On the two-dimensional viscous counterpart of the one-dimensional sonic throat

Abstract

An analysis of the full, compressible, non-adiabatic boundary-layer equations is presented to describe the so-called ‘throat’ formed when a two-dimensional viscous layer, interacting with a supersonic inviscid outer stream, is accelerated or decelerated through sonic velocity defined in some mean sense. The basic analysis differs from previous momentum integral theories in that the dynamics of the viscous layer is described by the exact local expressions for the streamwise gradients of the flow variables that obtain from the boundary-layer conservation equations, rather than on streamwise derivatives of integral properties of these equations. The theory is then used to develop an extensive analogy with the classical analysis of the throat in the inviscid quasi one-dimensional streamtube. The theory shows that a single integral constraint exists at the throat, which relates the velocity and temperature profiles in the viscous layer to the motion of the inviscid outer flow. One consequence of this constraint is that, for a one-parameter family of profile shapes, the solution can be started at the throat station by specifying only a single variable, the free stream Reynolds number based on the physical thickness of the viscous layer at the throat station. For the hypersonic near wake, this simplification permits one to obtain an approximate solution for the downstream flow without first solving the detailed motion in the base recirculation region. The paper ends with a discussion of the numerical results for the Stewartson family of wake-like profiles.