DIMOTAKIS, P. E., and CATRAKIS, H.J. 1997 "Turbulence, fractals, and mixing," NATO Advanced Studies Institute series, Mixing: Chaos and Turbulence (7--20 July 1996, Corsica, France), GALCIT Report FM91-1.


Proposals and experimental evidence, from both numerical simulations and laboratory experiments, regarding the behavior of level sets in turbulent flows are reviewed. Isoscalar surfaces in turbulent flows, at least in liquid-phase turbulent jets, where extensive experiments have been undertaken, appear to have a geometry that is more complex than (constant-D) fractal. Their description requires an extension of the original, scale-invariant, fractal framework that can be cast in terms of a variable (scale-dependent) coverage dimension, Dd(l). The extension to a scale-dependent framework allows level-set coverage statistics to be related to other quantities of interest. In addition to the pdf of point-spacings (in 1-D), it can be related to the scale-dependent surface-to-volume (perimeter-to-area in 2-D) ratio, as well as the distribution of distances to the level set. The application of this framework to the study of turbulent-jet mixing indicates that isoscalar geometric measures are both threshold and Reynolds-number dependent. As regards mixing, the analysis facilitated by the new tools, as well as by other criteria, indicates enhanced mixing with increasing Reynolds number, at least for the range of Reynolds numbers investigated. This results in a progressively less-complex level-set geometry, at least in liquid-phase turbulent jets, with increasing Reynolds number. In liquid-phase turbulent jets, the spacings in one-dimensional records, as well as the size distribution of individual "islands"; and "lakes"; in two-dimensional isoscalar slices, are found in accord with lognormal statistics in the inner-scale range. The coverage dimension, Dd(l), derived from such sets is also in accord with lognormal statistics, in the inner-scale range. Preliminary three-dimensional (2-D space + time) isoscalar-surface data provide further evidence of a complex level-set geometrical structure in scalar fields generated by turbulence, at least in the case of turbulent jets.

Fig. 3.10a Jet-fluid concentration in the far-field (z/dj=275) of a turbulent jet at Re=4.5x103.

Fig. 3.10b Jet-fluid concentration on the far-field (z/dj=275) of a turbulent jet at Re=9.0x103.

Fig. 3.10c Jet-fluid concentration on the far-field (z/dj=275) of a turbulent jet at Re=18x103.

Fig. 5.2 3-D rendering of an outer (low scalar threshold) isosurface portion, from (2+1)-D space-time measurements of the jet-fluid concentration in the far field of a turbulent jet.