A variational inequality formulation for transonic compressible steady potential flows: Radially symmetric transonic shock

TL;DR

This paper introduces a variational inequality approach to model steady transonic compressible potential flows with shocks, providing a new mathematical framework that aligns with physical jump conditions and supports numerical validation.

Contribution

It develops a novel variational inequality formulation for transonic shocks in steady compressible flows, linking critical points to physical shock conditions and establishing existence criteria.

Findings

Formulation captures transonic shocks accurately

Critical points satisfy the transonic equation and jump conditions

Numerical results validate the approach

Abstract

We establish a variational inequality formulation that captures the transonic shock for a steady compressible potential flow. Its critical point satisfies the transonic equation; moreover the associated jump conditions across its free boundary match the usual Rankine-Hugoniot jump conditions for a shock. By means of example we validate our formulation, and establish the necessary and sufficient condition for the existence of a transonic shock. Numerical results are also discussed.