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

Evan S. Gawlik, François Gay-Balmaz

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-Bénard 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.

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

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-Bénard 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.
Paper Structure (41 sections, 2 theorems, 150 equations, 5 figures, 2 tables)

This paper contains 41 sections, 2 theorems, 150 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Variational formulation for heat conducting viscous fluids
  3. Insulated boundaries
  4. Lagrangian description.
  5. Eulerian description.
  6. Standard Lagrangian.
  7. Energy and entropy balances.
  8. Dirichlet boundary conditions
  9. Lagrangian description.
  10. Eulerian description.
  11. Associated weak formulation
  12. Structure preserving variational discretization
  13. Discrete setting
  14. Discrete variational formulation
  15. Energy balance and mass conservation.
  16. ...and 26 more sections

Key Result

Proposition 3.1

The equations of motion that result from VP_NSF_spatial_discrete--VC_NSF_spatial_discrete and from the definition $\rho = \varrho _{0} \cdot g ^{-1}$ are where The variational principle VP_NSF_spatial_discrete--VC_NSF_spatial_discrete also yields the conditions

Figures (5)

  • Figure 1: The onset of Rayleigh-Bénard convection with Dirichlet boundary conditions.
  • Figure 2: The onset of Rayleigh-Bénard convection with nonhomogeneous Neumann boundary conditions.
  • Figure 3: Temperature contours at time $t=10$ during a simulation of Rayleigh-Bénard convection with $\mathrm{Re}=100$, $m=0$, $Z=2$, and $\mathrm{Pr}=2.5$, so that $\mathrm{Ra}= 90909.1$. Left column: Our scheme with $h \in \{\frac{\sqrt{2}}{4}, \frac{\sqrt{2}}{8}, \frac{\sqrt{2}}{16}, \frac{\sqrt{2}}{32} \}$, ordered from smallest $h$ to largest. Middle column: The scheme \ref{['FV_u']}-\ref{['FV_rho']} with $h \in \{\frac{1}{16}, \frac{1}{32}, \frac{1}{64}, \frac{1}{128} \}$, ordered from smallest $h$ to largest. Right column: Same as the middle column, but with the term proportional to $h^\xi$ excluded from \ref{['bh_FV']}.
  • Figure 4: Evolution of mass and energy during the three simulations in the top row of Fig. \ref{['fig:contours']}. The absolute deviations $|F(t)-F(0)|$ are plotted for each conserved quantity $F(t)$.
  • Figure 5: Evolution of total entropy in the purely reversible setting.

Theorems & Definitions (10)

  • Remark 2.1: Structure of the variational formulation
  • Remark 2.2: Prescribed heat flux
  • Proposition 3.1
  • Remark 3.1
  • Remark 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.2
  • proof