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Physica A 384 (2007) 719–724

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Parabolic scaling of tree-shaped constructal network

Diogo Queiros-Condea,�, Jocelyn Bonjourb,Wishsanuruk Wechsatolc, Adrian Bejanc

aEcole Nationale Superieure de Techniques Avancees, 32 Bd Victor, 75015 Paris, FrancebCETHIL-UMR 5008, CNRS INSA-Lyon Univ. Lyon 1, 9 rue de la Physique, F-69621 Villeurbanne, France

cDepartment of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, USA

Received 21 July 2006; received in revised form 3 April 2007

Available online 18 May 2007

Abstract

We investigate the multi-scale structure of a tree network obtained by constructal theory and we propose a new

geometrical framework to quantify deviations from scale invariance observed in many fields of physics and life sciences.

We compare a constructally deduced fluid distribution network and one based on an assumed fractal algorithm. We show

that: (i) the fractal network offers lower performance than the constructal object, and (ii) the constructal object exhibits a

parabolic scaling explained in the context of the entropic skins geometry based on a scale diffusion equation in the scale

space. Constructal optimization is equivalent to an equipartition of scale entropy production over scale space in the

context of entropic skins theory. The association of constructal theory with entropic skins theory promises a deterministic

theory to explain and build optimal arborescent structures.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Scale-dependent fractals; Constructal theory; Networks; Entropic skins geometry

1. Introduction

Networks and multi-scale approaches represent important tools in physics and life sciences. Mandelbrot’spioneering proposal of fractal geometry [1] has been widely applied, but in spite of its incontestableimportance in science, two main limitations have been identified: (i) the number of decades (between 0.5 and 2decades) where fractality is evidenced is small [2]; (ii) over this limited scale range, the fractal dimensionappears to be scale-dependent (which is paradoxical) in a significant number of studies [3–7]. In order toovercome these limitations, we propose here a new geometrical background combining constructal theory [8],which is a method to generate flow architectures such as tree-shaped networks, and entropic skins geometry [7](ESG), which describes deviations from scale invariance by a diffusion equation through scale space.

Constructal theory [8] is a theoretical view of how the configuration of flow systems would be generatedeverywhere, in nature and engineering. In this theory, the generation of flow configuration is reasoned on thebasis of a physics principle of maximization of flow access in flow systems that are free to morph. For example,

e front matter r 2007 Elsevier B.V. All rights reserved.

ysa.2007.05.037

ing author.

ess: diogo.queiros-conde@ensta.fr (D. Queiros-Conde).

ARTICLE IN PRESSD. Queiros-Conde et al. / Physica A 384 (2007) 719–724720

it is a design method to connect a point to a volume: it shows that this connection is optimized by a tree-shaped structure which provides the easiest possible access to the currents (heat, fluid,y) that flow throughthem. Constructal theory has been applied to a large number of systems [8–12] namely, mechanical andthermal structures, ducts and rivers, turbulent structure.

2. Departures from scale invariance

Even though some constructal objects exhibit tree-shaped structures, it must be emphasized thatthey are different from fractal trees. Constructal objects are obtained from deterministic physical equationsapplied from the smallest finite length scale lc applicable to the object to the largest scale l0. The cut-offscales lc and l0 are dictated by global constraints. Design and optimization are realized scale after scaleand are based on clear physical equations. Numerous studies [3–7] have pointed to clear deviations frompure fractality and indicated the existence of a sort of paradoxical ‘‘scale-dependent fractality’’. Thesedeviations have been often neglected but the interest in dealing with them is now established. Some studiesclaimed a linear variation of fractal dimension with scale logarithm for example, in astrophysics [5] andturbulent combustion [7]. In turbulent jets [4], considering experimental uncertainties, fractal dimensionevolution versus scale logarithm is consistent with linearity. Let us also mention the empirical proposal of a‘‘parabolic fractal’’ [13] which showed on numerous examples (geophysics, astrophysics) the parabolic form ofscale analyses which corresponds to a fractal dimension varying linearly with the logarithm of the scale. Agreat number of studies are thus pointing, in the context of deviations from fractality, to this specific parabolicscaling.

In this paper, we perform a scale analysis on an optimized constructal object, in order to highlight its multi-scale structure. Our aim is to show that constructal optimization corresponds to a parabolic scaling meaning alinear variation of fractal dimension. This parabolic scaling is interpreted in the context of ESG as anequipartition of scale entropy production in the scale space [7].

