3. An Asymptotic Approach to Separation and Stability Problems of a Transonic Boundary Layer
Oleg S. Ryzhov · Society for Industrial and Applied Mathematics eBooks · 1993
3.1. The State of the Art The asymptotic approach under discussion is based on the notion of an interacting boundary layer with the self-induced pressure gradient not known in advance. More specifically, an extension of the triple-deck theory invented by Neiland, Stewartson, and Messiter will be considered and employed to predict stability properties of the Blasius boundary layer on a thermally insulated flat plate in the transonic range of external velocities. However, before we set about to solve this problem it is instrumental to briefly outline the fundamental features of the triple deck with regard to subsonic and supersonic boundary layers. In these cases the unique small parameter —based on the Reynolds number where , , and are the density, viscosity, and velocity of an oncoming uniform stream, and is the characteristic length of the body, respectively—comes into play and, importantly, it can be ruled out from the final formulation of the boundary-value problem through a suitable definition of physical quantities. The sketch of the triple deck is shown in Fig. 3.1 along with the corresponding scalings of different sublayers in longitudinal and transverse directions in terms of ϵ. The outer region I is occupied by a potential flow; disturbances in the main deck II are inviscid in nature in spite of the essentially viscous pattern of the initial Blasius boundary layer; vorticity waves in the thin wall sublayer III are governed by the balance between the inertial forces, pressure, and viscous stresses. If the difference is properly included in scalings for both independent variables and unknown functions, the final formulation of the boundary value problem also becomes free from dependence on the Mach number .