Blasius Problem and Falkner-Skan model: Töpfer's Algorithm and its Extension
For researchers in fluid dynamics, this work provides an extended numerical method for solving boundary layer problems with scaling invariance, though it is an incremental extension of existing techniques.
This paper reviews Töpfer's algorithm for non-iterative numerical solution of the Blasius problem and extends it to a general class of problems, including the Falkner-Skan model. The extended algorithm successfully reproduces known reverse flow solutions, with numerical data in good agreement with previous studies.
In this paper, we review the so-called Töpfer 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 Töpfer'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.