Pressure-equilibrium-preserving and fully conservative discretization of compressible flow equations for real and thermally perfect gases

TL;DR

This paper introduces a novel numerical discretization method for compressible flow equations that preserves pressure equilibrium and total energy conservation simultaneously, applicable to real and thermally perfect gases.

Contribution

It is the first method to discretely preserve full conservation of linear invariants while enforcing pressure equilibrium exactly in compressible flow simulations.

Findings

Method effectively reduces spurious pressure oscillations.

Applicable to real and thermally perfect gases with arbitrary equations of state.

Shows excellent performance in supercritical and transcritical flow simulations.

Abstract

Numerical simulations of compressible real-fluid flows are notoriously plagued by spurious pressure oscillations arising in regions of abrupt flow variations. As a possible remedy, several numerical formulations enforce the pressure equilibrium condition for the compressible Euler equations, typically at the cost of spoiling the correct conservation of total energy or by overspecifying the thermodynamical variables. This study proposes for the first time a numerical discretization procedure which is able to discretely preserve the full conservation of the linear invariants (mass, momentum and total energy) and to exactly enforce the pressure equilibrium condition. The method also preserves the conservation of kinetic energy by convection, and is based on the specification of nonlinear numerical fluxes for mass and internal energy which depend on the details of the equation of state. Both thermally perfect and real gases with an arbitrary equation of state are considered, and a simplified approximate pressure equilibrium preserving formulation with excellent performances is also proposed. The effectiveness of the novel formulations is assessed through a series of numerical simulations in supercritical and transcritical conditions with some of the most popular cubic equations of state.