Part I. Vortex dynamics in wake models. Part II. Wave generation

Author: Hill, David J.

Year: 1998

Degree: Dissertation (Ph.D.)

Advisor: Saffman, Philip G.

Committee Members: Saffman, Philip G.; Leonard, Anthony; Pullin, Dale Ian; Meiron, Daniel I.

Option: Applied And Computational Mathematics

DOI: 10.7907/81N1-6X49

Abstract

In Part I, steady wakes in inviscid fluid are constructed and investigated using the techniques of vortex dynamics. As a generalization of Foppl's flow past a circular cylinder [5], a steady solution is given for flow past an elliptical cylinder of arbitrary aspect ratio (perpendicular or parallel to the flow at infinity) with a bound wake of two symmetric recirculating eddies in the form of a point vortex pair. Linear stability analysis predicts an asymmetric instability and the symmetric nonlinear evolution is discussed in terms of a Kirchhoff-Routh path function. The wake behind a sphere is represented by a thin cored vortex ring of arbitrary internal structure. Steady configurations are obtained and long-wavelength perturbations to the ring centerline identify a tilting instability. A generalization of the Kirchhoff Routh function to an axisymmetric flow consisting of vortex rings and a body is presented. Using conformal maps and point vortices, translating symmetric two-dimensional bubbles with a vortex pair wake are constructed. An instability in which the bubble and vortex pair tilt away from each other is found as well as a symmetric oscillatory instability. The cross-section of a trailing vortex pair immersed in a cross stream shear is represented by two counter-rotating vortex patches. Numerical and analytical analyses are provided. The method of Schwarz functions as introduced by Meiron, Saffman and Schatzman [13] is used in the computation and stability analysis of steady patch shapes. Excellent agreement is obtained using an elliptical patch model. An instability essentially isolated to a single patch is identified, the nonlinear evolution of the elliptical patch model suggests that the patch whose fluid elements rotate against the shear will be destroyed.

Part II examines a possible mechanism for the generation of water waves which arises from the instability of an initially planar free surface in the presence of a parallel, sheared, inviscid flow. A two-dimensional steady flow comprised of exponential profiles representing both wind and a drift layer in the water is infinitesimally perturbed. The resulting Rayleigh equation is analytically solved by mean of Hyper-geometric functions and the dispersion relation is implicitly defined as solutions of a transcendental equation; stability boundaries are determined and growth rates are calculated. Comparisons are made with the simpler model of Caponi et al. [2] which uses piecewise linear profiles.

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