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An Eulerian hyperbolic model for heat transfer derived via Hamilton's principle: analytical and numerical study

Firas Dhaouadi, Sergey Gavrilyuk

TL;DR

This paper develops a first-order Eulerian hyperbolic model for heat transfer in compressible flows derived from Hamilton's principle, with entropy evolution emerging as an Euler–Lagrange equation. It proves hyperbolicity via total-energy convexity and presents a Friedrichs symmetric form, revealing two wave families—acoustic and thermal—with the thermal fields capable of expansion shocks and compression fans. A relaxation mechanism links the model to Fourier's law asymptotically through $\tau = K c_V(\gamma-1) \rho^2 /( \varkappa^2 p)$, and a detailed Rankine–Hugoniot analysis shows nonexistence of contact discontinuities for $\varkappa\neq0$ and distinct Hugoniot branches. Numerical IMEX schemes corroborate the theoretical findings, showing agreement with Euler–Fourier in the long-wavelength limit and exposing nonclassical shock phenomena such as expansion shocks and shock splitting. The work provides a thermodynamically consistent, hyperbolic alternative to Fourier-based heat conduction with potential advantages for multidimensional curl-free discretizations.

Abstract

In this paper, we present a new model for heat transfer in compressible fluid flows. The model is derived from Hamilton's principle of stationary action in Eulerian coordinates, in a setting where the entropy conservation is recovered as an Euler--Lagrange equation. The governing system is shown to be hyperbolic. It is asymptotically consistent with the Euler equations for compressible heat conducting fluids, provided the addition of suitable relaxation terms. A study of the Rankine--Hugoniot conditions and the Clausius--Duhem inequality reveals that contact discontinuities cannot exist while expansion waves and compression fans are possible solutions to the governing equations. Evidence of these properties is provided on a set of numerical test cases.

An Eulerian hyperbolic model for heat transfer derived via Hamilton's principle: analytical and numerical study

TL;DR

This paper develops a first-order Eulerian hyperbolic model for heat transfer in compressible flows derived from Hamilton's principle, with entropy evolution emerging as an Euler–Lagrange equation. It proves hyperbolicity via total-energy convexity and presents a Friedrichs symmetric form, revealing two wave families—acoustic and thermal—with the thermal fields capable of expansion shocks and compression fans. A relaxation mechanism links the model to Fourier's law asymptotically through , and a detailed Rankine–Hugoniot analysis shows nonexistence of contact discontinuities for and distinct Hugoniot branches. Numerical IMEX schemes corroborate the theoretical findings, showing agreement with Euler–Fourier in the long-wavelength limit and exposing nonclassical shock phenomena such as expansion shocks and shock splitting. The work provides a thermodynamically consistent, hyperbolic alternative to Fourier-based heat conduction with potential advantages for multidimensional curl-free discretizations.

Abstract

In this paper, we present a new model for heat transfer in compressible fluid flows. The model is derived from Hamilton's principle of stationary action in Eulerian coordinates, in a setting where the entropy conservation is recovered as an Euler--Lagrange equation. The governing system is shown to be hyperbolic. It is asymptotically consistent with the Euler equations for compressible heat conducting fluids, provided the addition of suitable relaxation terms. A study of the Rankine--Hugoniot conditions and the Clausius--Duhem inequality reveals that contact discontinuities cannot exist while expansion waves and compression fans are possible solutions to the governing equations. Evidence of these properties is provided on a set of numerical test cases.
Paper Structure (29 sections, 1 theorem, 159 equations, 9 figures)

This paper contains 29 sections, 1 theorem, 159 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The Euler system for compressible heat conducting fluid flows
  3. Model description
  4. Lagrangian of the system
  5. Dissipation and asymptotic analysis
  6. Total energy convexity
  7. Friedrichs symmetric form
  8. Hyperbolic model analysis
  9. Eigenstructure study
  10. Structure of the eigenfields
  11. Dispersion relation
  12. Admissibility of shock waves and nonexistence of contact discontinuities
  13. Non-existence of contact discontinuities
  14. Hugoniot curve for hyperbolic thermal conductivity
  15. Properties of the Hugoniot curve
  16. ...and 14 more sections

Key Result

Theorem 1

Let $f(q_1,\cdots,q_n)$ be a convex function on a subset $\Omega\subset\mathbb{R_+}\times\mathbb{R}^{n-1}$. Then is also convex.

Figures (9)

  • Figure 1: log-linear plot of the real part of the phase velocities (Left) and log-log plot of the attenuation factors (Right) for both system of equations \ref{['eq:dissipative_hyp_heat_eq']} (Dashed lines) and system \ref{['eq:Euler-Fourier']} (Solid lines) as a function of the wavenumber $k$. Polytropic gas equation of state is used here with $\gamma=1.4$, $c_V=3/2$, $\rho_0=1$, $\eta_0=1$, $\varkappa=1$ and $K=0.1$.
  • Figure 2: Plot of the functions $\tilde{p}^+$ and $\tilde{p}^-$ as a function of $\tilde{v}$.
  • Figure 3: Plot of $\tilde{\Psi}^+$ (Left) and $\tilde{\Psi}^-$ (Right) as a function of $\tilde{v}$ for different values of $\tilde{\varkappa}$ that are $\varkappa_c= 1.0399...$, $\tilde{\varkappa}=0.8$ and $\tilde{\varkappa}=1.3$. $\gamma=2$. The dashed lines correspond to non-physical solutions that do not satisfy $\tilde{\Psi}\geq0$.
  • Figure 4: Schematic representation of the wave pattern in the $x-t$ plane, for the shock splitting solution. The shock propagates to the right, followed by a right facing compression fan. The Clausius-Duhem inequality is satisfied on the shock. The slope of the rightmost characteristic of the compression fan is equal to the velocity $\mathcal{D}_\star$ of a shock relating the right state to the $\star$ state.
  • Figure 5: Numerical Profiles at of the density (Left), velocity (middle) and pressure (Right) obtained after solving numerically the Euler equations supplemented with heat conduction for the initial condition \ref{['eq:IC_choctube']}. The solid line curves correspond to the numerical result at $t=0.2$ for different values of the heat conductivity $K=0$ (Blue), $K=10^{-3}$ (Red) and $K=10^{-2}$ (Green). The initial condition is plotted with black dashes lines. Here, $\gamma=1.4$ and $c_V =3/2$.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Remark
  • Remark
  • Definition
  • Remark