The steady compressible laminar boundary layer, with arbitrary pressure gradient and uniform wall temperature

Abstract

An approximate integral of the energy equation is derived by expressing the total temperature as a quadratic function of velocity, the Prandtl number being taken as unity. The three coefficients are chosen to satisfy the two temperature boundary conditions and to give the correct temperature profile in the outer part of the boundary layer. A transformation of the normal co-ordinate is applied, which partially reduces the momentum equation to an incompressible form. It is shown that in the transformed co-ordinates the functional relationships between the skin-friction, pressure gradient and shape parameters should be approximately given by their incompressible forms, the effects of compressibility appearing explicitly in certain additional terms in the equation and implicitly in the transformation. Using these correlations, together with the approximate temperature profile, the momentum equation is reduced to a form whose integration requires only two quadratures. The predictions of the theory are compared with three exact solutions. These indicate that the error in the predicted separation position increases with Mach number and is about 20 and 40 % at Mach numbers of 3 and 4 respectively. The detailed distribution of skin-friction is given with similar accuracy. A tentative empirical correction will probably yield greater accuracy in the prediction of separation, so that the error is only about 10 % at a Mach number of 4.