An approximate integral method for calculating the incompressible laminar boundary layer on an infinite swept wing on which velocity and suction distribution are arbitrary

Abstract

Summary. The method described in this Report gives an approximate solution for the cross-flow velocity in an incompressible laminar boundary layer on an infinite swept wing. The cross flow is defined as that velocity in the boundary layer in a direction normal to the outer flow streamline. The two-dimensional flow in the chordwise direction is assumed known. The boundary-layer equations are expressed in a non-dimensional form. Head\\'s method, which is based on the momentum and energy equations and is used to solve the two-dimensional flow, is briefly described. From the non-dimensional boundary-layer equations, an equation involving the cross-flow profile directly is obtained. This equation is then integrated throughout the boundary layer giving two integral equations. Typical cross-flow shapes have been used to calculate functions which when used in conjunction with the two integral equations and the boundary condition at the wall enable the cross flow to be determined for arbitrary velocity and suction distributions. The stability criterion for three-dimensional flow is expressed in the notation of this method and is a simple condition on the cross flow. The method is then applied to the upper and lower surfaces of a wing, suction distributions being calculated to maintain stability at each point of the wing. For the lower surface comparison is made with an independent method (Ref. 1) in the adverse pressure-gradient region. Two more sets of comparisons are made, each with an exact solution. The method is tested on a number of similar solutions, and on an exact solution obtained by American workers (Ref. 8). Consideration is also given to some difficulties encountered in the use of the method.