Chaotic Motion in Unsteady Vortical Flows

Abstract

Application of nonlinear dynamical systems theory to hydrodynamics started with the work by Henon (1966) on ABC flows (short for) Arnold, Beltrami & Childress, see an extensive study by Dombre et al. (1986 on this subject, who numerically studied the fluid particle motion in the ABC flow and showed that the flow contains the KAM tori and chaotic motion of the Smale horseshoe type. On a separate line, the investigation on chaotic motion of some vortical flows induced by point vortices initiated by the work of (1975, 1978, 1979), who exhibited global stochastic properties (ergodicity and mixing) of a four-vortex system, and continued by further work of (1979, 1980, 1983, 1985, 1989), who showed chaotic motion for two-dimensional vortical flows by numerically computing Poincare sections. In the case of three vortices, it is shown that the three-vortex problem is itself integrable but the induced vortical flow (motion of fluid particles) is in general a non-integrable problem. Application of homoclinic orbit theory to chaotic mixing and transport in some particular vortical flows, on the other hand, was first done by (1983), and later expounded in detail in the case for a vortex pair byRom-Kedar, Leonard and Wiggins (1990). For a systematic review on this aspect, one is referred to (1983) for background materials on nearly integrable Hamiltonian systems; (1989) for an overview and examples; (1992) where mathematical tools for chaotic transport (tangle dynamics) are summarized; and the monograph by (1991) where pattern formations are emphasized.KeywordsUnstable ManifoldChaotic MotionFluid ParticlePoint VortexIntegrable Hamiltonian SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.