Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Similarity Solutions of Two Dimensional Turbulent Boundary Layers

Version 1 : Received: 7 February 2022 / Approved: 8 February 2022 / Online: 8 February 2022 (10:37:08 CET)

How to cite: Sun, B. Similarity Solutions of Two Dimensional Turbulent Boundary Layers. Preprints 2022, 2022020105. https://doi.org/10.20944/preprints202202.0105.v1 Sun, B. Similarity Solutions of Two Dimensional Turbulent Boundary Layers. Preprints 2022, 2022020105. https://doi.org/10.20944/preprints202202.0105.v1

Abstract

The exact similarity solutions (also called as special exact solutions) of two dimensional laminar boundary layer were obtained by Blasius in 1908, however, no similarity solutions for two dimensional turbulent boundary layers have been reported in the literature. With the help of dimensional analysis and invariance principle, Prandtl mixing length $\ell=\kappa y$ for one dimensional turbulent boundary layer is extended to $\ell(x,y)=\kappa y (1-\frac{y}{\delta})\sqrt{\frac{\nu}{U\delta}} $ for the two dimensional turbulent boundary layers, furthermore, with a similarity transformation, we successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation $f'''+ ff''+\beta(1-f'^2)+\kappa^2[\eta^2(1-\eta)^2f''|f''|]'=0$. As an application, similarity solutions of the two dimensional turbulent boundary layer on a flat plate at zero incidence have been studied in detail. To solve the ordinary differential equation numerically, a complete Maple code is provided.

Keywords

Turbulent boundary layers; laminar boundary layers; similarity transformation; similarity solution; Prandtl mixing length; Reynolds number

Subject

Physical Sciences, Fluids and Plasmas Physics

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