Generalised methods for the calculation of the laminar compressible boundary-layer characteristics with heat transfer and non-uniform pressure distribution

Abstract

Three related methods are presented for calculating the skin friction and other characteristics in the laminar boundary layer on a non-adiabatic wall of constant temperature in the presence of pressure gradients. The first of these methods is a direct extension of the method given by Young for zero heat transfer. The assumptions made in the analysis restrict this 'first simple' method to flows with relatively small changes in free stream pressure and Mach number so that this method is applicable to the boundary layer on a thin sharp nosed wing at small angles of incidence. However, there is a need for a method of similar simplicity which can be applied to round nosed wings, that is to flows starting from a stagnation point and accelerating to supersonic velocities downstream and hence involving large changes in free stream pressure and Mach number. To meet this need the restrictive assumptions of the 'first simple' method have been relaxed and correction factors incorporated to allow for the effects of the pressure gradient on certain parameters previously assumed to be unaffected by the pressure gradient. The resulting 'complete' method involves the solution of a single quadrature in a step-by-step manner and is applicable with a high degree of accuracy to a very wide range of flows with Mach numbers from zero to five, with heat transfer to or from the surface, and with both favourable and adverse pressure gradients. For flows with favourable pressure gradients, the application of the correction factors may in part be relaxed and the 'complete' method then reduces to that which has been termed the 'second simple' method All three methods are applicable to values of the Prandtl number (σ) and the temperature-viscosity relationship index (ω) near, but not necessarily equal to unity. Several cases have been considered for which it is possible to compare the results given by the three methods with exact solutions and with the results given by other approximate methods. The 'complete' method is demonstrated to have advantages over other approximate methods on the grounds of accuracy, relative simplicity and breadth of application. It is also, in principle, applicable to cases with non-uniform wall temperatures although its accuracy in this application is not here examined.