The calculation of the compressible turbulent boundary layer in an arbitrary pressure gradient a correlation of certain previous methods

Abstract

Several methods are available for the calculation of the compressible turbulent boundary-layer. The earlier methods of Tucker (1951) and Young (1953) involve numerical integration, while the more recent methods of Englert (1957), Mager (1957), Reshotko and Tucker (1957), Culick and Hill (1958), Michel (1959) and Spence (1959) derive explicit formulae. It is found that the earlier methods may be developed analytically and that all the methods may then be correlated on the basis of Equation (1) of the Summary. The methods of Tucker, Young and Michel start directly from the momentum equation for the compressible turbulent boundary-layer. Tucker and Young assume the profile shape to be independent of pressure gradient, a simplification which has been found useful in incompressible flow; Michel assumes a constant profile in any given flow but discriminates between favourable and adverse gradients. On the other hand, subsequint to a suggestion by Van Le (1953), the later methods use, with some empiricism, a form of the theoretical transformation of Stewartson in order to convert existing solutions of the boundary layer equation from incompressible to compressible flow. In all the methods except Mager's the variation of skin friction with Mach number is based indirectly on the limited amount of available experimental evidence. All the methods except, those of Reshotko and Tucker and of Spence restrict consideration to an insulated wall, and the present paper is also thus restricted. Initial regions of laminar flow are omitted from the present paper for clarity, but may readily be treated by the standard techniques.