Topology, Dynamics and Multifractal Analysis of Turbulence near

a Turbulent/Non-Turbulent Interface

Rui Manuel Marques Jaulinorui.jaulino@ist.utl.pt

Instituto Superior Tecnico, Lisboa, Portugal

December 2014

Abstract

Two universal aspects of turbulence, the distribution of the invariants of the velocity gradient tensorand the multifractal spectrum of dissipation, are analysed near a Turbulent/Non-Turbulent Interfaceusing Direct Numerical Simulations of shear-free turbulence.The invariants of the velocity gradient tensor provide only local information, in contrast to eddy struc-tures which necessarily involve data from a given flow region. We show that in a turbulent/non-turbulentinterface, such as exists at the edges of wakes and jets, the invariants show the imprint of the small scaleeddies from the nearby turbulent region, and can be described by a simple model relating local flowtopology, dynamics and eddy structure, thus unifying the previous models for the viscous superlayerand the turbulent sublayer within the Turbulent/Non-Turbulent Interface.The recent result of non-conservativeness of the multiplicative cascade which describes the energy dis-sipation is obtained with the Wavelet Leaders method by analysing the flow both in the turbulentregion and in the Turbulent/Non-Turbulent Interface. The present results on the multifractal spec-trum of dissipation imply that the previous results in the literature on the fractal dimension of theTurbulent/Non-Turbulent Interface need to be revisited. Keywords: Turbulence, T/NT Interface,Multifractal Analysis

1. Introduction

In the real word turbulence is omnipresent,found in almost all natural flows, industrialprocesses and engineering applications, and isprobably the most challenging problem of fluidmechanics. Turbulence is Nature’s way of speed-ing up diffusion, transport and mixing in fluidsby the generation of self-similar multiscale motions.

1.1. Turbulent/non-Turbulent Interface

In turbulent shear flows, such as jets, shear layergrowth is promoted by the interaction between tur-bulent and non-rotational flow lying at oppositesides of the turbulent shear layer boundary, in a pro-cess denominated as Turbulent Entrainment. Thisentrainment process is characterized by significantexchanges of mass, momentum, passive or activescalars, like concentration or temperature, and en-ergy across a thin layer of turbulent fluid known asTurbulent/Non-Turbulent Interface.Fluid outside the interface, which is initially irrota-tional acquires vorticity in one of two ways: eitherby engulfment where there are large scale convolu-tions of the interface with negative curvature point-ing inward or by nibbling along the entire interface

by a viscous diffusion process.The vorticity build up mechanism across the TNTIis governed by two distinct mechanisms accordingto the different nature of the two layers that com-poses the TNTI [1]:

• at the so called, viscous super-layer (VSL),which is an external layer, close to the irrota-tional zone, where viscous processes dominatethe entrainment - diffusion of enstrophy;

• whereas in the turbulent sub-layer (TSL),where enstrophy production and vortexstretching dominates the entrainment.

From the geometrical point of view the interface isa very complex object with fractal characteristics[2].

1.2. ObjectivesThe present thesis is devoted to the investigation offundamentals in turbulence related to the univer-sal turbulent characteristics within free shear lay-ers which constitute canonical prototypes for fun-damental turbulence research being the goal of thiswork twofold: to investigate the evolution of thesmall scale universal features at Turbulent/Non-

1

Turbulent Interfaces, as exists in jets, wakes, mix-ing layers and boundary layers, and to develop andapply multifractal tools in this context.

2. Background

2.1. Invariants of the Velocity Gradient Tensor

The study of the invariants of the velocity gradienttensor present a clear geometric interpretation andresults based on these quantities shown some uni-versal characteristic of turbulence.By calculating the invariants of Aij = ∂ui

∂xjin any

flow, we can obtain information regarding the localgeometry because we have information of how theelements near a certain point are moving [3].As will be shown below, the characteristic equationof the velocity gradient tensor Aij , equation 1, de-termines the type of trajectories.

