Topics in Vortex Methods for the Computation of Three- and Two-Dimensional Incompressible Unsteady Flows

Citation

Winckelmans, Grégoire Stéphane (1989) Topics in Vortex Methods for the Computation of Three- and Two-Dimensional Incompressible Unsteady Flows. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/19HD-DF80. https://resolver.caltech.edu/CaltechETD:etd-11032003-112216

Abstract

Contributions to vortex methods for the computation of incompressible unsteady flows are presented. Three methods are investigated, both theoretically and numerically.

The first method to be considered is the inviscid method of vortex filaments in three dimensions, and the following topics are presented: (a) review of the method of regularized vortex filaments and of convergence results for multiple-filament computations, (b) modeling of a vortex tube by a single filament convected with the regularized Biot-Savart velocity applied on the centerline: velocity of the thin filament vortex ring and dispersion relation of the rectilinear filament, and (c) development of a new regularization of the Biot-Savart law that reproduces the lowest mode dispersion relation of the rectilinear vortex tube in the range of large to medium wavelengths.

Next the method of vortex particles in three dimensions is investigated, and the following contributions are discussed: (a) review of the method of singular vortex particles: investigation of different evolution equations for the particle strength vector and weak solutions of the vorticity equation, (b) review of the method of regularized vortex particles and of convergence results, and introduction of a new algebraic smoothing with convergence properties as good as those of Gaussian smoothing, (c) development of a new viscous method in which viscous diffusion is taken into account by a scheme that redistributes the particle strength vectors, and application of the method to the computation of the fusion of two vortex rings at Re = 400, and (d) investigation of the particle method with respect to the conservation laws and derivation of new expressions for the evaluation of the quadratic diagnostics: energy, helicity and enstrophy.

The third method considered is the method of contour dynamics in two dimensions. The particular efforts presented are (a) review of the classical inviscid method and development of a new regularized version of the method, (b) development of a new vector particle version of the method, both singular and regularized: the method of particles of vorticity gradient, (c) development of a viscous version of the method of regularized particles and application of the method to computation of the reconnection of two vortex patches of same sign vorticity, and (d) investigation of the particle method with respect to the conservation laws and derivation of new expressions for the evaluation of linear and quadratic diagnostics.

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