Shock polars for ideal non-polytropic gas

TL;DR

This paper analyzes shock polars for ideal non-polytropic gases, showing the existence of a unique critical shock angle under certain conditions and detailing the behavior of thermodynamic variables along shock curves.

Contribution

It demonstrates the existence and properties of a unique critical shock angle for non-polytropic gases with convex equations of state, extending shock polar theory.

Findings

Critical shock has a unique velocity angle maximum.

Temperature, pressure, energy, enthalpy, and entropy increase along the shock polar.

Speed decreases along the entire shock polar.

Abstract

We show that shock polars for ideal non-polytropic gas (thermally but not calorically perfect) have a unique velocity angle maximum, the critical shock, assuming convex equation of state (positive fundamental derivative) and other standard conditions. We also show that the critical shock is always transonic. In the process we show that temperature, pressure, energy, enthalpy, normal mass flux and entropy are strictly increasing along the forward Hugoniot curves, and hence along the polar from vanishing to normal shock; speed is strictly decreasing along the entire polar, mass flux and importantly Mach number are decreasing on subsonic parts of the polar. If the equation of state is ideal but not convex, or convex but not ideal, counterexamples can be given with multiple critical shocks, permitting more than two shocks attaining the same velocity angle, in particular more than one shock of weak type.