Theoretical and Computational Fluid Dynamics

, Volume 22, Issue 1, pp 37–64 | Cite as

Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape

  • Banavara N. Shashikanth
  • Artan Sheshmani
  • Scott David Kelly
  • Jerrold E. Marsden
Original Article

Abstract

We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot–Savart law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the Lie–Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.

Keywords

Ideal hydrodynamics Hydrodynamical interaction Vortex ring Vorticity Hamiltonian structure Poisson bracket Lie–Poisson bracket 

PACS

47.15.ki 47.10.Df 47.32.cb 

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References

  1. 1.
    Allen J.J., Jouanne, Y., Shashikanth, B.N.: Vortex interaction with a moving sphere. J. Fluid Mech. 587, 337–346Google Scholar
  2. 2.
    Amrouche C., Bernardi C., Dauge M. and Girault V. (1998). Vector potentials in three dimensional non-smooth domains. Math. Methods Appl. Sci. 21: 823–864 MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Arnold V.I. (1966). Sur la géométrie differentielle des groupes de Lie de dimenson infinie et ses applications à l’hydrodynamique des fluids parfaits. Ann. Inst. Fourier Grenoble 16: 319–361 Google Scholar
  4. 4.
    Arnold, V.I., Khesin, B.: Topological Methods in Hydrodynamics. Appl. Math. Sciences, vol. 125 Springer, Heidelberg (1998)Google Scholar
  5. 5.
    Arnold V.I. and Khesin B. (1992). Topological methods in hydrodynamics. Ann. Rev. Fluid Mech. 24: 145–166 CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Batchelor G.K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press, London MATHGoogle Scholar
  7. 7.
    Bendali A., Dominguez J.M. and Gallic S. (1985). A variational approache for the vector potential formulation of the Stokes and Navier-Stokes problems in three-dimensional domains. J. Math. Anal. Appl. 107: 537–560 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Blackmore D. and Knio O. (2001). Hamiltonian structure for vortex filament flows. ZAMM 81: 45–48 Google Scholar
  9. 9.
    Blackmore, D., Knio, O.: Hamiltonian formulation of the dynamics of interaction between a vortex ring and a rigid sphere. In: Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering. ECCOMAS 2000, Barcelona, Spain, 11–14 September (2000)Google Scholar
  10. 10.
    Cantor M. (1975). Perfect fluid flows over R n with asymptotic conditions. J. Funct. Anal. 18: 73–84 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Coutand D. and Shkoller S. (2005). Motion of an elastic solid inside an incompressible visoucs fluid. Arch. Ration. Mech. Anal. 176(1): 25–102 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Desjardins B. and Esteban M.J. (1999). Existence of weak solutions for the motion of a rigid body in a visocus fluid. Arch. Ration. Mech. Anal. 146(1): 59–71 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ebin D.G. and Marsden J.E. (1970). Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92: 102–163 CrossRefMathSciNetGoogle Scholar
  14. 14.
    Hirasaki G.J. and Hellums J.D. (1967). A general formulation of the boundary conditions on the vector potential in three-dimensional hydrodynamics. Quart. Appl. Math. 26(3): 331–342 MathSciNetGoogle Scholar
  15. 15.
    Kanso E., Marsden J.E., Rowley C.W. and Melli-Huber J.B. (2005). Locomotion of articulated bodies in a perfect fluid. J. Nonlinear Sci. 15: 255–289 MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Kelly, S.D.: The mechanics and control of robotic locomotion with applications to aquatic vehicles. Ph.D. Thesis, Institute of Technology, California (1998)Google Scholar
  17. 17.
    Klein R., Majda A.J. and Damodaran K. (1995). Simplified equations for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288: 201–248 MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Knio O.M. and Ting L. (1997). Vortical flow outside a body and sound generation. SIAM J. Appl. Math. 57(4): 972–981 MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lamb, H.: Hydrodynamics. Dover, NY, sixth edition (1932)MATHGoogle Scholar
  20. 20.
