*Abstract*

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,
D_{d}(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,
D_{d}(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/d_{j}=275) of a
turbulent jet at Re=4.5x10^{3}.

Fig. 3.10b
Jet-fluid concentration on the far-field (z/d_{j}=275) of a
turbulent jet at Re=9.0x10^{3}.

Fig. 3.10c
Jet-fluid concentration on the far-field (z/d_{j}=275) of a
turbulent jet at Re=18x10^{3}.

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.