard model
  3. Governing equations and properties
  4. Equivalent alternative formulation
  5. Numerical scheme & structural properties
  6. Time discretization
  7. Fully-discrete methodology
  8. Numerical results
  9. Phase separation
  10. Convergence test
  11. Rising bubble test cases
  12. Summary and outlook
  13. Non-dimensional formulation

Key Result

Lemma 4

The energy evolution of eq:sys2 follows from a linear combination of eq:sys2: div-eq:sys2: chem with the weights: $p+g y\left\{\rho\right\}$, $\mathbf{v}$, $\mu+ gy[\![\rho]\!]$, $-\partial_t \phi$.

Figures (11)

  • Figure 1: Phase separation: Snapshots of the volume fraction $\phi$ for the density ratios $\rho_1:\rho_2\in\{10^0:10^3,10^0:10^2,10^0:10^1,10^1:10^0,10^2:10^0,10^3:10^0\}$ from top to bottom at the times $\{0.1,0.3,1,2\}$ from left to right.
  • Figure 2: Phase separation: Evolution of the energy $\widetilde{\mathcal{E}}$ and the mass conservation error for the density ratios $\rho_1:\rho_2\in\{10^0:10^3,10^0:10^2,10^0:10^1,10^1:10^0,10^2:10^0,10^3:10^0\}$. Energy difference between the symmetric results is of order $10^{-6}$.
  • Figure 3: Rising bubble: Schematic representation of the problem setup
  • Figure 4: Case 1. Visualization of solution at final time $t=3$. (a) Phase field, (b) zero level set of the phase-field.
  • Figure 5: Case 2. Visualization of solution at final time $t=3$. (a) Phase field, (b) zero level set of the phase-field.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Remark 1: Mixture velocity
  • Remark 2: Mobility
  • Remark 3: Conservative form
  • Lemma 4: Energy evolution
  • proof
  • Lemma 5: Energy-stable variation formulation
  • proof
  • Theorem 7: Structure-preserving properties
  • proof
  • Remark 8: Finite element function spaces