theoretical advances on generalized fractals with ...theoretical advances on generalized fractals...

2007 IASME/WSEAS 5th International Conference on Fluid Mechanics & Aerodynamics (FMA’07)Vouliagmeni Beach, Athens, Greece, August 25 - 27, 2007

Theoretical Advances on Generalized Fractalswith Applications to Turbulence

Haris J. Catrakis ∗, Fazlul R. Zubair ∗∗, Siarhei Piatrovich ∗∗,Youssef N. Ibrahim ∗∗, Aaron P. Freeman ∗∗, and Jennifer Shockro ∗∗

Iracletos Flow Dynamics and Turbulence LaboratoriesMechanical and Aerospace Engineering, Henry Samueli School of Engineering

University of California, Irvine, 4200 Engineering Gateway, Irvine, California 92697United States of America

[email protected], http://flows.eng.uci.eduAbstract: - Theoretical advances toward a purely meshless framework for analysis of the gen-eralized fractal dimension, with applications to turbulence, are considered. The key basictheoretical idea is the formulation of the probability density function of the minimum distanceto the nearest part of the flow feature of interest, e.g. a turbulent interface, from any locationrandomly chosen within a reference flow region that contains the feature of interest. The prob-ability density function of the minimum-distance scales provides a means to define and evaluatethe generalized fractal dimension as a function of scale. This approach produces the generalizedfractal dimension in a purely meshless manner, in contrast to box-counting or other box-basedapproaches that require meshes. This enables the choice of a physical reference region whoseshape can be based on physical considerations, for example the region of fluid enclosed by theturbulent interface, in contrast to box-like boundaries necessitated by box-counting approaches.The purely meshless method is demonstrated on spiral interfaces as well as high-resolution ex-perimental turbulent jet interfaces. Examination of the generalized fractal dimension as afunction of scale indicates strong scale dependence, at the large energy-containing scales, thatcan be described theoretically using exponential Poisson analytical relations.Keywords: - Fractals, Dimensions, Distributions of Scales, Turbulence, Self-Similarity.

1. Introduction

Turbulent flows and other multiscale flows ex-hibit highly complex geometrical behaviour asis directly evident in quantitative and qualita-tive visualizations in nature as well as in labo-ratories [e.g. 1, 2]. Because of this geometricalcomplexity, which results directly from highlyirregular multiscale dynamics, advancing theknowledge of geometrical properties remainsa challenging task [e.g. 1–3]. In turbulence,geometrical aspects are useful both for funda-mental studies and for applications [e.g. 1–3],in a wide range of problems with phenomenaranging from molecular diffusion to electro-magnetic wave propagation. Fundamentallyas well as in applications, it is desirable to de-

velop new ideas that can facilitate the studyof geometrical multiscale properties of a va-riety of flow aspects such as fluid interfaces,vortices, or dissipation regions.

In previous studies of geometrical aspects ofturbulence and multiscale flows, a key con-cept that has emerged is the notion of frac-tional, i.e. non-integral, dimensions or fractaldimensions. The earliest suggestions of theuse of the fractal concept in fluid mechanicscan be traced to Richardson [4], who citedWeierstrass regarding what are now knownas fractal functions, i.e. self-similar func-tions. Extensive studies of fractals in tur-bulence started with the work of Mandelbrot[5] . The words fractal and self-similar, both

∗Associate Professor of Mechanical and Aerospace Engineering. ∗∗ Graduate Students.

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Proceedings of the 5th IASME / WSEAS International Conference on Fluid Mechanics and Aerodynamics, Athens, Greece, August 25-27, 2007 288

coined by Mandelbrot, reflect ideas concern-ing multiscale behaviour of direct interest toturbulence studies, with the former etymolog-ically meaning fragmentation and the latterdenoting repetition of structure at multiplescales, i.e. scale independence or scale invari-ance. However, other studies starting withthe work of Takayasu [6] indicate scale depen-dence as an additional, or alternative, con-cept for the multiscale geometrical nature ofturbulence [e.g. 2]. In fact, level-crossing mea-surements in turbulence [e.g. 7] in conjunctionwith general theoretical results on fractal di-mensions [8] indicate scale dependence ratherthan scale independence.

