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Entropy-conservative high-order methods for high-enthalpy gas flows

Georgii Oblapenko, Manuel Torrilhon

Abstract

A framework for numerical evaluation of entropy-conservative volume fluxes in gas flows with internal energies is developed, for use with high-order discretization methods. The novelty of the approach lies in the ability to use arbitrary expressions for the internal degrees of freedom of the constituent gas species. The developed approach is implemented in an open-source discontinuous Galerkin code for solving hyperbolic equations. Numerical simulations are carried out for several model 2-D flows and the results are compared to those obtained with the finite volume-based solver DLR TAU.

Entropy-conservative high-order methods for high-enthalpy gas flows

Abstract

A framework for numerical evaluation of entropy-conservative volume fluxes in gas flows with internal energies is developed, for use with high-order discretization methods. The novelty of the approach lies in the ability to use arbitrary expressions for the internal degrees of freedom of the constituent gas species. The developed approach is implemented in an open-source discontinuous Galerkin code for solving hyperbolic equations. Numerical simulations are carried out for several model 2-D flows and the results are compared to those obtained with the finite volume-based solver DLR TAU.
Paper Structure (20 sections, 4 theorems, 51 equations, 16 figures, 2 tables, 2 algorithms)

This paper contains 20 sections, 4 theorems, 51 equations, 16 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Flow equations and thermodynamic properties
  3. Internal energy models
  4. Implementation of internal energy models in a CFD solver
  5. Entropy-conservative flux
  6. Entropy and entropy variables
  7. Derivation of entropy-conservative flux
  8. Properties of the entropy-conservative flux
  9. Algorithm for pre-processing and flux computation
  10. Computational complexity
  11. Dissipative flux
  12. Numerical results
  13. Comparison to exact expressions for fluxes
  14. Calorically perfect gas
  15. Infinite harmonic oscillator
  16. ...and 5 more sections

Key Result

Theorem 1

The numerical flux given by Eqns. (eq:flux_fx_rho)--(eq:flux-ec-energy) is consistent if $\lim_{\substack{T_- \to T \\ T_+ \to T}} {c_v\left(T^\ast\right)}/{T^\ast} = {c_v(T)}/{T}$ and $\lim_{\substack{T_- \to T \\ T_+ \to T}}c_v\left(T^{\ast\ast}\right) = c_v(T)$.

Figures (16)

  • Figure 1: The heat capacity ratio $\gamma$ as a function of temperature for different models of the vibrational energy spectrum of oxygen (black lines) and nitrogen (blue lines).
  • Figure 2: Relative error in the energy flux for different temperature jumps as a function of the discretization step $\Delta T$ for a calorically perfect gas.
  • Figure 3: Relative error in the energy flux as a function of the discretization step $\Delta T$, infinite harmonic oscillator.
  • Figure 4: Absolute value of the entropy production rate over the course of the simulation.
  • Figure 5: Time-averaged absolute value of the entropy production rate as a function of the parameter $r$.
  • ...and 11 more figures

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • proof
  • Theorem 3
  • Lemma 1
  • proof
  • proof