Non-Ideal Compressible Fluid Dynamics: A Challenge for Theory

Abstract

The possibility that compression as well as rarefaction shocks may form in single phase vapours was envisaged first by Bethe (1942). However calculations based on the Van der Waals equation of state indicated that the latter type of shock is possible only if the specific heat at constant volume cv divided by the universal gas constant R is larger than about 17.5 which he considered too large to be satisfied by real fluids. This conclusion was contested by Thompson (1971) who showed that the type of shock capable of forming in arbitrary fluids is determined by the sign of the thermodynamic quantityto which he referred to as fundamental derivative of gas dynamics. Here v, p, s and c denote the specific volume, the pressure, the entropy and the speed of sound. Thompson and co-workers also showed that the required condition for the existence of rarefaction shocks, that Γ may take on negative values, is indeed satisfied for a number of hydrocarbon and fluorocarbon vapours. This finding spawned a burst of theoretical studies elaborating on the unusual and often counterintuitive behaviour of shocks with rarefaction shocks present. These produced both results of theoretical character but also results suggesting the practical importance of Non-Ideal Compressible Fluid Dynamics in general. The present paper addresses some of the challenges encountered in connection with the theoretical treatment of the associated flow behaviour. Weakly nonlinear acoustic waves of finite amplitude serve as a starting point. Here mixed rather than strictly positive nonlinearity generates a wealth of phenomena not possible in perfect gases. Examples of steady flows where these non-classical effects play a decisive role (and which may be useful also for future experimental work) are quasi one-dimensional nozzle flows and transonic two-dimensional flows past corners. The study of viscous effects concentrates on laminar flows of boundary layer type. Here non-classical phenomena are caused by the uncommon smallness of the Eckert number but also by the unconventional Mach number dependence on p in the external inviscid flow region.