Conservation laws of inviscid non-isentropic compressible fluid flow in n>1 spatial dimensions

TL;DR

This paper extends the classification of conservation laws in compressible fluid flow to non-isentropic cases, revealing new conserved quantities and the conditions under which classical vorticity integrals are preserved.

Contribution

It identifies new kinematic conserved integrals for specific non-isentropic equations of state and clarifies the conditions for vorticity conservation in higher dimensions.

Findings

Additional kinematic conserved integrals exist for specific non-isentropic equations of state.

Vorticity conserved integrals are limited to circulatory entropy, vanishing in irrotational flows.

Helicity and enstrophy are not conserved in non-isentropic flows in any dimension.

Abstract

Recent work giving a classification of kinematic and vorticity conservation laws of compressible fluid flow for barotropic equations of state (where pressure is a function only of the fluid density) in $n>1$ spatial dimensions is extended to general non-isentropic equations of state in which the pressure is also a function of the dynamical entropy (per unit mass) of the fluid. Two main results are obtained. First, we find that apart from the familiar conserved integrals for mass, momentum, energy, angular momentum and Galilean momentum, and volumetric entropy, additional kinematic conserved integrals arise only for non-isentropic equations of state given by a generalized form of the well-known polytropic equation of state with dimension-dependent exponent $\gamma=1+2/n$, such that the proportionality coefficient is an arbitrary function of the entropy (per unit mass). Second, we show that the only vorticity conserved integrals consist of a circulatory entropy (which vanishes precisely when the fluid flow is irrotational) in all even dimensions. In particular, the vorticity integrals for helicity in odd dimensions and enstrophy in even dimensions are found to be no longer conserved for any non-isentropic equation of state.