THE SELF-SIMILAR TOPOLOGY OF PASSIVE INTERFACES ADVECTED BY 2-DIMENSIONAL TURBULENT-LIKE FLOWS

We study the topology, and in particular the self-similar and space-fi lling properties of the topology of Line-interfaces passively advected by five different 2-D turbulent-like velocity fields. Special attenti on is given to three fundamental aspects of the flow: the time unstead iness, the classification of local spatial flow structure in terms of hyperbolic and elliptic points borrowed from the study of phase spaces in dynamical systems and a classification of flow structure in wavenu mber space derived from the studies of Weierstrass and related functio ns. The methods of analysis are based on a classification of interfaci al scaling topologies in term's of K- and H- fractals, and on two inte rfacial scaling exponents, the Kolmogorov capacity D-K and the dimensi on D introduced by Fung and Vassilicos [Phys. Fluids 11, 2725 (1991)] who conjectured that D>1 implies that the interface is H-fractal. An a rgument is presented (in the Appendix) to show that D>1 is a necessary condition for the evolving interface to be H-fractal through the acti on of the flow, and that D>1 is also sufficient provided that no isola ted regions exist where the flow velocity is either unbounded or undef ined in finite time. D is interpreted to be a degree of H-fractality a nd is different from the Hausdorff dimension D-H. In all our flows, st eady and unsteady, interfaces in particular realisations of the flow r each a non-space-filling steady self-similar state where D and D-K are both constant in time even though the interface continues to be advec ted and deformed by the flow. It is found that D is equal to 1 in 2-D steady flows and always increases with unsteadiness, that D-K generall y decreases with unsteadiness where the interfacial topology is domina ted by spirals, and that D-K increases with unsteadiness where the int erfacial topology is dominated by tendrils. In those flows with larger number of modes, D-K is a non-increasing function of unsteadiness and a decreasing function of the exponent p of the flow's self-similar en ergy spectrum E(k)similar to k(-p). D-K's decreasing dependences on un steadiness and the exponent p can be explained by the presence of spir als in the eddy regions of the flow. The values of D and D-K and their dependence on unsteadiness can change significantly only by changing other distribution of wavenumbers in wavenumber space while keeping th e phases and energy spectrum constant. (C) 1995 American Institute of Physics.