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Supersonic flow past an airfoil

Yannis Angelopoulos

Abstract

We consider a supersonic flow past an airfoil in the context of the steady, isentropic and irrotational compressible Euler equations. We show that for appropriate data, either a shock forms, in which case certain derivatives of the solution blow up, or a sonic line forms, which translates to the fact that the hyperbolic character of the equations degenerates.

Supersonic flow past an airfoil

Abstract

We consider a supersonic flow past an airfoil in the context of the steady, isentropic and irrotational compressible Euler equations. We show that for appropriate data, either a shock forms, in which case certain derivatives of the solution blow up, or a sonic line forms, which translates to the fact that the hyperbolic character of the equations degenerates.
Paper Structure (16 sections, 3 theorems, 195 equations, 1 figure)

This paper contains 16 sections, 3 theorems, 195 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The steady, isentropic and irrotational Euler equations in 2D
  3. The initial boundary problem for airfoils and some geometric assumptions
  4. The purely hyperbolic regime
  5. The equations
  6. Coordinate systems
  7. The equations in dynamical coordinates
  8. The main theorem
  9. Proof of the main theorem
  10. Construction of the initial data set
  11. The bootstrap assumptions
  12. Construction of the solution near spacelike infinity
  13. Construction of the solution in the interior region
  14. Formation of a shock
  15. Formation of the sonic line
  16. ...and 1 more sections

Key Result

Theorem 1.1

Consider a solution of the steady, isentropic and irrotational Euler equations in two dimensions (denoted by variables $x$ and $y$) given in the form of a potential flow $\phi$ emanating from data set on an initial smooth hypersurface $\Sigma$ as in figure supersonic-figure1, where we assume that a condition that makes the second order quasilinear equation satisfied by $\phi$ initially hyperbolic

Figures (1)

  • Figure 1: The initial hypersurface $\Sigma$, and the dotted area in red where the solution is constructed, and a shock and a sonic line form.

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 1.1
  • Remark 2.1
  • Definition 5.1
  • Theorem 5.1
  • Remark 5.1
  • Theorem 8.1