3. Constructal network

The constructal object used here is a point-to-circle water-supply network, developed on the basis of theoptimization algorithm reported in Ref. [12]. For sake of comparison, a purely fractal design based on thesame starting configuration, number of iterations and global size as the constructal object will also beanalyzed. The fluid-supply network distributes a given mass-flow rate of fluid _m from the center of a circulararea (radius R) to a given number of ports located on its periphery. The network consists of round tubes ofvarious diameters (di) and lengths (li) connected to form a tree-shaped structure by means of successivepairings (Fig. 1). The flow is assumed to be laminar and fully developed at any location. In Hagen–Poiseuilleflow, the pressure drop at the ith level (tubes di, li) is DPi ¼ 128 n _mili=ðpd4

i Þ so that the network overallpressure drop is DP ¼ 128 n _m

Ppi¼0li=ðpd4

i Þ, with i is varying from 0 (center of the disc) to p (periphery); n is thekinematic viscosity. Murray’s law [14] holds for the ratio of successive diameters: di+1/di ¼ 2�1/3. The overall

pressure drop in the network is then DP ¼ f 8pn _mR3=V2, where f is a dimensionless friction factor defined by

f ¼ n0

Ppi¼0ð2

i=3 ~liÞ

h i3where ~li ¼ li=R and n0 is the number of branches at the starting step [12]. The

dimensionless friction factor f can be minimized by morphing the geometry, the degrees of freedom on whichthe optimization is performed are the angles and duct lengths. One will thus obtain the minimum overallpressure drop for this optimization. This minimization was performed to obtain the optimized design in Fig. 1;the global flow resistance obtained in this case is f ¼ 21.6. A fractal design alternative is shown in Fig. 2; thisobject is built with the same large-scale features as the constructal one: the same number n0 of branches at thecenter and a constant ratio between successive duct lengths (a ¼ li+1/li) which corresponds to the first scalingratio used in constructal design (a ¼ l1/l0 ¼ 0.926). It gives a much higher dimensionless friction factorf ¼ 71.5 meaning a higher pressure loss and finally a higher power requirement for the fluid distribution thanwith the constructal network. The fractal dimension can be estimated analytically as Df ¼ ln(n0)/ln(2+2/Lf),with Lf ¼ 1.2 cos(y/2)(a�a7)/(1–a) and y ¼ (360/n0)/2 ¼ 361, yielding Df ¼ 1.72.

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~

~l0= 0.208

l1= 0.193

l2= 0.178

l3= 0.165

l4= 0.153

l5= 0.142

l6= 0.131

~

~

~

~

~

Fig. 2. Fractal network with five ducts in the center and six levels of pairing.

~

~

~

l0= 0.394

l1= 0.365 - α1=37.5°l2= 0.209 - α2= 41.6°l3= 0.101 - α3=48.0°l4= 0.047 - α4=54.8°l5= 0.022 - α5=61.2°l6= 0.011 - α6=66.5°

~

~

~

~

~α1

α2

~l0 ~~

l1

~l2

Fig. 1. Constructal network with five ducts in the center and six levels of pairing.

D. Queiros-Conde et al. / Physica A 384 (2007) 719–724 721

4. Scale analysis of a constructal network: parabolic scaling

Let us perform a scale analysis (Fig. 3) of these two objects using a box-counting method. The tube diameteris assumed negligible with respect to the tube length: here we are interested in the tree-shaped structure.Results are presented in Fig. 3. At the scale li, direct scale measurement yields the number noted Ni,R of boxesof size li necessary to cover an object the largest scale of which is R; Ni,R thus gives the contribution of therange [li;R] to scale development. Since we are interested only in the effect of scales ranged in [lc;l0], we usescale multiplicativity (Ni,0N0,R ¼ Ni,R) to determine Ni,0. To compare easily the two objects that are extendedover different scale ranges, we choose to represent the dimensionless function lnNi,0/ln(l0/lc) as a function offc,i ¼ ln(li/lc)/ln(l0/lc); these are in fact the coordinates used in Fig. 3. In case of pure scale invariance, a straightline (i.e., a power-law Ni;0 / ðli=lcÞ

�Df where Df is the fractal dimension) would be obtained. The fractal objectof Fig. 2 appears indeed scale invariant giving by linear fit DfE1.73 which is very close to the theoretical value.In contrast with this behavior, the constructal network is clearly scale-dependent its curve on Fig. 2 displayinga regular curvature. A range of fractal dimensions Di varying from around 1.72 at l0 to a value around 1 forscales close to lc is needed to describe its geometry. Nevertheless, as it can also be observed in Fig. 3, fractal

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Fig. 3. Scale analysis for fractal (solid squares) and constructal (open diamonds) objects: lnNi,0/ln(l0/lc) vs. fc,i. Linear fit corresponding to

fractal object: �1.73fc,i+1.716. Parabolic fit (over a range of fc,i values belonging to [0; 0.9] to avoid uncertainty at large scales)

corresponding to constructal network: �0:351f 2c;i � 1:009f c;i þ 1:383. Dashed line would correspond to a fractal object having fractal

dimension Dc ¼ 1.