As an idea as well as a tool, the fractal di-mension is invaluable in light of the need toquantify the multiscale geometry of turbu-lence. Reviews of some of the key develop-ments on fractals in turbulence are availablein a number of articles [e.g. 2, 3]. Findings of aconstant or nearly-constant fractal dimensionin a range of scales imply self-similarity [e.g.3]. Observations of scale dependence of thefractal dimension [e.g. 2, 8] can imply eitherintrinsic variation of structure with scale, orintermittently self-similar behaviour, or finite-size effects of upper and lower cutoff scales. Inpractice, the conventional approach to frac-tal aspects of turbulence is in terms of box-counting methods that require partitioning ofbox-shaped regions that are chosen to containthe turbulent features of interest [e.g. 3].

In the present work, theoretical advances to-ward a purely meshless approach to the ana-lysis of generalized fractal dimensions are in-vestigated. The probability density functionof the shortest distance from random pointlocations to the flow feature of interest is con-sidered. The method is demonstrated and in-vestigated using spiral interfaces as well asfully-developed turbulent jet interfaces. Re-sults are presented concerning the generalizedfractal behaviour and demonstrating the util-ity of the meshless method for the study ofthe multiscale geometry of turbulent flows.

2. The Minimum DistanceMethod and Purely MeshlessAnalytical Framework

Throughout most previous studies of fractalaspects of turbulence [e.g. 1, 3, 5], statisticalself-similarity has been assumed in a range ofscales in which a turbulent feature of inter-est has a fractal, i.e. fractional, dimension Dd

that is scale-independent and quantifies themultiscale geometrical behaviour:

0 ≤ Dd ≤ d , (1)

within a reference region of Euclidean dimen-sion d. In other studies, however, it has beensuggested to generalize the original notion of afractal dimension to allow for possible depen-dence on scale, i.e. a dimension Dd(λ) thatcan be a function of scale λ [e.g. 2, 6, 8]. Thishas been denoted as a differential or scale-dependent fractal dimension. We denote ithere as a generalized fractal dimension.

Motivated by previous theoretical work [8], weconsider as a basic idea toward a purely mesh-less framework, as shown in figure 1, the iden-tification of any random location from whichthe minimum distance to the nearest part ofthe turbulent feature is identified. This canbe performed for several random locations, asshown for example in figure 1, that are se-lected from within a reference boundary.

Theoretically, the shortest distances from allpossible such locations, i.e. an infinity of suchlocations, determine the probability densityof these distances. In practice, a finite num-ber of such locations depending on the scaleresolution desired for the probability densityfunction can be expected to be sufficient.

The method, therefore, provides an intrinsi-cally Monte Carlo approach. The procedureis repeated for as many locations as needed de-pending on the scale resolution desired for theresulting generalized fractal dimension func-tion. As will be shown below, the shortest-distance scales can be used directly to com-pute the generalized fractal dimension func-tion.

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Figure 1: In the purely meshless approach,a boundary region of any shape can be cho-sen. In this schematic, a circular boundaryis shown (dotted). Multiple random loca-tions (crosses) selected within this bound-ary are utilized to evaluate the minimumdistance from each location to the nearestflow feature.

The shortest-distance probability densityfunction, gd(λ), in general d-dimensionalspace, is defined as follows:

Minimum-Distance PDF gd(λ):

We define the quantity gd(λ)as the probability density func-tion (pdf) of the minimum dis-tance λ from a randomly lo-cated point, within the refer-ence boundary, to the nearestpart of the flow feature of in-terest.

By minimum distance, in d dimensions, wemean the shortest distance in a general sense,i.e. in space, in time, or in space–time, asappropriate, from each point randomly cho-sen but contained within a (d−1)-dimensionalreference boundary, to the flow feature of in-

terest. Along with the shortest-distance prob-ability density function, i.e.

gd(λ) = P{λ is min. distance to flow feature} ,(2)

we therefore have the associated shortest-distance probability gd(λ) dλ. The normaliza-tion integral, over the region contained withinthe reference boundary, is:∫ λC

0gd(λ)dλ = 1 , (3)

where λC denotes a scale, with λ ≤ L, and Ldenotes the largest characteristic scale of theflow. In general, i.e. for multiscale flow fea-tures, we can expect that there will be a scaleabove which gd(λ) = 0 yet this scale need notbe as large as the largest characteristic scale Lof the flow. This is because the convolutionsof the flow feature directly limit the maximumpossible value of the shortest distance. Wecan define λC as the maximum shortest dis-tance scale, i.e. such that gd(λ > λC) = 0 orgd(λ ≤ λC) > 0:

λC = maxλ{λ : gd(λ) > 0} . (4)

We note that this scale can also be effec-tively identified using minλ {λ : gd(λ) =0}. The cumulative minimum-distance dis-tribution function Gd(λ) associated with theshortest-distance probability density functiongd(λ) is, i.e.

