Application of discrete mechanics model to jump conditions in two-phase flows

TL;DR

This paper introduces a discrete mechanics model as an alternative to Navier-Stokes equations for two-phase flows, offering improved handling of surface discontinuities and jump conditions without sacrificing accuracy.

Contribution

It develops a novel discrete mechanics framework that modifies jump condition treatment in two-phase flows, enhancing the modeling of surface discontinuities compared to traditional methods.

Findings

Discrete mechanics yields solutions consistent with Navier-Stokes for constant properties.

The new formulation improves the treatment of jump conditions in two-phase flows.

Method maintains accuracy while better handling surface discontinuities.

Abstract

Discrete mechanics is presented as an alternative to the equations of fluid mechanics, in particular to the Navier-Stokes equation. The derivation of the discrete equation of motion is built from the intuitions of Galileo, the principles of Galilean equivalence and relativity. Other more recent concepts such as the equivalence between mass and energy and the Helmholtz-Hodge decomposition complete the formal framework used to write a fundamental law of motion such as the conservation of accelerations, the intrinsic acceleration of the material medium, and the sum of the accelerations applied to it. The two scalar and vector potentials of the acceleration resulting from the decomposition into two contributions, to curl-free and to divergence-free, represent the energies per unit of mass of compression and shear. The solutions obtained by the incompressible Navier-Stokes equation and the discrete equation of motion are the same, with constant physical properties. This new formulation of the equation of motion makes it possible to significantly modify the treatment of surface discontinuities, thanks to the intrinsic properties established from the outset for a discrete geometrical description directly linked to the decomposition of acceleration. The treatment of the jump conditions of density, viscosity and capillary pressure is explained in order to understand the two-phase flows. The choice of the examples retained, mainly of the exact solutions of the continuous equations, serves to show that the treatment of the conditions of jumps does not affect the precision of the method of resolution.