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On a class of unsteady boundary layers of finite extent

Published online by Cambridge University Press:  26 April 2006

W. R. C. Phillips
W. R. C. Phillips
Affiliation:
Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699-5725, USA Present address: Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2935, USA.
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Abstract

A class of unsteady boundary layers that form on flat extensible surfaces of finite but increasing length in otherwise stagnant surroundings is considered. The surface length $\overline{R}$ is assumed to grow with time as tp where p > 0 and the velocity at any location $0 \le \overline{r} \le \overline{R}$ on the surface as tp-np-1r-n, where n [ges ] 0. The problem is cast into similarity variables and the governing parabolic differential equation shown to exhibit, for various combinations of n and p, regions of mixed mathematical diffusivity and reversals in the direction of convection of vorticity. Equations depicting such behaviour are usually termed singular parabolic and are here classified as follows: type-0, in which the mathematical diffusivity may be either positive or mixed but in which there are no reversals in the direction of convection of vorticity; type-1, in which the mathematical diffusivity may be either positive or mixed but in which there are reversals in the direction of convection of vorticity. Both types are shown to occur. Moreover while type-0 flows occur only when n = 1 and form with an unsteady separated stagnation point at the origin, type-1 flows occur only for 0 [les ] n < 1 and form with a steady stagnation point at the origin. Type-1 flows are further characterized by boundary layers with zero displacement thickness both at the origin and leading edge. Because singular parabolic equations require two initial conditions plus boundary conditions to ensure uniqueness, they are here treated numerically in a manner akin to elliptic boundary value problems. A successive-approximation implicit scheme was thus used and a wide range of cases solved in the parameter range n ∈ [0,1], p ∈ (0,2]. Amongst other things, it is shown that type-0 flows have lower drag than their type-1 counterparts. It is further shown that the drag on a flat rigid surface of finite length moving in its own plane at constant velocity and being continuously produced at the origin is higher than on a corresponding length of either a semi-infinite surface likewise produced or a semi-infinite plate in an aligned uniform stream; however if the surface is extensible and $n > \frac{1}{2}$ the converse is true.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 319 , 25 July 1996 , pp. 151 - 170
Copyright
© 1996 Cambridge University Press

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