Jc. Vassilicos et Jch. Fung, THE SELF-SIMILAR TOPOLOGY OF PASSIVE INTERFACES ADVECTED BY 2-DIMENSIONAL TURBULENT-LIKE FLOWS, Physics of fluids, 7(8), 1995, pp. 1970-1998
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.