Compressible and immiscible fluids with arbitrary density ratio

TL;DR

This paper introduces a new theoretical model for simulating immiscible fluids with arbitrary density ratios, overcoming limitations of traditional Navier-Stokes and Euler equations, based on energy minimization principles.

Contribution

The authors develop a novel density evolution equation for immiscible fluids that generalizes Bernoulli's principle and applies to high density ratio systems, expanding CFD capabilities.

Findings

Model applicable to fluids with high density ratios

Provides a generalized energy conservation framework

Enables simulation of both compressible and incompressible fluids

Abstract

For the water-air system, the bulk density ratio is as high as about 1000; no model can fully tackle such a high density ratio system. In the Navier-Stokes and Euler equations, the density $\rho$ within the water-air interface is assumed to be a constant based on the Boussinesq approximation namely $\rho (\mathrm{d} \mathbf u/\mathrm{d} t)$, which does not account for the true momentum evolution $\mathrm{d} (\rho \mathbf u)/\mathrm{d} t$ ($\mathbf u$-fluid velocity). Here, we present an alternative theory for the density evolution equations of immiscible fluids in computational fluid dynamics, differing from the concept of Navier-Stokes and Euler equations. Our derivation is built upon the physical principle of energy minimization from the aspect of thermodynamics. The present results provide a generalization of Bernoulli's principle for energy conservation and a general formulation for the sound speed. The present model can be applied for immiscible fluids with arbitrarily high density ratios, thereby, opening a new window for computational fluid dynamics both for compressible and incompressible fluids.