Thermodynamically consistent semi-compressible fluids: a variational perspective

TL;DR

This paper develops a unified, thermodynamically consistent variational framework for semi-compressible fluids, including Boussinesq, anelastic, and pseudoincompressible models, applicable to diverse geometries and thermodynamic conditions.

Contribution

It introduces a unified variational formulation for semi-compressible fluids that ensures thermodynamic consistency and extends existing models to arbitrary manifolds and thermodynamic potentials.

Findings

Unified variational formulations for Boussinesq, anelastic, and pseudoincompressible fluids.

Models obey the 1st and 2nd laws of thermodynamics by design.

Derived elliptic equations for reversible and some irreversible dynamics.

Abstract

This paper presents (Lagrangian) variational formulations for single and multicomponent semi-compressible fluids with both reversible (entropy-conserving) and irreversible (entropy-generating) processes. Semi-compressible fluids are useful in describing \textit{low-Mach} dynamics, since they are \textit{soundproof}. These models find wide use in many areas of fluid dynamics, including both geophysical and astrophysical fluid dynamics. Specifically, the Boussinesq, anelastic and pseudoincompressible equations are developed through a unified treatment valid for arbitrary Riemannian manifolds, thermodynamic potentials and geopotentials. By design, these formulations obey the 1st and 2nd laws of thermodynamics, ensuring their thermodynamic consistency. This general approach extends and unifies existing work, and helps clarify the thermodynamics of semi-compressible fluids. To further this goal, evolution equations are presented for a wide range of thermodynamic variables: entropy density $s$, specific entropy $\eta$, buoyancy $b$, temperature $T$, potential temperature $\theta$ and a generic entropic variable $\chi$; along with a general definition of buoyancy valid for all three semicompressible models and arbitrary geopotentials. Finally, the elliptic equation is developed for all three equation sets in the case of reversible dynamics, and for the Boussinesq/anelastic equations in the case of irreversible dynamics; and some discussion is given of the difficulty in formulating the elliptic equation for the pseudoincompressible equations with irreversible dynamics.