Variational and thermodynamically consistent finite element discretization for heat conducting viscous fluids

TL;DR

This paper introduces a finite element method that respects thermodynamic laws for simulating heat conducting viscous fluids, ensuring physically accurate results by preserving energy, mass, and entropy principles at discrete levels.

Contribution

It develops a variational, thermodynamically consistent discretization framework for fluids with irreversible processes, extending Hamilton's principle to include entropy production.

Findings

The scheme preserves energy and mass to machine precision.

It applies to various boundary conditions including insulated and prescribed heat flux.

The method is demonstrated through Rayleigh-Bénard thermal convection simulations.

Abstract

Respecting the laws of thermodynamics is crucial for ensuring that numerical simulations of dynamical systems deliver physically relevant results. In this paper, we construct a structure-preserving and thermodynamically consistent finite element method and time-stepping scheme for heat conducting viscous fluids, with general state equations. The method is deduced by discretizing a variational formulation for nonequilibrium thermodynamics that extends Hamilton's principle for fluids to systems with irreversible processes. The resulting scheme preserves the balance of energy and mass to machine precision, as well as the second law of thermodynamics, both at the spatially and temporally discrete levels. The method is shown to apply both with insulated and prescribed heat flux boundary conditions, as well as with prescribed temperature boundary conditions. We illustrate the properties of the scheme with the Rayleigh-B\'enard thermal convection. While the focus is on heat conducting viscous fluids, the proposed discrete variational framework paves the way to a systematic construction of thermodynamically consistent discretizations of continuum systems.