Shape and Stability of Two-Dimensional Uniform Vorticity Regions

Bifurcation;Schwarz function;stability;vorticity

Kamm, James Russell ; Saffman, Philip G.

University:California Institute of Technology

Department:Engineering and Applied Science

关键词: Bifurcation;    Schwarz function;    stability;    vorticity;   

Others  :  https://thesis.library.caltech.edu/2782/1/Kamm_jr_1987.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

The steady shapes, linear stability, and energetics of regions of uniform, constant vorticity in an incompressible, inviscid fluid are investigated. The method of Schwarz functions as introduced by Meiron, Saffman and Schatzman [1984] is used in the mathematical formulation of these problems.

Numerical and analytical analyses are provided for several configurations. For the single vortex in strained and rotating flow fields, we find new solutions that bifurcate from the branch of steady elliptical solutions. These nonelliptical steady states are determined to be linearly unstable. We examine the corotating vortex pair and numerically confirm the theoretical results of Saffman and Szeto [1980], relating linear stability characteristics to energetics. The stability properties of the infinite single array of vortices are quantified. The pairing instability is found to be the most unstable subharmonic disturbance, and the existence of an area-dependent superharmonic instability (Saffman and Szeto [1981]) is numerically confirmed. These results are exhibited qualitatively by an elliptical vortex model. Lastly, we study the effects of unequal area on the stability of the infinite staggered double array of vortices. We numerically verify the results of the perturbation analysis of Jiménez [1986b] by showing that the characteristic subharmonic stability "cross" persists for vortex streets of finite but unequal areas.