D. Queiros-Conde et al. / Physica A 384 (2007) 719–724722

and constructal objects display the same scaling for large scales owing to their same initial branching near thecenter.

5. Entropic skins geometry and constructal network

ESG has been initially developed to describe the phenomenon of intermittency in fully developed turbulence[15], but it is also adequate to describe scale deviations from scale invariance. Let us consider a multi-scalesystem having scales ranged from an inner cut-off lc to an outer length l0. To the scales lc and l0 correspond thefractal dimensions Dc and D0, with DcoD0. Dc characterizes the most localized structure in the system (calledthe crest) and D0 is the dimension of the most extended structure (the bulk). Here lc and l0 are, respectively, thesize of the smallest branches and the largest branches of network. The bulk dimension D0 depends on largescale characteristics. The crest dimension is for the constructal object, Dc ¼ 1. In the context of ESG, the bulkis linked to the injection of energy and the crest to its dissipation. For the constructal object of Fig. 1, thiscorresponds, respectively, to the injection of fluid and to the discharge of fluid.

In ESG, scale invariance is assumed only locally in scale space: for two scales li and li+1 (with li+1oli) closeenough in scale space, one can define a local dimension Di assuming that, for scales l with li+1plpli, one hasN(l)�l�Di where N(l) is the number of boxes of size l necessary to cover the structure. At each scale li, one candefine Vi as the volume occupied by the system at scale li: it can be written Vi ¼ Ni,0li

d where Ni,0 is the numberof boxes of size li needed to cover the system up to the scale l0, and d is the embedding dimension. BecauseN0,0 ¼ 1, we have V0 ¼ l0

d. We then introduce the scale entropy Si,0 between scales li and l0 which is defined asSi,0 ¼ ln(V0/Vi). Denoting x ¼ ln(li/l0), Sx ¼ Si,0, we introduce the scale entropy flux fx ¼ dSx/dx withfx ¼ Di�d. By an argument of scale entropy flux conservation, one can write d2Sx/dx2

�o(x) ¼ 0, where o(x)is a sink of scale entropy in the scale space interval dx. Using the notion of scale diffusivity [7], this equationcan be generalized to a time behavior, but here we restrict the discussion to the stationary case. Assuming auniform scale entropy sink [o(x) ¼ b with bX0] through scale space (equipartition), we arrive at a parabolicform Sx ¼ (b/2)x2+(D0�d)x, where b ¼ (D0�Dc)/ln(l0/lc). The classical fractal case is recovered for b ¼ 0.

The fractal dimension depends thus linearly on the scale logarithm x ¼ ln(li/l0): Di ¼ D0+bx. Scale analysisthen leads to a parabolic form lnNi,0 ¼ �(b/2)x

2+(D0�d)x since, by definition, Sx ¼ �xd�lnNi,0. Let us alsoobserve that Ni;0 ¼ ðl0=liÞ

ðDiþD0Þ=2. In order to compare objects displaying different scale ranges, it isconvenient to use the dimensionless variable fc,i ¼ ln(li/lc)/ln(l0/lc). Introducing Dc;0 ¼ ðD0 þ DcÞ=2 as the mean

ARTICLE IN PRESSD. Queiros-Conde et al. / Physica A 384 (2007) 719–724 723

dimension between bulk and crest, then it can be easily shown that

lnðNi;0Þ

lnðl0=lcÞ¼ �ðD0 � Dc;0Þf

2c;i � Dcf c;i þ Dc;0. (1)

We also note that Dc;0 ¼ ln Nc;0= lnðl0=lcÞ and D0 � Dc;0 ¼ ðb=2Þ lnðl0=lcÞ. It is worth noting that in thesedimensionless coordinates the vertical axis (i.e. fc,i ¼ 0) gives directly the values of the fractal dimensionsinvolved i.e. D0E1.73 and Dc;0 � 1:36. A parabolic drift is changing the fractal dimension Di from D0 to Dc.The dimension Dc;0 can be interpreted as the dimension of an equivalent fractal object that would give thesame global quantity Nc,0 for the same scale range.

