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Thermodynamically consistent semi-compressible fluids: a variational perspective

Journal Article · · Journal of Physics. A, Mathematical and Theoretical
 [1];  [2]
  1. Sandia National Laboratories (SNL), Albuquerque, NM, and Livermore, CA (United States)
  2. Ecole Normale Superieure de Paris (France)
+ Show Author Affiliations

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 low-Mach dynamics, since they are 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 η, buoyancy b, temperature T, potential temperature θ and a generic entropic variable χ; along with a general definition of buoyancy valid for all three semicompressible models and arbitrary geopotentials. Finally, the elliptic equation for the pressure perturbation (the Lagrange multiplier that enforces semi-compressibility) 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 it for the pseudoincompressible equations with irreversible dynamics.

Research Organization:
Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Biological and Environmental Research (BER); French National Research Agency (ANR)
Grant/Contract Number:
NA0003525
OSTI ID:
1827611
Alternate ID(s):
OSTI ID: 23136553
Report Number(s):
SAND--2021-9814J; 700605
Journal Information:
Journal of Physics. A, Mathematical and Theoretical, Journal Name: Journal of Physics. A, Mathematical and Theoretical Journal Issue: 34 Vol. 54; ISSN 1751-8113
Publisher:
IOP PublishingCopyright Statement
Country of Publication:
United States
Language:
English

References (25)

Thermodynamic Consistency of a Pseudoincompressible Approximation for General Equations of State journal March 2012
Variational integrators for anelastic and pseudo-incompressible flows journal January 2019
Thermodynamics/Dynamics Coupling in Weakly Compressible Turbulent Stratified Fluids journal March 2012
Energy Conservation and Gravity Waves in Sound-Proof Treatments of Stellar Interiors. ii. Lagrangian Constrained Analysis journal August 2013
Geometric, variational discretization of continuum theories journal October 2011
A variational derivation of the thermodynamics of a moist atmosphere with rain process and its pseudoincompressible approximation journal September 2018
A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: Continuum systems journal January 2017
Variational integrator for the rotating shallow‐water equations on the sphere
  • Brecht, Rüdiger; Bauer, Werner; Bihlo, Alexander
  • Quarterly Journal of the Royal Meteorological Society, Vol. 145, Issue 720 https://doi.org/10.1002/qj.3477
journal March 2019
Improving the Anelastic Approximation journal June 1989
Semicompressible Ocean Dynamics journal January 2015
A Variational Finite Element Discretization of Compressible Flow journal November 2020
Thermodynamic Consistency of the Anelastic Approximation for a Moist Atmosphere journal August 2008
Scale Analysis of Deep and Shallow Convection in the Atmosphere journal March 1962
A Scale Analysis of Deep Moist Convection and Some Related Numerical Calculations journal October 1982
Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits journal January 1966
Variational formulations of sound-proof models: Variational Sound-Proof Models journal April 2014
An efficient method for the incompressible Navier–Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary journal September 2011
The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories journal July 1998
A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems journal January 2017
Dynamic Enthalpy, Conservative Temperature, and the Seawater Boussinesq Approximation journal February 2010
Semicompressible Ocean Thermodynamics and Boussinesq Energy Conservation journal April 2016
Variational discretization for rotating stratified fluids journal August 2013
Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations journal January 2018
From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective journal December 2018
Single and double generator bracket formulations of multicomponent fluids with irreversible processes journal September 2020

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97 MATHEMATICS AND COMPUTING
variational principle
irreversible thermodynamics
semi-compressible fluids
geometric mechanics