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Shallow flow past a cylinder: transition phenomena at low Reynolds number

Published online by Cambridge University Press:  27 September 2005

H. FU
Affiliation:
Lehigh University, Department of Mechanical Engineering and Mechanics, 354 Packard Laboratory, 19 Memorial Drive West, Bethlehem, PA 18015, USA dor0@lehigh.edu
D. ROCKWELL
Affiliation:
Lehigh University, Department of Mechanical Engineering and Mechanics, 354 Packard Laboratory, 19 Memorial Drive West, Bethlehem, PA 18015, USA dor0@lehigh.edu

Abstract

The unstable structure of the near wake of a vertical cylinder, in a fully developed laminar free-surface layer, is characterized in relation to the unsteadiness of the horseshoe (necklace) vortex system about the upstream surface of the cylinder. A cinema technique of high-image-density particle image velocimetry allows space–time imaging of the critical regions of the flow and thereby wholefield representations of patterns of the flow structure, in conjunction with spectra and cross-spectra at a large number of points over the flow domain.

The unsteadiness of the near wake was examined over a range of wake stability parameter $S \,{=}\, c_{f}{D/h}_{w}$, in which $c_{f}$ is the bed friction coefficient, $D$ is the cylinder diameter, and $h_{w}$ is water depth; this range of $S$ was selected such that the classical Káarmán mode of vortex formation remained completely suppressed. Within this range, increase of the Reynolds number, based on depth $h_{w}$ of the shallow layer and $D$ of the cylinder, yielded the onset and development of an instability mode that takes the form of a varicose, as opposed to a sinuous, pattern of vortices. It is related to the unsteadiness of the horseshoe (necklace) vortex system on the upstream side of the cylinder. The process of vortex formation in the near wake is interpreted in terms of multiple, coexisting layers of vorticity due to both the horseshoe vortices and the vorticity layer associated with separation from the cylinder.

Furthermore, it is demonstrated that when the near wake is stable at sufficiently low values of the Reynolds number, based on depth $h_{w}$ and cylinder diameter $D$, application of external perturbations via small-amplitude rotational oscillations of the cylinder, at the most unstable frequency of the separating shear layers, can lead to destabilization of the near wake in a sinuous mode of small-scale vortical structures. Moreover, this type of rotational perturbation of the cylinder, applied at the expected frequency of large-scale Kármán vortex formation, can also yield destabilization of the near wake in this mode. These types of perturbations lead to substantial alterations of the patterns of vorticity and streamline topology, as well as Reynolds stresses and entrainment velocities of the separating shear layers, along the bed, relative to patterns above the bed.

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© 2005 Cambridge University Press

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