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

Citation

Hill, David J. (1998) Part I. Vortex dynamics in wake models. Part II. Wave generation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/81N1-6X49. https://resolver.caltech.edu/CaltechETD:etd-04052007-141032

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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