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On a Doubly Reduced Model for Dynamics of Heterogeneous Mixtures of Stiffened Gases, its Regularizations and their Implementations

A. Zlotnik, T. Lomonosov

TL;DR

The paper tackles the dynamics of heterogeneous binary mixtures of compressible fluids governed by stiffened-gas equations of state using a doubly reduced four-equation model with a common pressure, reduced further to a quasi-homogeneous form. It derives the physically correct root for the common pressure, provides compact formulas for the squared speed of sound $c_s^2$ and its relation to the Wood speed, and introduces two regularizations (QGD and QHD) adapted to the heterogeneous, stiffened-gas setting. For 1D problems, it constructs explicit two-level-in-time, symmetric three-point-in-space finite-difference schemes without limiters and demonstrates their accuracy and robustness through shock-tube tests involving air–water, vapor–liquid, and CO$_2$ depressurization scenarios. The results establish a stable, efficient framework for simulating complex multiphase flows with phase transitions and lay groundwork for higher-dimensional extensions. The work provides practical numerical tools and theoretical insights that can be applied to phase-change problems and multiphase flow simulations in engineering and physics.

Abstract

We deal with the reduced four-equation model for the dynamics of heterogeneous compressible binary mixtures with the stiffened gas equations of state. We study its further reduced form, with the excluded volume concentrations, and with a quadratic equation for the common pressure of the components; this form can be called a quasi-homogeneous form. We prove new properties of the equation, derive simple formulas for the squared speed of sound and present an alternative proof for a formula that relates it to the squared Wood speed of sound; also, a short derivation of the pressure balance equation is given. For the first time, we introduce regularizations of the heterogeneous model (in the quasi-homogeneous form). Previously, regularizations of such type were developed only for the homogeneous mixtures of perfect polytropic gases, and it was unclear how to cover the case considered here. In the 1D case, based on these regularizations, we construct new explicit two-level in time and symmetric three-point in space finite-difference schemes without limiters, and provide numerical results for various flows with shock waves.

On a Doubly Reduced Model for Dynamics of Heterogeneous Mixtures of Stiffened Gases, its Regularizations and their Implementations

TL;DR

The paper tackles the dynamics of heterogeneous binary mixtures of compressible fluids governed by stiffened-gas equations of state using a doubly reduced four-equation model with a common pressure, reduced further to a quasi-homogeneous form. It derives the physically correct root for the common pressure, provides compact formulas for the squared speed of sound and its relation to the Wood speed, and introduces two regularizations (QGD and QHD) adapted to the heterogeneous, stiffened-gas setting. For 1D problems, it constructs explicit two-level-in-time, symmetric three-point-in-space finite-difference schemes without limiters and demonstrates their accuracy and robustness through shock-tube tests involving air–water, vapor–liquid, and CO depressurization scenarios. The results establish a stable, efficient framework for simulating complex multiphase flows with phase transitions and lay groundwork for higher-dimensional extensions. The work provides practical numerical tools and theoretical insights that can be applied to phase-change problems and multiphase flow simulations in engineering and physics.

Abstract

We deal with the reduced four-equation model for the dynamics of heterogeneous compressible binary mixtures with the stiffened gas equations of state. We study its further reduced form, with the excluded volume concentrations, and with a quadratic equation for the common pressure of the components; this form can be called a quasi-homogeneous form. We prove new properties of the equation, derive simple formulas for the squared speed of sound and present an alternative proof for a formula that relates it to the squared Wood speed of sound; also, a short derivation of the pressure balance equation is given. For the first time, we introduce regularizations of the heterogeneous model (in the quasi-homogeneous form). Previously, regularizations of such type were developed only for the homogeneous mixtures of perfect polytropic gases, and it was unclear how to cover the case considered here. In the 1D case, based on these regularizations, we construct new explicit two-level in time and symmetric three-point in space finite-difference schemes without limiters, and provide numerical results for various flows with shock waves.
Paper Structure (14 sections, 9 theorems, 117 equations, 7 figures, 3 tables)

This paper contains 14 sections, 9 theorems, 117 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A reduced system of PDEs for the dynamics of heterogeneous mixtures of stiffened gases and its further reduction
  3. Regularized systems of PDEs for the dynamics of quasi-homogeneous mixtures of stiffened gases
  4. Finite-difference schemes for the 1D regularized systems of PDes
  5. Numerical experiments
  6. Air-to-water shock tube problem
  7. Water-to-air shock tube problem
  8. Shock tube test with a mixture containing mainly water vapor
  9. Shock tube test with a vanishing liquid phase
  10. Shock tube test with a mixture containing mainly liquid water
  11. Dodecane vapor-to-liquid shock tube
  12. Carbon dioxide depressurization
  13. CONCLUSION
  14. Proofs of Propositions from Section \ref{['Section II']}

Key Result

Proposition 1

Let $\Delta_*:=p_{*2}-p_{*1}$. The following formulas hold where Consequently, $d>0$, thus, these $p_\pm$ and those given by formula p pm roots are the same.

Figures (7)

  • Figure 1: Numerical results for air-to-water shock tube problem (test A for $N= 300$ (dark magenta), $2000$ (blue), $a=0.3$ and $\beta=0.2$ (the QGD regularization)
  • Figure 2: Numerical results for water-to-air shock tube (test B) for $N= 500$ (dark magenta), $2500$ (blue), $a=2$ and $\beta=0.1$ (the QGD regularization)
  • Figure 3: Numerical results for shock tube with a mixture containing mainly water vapor (test C) for $N= 200$ (dark magenta), $500$ (blue), $a=0.8$ and $\beta=0.2$ (the QHD regularization)
  • Figure 4: Numerical results for shock tube with a vanishing liquid phase (test D) for $N=100$ (dark magenta), $500$ (blue), $a=0.2$ and $\beta=0.2$ (the QGD regularization)
  • Figure 5: Numerical results for shock tube with a mixture containing mainly liquid water (test E) for $N= 500$ (dark magenta), $1500$ (blue), $a=0.8$ and $\beta=0.3$ (the QHD regularization)
  • ...and 2 more figures

Theorems & Definitions (18)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • proof
  • Proposition 9
  • ...and 8 more