Stability of Transonic Shock-Fronts in Three-Dimensional Conical Steady Potential Flow past a Perturbed Cone

TL;DR

This paper investigates the stability of transonic shock-fronts in three-dimensional steady potential flow past a cone, showing that these shock-fronts remain stable under certain boundary and flow perturbations, with their slope approaching the unperturbed state at infinity.

Contribution

The paper proves the conditional stability of self-similar transonic shock-fronts in 3D potential flow with respect to boundary and upstream flow perturbations.

Findings

Shock-front slope converges to the unperturbed slope at infinity.

Transonic shock-fronts are conditionally stable under conical boundary perturbations.

Stability is established in appropriate function spaces.

Abstract

For an upstream supersonic flow past a straight-sided cone in $\R^3$ whose vertex angle is less than the critical angle, a transonic (supersonic-subsonic) shock-front attached to the cone vertex can be formed in the flow. In this paper we analyze the stability of transonic shock-fronts in three-dimensional steady potential flow past a perturbed cone. We establish that the self-similar transonic shock-front solution is conditionally stable in structure with respect to the conical perturbation of the cone boundary and the upstream flow in appropriate function spaces. In particular, it is proved that the slope of the shock-front tends asymptotically to the slope of the unperturbed self-similar shock-front downstream at infinity.