CRITICAL FLOW AND SPURIOUS SOLUTIONS .1. THE CASE OF A SINGLE EQUATION

This paper presents a detailed analysis of the various flow regimes wh ich may occur in channels of variable cross-sectional area through whi ch there flows a compressible fluid. The analysis uses topological met hods which make it possible to draw wide-ranging physical conclusions without actually solving the governing equation. The present paper con stitutes Part I of a wider study aimed at extending considerations in the case of two-phase flow. The flows which occur in practice are mode led 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 equation s which, in the case of critical flow, can be reduced to a set of n co upled ordinary differential equations for the local tangent V = d sigm a/dz to a solution (trajectory) in the phase space sigma boolean OR z. Here sigma is the state-velocity vector of the dependent quantities o f the problem. This reduces the analysis to the well-known methodology used in the field of dynamical systems. Profiting from the fact, prov ed earlier, that the basic topological relations are the same regardle ss of the complexity of the problem expressed by the number n of compo nents of sigma, the present paper concentrates on the simplest possibl e case when n = 1. This corresponds to the adiabatic, one-dimensional flow of a perfect gas with constant specific heats which is governed b y 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 ensemb le of solutions of the problem is induced by the singular points of th e differential equation for V. In the most popular case, this is a sad dle point S. Critical flow is described by the two trajectories crossi ng the saddle point. A saddle point has the property that the directio ns of V at S are radically different from those in its immediate neigh borhood. As a consequence, numerical solutions can never reach the sad dle point by a forward-marching algorithm which, under certain conditi ons, may produce spurious solutions. The latter may be either grossly inaccurate or plain wrong. Prescriptions for avoiding spurious branche s are discussed. The paper further examines the flow regimes which may occur in channels of less conventional profiles. In particular, a cas e of two singular points, a saddle followed by a spiral, leads to a nu mber of flow regimes whose occurrence is not intuitively obvious.