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An unsteady point vortex method for coupled fluid–solid problems

  • Sébastien Michelin
  • Stefan G. Llewellyn Smith
Open Access
Original Article

Abstract

A method is proposed for the study of the two-dimensional coupled motion of a general sharp-edged solid body and a surrounding inviscid flow. The formation of vorticity at the body’s edges is accounted for by the shedding at each corner of point vortices whose intensity is adjusted at each time step to satisfy the regularity condition on the flow at the generating corner. The irreversible nature of vortex shedding is included in the model by requiring the vortices’ intensity to vary monotonically in time. A conservation of linear momentum argument is provided for the equation of motion of these point vortices (Brown–Michael equation). The forces and torques applied on the solid body are computed as explicit functions of the solid body velocity and the vortices’ position and intensity, thereby providing an explicit formulation of the vortex–solid coupled problem as a set of non-linear ordinary differential equations. The example of a falling card in a fluid initially at rest is then studied using this method. The stability of broadside-on fall is analysed and the shedding of vorticity from both plate edges is shown to destabilize this position, consistent with experimental studies and numerical simulations of this problem. The reduced-order representation of the fluid motion in terms of point vortices is used to understand the physical origin of this destabilization.

Keywords

Fluid–solid interaction Point vortex Vortex shedding 

PACS

47.15.ki 47.63.mc 

Notes

Acknowledgments

This work was funded by NSF award CTS-0133978. We are grateful to Professor J. B. Keller for introducing us to this problem and to Professors H. K. Cheng and D. I. Pullin for helpful conversations. We are also grateful to the referees for their helpful comments.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Sébastien Michelin
    • 1
    • 2
  • Stefan G. Llewellyn Smith
    • 1
  1. 1.Department of Mechanical and Aerospace EngineeringJacobs School of Engineering, UCSDLa JollaUSA
  2. 2.Ecole Nationale Supérieure des Mines de ParisParis Cedex 06France

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