Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Oblique wave interactions in 2D steady supersonic flows of Bethe-Zel'dovich-Thompson fluids

Geng Lai

TL;DR

This work develops a rigorous analysis of steady, two-dimensional, oblique wave interactions in a semi-infinite divergent duct for Bethe-Zel'dovich-Thompson (BZT) fluids with a nonconvex equation of state. By combining characteristic decomposition, Riemann invariants, and a hodograph transformation, the authors classify thirteen oblique interaction types and construct global, piecewise-smooth, supersonic solutions inside the duct, including complex fan–shock–fan and shock–fan–shock structures. They formulate and solve multiple hyperbolic boundary-value problems (Goursat, discontinuous Goursat, mixed boundary value, singular Cauchy) to serve as building blocks for the full duct flow, and they derive detailed solution structures and envelopes such as sonic shocks. The results extend classical compressible-flow theory to nonconvex EOS contexts, with implications for 2D Riemann problems and for understanding nonclassical wave phenomena in BZT fluids, including rarefaction shocks and post-sonic shocks.

Abstract

This paper studies steady supersonic flow in a 2D semi-infinite divergent duct. We assume that the flow satisfies the slip boundary condition on the walls of the duct, and the state of the flow is given at the inlet of the divergent duct. When the fluid is a polytropic ideal gas, the problem can be reduced to some interactions of rarefaction simple waves, and the existence of a global classical solution inside the divergent duct can be established using the method of characteristics. In this paper we assume that the fluid is a nonconvex Bethe-Zel'dovich-Thompson (BZT) fluid. This type of fluid may significantly differ from polytropic ideal gases. For instance, physically admissible rarefaction shocks can occur. Depending on the oncoming flow state and the flare angles of the divergent duct, thirteen distinct types of oblique wave interactions may occur, including oblique composite waves consisting of shocks and centered simple waves. This paper systematically studies these oblique wave interactions and constructs global, piecewise smooth, supersonic solutions within the divergent duct using characteristic decomposition and hodograph transformation methods. We also obtain the detailed structures of these solutions in addition to their existence. The results and methods of this paper are also applicable to some 2D Riemann problems for gases with nonconvex equations of state.

Oblique wave interactions in 2D steady supersonic flows of Bethe-Zel'dovich-Thompson fluids

TL;DR

This work develops a rigorous analysis of steady, two-dimensional, oblique wave interactions in a semi-infinite divergent duct for Bethe-Zel'dovich-Thompson (BZT) fluids with a nonconvex equation of state. By combining characteristic decomposition, Riemann invariants, and a hodograph transformation, the authors classify thirteen oblique interaction types and construct global, piecewise-smooth, supersonic solutions inside the duct, including complex fan–shock–fan and shock–fan–shock structures. They formulate and solve multiple hyperbolic boundary-value problems (Goursat, discontinuous Goursat, mixed boundary value, singular Cauchy) to serve as building blocks for the full duct flow, and they derive detailed solution structures and envelopes such as sonic shocks. The results extend classical compressible-flow theory to nonconvex EOS contexts, with implications for 2D Riemann problems and for understanding nonclassical wave phenomena in BZT fluids, including rarefaction shocks and post-sonic shocks.

Abstract

This paper studies steady supersonic flow in a 2D semi-infinite divergent duct. We assume that the flow satisfies the slip boundary condition on the walls of the duct, and the state of the flow is given at the inlet of the divergent duct. When the fluid is a polytropic ideal gas, the problem can be reduced to some interactions of rarefaction simple waves, and the existence of a global classical solution inside the divergent duct can be established using the method of characteristics. In this paper we assume that the fluid is a nonconvex Bethe-Zel'dovich-Thompson (BZT) fluid. This type of fluid may significantly differ from polytropic ideal gases. For instance, physically admissible rarefaction shocks can occur. Depending on the oncoming flow state and the flare angles of the divergent duct, thirteen distinct types of oblique wave interactions may occur, including oblique composite waves consisting of shocks and centered simple waves. This paper systematically studies these oblique wave interactions and constructs global, piecewise smooth, supersonic solutions within the divergent duct using characteristic decomposition and hodograph transformation methods. We also obtain the detailed structures of these solutions in addition to their existence. The results and methods of this paper are also applicable to some 2D Riemann problems for gases with nonconvex equations of state.
Paper Structure (54 sections, 49 theorems, 441 equations, 45 figures)

This paper contains 54 sections, 49 theorems, 441 equations, 45 figures.

Table of Contents

  1. Introduction
  2. Systems of potential flow
  3. Characteristic equation
  4. Characteristic decompositions
  5. Hodograph transformation
  6. Riemann invariants
  7. Systems in $(r, s)$-plane
  8. Hyperbolicity and invariant region
  9. Notation
  10. Oblique waves around the sharp corner
  11. Oblique shocks
  12. Centered simple waves
  13. Oblique waves around the corners
  14. $\tau_0>\tau_2^i$
  15. $\tau_1^{i}<\tau_0\leq\tau_2^{i}$
  16. ...and 39 more sections

Key Result

Proposition 2.2.1

(Commutator relation) For the directional derivatives $\bar{\partial}_{\pm}$ defined in (41404), we have

Figures (45)

  • Figure 1: Supersonic flow in a 2D semi-infinitely long divergent duct.
  • Figure 2: The graph of the equation of state $p=p(\tau)$.
  • Figure 3: Twelve additional types of oblique wave interactions in the divergent duct; where the red lines represent oblique shocks and the black lines represent centered simple waves.
  • Figure 4: Characteristic curves, characteristic directions, and characteristic angles.
  • Figure 5: Invariant region in the $(r, s)$-plane.
  • ...and 40 more figures

Theorems & Definitions (86)

  • Proposition 2.2.1
  • proof
  • Proposition 2.2.2
  • proof
  • Theorem 2.2.3
  • Lemma 2.4.1
  • proof
  • Proposition 3.1.1
  • proof
  • Proposition 3.1.2
  • ...and 76 more