A Numerical Solution of the Three Dimensional Turbulent Boundary Layer Equations

Abstract

The present investigation is concerned with the computation of three-dimensional turbulent boundary layers. A numerical method has been developed to solve the three-dimensional boundary layer equations using an iterative scheme based essentially on the Crank-Nicolson finite difference approximation. The scheme also employs a streamline-type transformation which enables the individual velocity profiles to be iterated independently of each other and improving the efficiency of the calculation. The effective viscosity is computed from the mixing length concept and an empirical correlation for the outer layer. The logarithmic law of the wall is used as the effective wall condition. A listing of a computer program written in Fortran IV, to calculate boundary layer development using this method is also included. Extensive comparisons -the present theory with both experiment and alternative theories have been included. Two-dimensional flows have been calculated with reasonable success, predictions for which compare favourably with calculations based on Head’s entrainment approach, and two severe cases were treated completely. In the first the pressure gradient was suddenly removed from an equilibrium layer, and in second the flow was maintained in a near-separating condition. The pseudo-three-dimensional flows considered show that crossflow angles can be treated quite successfully while in three-dimensional comparison, even though the crossflow is predicted well, the crossflow angle tends to be significantly underestimated. This two tree-dimensional turbulent boundary layers calculated provide good overall agreement with experiment. The present work provides a firm basis on which to further investigate the three-dimensional turbulent boundary layer and the enclosed program will provide a useful tool for predicting such flows. It is felt however that this effective viscosity model used in the outer layer should be more broadly based by considering more experimental configurations for the purpose of empirical correlation. A great benefit will be obtained overall by considering this problem even on a two-dimensional basis. Nevertheless, the present scheme is capable of coping adequately with varying types of boundary layer development in both two and three dimensions.