Fully implicit and accurate treatment of jump conditions for two-phase incompressible Navier-Stokes equation

TL;DR

This paper introduces a fully implicit numerical method for two-phase incompressible Navier-Stokes equations that accurately captures jump discontinuities in stress and material properties, ensuring sharp interface representation.

Contribution

The method uniquely handles all components of jump conditions implicitly, using a linear combination of singular forces and derivatives, and demonstrates convergence even with non-smooth velocities and high density ratios.

Findings

Captures jump discontinuities sharply without neglecting components.

Converges in $L^inity$ norms despite non-smooth velocities and pressures.

Effective for large density ratios in realistic simulations.

Abstract

We present a numerical method for two-phase incompressible Navier-Stokes equation with jump discontinuity in the normal component of the stress tensor and in the material properties. Although the proposed method is only first-order accurate, it does capture discontinuity sharply, not neglecting nor omitting any component of the jump condition. Discontinuities in velocity gradient and pressure are expressed using a linear combination of singular force and tangential derivatives of velocities to handle jump conditions in a fully implicit manner. The linear system for the divergence of the stress tensor is constructed in the framework of the ghost fluid method, and the resulting saddle-point system is solved via an iterative procedure. Numerical results support the inference that the proposed method converges in $L^\infty$ norms even when velocities and pressures are not smooth across the interface and can handle a large density ratio that is likely to appear in a real-world simulation.