The Compressible Laminar Boundary Layer with Arbitrary Pressure and Surface Temperature Gradients

Abstract

The Karman-Pohlhausen method is extended systematically to cover the hitherto unsolved problem of two-dimensional compressible laminar flow with arbitrary surface temperature and pressure gradients and arbitrary but constant Prandtl Number. The main features of this development are clearly distinguished from special numerical assumptions (e.g., particular boundarylayer profile representations), which are evaluated a posteriori and are subject to modification and refinement. Application of the present method to the calculation of heat transfer and skin friction in a given case involves the numerical integration of two simultaneous ordinary differential equations, which is readily accomplished by hand or with an automatic digital calculator. The theory is well adapted to an investigation of the combined effect of pressure and surface temperature gradients in the compressible boundary layer, for which no other practical method is available. Solutions by the present method are compared with more exact solutions in the special cases for which the latter are available, and the agreement is good in all cases. The special cases used for comparison include skin friction for incompressible retarded flow, skin friction and heat transfer for incompressible flow over a special cylindrical shape, and skin friction and heat transfer for compressible flow over a flat plate, with and without a surface temperature gradient, A sample problem is presented for a circular-arc airfoil at zero angle of attack in supersonic flow, with a specified surface temperature distribution. The local skin-friction and heat-transfer coefficients are determined as a function of distance along the surface and are compared with the results obtained by application of the common flat-plate relations.