Critical flow and spurious solutions Part II. The case of n equations

Abstract

This paper constitutes a continuation of a paper which dealt with the occurrence of spurious solutions in adiabatic two-phase flows through channels of variable cross-sectional area. These are likely to occur in critical, choked conditions. The problem was straightforward when the mathematical model consisted of a single, ordinary, nonlinear differential equation—which is rare and valid only when the fluid is a perfect gas with constant specific heats. In such cases the solution was sought in the form of an initial-value problem. In the more general, realistic case the canonical form consists of n ⩾ 2 coupled nonlinear equations. This introduces considerable complexity, though the basic topological pattern of the portrait of solutions remains unchanged. Now it becomes necessary to replace the initial-value problem by one with given boundary conditions. Since critical flows always occur in the presence of a singular point in phase space (a saddle point in the case considered), the boundary there becomes movable. In all cases, spurious solutions are avoided by starting the numerical code at the critical point with analytically determined slopes. However, when n ⩾ 2 it becomes necessary to apply an iterative process (“shooting” method) whose details are described.