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The motion of thin-cored vortex filaments : the equations of motion and their solution for some special cases

Citation

Lough, Michael F. (1995) The motion of thin-cored vortex filaments : the equations of motion and their solution for some special cases. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-10172007-093152

Abstract

This thesis looks at the motion of vortex filaments, which are regions in a fluid flow where the vorticity field, equal to the curl of the velocity field, is negligible outside a cylindrical type tube -- the filament. The vortex filament is said to be thin-cored if the radius of the tube is much smaller than any axial length scale along the filament. This thin core assumption allows the motion of the filament to be described by the motion of the centerline of the tube, when it is coupled to the internal core dynamics via an asymptotic matching procedure. Our studies of the motion and dynamics of such structures can be grouped into three topics: (1) analyses of equations of motion for thin-cored vortex filaments, (2) an analysis of the linear stability of a vortex ring moving along the axis of a pipe and (3) the construction of finite amplitude wave solutions for the shape of the centerline of planar vortex filament.

Our main thrust in the first topic is to show that the "new" equations derived by Klein and Majda (1991) are merely a reformulation of some of the more well known equations for vortex filament motion; in particular we show that their equations can be obtained by a linearization of the well known cut-off equation. The cut-off equation has been used by a number of authors (e.g., Crow (1970), Widnall and Sullivan (1973), etc.) to analyse problems in vortex motion, and a systematic justification for the equation was provided by Moore and Saffman (1972), who showed the equation was asymptotically correct.

With regard to the second topic, questions of stability naturally arise when one considers coherent structures, such as vortex rings. The stability of a vortex ring in an unbounded fluid was first examined by Thomson (1883) who found that the ring was stable to infinitesimal perturbations (the vortex ring being a model for the so-called indestructible atom). Widnall and Sullivan (1973) reconsidered the stability problem, but retained terms of higher order in the small parameter (i.e., the ratio of core radius to ring radius). Their results indicated a spurious instability where theonly unstable perturbations were those with wavelengths that were comparable with the core radius (a situation for which the cut-off equation is inapplicable). For the case where the ring moves inside a cylindrical pipe, along its axis, only the effect of the wall on the speed of the ring had been computed to date (Raja Gopal (1963) and Brasseur (1979)). From the theoretical point of view, the effect of the wall on the stability of the ring has, until now, been unknown. We show that the wall induces an instability on the vortex ring characterized by a tendancy for the ring to tilt out of its original plane.

The so-called local induction equation, for vortex filament motion, corresponds to the zero core radius limit of the cut-off equation. Hasimoto (1972) showed that there was a direct relationship between the local induction equation and the cubic Schrodinger equation, a completely integrable equation. He also computed the corresponding soliton solution of the equation. Kida (1981) computed a general solution of the equation having periodic shape parameters. Both of these solutions are finite amplitude wave solutions for vortex filaments whose motion is governed by the local induction equation. Unfortunately, the local induction equation is known to admit some solutions that are unphysical, a fact which lends credence to the belief that the general solutions are also unphysical. Accordingly, it is important to see whether equations containing more physics also admit finite amplitude wave solutions. Kelvin (1880) obtained a solution of the linearized problem for periodic waves of infinitesimal amplitude that lay in a rotating plane. (In fact, Kelvin's solutions described infinitesimal filament waves having a general three-dimensional structure.) We numerically compute such plane wave solutions to the full nonlinear equations and continue the solution to finite amplitude.

Item Type:Thesis (Dissertation (Ph.D.))
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Thesis Availability:Restricted to Caltech community only
Research Advisor(s):
  • Saffman, Philip G.
Thesis Committee:
  • Unknown, Unknown
Defense Date:24 August 1994
Record Number:CaltechETD:etd-10172007-093152
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-10172007-093152
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4137
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:26 Oct 2007
Last Modified:26 Dec 2012 03:05

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