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A fully-decoupled second-order-in-time and unconditionally energy stable scheme for a phase-field model of two phase flow with variable density

Jinpeng Zhang, Li Luo, Xiaoping Wang

Abstract

In this paper, we develop a second-order, fully decoupled, and energy-stable numerical scheme for the Cahn-Hilliard-Navier-Stokes model for two phase flow with variable density and viscosity. We propose a new decoupling Constant Scalar Auxiliary Variable (D-CSAV) method which is easy to generalize to schemes with high order accuracy in time. The method is designed using the "zero-energy-contribution" property while maintaining conservative time discretization for the "non-zero-energy-contribution" terms. Our algorithm simplifies to solving three independent linear elliptic systems per time step, two of them with constant coefficients. The update of all scalar auxiliary variables is explicit and decoupled from solving the phase field variable and velocity field. We rigorously prove unconditional energy stability of the scheme and perform extensive benchmark simulations to demonstrate accuracy and efficiency of the method.

A fully-decoupled second-order-in-time and unconditionally energy stable scheme for a phase-field model of two phase flow with variable density

Abstract

In this paper, we develop a second-order, fully decoupled, and energy-stable numerical scheme for the Cahn-Hilliard-Navier-Stokes model for two phase flow with variable density and viscosity. We propose a new decoupling Constant Scalar Auxiliary Variable (D-CSAV) method which is easy to generalize to schemes with high order accuracy in time. The method is designed using the "zero-energy-contribution" property while maintaining conservative time discretization for the "non-zero-energy-contribution" terms. Our algorithm simplifies to solving three independent linear elliptic systems per time step, two of them with constant coefficients. The update of all scalar auxiliary variables is explicit and decoupled from solving the phase field variable and velocity field. We rigorously prove unconditional energy stability of the scheme and perform extensive benchmark simulations to demonstrate accuracy and efficiency of the method.

Paper Structure

This paper contains 14 sections, 65 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. The model and discretization
  3. The model
  4. Numerical schemes
  5. The modified system
  6. Numerical schemes
  7. The first-order scheme
  8. The second-order scheme
  9. Numerical results
  10. Accuracy and stability test
  11. Bubble rising
  12. Bubble merging
  13. Rayleigh–Taylor instability
  14. Conclusions

Figures (13)

  • Figure 1: First-order temporal convergence tests: $L^{2}$ errors at $t=0.64$ for the phase-field variable $\phi$, the average of the two velocity components $(u,v)$, and the pressure $p$.
  • Figure 2: Second-order temporal convergence tests: (a) $L^{2}$ errors of the phase-field variable $\phi$, the average of the two velocity components $(u,v)$, and the pressure $p$ at $t=0.64$. (b) $L^{2}$ errors of the scalar variables $r, Q, R, T, K$ at $t=0.64$.
  • Figure 3: (a) Evolution of the original energy \ref{['EO']} computed using $\delta t=10^{-5}$ and the modified total free energy \ref{['EM_sed']} computed using different time step sizes. (b) Evolution of the volume $\int_{\Omega}\phi dx$.
  • Figure 4: (a) Shapes of the rising bubble at different times for Case 1. (b) The shape and velocity vector distribution of the rising bubble at final time $t=3$ for Case 1.
  • Figure 5: Shapes of the rising bubble and the velocity vector distribution at different times for Case 2.
  • ...and 8 more figures