An extension of the transpired skin-friction equation to compressible turbulent boundary layers

Abstract

W INJECTION or suction of fluid into the turbulent boundary layer has proved over the years to be an effective means of controlling the properties of fluid flowing over a surface. The result of this injection or suction of fluid is to modify the velocity and temperature distributions through the boundary layer so that the drag and the heat transfer are either reduced or increased. Most of the studies on this topic, however, deal with incompressible flow. The majority of the results on supersonic flow comes from the work of Squire and his students at Cambridge University [l-S]. These results are given for Mach mtmbers up to 3.6 at various injection rates. Full tables of measured profiles are presented and expressions for the law of the wall and the law of wake are proposed. As for the incompressible case, the law of the wall is obtained by straightforward application of the mixing-length theory [l]. Unfortunately, in those works the skin-friction coefficients are evaluated by means of the momentum-integral equation which tends to be very inaccurate, thus definitive checks on the proposed law of the wall are impossible. In a previous paper [6], the present author has proposed a skin-friction equation for transpired incompressible turbulent boundary layers. This equation is much less sensitive than the momentum-integral equation to small variations in the flow parameters so providing much more reliable results. The aim of this work is to extend this skin-friction equation to compressible flow. The approach of transforming a compressible turbulent boundary layer into a corresponding incompressible flow has been pursued by several authors in the past with reasonable success. The underlying idea is to reduce the complex system of partial differential equations which governs the motion of a compressible flow into a simpler system, such as the system of equations for an incompressible flow. Solutions of the simpler system can then be transformed back to predict the behaviour of the solutions of the complex one. This procedure, however, still presents the diSlcult problem of selecting the right transformation parameters. An alternative approach is to use the concept of generalized velocity. Using the assumption that the mixing length is proportional to the wall distance and that the shear stress in the fluid is constant and equal to its value at the wall, Van Driest [7] solved the equations of motion and obtained, for an adiabatic flow urn