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

Shashikanth, Banavara N. and Sheshmani, Artan and Kelly, Scott David and Marsden, Jerrold E. (2008) Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape. Theoretical and Computational Fluid Dynamics, 22 (1). pp. 37-64. ISSN 0935-4964. http://resolver.caltech.edu/CaltechAUTHORS:20101005-085628821

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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 themethod of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00162-007-0065-y DOIUNSPECIFIED
http://www.springerlink.com/content/d72p0848j4602755/PublisherUNSPECIFIED
Additional Information:© 2007 Springer-Verlag. Received: 19 December 2006; Accepted: 14 August 2007; Published online: 21 September 2007. Communicated by D.D. Holm. Research of SDK was partially supported by the NSF grants CMMI 04-49319 and ECCS 05-01407 and the research of JEM was partially supported by NSF grant IIS-0413078 and the California Institute of Technology.
Funders:
Funding AgencyGrant Number
NSFCMMI 04-49319
NSFECCS 05-01407
NSFIIS-0413078
CaltechUNSPECIFIED
Subject Keywords:ideal hydrodynamics; hydrodynamical interaction; vortex ring; Vorticity; Hamiltonian structure; Poisson bracket; Lie-Poisson bracket
Classification Code:PACS: 47.15.ki, 47.10.Df, 47.32.cb
Record Number:CaltechAUTHORS:20101005-085628821
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20101005-085628821
Official Citation:Shashikanth, B., A. Sheshmani, et al. (2008). "Hamiltonian structure for a neutrally buoyant rigid body interacting with <i>N</i> vortex rings of arbitrary shape: the case of arbitrary smooth body shape." Theoretical and Computational Fluid Dynamics 22(1): 37-64.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20292
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:16 Nov 2010 23:41
Last Modified:26 Dec 2012 12:30

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