The self-similar topology of passive interfaces advected by two-dimensional turbulent-like flows

We study the topology, and in particular the self-similar and space-filling properties of the topology of line-interfaces passively advected by five different 2-D turbulent-like velocity fields. Special attention is given to three fundamental aspects of the flow: the time unsteadiness, 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 wavenumber space derived from the studies of Weierstrass and related functions. The methods of analysis are based on a classification of interfacial scaling topologies in terms of K- and H- fractals, and on two interfacial scaling exponents, the Kolmogorov capacity DK and the dimension D introduced by Fung and Vassilicos [Phys. Fluids 11, 2725 (1991)] who conjectured that D≳1 implies that the interface is H-fractal. An argument is presented (in the Appendix) to show that D≳1 is a necessary condition for the evolving interface to be H-fractal through the action of the flow, and that D≳1 is also sufficient provided that no isolated regions exist where the flow velocity is either unbounded or undefined in finite time. D is interpreted to be a degree of H-fractality and is different from the Hausdorff dimension DH. In all our flows, steady and unsteady, interfaces in particular realisations of the flow reach a non-space-filling steady self-similar state where D and DK are both constant in time even though the interface continues to be advected 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 DK generally decreases with unsteadiness where the interfacial topology is dominated by spirals, and that DK increases with unsteadiness where the interfacial topology is dominated by tendrils. In those flows with larger number of modes, DK is a non-increasing function of unsteadiness and a decreasing function of the exponent p of the flow’s self-similar energy spectrum E(k)∼k−p. DK’s decreasing dependences on unsteadiness and the exponent p can be explained by the presence of spirals in the eddy regions of the flow. The values of D and DK and their dependence on unsteadiness can change significantly only by changing the distribution of wavenumbers in wavenumber space while keeping the phases and energy spectrum constant.