Global non-isentropic rotational supersonic flows in a semi-infinite divergent duct

TL;DR

This paper constructs and analyzes a global non-isentropic rotational supersonic flow in a semi-infinite divergent duct, addressing existence, vacuum formation, and flow structure using characteristic methods.

Contribution

It develops a novel method of characteristic decompositions to establish uniform a priori estimates for 2D steady Euler flows in divergent ducts.

Findings

Existence of global non-isentropic rotational supersonic flows

Conditions for vacuum formation and its location

Flow interface between gas and vacuum is straight

Abstract

Supersonic flows for the two-dimensional (2D) steady full Euler system are studied. We construct a global non-isentropic rotational supersonic flow in a semi-infinite divergent duct. The flow satisfies the slip condition on the walls of the duct, and the state of the flow is given at the inlet of the duct. The solution is constructed by the method of characteristics. The main difficulty for the global existence is that uniform a priori $C^1$ norm estimate of the solution is hard to obtain, especially when the solution tends to vacuum state. We derive a group of characteristic decompositions for the 2D steady full Euler system. Using these decompositions, we obtain the uniform a priori estimates of the derivatives of the solution. A sufficient condition for the appearance of vacuum is given. We also show that if there is a vacuum then the vacuum is always adjacent to one of the walls, and the interface between gas and vacuum must be straight.