λ3i + Pλ2i +Qλi +R = 0 (1)

where λi are the eigenvalues of Aij and P,Q andR are the invariants of Aij . Since P = 0 in in-compressible flows the other two invariants are usu-ally analysed in a joint probability distribution map(Q,R map). Here, depending on the location, theideas presented can be associated with zones in thismap (figure 1).

R

Q

Vortexstretching

Vortexcompression

Sheetstructure

Tubestructure

DA>0

DA<0

DA=0DA=0

0

Figure 1: Interpretation of the Q, R map.

The (Q,R) map (figure 1) allows to infer aboutthe relation between the local flow topology (en-strophy or strain dominated) and the enstrophyproduction term (vortex stretching or vortexcompression).In many turbulent flows the (R,Q) map displaysa correlation between R and Q in the regionR < 0, Q > 0 associated with a predominanceof vortex stretching and a strong (anti) correla-tion between R and Q is present in the regionR > 0, Q < 0 associated with sheet like structures.

Figure 2: Typical Q, R map observed in turbulence.

This gives the (Q,R) map its characteristic”teardrop” shape that has been observed in a greatvariety of different turbulent flows such as isotropicturbulence, mixing layers and channel flows.

2.2. Fractals

A fractal (from the Latin word fractus, which meansbroken or fractured) can be defined as fragmentedgeometric shape which can be split into parts thatare a scaled copy of the whole. While a fractal isa strictly mathematical construction, it is found invarious non-mathematical models such as naturalsystems.To understand fractals, it is important to knowwhat their characteristics are. Its first characteris-tic is that its structure is defined by fine and smallscales and substructures. Another characteristic isthat its shape cannot be defined with Euclidean ge-ometry. In addition it is recursive and shows iter-ation to some degree. In addition, fractals are in-formally considered to be infinitely complex as theyappear similar in all levels of magnification.Studying fractals is both a complicated yet inter-esting branch of mathematics, and, despite all itsintricacies, it proves itself a useful tool.

2.2.1 Fractal Dimension

The fractal dimension of a given object can be de-fined as [4]:

df = liml→0− log(N(l))

log(l)(2)

where N(l) is the number of elements of size lneeded to cover the object. In practical terms d isthe slope of log(N(l)) versus log(l). This definitiongives the euclidean dimension for regular objects.A common example of a fractal object is the Kochcurve. It is the limit of the iterative process repre-sented in figure [3].

2

Figure 3: The Koch curve. This object is generatediteratively by replacement of the straight lines withthe original object (at each iteration).

To determine it’s fractal dimension one only hasto notice that, in the top of the figure (the shapethat is successively reproduced) are needed 4 ele-ments with length 1

3 . In the next iteration it is com-posed of 16 = 42 elements of length 1

9 = 132 . In some

iteration k, this idea can be generalized and so onehas 4k elements with length 1

3k. Then by equation

[2] is possible to obtain that the fractal dimension

of the Koch curve is just df = liml→0− log(N(l))log(l) =

limk→∞− log(4k)

log( 1

3k)

= log(4)log(3) = 1.26.

2.3. Multifractals

Multifractal systems are common in nature. Theyinclude fully developed turbulence, stock markettime series, real world scenes, the Suns magneticfield time series, heartbeat dynamics and naturalluminosity time series.In a multifractal system, the behaviour near anypoint x can be described as [4]:

f(x+ l)− f(x) ∼ lh(x) (3)

where h(x) is the holder exponent at the point xwhich can interpreted as a generalization of thenotion of Taylor series. Hence any function canbe expanded as a Taylor series where the expo-nent h is present in the higher order term i.e,f(x) = a0 + a1x+ · · ·+ ahx

h.The fundamental quantity that characterizes a mul-tifractal is the multifractal spectrum, D(h). Typi-callyD(h) is bell-shaped [4 and it can be interpretedin geometrically.

Figure 4: Typical example of a multifractal spec-trum.

