Skip to main content Accessibility help
×
Home

Stability of a coupled body–vortex system

  • EVA KANSO (a1) and BABAK GHAEMI OSKOUEI (a1)

Abstract

This paper considers the dynamics of a rigid body interacting with point vortices in a perfect fluid. The fluid velocity is obtained using the classical complex variables theory and conformal transformations. The equations of motion of the solid–fluid system are formulated in terms of the solid variables and the position of the point vortices only. These equations are applied to study the dynamic interaction of an elliptic cylinder with vortex pairs because of its relevance to understanding the swimming of fish in an ambient vorticity field. Two families of relative equilibria are found: moving Föppl equilibria; and equilibria along the ellipse's axis of symmetry (the axis perpendicular to the direction of motion). The two families of relative equilibria are similar to those present in the classical problem of flow past a fixed body, but their stability differs significantly from the classical ones.

Copyright

References

Hide All
Batchelor, G. K. 1970 An Introduction to Fluid Dynamics. Cambridge University Press.
Beal, D. N., Hover, F. S., Triantafyllou, M. S., Liao, J. C. & Lauder, G. V. 2006 Passive propulsion in vortex wakes. J. Fluid Mech. 549, 385402.
Borisov, A. V. & Mamaev, I. S. 2003 An integrability of the problem on motion of cylinder and vortex in the ideal fluid, Regular Chaotic Dyn. 8, 163166.
Borisov, A. V., Mamaev, I. S. & Ramodanov, S. M. 2007 Dynamic interaction of point vortices and a two-dimensional cylinder. J. Math. Phy. 48, 19.
Crowdy, D. & Marshall, J. 2005 Analytical formulae for the Kirchhoff–Routh path function in multiply connected domains. Proc. R. Soc. Lond. A 461, 24772501.
Crowdy, D. G., Surana, A. & Yick, K.-Y. 2007 The irrotational flow generated by two planar stirrers in inviscid fluid. Phys. Fluids 19, 018103.
Hill, D. 1998 Vortex Dynamics in Wake Models. PhD thesis, California Institute of Technology.
Kanso, E., Marsden, J. E., Rowley, C. W. & Melli-Huber, J. B. 2005 Locomotion of articulated bodies in a Perfect fluid. Intl J. Nonlinear Sci. 15, 255289.
Lamb, H. 1932 Hydrodynamics. Dover.
Liao, J. C., Beal, D. N., Lauder, G. V. & Triantafyllou, M. S. 2003 Fish exploiting vortices decrease muscle activity. Science 302, 15661569.
Lin, C. C. 1941 a On the motion of vortices in two-dimensions - I. existence of the Kirchhoff–Routh function. Proc. Natl Acad. Sci. 27, 570575.
Lin, C. C. 1941 b On the motion of vortices in two-dimensions - II. some further investigations on the Kirchhoff–Routh function. Proc. Natl Acad. Sci. 27, 575577.
Muskhelishvili, N. I. 1953 Singular Integral Equations. Noordhoff, Groningen, Holland.
Nair, S. & Kanso, E. 2007 Hydrodynamically coupled rigid bodies. J. Fluid Mech. 592, 393411.
Saffman, P. G. 1992, Vortex Dynamics. Cambridge.
Shashikanth, B. N. 2005 Poisson brackets for the dynamically interacting system of a 2D rigid cylinder and N point vortices: the case of arbitrary smooth cylinder shapes. Regular Chaotic Dyn. 10, 110.
Shashikanth, B. & Marsden, J. E. 2003 Leapfrogging vortex rings: Hamiltonian structure, geometric phases and discrete reduction. Fluid Dyn. Res. 33, 333356.
Shashikanth, B. N., Marsden, J. E., Burdick, J. W. & Kelly, S. D. 2002 The Hamiltonian structure of a 2D rigid circular cylinder interacting dynamically with N Point vortices. Phys. Fluids 14, 12141227.
Silverman, R. 1974 Introductory Complex Analysis. Dover.
Wang, Q. X. 2004 Interaction of two circular cylinders in inviscid fluid. Phys. Fluids 16, 4412.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed