Blasius Problem and Falkner-Skan model: T\"opfer's Algorithm and its Extension

TL;DR

This paper reviews and extends T"opfer's algorithm for solving boundary layer problems like the Blasius and Falkner-Skan models, providing a non-iterative solution method and new numerical results for complex cases.

Contribution

The paper introduces an extended iterative version of T"opfer's algorithm applicable to a broader class of boundary layer problems with non-homogeneous boundary conditions.

Findings

Successful application to Falkner-Skan model with multiple solutions

Numerical results agree with previous studies

Extended algorithm handles non-homogeneous boundary conditions

Abstract

In this paper, we review the so-called T\"opfer algorithm that allows us to find a non-iterative numerical solution of the Blasius problem, by solving a related initial value problem and applying a scaling transformation. Moreover, we remark that the applicability of this algorithm can be extended to any given problem, provided that the governing equation and the initial conditions are invariant under a scaling group of point transformations and that the asymptotic boundary condition is non-homogeneous. Then, we describe an iterative extension of T\"opfer's algorithm that can be applied to a general class of problems. Finally, we solve the Falkner-Skan model, for values of the parameter where multiple solutions are admitted, and report original numerical results, in particular data related to the famous reverse flow solutions by Stewartson. The numerical data obtained by the extended algorithm are in good agreement with those obtained in previous studies.