On the geometry of two-dimensional slices of irregular level sets in turbulent flows Page: 4 of 16


The successes of these proposals, however, must be tempered by the host
of turbulent-flow issues that cannot be addressed by correlation/spectral/moment
analyses that classical descriptions have provided, that are also typically limited to
uniform and isotropic flows. Part of the difficulty can be traced to the fact that in-
formation offered by such analyses is not invertible. Given a process, its spectrum,
for example, is specified. Knowledge of the spectrum, alone, yields only limited
other information about the process.
Such turbulent-flow issues often pose questions regarding the geometrical prop-
erties of turbulence-generated fields. Examples of such issues include, heat and mass
transfer in turbulent flows; mixing and chemically-reacting turbulent flows, requir-
ing information about the surface-to-volume ratio of scalar level sets; aerooptics
and optical-beam propagation through a turbulent medium, which (absent addi-
tional modeling and assumptions) require geometrical information about index-of-
refraction gradients; aeroacoustics and weak- and strong-wave propagation through
turbulence, which rely on the geometrical properties of both scalar and velocity
fields; and many others. While important progress has been made in these phe-
nomena as well, which has derived considerable benefit from classical turbulence
theory, in almost all cases, additional, often ad hoc, assumptions, variations, and
models are employed, often implicitly.
More recently, the realization that Direct Numerical Simulation (DNS) methods
cannot hope to represent turbulent phenomena at the high Reynolds numbers of
interest, especially when coupled to other physical processes that must be computed
in parallel, has led to the quest for sub-grid-scale (SGS) models that describe the
behavior of scales smaller than those that can potentially be numerically resolved.
Significantly, classical models do not yield the necessary SGS models, which also
require additional structure and assumptions, as would be employed in Large Eddy
Simulations (LES) calculations. Geometrical scaling information that would permit
an extension of descriptions founded on an underresolved range of scales would
facilitate this quest.
An important contribution that addressed some geometric-scaling issues was
made by B. Mandelbrot (1975, 1982), who proposed that (power-law/self-similar)
fractals could be used to describe level-set behavior of scalar and other turbulence-
generated fields. A considerable body of experimental and modeling work by many
investigators followed these proposals, as discussed by Sreenivasan (1991, 1994).
An extension of these proposals, necessitated by recent experiments and direct
numerical simulations, will be discussed below.

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Catrakis, H.J.; Cook, A.W.; Dimotakis, P.E. & Patton, J.M. On the geometry of two-dimensional slices of irregular level sets in turbulent flows, article, March 20, 1998; California. (digital.library.unt.edu/ark:/67531/metadc664330/m1/4/ocr/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.

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