Critical flow and spurious solutions Part I. The case of a single equation

Abstract

This paper presents a detailed analysis of the various flow regimes which may occur in channels of variable cross-sectional area through which there flows a compressible fluid. The analysis uses topological methods which make it possible to draw wide-ranging physical conclusions without actually solving the governing equation. The present paper constitutes Part I of a wider study aimed at extending considerations in the case of two-phase flow. The flows which occur in practice are modeled on a one-dimensional version of the conservation laws augmented by appropriate closure conditions and equations of state. The resulting mathematical model is a set of n coupled partial differential equations which, in the case of critical flow, can be reduced to a set of n coupled ordinary differential equations for the local tangent V = dσ/d z to a solution (trajectory) in the phase space σ ∪ z. Here σ is the state-velocity vector of the dependent quantities of the problem. This reduces the analysis to the well-known methodology used in the field of dynamical systems. Profiting from the fact, proved earlier, that the basic topological relations are the same regardless of the complexity of the problem expressed by the number n of components of σ, the present paper concentrates on the simplest possible case when n = 1. This corresponds to the adiabatic, one-dimensional flow of a perfect gas with constant specific heats which is governed by a single uncoupled equation in the ( M 2, z) projection, where M is the Mach number. The essential thesis of this paper is that the ensemble of solutions of the problem is induced by the singular points of the differential equation for V. In the most popular case, this is a saddle point S. Critical flow is described by the two trajectories crossing the saddle point. A saddle point has the property that the directions of V at S are radically different from those in its immediate neighborhood. As a consequence, numerical solutions can never reach the saddle point by a forward-marching algorithm which, under certain conditions, may produce spurious solutions. The latter may be either grossly inaccurate or plain wrong. Prescriptions for avoiding spurious branches are discussed. The paper further examines the flow regimes which may occur in channels of less conventional profiles. In particular, a case of two singular points, a saddle followed by a spiral, leads to a number of flow regimes whose occurrence is not intuitively obvious.