A Three-Dimensional Hybrid Spectral Element-Fourier Spectral Method for Wall-Bounded Two-Phase Flows

TL;DR

This paper introduces a hybrid spectral element-Fourier spectral method for simulating wall-bounded two-phase flows in three dimensions, efficiently handling variable densities and viscosities by transforming 3D problems into decoupled 2D computations.

Contribution

The paper develops a novel hybrid spectral method that combines spectral element and Fourier expansions, enabling efficient 3D two-phase flow simulations with variable properties and wall boundaries.

Findings

Method accurately simulates two-phase flows with moving contact lines.

Transforming 3D problems into 2D planes reduces computational complexity.

Numerical results demonstrate high accuracy and efficiency.

Abstract

We present a hybrid spectral element-Fourier spectral method for solving the coupled system of Navier-Stokes and Cahn-Hilliard equations to simulate wall-bounded two-phase flows in a three-dimensional domain which is homogeneous in at least one direction. Fourier spectral expansions are employed along the homogeneous direction and $C^0$ high-order spectral element expansions are employed in the other directions. A critical component of the method is a strategy we developed in a previous work for dealing with the variable density/viscosity of the two-phase mixture, which makes the efficient use of Fourier expansions in the current work possible for two-phase flows with different densities and viscosities for the two fluids. The attractive feature of the presented method lies in that the two-phase computations in the three-dimensional space are transformed into a set of de-coupled two-dimensional computations in the planes of the non-homogeneous directions. The overall scheme consists of solving a set of de-coupled two-dimensional equations for the flow and phase-field variables in these planes. The linear algebraic systems for these two-dimensional equations have constant coefficient matrices that need to be computed only once and can be pre-computed. We present ample numerical simulations for different cases to demonstrate the accuracy and capability of the presented method in simulating the class of two-phase problems involving solid walls and moving contact lines.