Subsonic flows past a profile with a vortex line at the trailing edge

TL;DR

This paper proves the existence, uniqueness, and stability of subsonic flows around an airfoil with a vortex line at the trailing edge, modeled by the steady compressible Euler equations, using advanced mathematical techniques.

Contribution

It introduces a novel mathematical framework for analyzing subsonic flows with vortex lines at the trailing edge of an airfoil, addressing stability and boundary fitting issues.

Findings

Established existence and uniqueness of the flow solution.

Proved stability of the vortex line at the trailing edge.

Developed methods to handle boundary fitting and instability challenges.

Abstract

We established the existence, uniqueness and stability of subsonic flows past an airfoil with a vortex line at the trailing edge. Such a flow pattern is governed by the two dimensional steady compressible Euler equations. The vortex line attached to the trailing edge is a contact discontinuity for the Euler system and is treated as a free boundary. The problem is formulated and solved by using the implicit function theorem. The main difficulties are due to the fitting of the vortex line with the profile at the trailing edge and the possible subtle instability of the vortex line at the far field. Suitable choices of the weights and elaborate barrier functions are found to deal with such difficulties.