Gd(λ) =∫ λ

0gd(λ′) dλ′ , or , gd(λ) =

dGd(λ)dλ

,

(5)with limiting values Gd(λ → 0) → 0 andGd(λ → λC) → 1 at the smallest andlargest scales, respectively. The minimum-distance probability density function gd(λ)and the minimum-distance cumulative distri-bution function Gd(λ) can now be utilized di-rectly, in analogy with previous theory [8],to obtain the generalized fractal dimensionDd(λ) and scaling exponent αd(λ):

Dd(λ) ≡ d − d log Gd(λ)d log λ

, (6)

andαd(λ) ≡ − d log gd(λ)

d log λ. (7)

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Substituting for Gd(λ), using equation 5 andthe normalization equation 3, we have:

Dd(λ) = d − λ gd(λ)∫ λ

0gd(λ′) dλ′

, (8)

or,

Dd(λ) = = d − λ gd(λ)

1 −∫ λC

λgd(λ′) dλ′

, (9)

and,

gd(λ) =Ed(λ)

λexp

{−

∫ λC

λ

[Ed(λ′)

] dλ′

λ′

},

(10)where Ed(λ) ≡ d − Dd(λ) and the improve-ment with respect to the earlier framework [8]is that this approach is purely meshless, i.e.it is free of any box-like constraints. Thus,the method enables the evaluation of the gen-eralized fractal dimension Dd(λ) from theminimum-distance probability density func-tion gd(λ).

3. Applications of the Min-imum Distance Framework toSpiral and Turbulent Inter-faces

Application to spiral interfaces, in figure 2,shows that the purely meshless method ismore accurate than box-based methods be-cause it captures very well the transition inthe dimension at the scale of the spacing be-tween spiral turns. As is evident in figure 1,the concept of the minimum distance is ap-plicable to all randomly-chosen points withinthe reference boundary. For each such point,the shortest distance λ ≥ 0. If the randomlocation is a part of the flow feature, thenλ = 0. If the random location is not partof the flow feature, then λ > 0. Because therandomly chosen locations must be within thereference boundary, the probability densityfunction gd(λ) is therefore a conditional prob-ability density function of shortest-distancescales.

Figure 2: Spiral of Archimedes, i.e. r ∼ θ,and the generalized fractal dimension as afunction of scale for a circular boundary us-ing the purely meshless approach (dashed)and for a box boundary using box counting(solid). The purely meshless approach cap-tures accurately the transition in the gen-eralized fractal dimension, which occurs atthe scale given by the spacing between spi-ral turns, as shown by the arrows.

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Figure 2. Example of a 2D image of a turbulent jet scalar interface beforecoarse-graining. Resolution of the image is 1024 x 1024.

Figure 3. The region of fluid contained within the scalar interface, shownin figure 2, used as the reference region for randomly chosen locations.

Consider a point x∈ [xm,xm+1], from which the displacement δxmfrom the nearest data point on the left can be defined as δxm =x− xm, where δxm ∈ [0,1]. In order to find the interpolated valuebm in the region, bilinear interpolation is defined as

bm(x) = bm(δxm) = (1−δxm)bm +δxmbm+1 (15)

With any method of interpolation used for coarse-graining, thecontinuity across the boundary of each region [xm,xm+1] must be

Figure 4. Randomly chosen locations within the turbulent scalar inter-face of the jet.

Figure 5. A log-log plot showing the fractal dimension of the 2D scalarinterface as a function of scale.

preserved per the requirement

bm(δxm = b) = bm+1(δxm+1 = a), (16)

where a = 0, b = 1 for bilinear interpolation. This method canbe extended easily to two-dimensional interpolation, as is usedin this study, or to three-dimensional interpolation.

The same image of a turbulent scalar interface shown in fig-ure 2 is chosen to illustrate the process on the following pages.The image is increasingly coarse-grained with bilinear interpo-lation and a new random number of points is chosen within thereference boundary of the scalar interface. This is shown in fig-ures 6 through 13. Also shown are plots of the fractal dimensionversus scale for each increasingly coarse-grained image.

4 Copyright c© 2007 by ASME

Figure 3: Example of conditional randomly chosen locations and minimum-distance scaleswithin an outer turbulent jet scalar interface, with the interface as the reference boundary.The Reynolds number is Re ∼ 20, 000 and the Schmidt number is Sc ∼ 2, 000.