D(h) represents the fractal dimension of theset of intervals where the holder exponent is h;since h can take any value between hmin and hmaxthere are an infinite number of fractal sets residingwithin the same set - hence the term multifractal.

2.3.1 The Multifractal Formalism

Characterizing the multifractal spectrum from it’sdefinition is an almost impossible task - it wouldinvolve the determination of the point wise singu-larity exponent for every point in the function. Toovercome this issue the multifractal formalism wasdeveloped allowing to determine the multifractalspectrum from quantities that are relatively easyto compute.Parisi and Frisch in [5] developed a multifractal for-malism to study the behaviour of velocity incre-ments in fully-developed turbulence (a key quantityin Kolmogorov’s theory). The idea was to studyasymptotic power law behaviour of moments of ve-locity increments, when the distance l → 0 (hereit should be considered the velocity signal as a 1Dfunction). Mathematically this is done by definingthe structure function S(l, q) as

S(l, q) =1

n(l)

∑x0

|f(x0 + l)− f(x0)|q (4)

where n(l) is the number of points separated by adistance l. The quantity τq is called scaling functionand it is defined as

τq = liml→0

logS(l, q)

log(l)(5)

To determine D(h) the following relation holds

D(h) = minq

(1− τq + qh) (6)

In reference [6], the multifractal spectrum is associ-ated with a measure µ(x) which is a function thatassigns a value to an interval of the space. For each

3

point x a local singularity exponent α(x) is definedas (in analogy with the holder exponent)

α(x) = liml→0

log(µ(Il))

log(l)(7)

where Il is the interval where the measure is de-fined. In the same vein as in the definition of themultifractal spectrum, it is possible to define the socalled singularity spectrum f(α) as the fractal di-mension of the set where the singularity exponentis α. An analogue to the structure function definedin [4] is

Sm(l, q) =

∫Rµ(Bl(x))qdµ(x) (8)

where Bl(x) is the ball of radius l centred in x. Ascaling exponent τMq can be defined as (in analogywith τq)

τMq = liml→0

∫R µ(Bl(x))qdµ(x)

log(l)(9)

The analogy between τq and τMq can be made [7];if one works in one dimension, is easy to note thatBl(x) = [x− l, x+ l], and then τq can be rewrittenas

τq = liml→0

log(∫R |ψ(x+ l)− ψ(x− l)|qdµ(x))

log(l)(10)

where

ψ(x) =

∫ x

−∞dµ(y) (11)

Then τq and τMq have the same meaning if one con-siders the cumulative integral of the measure understudy. Another point of interest is the following: ifone considers the holder and the scaling exponentsis possible to conclude from the previous analogythat α = h+1; by integration of the measure troughequation 11 we are increasing the exponent in one(for one dimension) and so τMq = τq + dq.

2.3.2 Multifractal Processes

The binomial cascade is one the simplest process toexhibit multifractal properties. This (iterative) pro-cess starts with the unit interval I0 = [0, 1], wherea constant mass is assigned, divides it into consecu-tive smaller pieces and at the same time divides themass attributed at each smaller interval.The iteration begins with a uniform distributionµ0[0, 1] = m0 (with 0 < m0 < 1), subdivides it intoa distribution with µ1[0, 12 ] = m0 and µ1[ 12 , 1] =m1 = 1 −m0, further subdivides it into µ2[0, 14 ] =m1m1, µ2[ 14 ,

12 ] = m0m1, µ2[ 12 ,

34 ] = m0m1 and

µ2[ 34 , 1] = m0m0 and so on. As shown in figure 5,successive iterations of this produces a multiplica-tive cascade that generates an infinite sequence of

measures whose limit measure is the binomial cas-cade.The scaling exponent is given by

τMq = − log2(mq0 +mq

1) (12)

(a) µ1 (b) µ2

(c) µ4 (d) µ10

Figure 5: The generation of the binomial measurewith (m0 = 0.25), µi, where i is the number of theiteration - from The Wolfram Project

