Vortex dynamics in a viscous incompressible fluid
Abstract
An asymptotic theory of weakly interacting vortices is derived from the two dimensional incompressible Navier-Stokes equations. The scheme extends current perturbation theories by including arbitrary unperturbed core vorticity distributions and viscous dissipation of interaction energy. Viscous effects are shown to provide a radial component of induced velocity between two cores of the same sign, as well as the usual diffusive spreading of the cores. The new model is illustrated by determining the evolution for several elementary vortex flows: viz., the vortex pair, the vortex couple, the polygonal vortex array, the linear array and the vortex street. Also the stability of several of these arrangements is investigated: viz., the polygonal vortex array, the linear array and the Karman vortex street. The vortex pair is not an equilibrium configurations in the viscous dynamics. However the vortex couple, linear vortex array, and vortex street with arbitrary stagger are equilibrium configurations in the viscous dynamics. The polygonal vortex array is an equilibrium configuration on when five vortices are considered. The linear vortex array is unstable and the growth rate is amplified in the viscous dynamics. The pentagonal array is also unstable. When core growth is suppressed, for the Karman vortex street, the inclusion of viscous effects provides asymptotic stability to any street whose spacing ratio is sufficiently close to that identified by von Karman as the only linearly neutrally stable arrangement.
Degree
Ph.D.
Advisors
Williams, Purdue University.
Subject Area
Aerospace materials|Fluid dynamics|Gases
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