J. Fluid Mech. (1984), vol. 149, pp. 339-363 Printed in Great Britain 339 On a class of compressible laminar boundary-layer flows and the solution behaviour near separation By ANTONIOS LIAKOPOULOS AND CHEN-CHI HSU Department of Engineering Sciences, University of Florida, Gainesville, Florida 3261 1 (Received 16 December 1983 and in revised form 29 June 1984) A class of compressible laminar boundary-layer flows subject to adverse pressure gradients of different magnitude is studied using a finite-element-differential method in which the assumed solutions are represented by classical cubic spline functions. The numerical integration process for the reduced initial-value problem has been carried out directly to at least one integration step upstream of the separation point, and very accurate numerical results have been obtained for a large number of integration steps extremely close to separation. The skin-friction and heat-transfer coefficients for nearly zero-heat-transfer, cooled-wall and heated-wall cases, computed under the assumption of constant Prandtl number Pr = 1 .O as well as Pr = 0.72, have clearly exhibited the same distinctive behaviour near separation. It is deduced that Buckmaster’s series expansions for the solution near separation, derived on the assumptions of cooled wall and Pr = 1.0, are valid for all the cases considered. By matching the numerical results with Buckmaster’s expansions, accurate distributions of skin friction and heat transfer have been obtained up to the separation point. Moreover, the importance of Prandtl number on the solution is evidenced from the numerical results presented. 1. Introduction The development of a boundary layer under the action of a sharp adverse pressure gradient has been of special interest in compressible-flow analysis mainly because of its connection to shock-wave-boundary-layer interaction problems. It is well known that numerical methods for parabolic problems can provide accurate solutions of the governing equations for most of the boundary-layer flow region. However, close to separation the accuracy of the numerical results deteriorates to the extent that the true solution characteristics are lost and an interactive boundary-layer approach is necessary for a complete solution. Consequently, the accurate computation of the skin friction and heat transfer close to separation is a challenging problem. Moreover, an accurate solution near separation may provide valuable information for the solution of the Navier-Stokes equations involving a point of zero skin friction. Therefore a great deal of attention has been focused on resolving the true characteristics of the boundary -layer solutions near the separation point. Two decades ago, Stewartson (1962) investigated the solution of a laminar compressible boundary layer near a point of zero skin friction. Following closely the approach introduced for the incompressible case (Goldstein 1948 ; Stewartson 1958), Stewartson concluded that a general compressible laminar boundary layer can develop a singularity at a point of zero skin friction only if the heat transfer at that point is zero. This conclusion was supported by the numerical results reported earlier by Poots (1960), and was not contradicted by those of Curle (1958). However, later