An asymptotic outer solution applied to the Keller box method

Abstract

The Keller box method (“Numerical Solutions of Partial Differential Equations, Vol. 2” (B. Hubbard Ed.), pp. 327–350, Academic Press, New York, 1970) was applied to incompressible flow past a flat plate to demonstrate that the basic computation region must extend outward from the wall until the outer boundary conditions are effectively obtained. The Keller box method was modified to include an asymptotic outer solution for the case of the self-similar solution for compressible flow in a boundary layer. Initial application of the basic and modified Keller box methods to incompressible flow past a flat plate showed similar rates of convergence but smaller RMS error for the same basic range of the independent variable when the asymptotic outer solution is applied. Furthermore, extension of the solution beyond the range of the independent variable for the numerical solution using the resulting asymptotic solution produced RMS error at least as small as the RMS error on the range of the numerical solution. Also, when the asymptotic solution was applied, a smaller range of independent variables could be used in the numerical solution to obtain the same RMS error. Numerical results for compressible flow were qualitatively the same as for the case with the incompressible velocity profile except the rate of iterative convergence was slightly slower. Application of asymptotic outer solution for incompressible flow at a two dimensional stagnation point produced similar results with smaller relative improvements. For compressible flow with smaller favorable pressure gradients than the stagnation point and with adverse pressure gradients, significant improvements were again obtained. Examination of the errors associated with the asymptotic solution reveals that greatest success is obtained for flows with thicker boundary layers and shows that the boundary layer at a two dimensional stagnation point is too thin for small error in the asymptotic solution. Despite relatively large errors in the asymptotic solutions for boundary layer in strong favorable pressure gradients where the boundary layer is thin, the boundary layer solutions generally showed improvement in error and reduction in computation times.