    Langer J. and Perline R. (1991). Poisson geometry of the filament equation. J. Nonlinear Sci. 1: 71–94 MATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Lewis D., Marsden J.E., Montgomery R. and Ratiu T.S. (1986). The Hamiltonian structure for dynamic free boundary problems. Phys. D 18: 391–404 MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lighthill M.J. (1956). The image system of a vortex element in a rigid body. Proc. Camb. Phil. Soc. 52: 317–321 MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Ma, Z., Shashikanth, B.N.: Dynamics and control of the system of a 2D rigid circular cylinder and point vortices. In: Proceedings of the American Control Conference. Minneapolis, Minnesota, USA, 14–16 June (2006)Google Scholar
  24. 24.
    Marsden, J.E.: Lectures on Mechanics, vol. 174. London Math. Soc. Lecture Note Ser., vol. 174. Cambridge University Press, London (1992)Google Scholar
  25. 25.
    Marsden J.E., Montgomery R. and Ratiu T. (1984). Gauged Lie–Poisson structures. Contempl. Math. 28: 101–114 MathSciNetGoogle Scholar
  26. 26.
    Marsden, J.E., Montgomery, R., Ratiu, T.S.: Reduction, symmetry and phases in mechanics, vol. 436. Memoirs of the AMS, vol. 436. Am. Math. Soc. Providence (1990)Google Scholar
  27. 27.
    Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, vol. 17. Texts in Applied Mathematics, vol. 17; 1994, 2nd edn. Springer, Heidelberg (1999)Google Scholar
  28. 28.
    Marsden J.E. and Weinstein A. (1983). Coadjoint orbits, vortices and Clebsch variables for incompressible fluids. Phys. D 7: 305–323 CrossRefMathSciNetGoogle Scholar
  29. 29.
    Milne-Thomson L.M. (1996). Theoretical Hydrodynamics, 5th edn. Dover, New York Google Scholar
  30. 30.
    Nirenberg L. and Walker H.F. (1973). The null spaces of elliptic partial differential operators in \({\mathbb{R}}^n\) J. Math. Anal. Appl. 42: 271–301 MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Radford, J.: Symmetry, reduction and swimming in a perfect fluid. Ph.D. Thesis, California Institute of Technology (2003)Google Scholar
  32. 32.
    Saffman P.G. (1992). Vortex Dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge Univ. Press., London Google Scholar
  33. 33.
    Shashikanth B.N. (2005). Poisson brackets for the dynamically interacting system of a 2D rigid boundary and N point vortices: The case of arbitrary smooth cylinder shapes. Reg. Chaotic Dyn. 10(1): 1–14 MATHCrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Shashikanth B.N. (2006). Symmetric pairs of point vortices interacting with a neutrally buoyant 2D circular cylinder. Phys. Fluids 18: 127103 CrossRefADSGoogle Scholar
  35. 35.
    Shashikanth B.N. and Marsden J.E. (2003). Leapfrogging vortexrings: Hamiltonian structure, geometricphases and disete reduction. Fluid Dyn. Res. 33: 333–356 MATHCrossRefADSMathSciNetGoogle Scholar
  36. 36.
    Shashikanth, B.N., Marsden, J.E., Burdick, J.W., Kelly, S.D.: The Hamiltonian structure of a 2D rigid circular cylinder interacting dynamically with N point vortices. Phys. Fluids 14(3), 1214–1227 (2002) (see also Erratum. Phys. of Fluids 14(11), 4099)Google Scholar
  37. 37.
    Ting, L., Klein, R.: Viscous Vortical Flows. Lecture Notes in Physics, vol. 374. Springer, Berlin (1991)Google Scholar
  38. 38.
    Truesdell C. (1954). The Kinematics of Vorticity. Indiana University Press, Bloomington MATHGoogle Scholar
  39. 39.
    Vladimirov, V.A.: Vortical momentum of bounded ideal incompressible fluid flows. J. Appl. Mech. Tech. Phys. 20, 157–163 (1979) (Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No.2, 53–61, (1979))Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Banavara N. Shashikanth
    • 1
  • Artan Sheshmani
    • 2
  • Scott David Kelly
    • 3
  • Jerrold E. Marsden
    • 4
  1. 1.Department of Mechanical EngineeringNew Mexico State UniversityLas CrucesUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Department of Mechanical Engineering and ScienceUniversity of North Carolina at CharlotteCharlotteUSA
  4. 4.Department of Control and Dynamical SystemsCalifornia Institute of TechnologyPasadenaUSA

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