Quasi-conservation laws for compressible 3D Navier-Stokes flow

TL;DR

This paper introduces a quasi-Lagrangian framework for analyzing the transport of mass density and vorticity projection in 3D compressible Navier-Stokes flows, revealing that density level sets are impermeable to certain vorticity transport.

Contribution

It formulates a novel quasi-Lagrangian perspective on compressible flow dynamics, showing that density level sets act as barriers to vorticity projection transport.

Findings

Level sets of density are impermeable to the transport of the vorticity projection q.

The quasi-Lagrangian formulation applies broadly to ideal gases and arbitrary equations of state.

Transport dynamics of mass density and vorticity projection are constrained by these quasi-conservation laws.

Abstract

We formulate the quasi-Lagrangian fluid transport dynamics of mass density $\rho$ and the projection $q=\bom\cdot\nabla\rho$ of the vorticity $\bom$ onto the density gradient, as determined by the 3D compressible Navier-Stokes equations for an ideal gas, although the results apply for an arbitrary equation of state. It turns out that the quasi-Lagrangian transport of $q$ cannot cross a level set of $\rho$. That is, in this formulation, level sets of $\rho$ (isopychnals) are impermeable to the transport of the projection $q$.