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Physics of Fluids / Volume 24 / Issue 2 / ARTICLES / Laminar Flows

Phys. Fluids 24, 023601 (2012); http://dx.doi.org/10.1063/1.3682097 (12 pages)

Strain-vorticity induced secondary motion in shallow flows

Leon P. J. Kamp

Eindhoven University of Technology, Department of Applied Physics, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands

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(Received 23 December 2010; accepted 3 January 2012; published online 6 February 2012)

Deviations from two-dimensionality of a shallow flow that is dominated by bottom friction are quantified in terms of the spatial distribution of strain and vorticity as described by the Okubo-Weiss function. This result is based on a Poisson equation for the pressure in a quasi-horizontal (primary) flow. It is shown that the Okubo-Weiss function specifies vertical pressure gradients, which for their part drive vertical (secondary) motion. An asymptotic expansion of these gradients based on the smallness of the vertical to horizontal scale ratio demonstrates that the sign and magnitude of secondary circulation inside the fluid layer is dictated by the signs and magnitude of the Okubo-Weiss function. As a consequence of this, secondary motion as well as nonzero horizontal divergence do also depend on the strength, i.e., the Reynolds number of the primary flow. The theory is exemplified by two generic vortical structures (monopolar and dipolar structures). Most importantly, the theory can be applied to more complicated turbulent shallow flows in order to assess the degree of two-dimensionality using measurements of the free-surface flow only.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. QUASI-LINEAR THEORY OF SECONDARY MOTION IN Q2D SHALLOW FLOWS
  3. VERTICAL MOTION INSIDE AN AXISYMMETRIC MONOPOLAR VORTEX
  4. VERTICAL MOTION INSIDE A LAMB-CHAPLYGIN DIPOLE
  5. CONCLUSIONS

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KEYWORDS and PACS

Keywords

friction, Poisson equation, turbulence, vortices

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3682097

PUBLICATION DATA

ISSN

1070-6631 (print)  
1089-7666 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    J. Schumacher and B. Eckhardt, “Clustering dynamics of Lagrangian tracers in free-surface flows,” Phys. Rev. E 66, 017303 (2002).

    R. A. D. Akkermans, A. R. Cieslik, L. P. J. Kamp, H. J. H. Clercx, and G. J. F. van Heijst, “The three-dimensional structure of an electromagnetically generated dipolar vortex in a shallow fluid layer,” Phys. Fluids 20, 116601 (2008)PHFLE6000020000011116601000001.

    M. P. Satijn, A. W. Cense, R. Verzicco, H. J. H. Clercx, and G. J. F. van Heijst, “Three-dimensional structure and decay properties of vortices in shallow fluid layers,” Phys. Fluids 13, 1932 (2001)PHFLE6000013000007001932000001.

    N. T. Ouellette and J. P. Gollub, “Curvature fields, topology, and dynamics of spatiotemporal chaos,” Phys. Rev. Lett. 99, 194502 (2007).

    M. Duran-Matute, J. Albagnac, L. P. J. Kamp, and G. J. F. van Heijst, “Dynamics and structure of decaying shallow dipolar vortices,” Phys. Fluids 22, 116606 (2010)PHFLE6000022000011116606000001.

    D. Sous, N. Bonneton, and J. Sommeria, “Turbulent vortex dipoles in a shallow water layer,” Phys. Fluids 16, 2886 (2004)PHFLE6000016000008002886000001.


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