Investigation of the fractal structure of jets and plumes

Sedimentology

For several decades, sedimentologists have had difficulty in obtaining an efficient index of particle form that can be used to specify adequately irregular morphology of sedimentary particles. Mandelbrot has suggested the use of the fractal dimension as a single value estimate of form, in order to characterize morphologically closed loops of an irregular nature. The concept of fractal dimension derives from Richardson's unpublished suggestion that a stable linear relationship appears when the logarithm of the perimeter estimate of an irregular outline is plotted against the logarithm of the unit of measurement (step length). Decreases in step length result in an increase in perimeter by a constant weight (b) for particles whose morphological variations are the same at all measurement scales (self-similarity). The fractal dimension (D) equals 1.0-(b), where b is the slope coefficient of the best-fitting linear regression of the plot. The value of D lies between 1.0 and 2.0, with ...

Nonlinear Dynamics, 2019

In this work, we reported there are two kinds of independent complexities in mono-scale-invariance, namely to be behavioral complexity determined by fractal behavior and original one wrapped in scaling object. Quantitative characterization of the complexities is fundamentally important for mechanism exploration and essential understanding of nonlinear systems. Recently, a new concept of fractal topography (Ω) was emerged to define the fractal behavior in self-similarities by scaling lacunarity (P) and scaling coverage (F) as Ω(P, F). It is, however, a “special fractal topography” because fractals are rarely deterministic and always appear stochastic, heterogeneous, and even anisotropic. For that, we reviewed a novel concept of “general fractal topography”, Ω(P, F). In Ω, P is generalized to a set accounting for directiondependent scaling behaviors of the scaling object G, while F is extended to consider stochastic and heterogeneous effects. And then, we decomposed G into G(G+, G−) to ease type controlling and measurement quantification, with G+ wrapping the original complexity while G− enclosing behavioral complexity. Together with Ω and G, a mathematical model F3S (Ω,G) was then established to unify the definition of deterministic or statistical, self-similar or selfaffine, single- or multi-phase properties. For demonstration, algorithms are developed to model natural scale-invariances. Our investigations indicated that the general fractal topography is an open mathematic framework which can admit most complex scaling objects and fractal behaviors.

Chaos, Solitons & Fractals, 2008

Multi-scale structure of spray images is investigated for varying ranges of pressure and temperature in quiescent air. For spray images a standard PIV set is used consisting basically on a CCD camera and a laser sheet. A deviation to fractality is evidenced: the scale analysis has a parabolic form. A scale-dependent fractal dimension is measured which displays a linear variation with scale-logarithm. The classical fractal dimension usually measured so far is reinterpreted as a mean slope for scales close to the outer cut-off scale. This multi-scale behaviour is described by a diffusion equation of a new geometrical quantity called scale entropy related to the wrinkling of a set over scales. This equation is based on the conservation of a scale entropy flux through scale-space which is interpreted as the evolutive potential of the system at a given scale. This gives access to the scale-dependency of fractal dimension and points to the importance of the variations through scale space of this evolutive potential and namely its gradient. It has been shown that for sprays, the evolution of the evolutive potential gradient is constant through scale space which corresponds to a parabolic behaviour for scale analysis.

2007

We consider three developments in high number of degrees of freedom approaches to nonlinear geophysics: a) the transition from fractal geometry to multifractal processes, b) the Twenty Years on 18

The Professional Geographer, 1992

tionships between land-surface processes and clio mate, and the statistical analysis of large-scale clio mate fields.

Monthly Notices of the Royal …, 2010

In the multifractal analysis of the large-scale matter distribution, the scale of the transition to homogeneity is defined as the scale above which the fractal dimension (D q) of the underlying point distribution is equal to the ambient dimension (D) of the space in which points are distributed. With the finite sized weakly clustered distribution of tracers obtained from galaxy redshift surveys it is difficult to achieve this equality. Recently Bagla et al. have defined the scale of homogeneity to be the scale above which the deviation (D q) of the fractal dimension from the ambient dimension becomes smaller than the statistical dispersion of D q , i.e. σ D q. In this paper we use the relation between the fractal dimensions and the correlation function to compute σ D q for any given model in the limit of weak clustering amplitude. We compare D q and σ D q for the cold dark matter (CDM) model and discuss the implication of this comparison for the expected scale of homogeneity in the concordant model of cosmology. We estimate the upper limit to the scale of homogeneity to be close to 260 h −1 Mpc for the CDM model. Actual estimates of the scale of homogeneity should be smaller than this as we have considered only the statistical contribution to σ D q and we have ignored cosmic variance and contributions due to survey geometry and the selection function. Errors arising due to these factors enhance σ D q and as D q decreases with increasing scale, we expect to measure a smaller scale of homogeneity. We find that as long as non-linear corrections to the computation of D q are insignificant, the scale of homogeneity does not change with epoch. The scale of homogeneity depends very weakly on the choice of tracer of the density field. Thus the suggested definition of the scale of homogeneity is fairly robust.

Physical Review A, 1992

A turbulent flow lacking homogeneity and isotropy has been studied using the technique of photoncorrelation homodyne spectroscopy to measure velocity differences in different directions and at various length scales. The measured intensity correlation function G(qt, L) is found to be of the scaling form G(qtu(L)) with u (L)-L~, where u (L) is the characteristic turbulent velocity at the length scale L and q is the scattering vector. The exponent g varies with scattering geometries and with flow cells. The scaling behavior of G(qt, L) is found to be independent of spatial positions and orientations. A scaling G(qtu(L)) indicates that the velocity-density function P(V, R) has a scaling form Q(V/u(R))/u(R). The experiment suggests that the scaling argument can still be used to describe anisotropic and inhomogeneous turbulent flows, and that the exact functional form of P(V, R) (which is of scaling form) may vary with spatial positions and orientations in turbulent flows, reflecting the inhomogeneity and anisotropy of the turbulence.

Chaos, Solitons & Fractals, 2009

Phenomenological Universality (PUN) represents a new tool for the classification and interpretation of different non-linear phenomenologies in the context of cross-disciplinary research. Also, they can act as a ''magnifying glass" to finetune the analysis and quantify the difference among similarly looking datasets. In particular, the class U2 is of special relevance since it includes, as subcases, most of the commonly used growth models proposed to date. In this contribution we consider two applications of special interest in two subfields of Elasto-dynamics, i.e. Fast-and Slow-Dynamics, respectively. The results suggest that new equations should be adopted for the fitting of the experimental results and that fractal-dimensioned variables should be used to recover the scaling invariance, which is invariably lost due to non-linearity.

1997