Structural stability of three-dimensional vortex flows

Topological approaches which permit the description of three-dimensional flowfields in terms of nonlinear dynamical systems are discussed. The basic idea is to treat unsteady flow phenomena as steady topological flow structures (TPS). This requires tracking the evolution of the TPS over time by means of empirical measurements of locally measured quantities which can be correlated with changes in the TPS. For two-dimensional flows, the changes that occur are either a local or a global bifurcation, forming, e.g., a separation bubble, vortex flows and interactions of vortices. Two-dimensional flows are actually structurally unstable, transient states of three-dimensional flows. Analytical techniques are defined for modeling the formation of elementary topological structures and applied to describing TPS which appear in Rayleigh-Benard convection. It is noted that the comparison of results for different nonlinear dynamical systems depends on the characterization of topologically equivalent structures and structural changes and their relationship to an invariant flow quantity.