Accurate benchmark results of Blasius boundary layer problem using a leaping Taylors series that converges for all real values

TL;DR

This paper introduces a novel leaping Taylor series method with shifted centers to accurately solve the Blasius boundary layer problem across all relevant ta values, overcoming convergence issues of traditional series.

Contribution

The paper presents a new approach using shifted Taylor series expansions with iterative IVP solutions to achieve convergence for high ta, providing benchmark results for boundary layer parameters.

Findings

Achieved convergent solutions for high ta values

Provided benchmark results for boundary layer parameters

Demonstrated effectiveness of leaping Taylor series method

Abstract

Blasius boundary layer solution is a Maclaurin series expansion of the function \(f(\eta)\), which has convergence problems when evaluating for higher values of \(\eta\) due to a singularity present at \(\eta\approx-5.69\). In this paper we are introducing an accurate solution to \(f(\eta)\) using Taylor's series expansions with progressively shifted centers of expansion(Leaping centers). Each series is solved as an IVP with the three initial values computed from solution of the previous series, so the gap between the centers of two consecutive expansions is selected from within the convergence disc of the first series. The last series is formed such that it is convergent for a reasonable high \(\eta\) value needed for implementing the boundary condition at infinity. The present methodology uses Newton-Raphson method to compute the value of the unknown initial condition viz. \(f''(0)\) in an iterative manner. Benchmark accurate results of different parameters of flat plate boundary layer with no slip and slip boundary conditions have been reported in this paper.