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On the expansion of a wedge of van der Waals gas into vacuum III: interaction of fan-shock-fan composite waves

Geng Lai

TL;DR

This work develops a rigorous framework for the expansion into vacuum of a wedge of a nonconvex van der Waals gas, focusing on the complex interaction of fan and shock waves in a 2D Riemann-type setting. By combining hodograph transformation with characteristic decomposition, the authors cast the mixed-type, shock-containing problem into a sequence of tractable boundary-value problems, including a discontinuous Goursat-type problem and centered-wave analysis. They prove that the shocks in the interaction region must be post-sonic and that the overall solution is a global, piecewise smooth function in a suitably defined interaction domain, under small perturbations of the initial EOS parameter. The results advance the understanding of nonconvex EOS gas dynamics and provide a robust construction technique with potential applications to multi-wave interactions in nonideal fluids.

Abstract

This paper studies the expansion into vacuum of a wedge of gas at rest. This problem catches several important classes of wave interactions in the context of 2D Riemann problems. When the gas at rest is a nonideal gas, the gas away from the sharp corner of the wedge may expand into the vacuum as two symmetrical planar rarefaction fan waves, shock-fan composite waves, or fan-shock-fan composite waves. Then the expansion in vacuum problem can be reduced to the interactions of these elementary waves. Global existences of classical solutions to the interaction of the fan waves and the interaction of the shock-fan composite waves were obtained by the author in [21,22]. In the present paper we study the third case: interaction of fan-shock-fan composite waves. In contrast to the first two cases, the third case involves shock waves in the interaction region and is actually a shock free boundary problem. Differing from the transonic shock free boundary problems arising in 2D Riemann problems for ideal gases, the type of the shocks for this shock free boundary problem is also a priori unknown. This results in the fact that the formulation of the boundary conditions on the shocks is also a priori unknown. By calculating the curvatures of the shocks and using the Liu's extended entropy condition, we prove that the shocks in the interaction region must be post-sonic (in the sense of the flow velocity relative to the shock front). We also prove that the shocks are envelopes of one out of the two families of wave characteristics of the flow behind them, and not characteristics. By virtue of the hodograph transformation method and the characteristic decomposition method, we construct a global-in-time piecewise smooth solution to the expansion in vacuum problem for the third case.

On the expansion of a wedge of van der Waals gas into vacuum III: interaction of fan-shock-fan composite waves

TL;DR

This work develops a rigorous framework for the expansion into vacuum of a wedge of a nonconvex van der Waals gas, focusing on the complex interaction of fan and shock waves in a 2D Riemann-type setting. By combining hodograph transformation with characteristic decomposition, the authors cast the mixed-type, shock-containing problem into a sequence of tractable boundary-value problems, including a discontinuous Goursat-type problem and centered-wave analysis. They prove that the shocks in the interaction region must be post-sonic and that the overall solution is a global, piecewise smooth function in a suitably defined interaction domain, under small perturbations of the initial EOS parameter. The results advance the understanding of nonconvex EOS gas dynamics and provide a robust construction technique with potential applications to multi-wave interactions in nonideal fluids.

Abstract

This paper studies the expansion into vacuum of a wedge of gas at rest. This problem catches several important classes of wave interactions in the context of 2D Riemann problems. When the gas at rest is a nonideal gas, the gas away from the sharp corner of the wedge may expand into the vacuum as two symmetrical planar rarefaction fan waves, shock-fan composite waves, or fan-shock-fan composite waves. Then the expansion in vacuum problem can be reduced to the interactions of these elementary waves. Global existences of classical solutions to the interaction of the fan waves and the interaction of the shock-fan composite waves were obtained by the author in [21,22]. In the present paper we study the third case: interaction of fan-shock-fan composite waves. In contrast to the first two cases, the third case involves shock waves in the interaction region and is actually a shock free boundary problem. Differing from the transonic shock free boundary problems arising in 2D Riemann problems for ideal gases, the type of the shocks for this shock free boundary problem is also a priori unknown. This results in the fact that the formulation of the boundary conditions on the shocks is also a priori unknown. By calculating the curvatures of the shocks and using the Liu's extended entropy condition, we prove that the shocks in the interaction region must be post-sonic (in the sense of the flow velocity relative to the shock front). We also prove that the shocks are envelopes of one out of the two families of wave characteristics of the flow behind them, and not characteristics. By virtue of the hodograph transformation method and the characteristic decomposition method, we construct a global-in-time piecewise smooth solution to the expansion in vacuum problem for the third case.
Paper Structure (36 sections, 33 theorems, 413 equations, 22 figures)

This paper contains 36 sections, 33 theorems, 413 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Expansion into vacuum of a wedge of gas
  3. Systems for potential flow
  4. Oblique shocks in 2D pseudo-steady potential flows of non-convex gases
  5. Interaction of fan-shock-fan composite waves
  6. Main theorem
  7. Overview
  8. Characteristic and characteristic decomposition
  9. Characteristic directions and characteristic angles
  10. Characteristic equations and characteristic decompositions
  11. Hodograph transformation
  12. Concept of hodograph transformation
  13. Systems in the holograph plane
  14. Characteristic decompositions in the hodograph plane
  15. Interaction of fan-shock-fan composite waves
  16. ...and 21 more sections

Key Result

Proposition 1.1

( Double-sonic) There exists a unique pair $\tau_1^e$ and $\tau_2^e$ where $1<\tau_1^e<\tau_2^e$, such that and for all $\tau\in(\tau_1^e, \tau_2^e)$. Moreover, $\tau_1^e\in (\tau_1^a, \tau_1^i)$ and $\tau_2^e\in (\tau_2^i, \tau_2^a)$.

Figures (22)

  • Figure 1: Initial data.
  • Figure 2: An isentrope of a van der Waals gas.
  • Figure 3: Wave interactions in the self-similar $(\xi,\eta)$-plane: (1) interaction of fan waves; (2) interaction of shock-fan composite waves; (3) interaction of fan-shock-fan composite waves. Here, the solid lines represent characteristic lines; the thick lines represent shock waves; the dashed lines represent vacuum boundaries.
  • Figure 4: A 2D self-similar shock curve.
  • Figure 5: Left: the isentrope $p=p(\tau)$; right: the graph of $-p'(\tau)$.
  • ...and 17 more figures

Theorems & Definitions (64)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • proof
  • ...and 54 more