Nonsimilar Solutions of the Compressible-Laminar Boundary-Layer Equations With Applications to the Upstream-Transpiration Cooling Problem

A new method is presented for predicting the boundary-layer characteristics downstream of the porous region of an injectioncooled surface. The method consists of a general scheme for obtaining nonsimilar solutions of the compressible-laminarboundary-layer equations and is formulated along the following lines. The viscous domain is divided into N curvilinear strips. The governing equations are then integrated along the coordinate normal to the body from the surface to the boundary of each strip. As a result, one obtains a set of independent integrodifferential relations. The integration is carried out by expressing the integrands as polynomials, the coefficients of which are functions of the unknown values of the velocity and temperature on the strip boundaries as well as of the imposed boundary condition a t the wall and at the outer edge. After the integration is performed, a set of ordinary first-order differential equations is obtained. The set of equations may be solved for given initial conditions by a numerical integration scheme such as the Runge-Kutta method. Several numerical examples of interest are presented.