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Supersonic flow of a Chaplygin gas past a conical wing with $Λ$-shaped cross sections

Minghong Han, Bingsong Long, Hairong Yuan

Abstract

In this paper, by considering the anhedral angle, we for the first time study the problem of supersonic flow of a Chaplygin gas over a conical wing with $Λ$-shaped cross sections, where the flow is governed by the three-dimensional steady isentropic irrotational compressible Euler equations. This work is motivated by the design of the Nonweiler wing, which is one of the simplest waveriders. Mathematically, the problem can be reformulated as a boundary value problem for a nonlinear mixed-type equation in conical coordinates. By introducing a viscosity parameter to treat the degenerate boundary, we use the continuity method to establish the existence of a piecewise smooth self-similar solution to the problem, in the case that the shock is attached to the leading edge of the conical wing. Our results verify part of Küchemann's speculation on the conical flow field structures of this type, and also find a new conical flow field structure.

Supersonic flow of a Chaplygin gas past a conical wing with $Λ$-shaped cross sections

Abstract

In this paper, by considering the anhedral angle, we for the first time study the problem of supersonic flow of a Chaplygin gas over a conical wing with -shaped cross sections, where the flow is governed by the three-dimensional steady isentropic irrotational compressible Euler equations. This work is motivated by the design of the Nonweiler wing, which is one of the simplest waveriders. Mathematically, the problem can be reformulated as a boundary value problem for a nonlinear mixed-type equation in conical coordinates. By introducing a viscosity parameter to treat the degenerate boundary, we use the continuity method to establish the existence of a piecewise smooth self-similar solution to the problem, in the case that the shock is attached to the leading edge of the conical wing. Our results verify part of Küchemann's speculation on the conical flow field structures of this type, and also find a new conical flow field structure.
Paper Structure (12 sections, 7 theorems, 92 equations, 14 figures)

This paper contains 12 sections, 7 theorems, 92 equations, 14 figures.

Table of Contents

  1. Introduction and main results
  2. Analysis of the flow outside Mach cone and the shock structures
  3. Uniform downstream flow outside the Mach cone
  4. Structures of shocks in conical coordinates
  5. Simplification of Problem A
  6. Analysis of the flow inside Mach cone
  7. Lipschitz estimate
  8. The proof of main theorem
  9. Further discussions
  10. On the possible flow field structure of the --wing
  11. Conical wings with asymmetric cross-sections
  12. Shock polar for Chaplygin gas

Key Result

Theorem 1.1

Assume that the wing $\mathcal{W}^\beta_\sigma$ and incoming flow $U_{\infty}$ given by $(sweep wing)$ and $(incoming flow)$, respectively. Then, we can find a critical attack angle $\alpha_0 = \alpha_0(\rho_\infty, q_\infty) \in (0, \pi/2)$ such that for any fixed $\alpha \in (0, \alpha_0)$, there

Figures (14)

  • Figure 1: Nonweiler or caret wing.
  • Figure 2: A conical wing with $\Lambda$-shaped cross section.
  • Figure 3: View of a conical wing from the $x_1$--direction, where $\sigma$ is the sweep angle.
  • Figure 4: View of a conical wing from the $x_3$--direction, where $\beta$ is the anhedral angle.
  • Figure 5: View of a conical wing from the $x_2$--direction, where $\alpha$ is the attack angle.
  • ...and 9 more figures

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • ...and 15 more