An Investigation of the Bursting of Trailing Vortices Using Numerical Simulation

Citation

Beran, Philip Stewart (1989) An Investigation of the Bursting of Trailing Vortices Using Numerical Simulation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/chyb-nk54. https://resolver.caltech.edu/CaltechETD:etd-02012007-105641

Abstract

Solutions of the Navier-Stokes equations are obtained for the flow of an isolated, trailing vortex, and for the swirling flow through a frictionless pipe. In both cases, the flow is assumed to be steady, incompressible and rotationally symmetric. Solutions are computed using Newton's method and Gaussian elimination for a wide range of values of two parameters: Reynolds number, Re, and vortex strength, V. Pseudo-arclength continuation is employed to facilitate the computation of solution points in the parameter space. The numerical procedure is validated through comparison of solutions with solutions obtained in previous investigations for the case of a trailing vortex. Solutions are also compared with results reported by Brown and Lopez (1988) for the case of flow through a pipe.

Solutions of the quasi-cylindrical equations are obtained for the flow of a trailing vortex. Solutions are computed using an explicit, space-marching scheme, and are compared with solutions of the Navier-Stokes equations.

Provided that Re is about 200, or larger, four vortex states are observed.

1. When V is sufficiently small, the flow is entirely supercritical.

2. As V is increased, the flow at an axial station becomes critical and a transition point forms. At the point, the flow departs from an upstream state that is supercritical to a downstream state that is marked by large-amplitude, spatial oscillations of core radius. When Re is large, the downstream state is nearly periodic. The general features of transition are well described by the conjugate-flow theory of Benjamin 1967). Failure of the quasi-cylindrical equations is found to be a necessary and sufficient condition for the existence of a transition point. As V is further increased, the transition point moves upstream. Reversed flow is not observed.

3. Over a narrow range of vortex strengths, a small bubble of reversed flow is observed downstream of the transition point.

4. When V is large, the entire flow is marked by large-amplitude, spatial oscillations of core radius. A transition point is not evident within the computational domain. Typically, large regions of reversed flow are observed.

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