Application to experimental fully-developedturbulent jet interfaces is shown in figure 3.The minimum-distance scales from condition-ally random locations chosen within the outerturbulent jet scalar regions are indicated infigure 3. In order to focus on the behav-ior of the generalized fractal dimension at theenergy-containing large scales, coarse grainingwas performed to retain large scales and filtersmall scales at the Taylor scale threshold.

The corresponding results on the generalizedfractal dimension as a function of scale areshown in figure 4. The results clearly indi-cate a strong scale dependence of the gener-alized fractal dimension as a function of scaleat large scales, which in particular appears tocorrespond to an exponential distribution ofscales, i.e., Poisson statistics, as is evident bycomparing the behavior in figure 4 to previoustheoretical studies of exponential probabilitydensity functions of scales [8].

Figure 4: Generalized fractal dimension asa function of logarithmic scale, evaluatedusing the shortest-distance scales from con-ditionally random locations in the coarsegrained scalar turbulent jet regions, cre-ated by spatial averaging combined withbilinear interpolation.

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The experimental data shown in figure 4,which are for coarse-grained turbulent jetscalar interfaces, are in very close agreementwith exponential statistics [8]. Specifically,the theoretical formula for the generalizedfractal dimension as a function of scale for ex-ponential statistics is as follows:

D2(λ) = 2 − λ/lm

eλ/lm − 1. (11)

Thus, this result shows that the large-scalestatistical geometrical behavior in turbulentjets can be quantified in terms of exponen-tial Poisson statistics for the generalized frac-tal dimension as a function of scale. This isspecifically the case in the range of energy-containing scales which, for the present data,is 7.8 × 10−3 ∼ λ/L ∼ 1. We note that thisscale is consistent with an estimate based onlarge-scale dynamics, i.e.λL

L∼ Re−1/2 ∼ (2×104)−1/2 ∼ 7.1×10−3 ,

(12)for the present conditions of Reynolds numberRe ∼ 20, 000.

4. Conclusions

The present findings show that the purelymeshless minimum-distance theoreticalframework provides a practical method fordetermining the generalized fractal dimensionas a function of scale. We have demonstratedthe use of the method for spiral interfaces andfor fully-developed turbulent interfaces. Forexample, the present observation of Poissonbehavior of the generalized fractal dimensionas a function of scale, at large scales, quan-tifies the scale dependence in turbulent jets.This is crucial in several applications, e.g. formodeling the mixture fraction, because recentwork has shown that the large-scale geome-try of the outer scalar interfaces provides thedominant contribution to the mixture fraction[2]. We note that the large-scale geometricalaspects of turbulent scalar interfaces are of di-rect practical interest in various applicationssuch as the mixing efficiency of fluids engi-neering devices, laser propagation throughturbulence, and flow optimization.

Acknowledgments:

This work is part of a research program onturbulence and the dynamics of flows. Thecontributions of J. Salvans-Tort, P. J. Garcia,J. C. Nathman, R. C. Aguirre, R. D. Thayne,B. A. McDonald, and J. W. Hearn to earlierphases of this work are gratefully acknowl-edged.

References

[1] S. B. Pope. Turbulent Flows. CambridgeUniversity Press, 2000.

[2] H. J. Catrakis. Turbulence and the dy-namics of fluid interfaces with applica-tions to mixing and aero-optics. In RecentResearch Developments in Fluid Dynam-ics Vol. 5, pages 115–158. Transworld Re-search Network Publishers, ISBN 81-7895-146-0, 2004.

[3] K. R. Sreenivasan. Fractals and multifrac-tals in fluid turbulence. Annu. Rev. Fluid.Mech., 23:539–600, 1991.

[4] L. F. Richardson. Atmospheric diffu-sion shown on a distance-neighbour graph.Proc. R. Soc. Lond. A, 110:709–737, 1926.

[5] B. B. Mandelbrot. On the geometry ofhomogeneous turbulence, with stress onthe fractal dimension of the isosurfaces ofscalars. J. Fluid Mech., 72(2):401–416,1975.

[6] H. Takayasu. Differential fractal dimen-sion of random walk and its applicationsto physical systems. J. Phys. Soc. Jap.,51:3057–3064, 1982.

[7] K. R. Sreenivasan, A. Prabhu, andR. Narasimha. Zero-crossings in turbu-lent signals. J. Fluid Mech., 137:251–272,1983.

[8] H. J. Catrakis. Distribution of scales inturbulence. Phys. Rev. E, 62:564–578,2000.

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