Scale distributions and fractal dimensions in turbulence

A new geometric framework connecting scale distributions to coverage statistics is employed to analyze level sets arising in turbulence as well as in other phenomena. A 1D formalism is described and applied to Poisson, lognormal, and power-law statistics. A d-dimensional generalization is also presented. Level sets of 2D spatial measurements of jet-fluid concentration in turbulent jets are analyzed to compute scale distributions and fractal dimensions. Lognormal statistics are used to model the level sets at inner scales. The results are in accord with data from other turbulent flows

Shape Complexity in Turbulence

The shape complexity of irregular surfaces is quantified by a dimensionless area-volume measure. A joint distribution of shape complexity and size is found for level-set islands and lakes in two-dimensional slices of the scalar field of liquid-phase turbulent jets, with complexity values increasing with size. A well-defined power law, over 3 decades in size (6 decades in area), is found for the shape complexity distribution. Such properties are important in various phenomena that rely on large area-volume ratios of surfaces or interfaces, such as turbulent mixing and combustion

Mixing in turbulent jets: scalar measures and isosurface geometry

Experiments have been conducted to investigate mixing and the geometry of scalar isosurfaces in turbulent jets. Specifically, we have obtained high-resolution, high-signal-to-noise-ratio images of the jet-fluid concentration in the far field of round, liquid-phase, turbulent jets, in the Reynolds number range 4.5 × 10^3 ≤ Re ≤ 18 × 10^3, using laser-induced-fluorescence imaging techniques. Analysis of these data indicates that this Reynolds-number range spans a mixing transition in the far field of turbulent jets. This is manifested in the probability-density function of the scalar field, as well as in measures of the scalar isosurfaces. Classical as well as fractal measures of these isosurfaces have been computed, from small to large spatial scales, and are found to be functions of both scalar threshold and Reynolds number. The coverage of level sets of jet-fluid concentration in the two-dimensional images is found to possess a scale-dependent-fractal dimension that increases continuously with increasing scale, from near unity, at the smallest scales, to 2, at the largest scales. The geometry of the scalar isosurfaces is, therefore, more complex than power-law fractal, exhibiting an increasing complexity with increasing scale. This behaviour necessitates a scale-dependent generalization of power-law-fractal geometry. A connection between scale-dependent-fractal geometry and the distribution of scales is established and used to compute the distribution of spatial scales in the flow

Turbulence, fractals, and mixing

Proposals and experiment 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(λ). 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(λ), 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

Mixing and the Geometry of Isosurfaces in Turbulent Jets

Experiments have been conducted to investigate mixing and the geometry of scalar isosurfaces in turbulent jets. Specifically, images of the jet-fluid concentration in the far-field of round, liquid-phase, turbulent jets have been recorded at high resolution and signal-to-noise ratio using laser-induced-fluorescence digital-imaging techniques, in the Reynolds number range 4.5 x 10³ ≤ Re ≤ 18 x 10³. Analysis of these data indicates that this Reynolds-number range spans a mixing transition in the far field of turbulent jets. This is manifested in the probability-density function of the scalar field, as well as in other scalar-field and scalar-isosurface measures. Classical as well as fractal measures of the isosurfaces have been computed, from small to large spatial scales, and are found to be functions of both scalar threshold and Reynolds number. The coverage of level sets of jet-fluid concentration in the two-dimensional images is found to possess a scale-dependent-fractal dimension that increases continuously with increasing scale, from near unity, at the smallest scales, to 2, at the largest scales. The geometry of the scalar isosurfaces is, therefore, more complex than power-law fractal, exhibiting an increasing complexity with increasing scale. This behavior necessitates a scale-dependent generalization of power-law-fractal geometry. A connection between scale-dependent-fractal geometry and the distribution of scales is established and used to compute the distribution of spatial scales in the flow. A lognormal model of scales is proposed. The data also indicate a lognormal distribution of size of the isoscalar islands and lakes, and a powerlaw distribution of shape complexity, with values of the latter that increase with increasing size.</p

IMECE2004-59917 TURBULENT FLUID INTERFACES AND MIXING EFFICIENCY: DYNAMICS, OPTIMIZATION, AND REGULARIZATION

ABSTRACT In many fluid engineering devices, improved design and performance require knowledge of the dynamics of turbulent fluid interfaces. Depending on the device, and the application, the performance may be more sensitive to the large scales or the small scales of the interfaces. We distinguish between cases depending on whether the surface area of, or the volume enclosed by, the interfaces is practically more relevant. For turbulent interfaces, the surface area is dominated by the small scales whereas the volume enclosed is dominated by the large scales. We examine the interfacial dynamics in separated flows and we demonstrate the differences in the area-volume behavior in the context of mixing. Resolution effects on the interfacial behavior reveal that the mixing efficiency is strongly dominated by the large-scale interfacial properties. This has broad implications, at least for fluid-mixing devices, for the development of flow-prediction and flow-control techniques