A Parabolic Relaxation Model for the Navier-Stokes-Korteweg Equations

TL;DR

This paper introduces a parabolic relaxation model for the Navier-Stokes-Korteweg equations that simplifies numerical solutions of two-phase flow by avoiding higher-order derivatives and ensuring hyperbolicity, validated through various simulations.

Contribution

A lower-order parabolic relaxation model for the NSK equations that is compatible with standard flow solvers and validated with 1D, 2D, and 3D test cases.

Findings

The relaxation model accurately approximates the classical NSK solutions.

The approach enables efficient 3D simulations of droplet collisions.

The modified pressure function maintains hyperbolicity of the fluxes.

Abstract

The isothermal Navier-Stokes-Korteweg system is a classical diffuse interface model for compressible two-phase flow. However, the numerical solution faces two major challenges: due to a third-order dispersion contribution in the momentum equations, extended numerical stencils are required for the flux calculation. Furthermore, the equation of state given by a Van-der-Waals law, exhibits non-monotone behaviour in the two-phase state space leading to imaginary eigenvalues of the Jacobian of the first-order fluxes. In this work a lower-order parabolic relaxation model is used to approximate solutions of the classical NSK equations. Whereas an additional parabolic evolution equation for the relaxation variable has to be solved, the system involves no differential operator of higher as second order. The use of a modified pressure function guarantees that the first-order fluxes remain hyperbolic. Altogether, the relaxation system is directly accessible for standard compressible flow solvers. We use the higher-order Discontinuous Galerkin spectral element method as realized in the open source code FLEXI. The relaxation model is validated against solutions of the original NSK model for a variety of 1D and 2D test cases. Three-dimensional simulations of head-on droplet collisions for a range of different collision Weber numbers underline the capability of the approach.