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Wakes behind two-dimensional surface obstacles in turbulent boundary layers

Published online by Cambridge University Press:  29 March 2006

J. Counihan ,
J. C. R. Hunt  and
P. S. Jackson
J. Counihan
Affiliation:
Central Electricity Research Laboratories, Leatherhead, Surrey
J. C. R. Hunt
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge Present address: Central Laboratories, Ministry of Works, Lower Hutt, New Zealand.
P. S. Jackson
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

By making simple assumptions, an analytical theory is deduced for the mean velocity behind a two-dimensional obstacle (of height h) placed on a rigid plane over which flows a turbulent boundary layer (of thickness δ). It is assumed that h [Gt ] δ, and that the wake can be divided into three regions. The velocity deficit − u is greatest in the two regions in which the change in shear stress is important, a wall region (W) close to the wall and a mixing region (M) spreading from the top of the obstacle. Above these is the external region (E) in which the velocity field is an inviscid perturbation on the incident boundary-layer velocity, which is taken to have a power-law profile U(y) = U∞(y − y1)n/δn, where n [Gt ] 1. In (M), assuming that an eddy viscosity (= KhU(h)) can be defined for the perturbed flow in terms of the incident boundary-layer flow and that the velocity is self-preserving, it is found that u(x,y) has the form $\frac{u}{U(h)} = \frac{ C }{Kh^2U^2(h)} \frac{f(n)}{x/h},\;\;\;\; {\rm where}\;\;\;\; \eta = (y/h)/[Kx/h]^{1/(n+2)}$, and the constant which defines the strength of the wake is $C = \int^\infty_0 y^U(y)(u-u_E)dy$, where u = uE(x, y) as y → 0 in region (E).

In region (W), u(y) is proportional to In y. By considering a large control surface enclosing the obstacle it is shown that the constant of the wake flow is not simply related to the drag of the obstacle, but is equal to the sum of the couple on the obstacle and an integral of the pressure field on the surface near the body.

New wind-tunnel measurements of mean and turbulent velocities and Reynolds stresses in the wake behind a two-dimensional rectangular block on a roughened surface are presented. The turbulent boundary layer is artificially developed by well-established methods (Counihan 1969) in such a way that δ = 8h. These measurements are compared with the theory, with other wind-tunnel measurements and also with full-scale measurements of the wind behind windbreaks.

It is found that the theory describes the distribution of mean velocity reasonably well, in particular the (x/h)−1 decay law is well confirmed. The theory gives the correct self-preserving form for the distribution of Reynolds stress and the maximum increase of the mean-square turbulent velocity is found to decay downstream approximately as $ (\frac{x}{h})^{- \frac{3}{2}} $ in accordance with the theory. The theory also suggests that the velocity deficit is affected by the roughness of the terrain (as measured by the roughness length y0) in proportion to In (h/y0), and there seems to be some experimental support for this hypothesis.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 64 , Issue 3 , 8 July 1974 , pp. 529 - 564
Copyright
© 1974 Cambridge University Press

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Footnotes

Also: Department of Engineering.

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