So far, our scale of reference was l0, but since constructal theory gives to the inner cut-off scale afundamental meaning, let us now work with lc as the scale of reference. For scales in the scale range [lc; l0], ausual scale analysis at scale li gives in fact Ni,0, i.e. the number of boxes needed to cover a part of the systemhaving a size l0. Since Nc,0 is the number of boxes of size lc, the effect of the range [lc; li] is given by Nc,i, whichcorresponds to the average number of boxes of size lc in a larger box of size li. By scale multiplicativity(Nc,i ¼ Nc,0/Ni,0) we obtain lnNc,i ¼ (b/2)fc,i

2+Dcfc,i and, consequently, using ðDc;0 � DcÞ ¼ b lnðl0=lcÞ=2; itleads to the simple parabolic form:

ln½Nc;i=ðli=lcÞDc �

lnðl0=lcÞ¼ ðDc;0 � DcÞf

2c;i. (2)

In the case of the constructal network (Fig. 1), using D0 ¼ Df ( ¼ 1.72), Dc ¼ 1 and l0=lc ¼~l0=~l6,

the theoretical value obtained is bth ¼ 0.20; this also implies the theoretical values ðD0 � Dc;0Þth ¼

ðDc;0 � DcÞth ¼ 0:36. Let us now compare these theoretical features with direct scale measurements. For thiswe return to Figs. 1 and 2, which show the scale analysis of the constructal and fractal objects. A parabolic fitof lnNi,0/ln(l0/lc) as a function of fc,i yields D0 � Dc;0 ¼ 0:35, which means b ¼ 0.196, i.e. very close totheoretical values. The measured quantity ln ðNc;i=Ncrest

c;i Þ as a function of f 2c;i is given in Fig. 4. This quantity

follows very closely the theoretical linear behavior, the slope of which is ðDc;0 � DcÞth ¼ 0:36. These resultsimply that the constructal object follows the case of a uniform scale entropy sink over scale space leading to aparabolic scaling while the fractal object, as expected, corresponds to a zero scale entropy sink (b ¼ 0).

To simplify and interpret the formalism of Eq. (2), let us introduce Ncrestc;i ¼ ðli=lcÞ

Dcwhich gives the numberof boxes necessary to cover the object if it displayed through scale space only its crest dimension. For a purefractal Dc ¼ D0 ¼ Df and, for the constructal of Fig. 1, we have DcoD0 with Dc ¼ 1. Let us then introduce thequantity Nc;i=Ncrest

c;i which gives the ‘‘number of crests’’ contained in the object at scale li. Then we can simplywrite lnðNc;i=Ncrest

c;i Þ ¼ lnðNc;0=Ncrestc;0 Þf

2c;i. The quantity f 2

c;i (with 0pf 2c;ip1) appears to play the role of an

efficiency of construction through scale space. Fractality corresponds to the very specific case where bulk and

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

f2c,i

In[Nc,i/(l i/lc)Δ

c]/ln(l0/lc)

Fig. 4. ln½Nc;i=ðli=lcÞDc �= lnðl0=lcÞ as a function of f 2

c;i for fractal (solid squares) and constructal (open diamonds) objects. The dashed line

corresponds to theoretical behavior (Dth ¼ 0.20); its slope is (ðDc;0 � DcÞth ¼ 0:36.

ARTICLE IN PRESSD. Queiros-Conde et al. / Physica A 384 (2007) 719–724724

crest follow the same scaling. For the constructal object, bulk and crest display different scalings, preciselybecause of the existence of the physical cut-off length scales lc and l0. Fractality corresponds to the limit l0/lc-N, i.e. b-0. The optimization of intermediate scales [lc; l0] is possible by constructal theory, leading in thecase of a parabolic scaling to a constant value b of the scale entropy sink through scale space.

6. Conclusion

To conclude, our numerical results suggest that constructal theory leads to tree-shaped networks displayingmulti-scale features very close to the parabolic scaling introduced in the context of entropic skins geometry. Itcorresponds to an equipartition of scale entropy sink over scale space. The connection between constructaltheory and entropic skins theory leads to a scale dynamics that can be used to build and optimize tree-shapednetworks and to understand their behavior. We think that this work could be useful for many other fieldsdealing with fluid or energy distribution where multi-scale aspects are crucial such as biological systemsthrough transportation networks, mixing and chaotic advection, microfluidic networks and turbulence.

Acknowledgment

D. Queiros-Conde expresses all his gratitude to Sezin Topc-u for her help and to Walter Furst for hisencouraging support.

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