The fractal facets of turbulence
Abstract
Speculations abound that several facets of fully developed turbulent flows are fractals. Although the earlier leading work of Mandelbrot (1974, 1975) suggests that these speculations, initiated largely by himself, are plausible, no effort has yet been made to put them on firmer ground by, resorting to actual measurements in turbulent shear flows. This work is an attempt at filling this gap. In particular, we examine the following questions: (a) Is the turbulent/nonturbulent interface a selfsimilar fractal, and (if so) what is its fractal dimension ? Does this quantity differ from one class of flows to another? (b) Are constantproperty surfaces (such as the isovelocity and isoconcentration surfaces) in fully developed flows fractals? What are their fractal dimensions? (c) Do dissipative structures in fully developed turbulence form a fractal set? What is the fractal dimension of this set? Answers to these questions (and others to be less fully discussed here) are interesting because they bring the theory of fractals closer to application to turbulence and shed new light on some classical problems in turbulence  for example, the growth of material lines in a turbulent environment. The other feature of this work is that it tries to quantify the seemingly complicated geometric aspects of turbulent flows, a feature that has not received its proper share of attention. The overwhelming conclusion of this work is that several aspects of turbulence can be described roughly by fractals, and that their fractal dimensions can be measured. However, it is not clear how (or whether), given the dimensions for several of its facets, one can solve (up to a useful accuracy) the inverse problem of reconstructing the original set (that is, the turbulent flow itself).
 Publication:

Journal of Fluid Mechanics
 Pub Date:
 December 1986
 DOI:
 10.1017/S0022112086001209
 Bibcode:
 1986JFM...173..357S
 Keywords:

 Computational Fluid Dynamics;
 Fractals;
 Turbulent Flow;
 Boundary Value Problems;
 Energy Dissipation;
 Flat Plates;
 Flow Visualization;
 Fluid Boundaries;
 Fluid Mechanics and Heat Transfer