Transonic flow of a fluid with positive and negative nonlinearity through a nozzle

Abstract

The one-dimensional transonic flow of an inviscid fluid, which at large values of the specific heats exhibits both positive (Γ̄>0) and negative (Γ̄<0) nonlinearity regions {Γ̄=(1/ρ)[∂(ρa)/∂ρ]s} and which remains in a single phase, is studied. By assuming that Γ̄ changes its sign in the small neighborhood of the throat of the nozzle where transonic flow exists and introducing a new scaling of the independent variables, an approximate first-order partial differential equation (PDE) with a nonconvex flux function is derived. It governs both the steady transonic flows and the upstream moving waves near sonic point. The existence of continuous and discontinuous steady transonic flows when the throat area is either a maximum or a minimum is shown. The existence of standing sonic discontinuities and rarefaction shocks in the transonic flow are noted for the first time. Unlike in the classical gas flows, there are two sonic points and continuous transonic flows are possible only through one of them. The numerical evolution of those transonic waves that have both positive and negative nonlinearity in the same pulse is studied and some comments are made on the local stability of the particular steady flows.