The growth of compressible turbulent boundary layers on isothermal and adiabatic walls

Abstract

A recent study of velocity profiles in the turbulent boundary layer on a flat plate at high Mach numbers has suggested the formula for relating the form parameter H to its incompressible value Hi. This formula is used in conjunction with a generalisation of the Stewartson-Illingworth transformation to reduce the left-hand side of the integral momentum equation (including pressure gradients) to incompressible form, for arbitrary values of the flat-plate recovery factor (or of the turbulent Prandtl number), which is about 0.89 for air. The intermediate temperature formula of Eckert is used to relate the skin friction to its incompressible value, for which a 1/nth power law is used. The momentum equation is then integrable when Hi is given a constant value, which may be chosen to secure agreement in the incompressible case with Maskell's quadrature for the momentum thickness θ, or with the author's modification of the latter. The integration is carried out for the cases of zero heat transfer and of a constant-temperature wall, the details of the Stewartson-type transformation being slightly different in the two cases, but the final forms are the same. The constant B depends in the second case on the ratio of wall to total stream temperature, but F(M) does not. This is of the same form as the equation obtained by Reshotko and Tucker in the special case of unit recovery factor. A numerical example shows results very close to those obtained by Young's method, which involves two quadratures, in a particular case at zero heat transfer. Some calculations have also been made to illustrate the effect of cooling the wall on the boundary-layer growth in a pressure gradient.