On the critical point of the Crocco-Lees mixing theory in the laminar near wake

Abstract

The application of the multimoment integral method to the study of the base pressure behind a supersonic vehicle is examined. The primary purpose is to understand the exact nature of the Crocco-Lees critical point, which provides a uniqueness condition in the problem. The analysis is carried out mainly in Poincare phase space, where the singularities of the differential equations are investigated. Two singular curves are found. The one which is physically meaningful for the near wake flow is located downstream of the rear stagnation point. This singular curve consists of saddle points which yield “wake” solutions, and focal points or saddle-foci. Only those saddle points which yield “wake” solutions correspond to Crocco-Lees critical points. Thus, the integration of the differential equations should be started from a saddle point both for the upstream and the downstream solutions. The current analysis then requires that one deal with only a one-parameter (freestream Mach number) family of solutions, rather than the two-parameter (freestream Mach number and Reynolds number based on thickness of the viscous shear layer at the rear stagnation point) family of solutions obtained in previous works. Finally, this paper clarifies anomalous details of previous numerical investigations carried out for the near wake by Webb, Golik, Vogenitz, and Lees.