2.3.3 Turbulence Energy Cascade as a Multiplica-tive Process

In 1962 Obukhov proposed that the 1941 Kol-mogorov theory should be generalized to include thefluctuations in the mean dissipation. In particular,he suggested that the velocity structure functionsshould consider the manner at which ε varies lo-cally. This yields

Sp(r) =< |u(x+r)−u(x)|p >∼< εp/3r > rp/3 (13)

where εr(x) is the mean dissipation over a domainof size r. The key issue should be to understand howεr behaves and to understand the moments < εqr >with some kind of mechanism. Evidences of multi-fractal behaviour of εr (i.e, deviation from a purefractal, like the previous examples) where shown in[8]. This picture of the energy viscous dissipationas a multiplicative process originated from the ar-gument of the energy cascade.A vital ingredient is the concept of a measure den-sity which, in the present context, is the rate of dis-sipation per unit volume εr. Establishing an anal-ogy between the total dissipation Er, in a certainpiece Ω of size r, Er =

∫ε(x)dx, when a piece Ω

decays into smaller ones, each smaller piece can bethought of as receiving a fraction of Er. The localsingularity exponent in agreement with equation 7is defined as:

ErEL∼( rL

)α(14)

or,εrεL∼( rL

)α−d(15)

4

where d is the euclidean dimension of the supportof the measure.Following the reasoning outlined for the study ofmultifractals one can describe the process by it’smoments, ∑

Eqr ∼ EqL

( rL

)τMq

(16)

and define the [Hent procc] generalized dimensionsDq as

Dq =τMqq − 1

(17)

With this underlying picture, reference [8] and [9]observed that a binomial cascade with m0 = 0.3agrees with the behaviour of dissipation in severalflows and conjectured that Dq and D(h+ 1) shouldbe universal parameter in turbulence.

2.3.4 Fractal Dimension of T/NT Interface

An estimation of the fractal dimension of the T/NTInterface is present in reference [2]:

df =7

3+

2

3

(3−D1/3

)(18)

where D1/3 is the generalized dimension at q = 1/3.This result depends on the multifractal spectrum ofthe dissipation in the interface.

2.3.5 Practical Multifractal Analysis

Wavelets have been used for a long time to studyscaling behaviour and irregularity of functions [10].It can be proven [11] that a partition function basedon the discrete wavelet coefficients of the functioncan be used to determine the multifractal spectrum.

Zd(j, q) =1

2j

∑k

d(j, k)q (19)

where d(j, k) are the wavelet coefficients at positionk and scale j. Also Zd(j, q) scales as

Zd(j, q) ∼ 2jτq (20)

and so a reasoning identical to equation 6 is valid[11].

3. Direct Numerical Simulations and Post-Processing Tools

3.1. Shear-Free TurbulenceThe present work makes use of direct numerical sim-ulations (DNS) of shear free turbulence in a periodicbox with sizes 2π with 5123 collocation points[12].The simulation starts by instantaneously insertinga velocity field from a previously run DNS of forcedisotropic turbulence into a field of quiescent fluid.The initial isotropic turbulence region then spreadsinto the irrotational region in the absence of mean

shear, developing a distinct TNTI layer. In thepresent shear free simulation the Reynolds num-ber based on the Taylor micro-scale is equal toReλ ≈ 115 and the resolution is ∆x/η ≈ 1.5. De-tails can be found in [12] and references therein.

3.1.1 Conditional Statistics

Conditional statistics of several quantities with re-spect to the distance from the TNTI have been em-ployed in recent works [1] and the procedure to ob-tain them is only briefly described here. The loca-tion of the outer edge of the TNTI is defined bythe surface where the vorticity magnitude is equalto a certain threshold ω = ωtr, whose level is ob-tained from topological considerations as describedin reference [1] and above. The conditional statis-tics are then computed as function of the distanceyI to the TNTI layer using one of three possible ori-entations to the interface (see Fig. 6): i) ’vertical’to the TNTI i.e. parallel to the y-axis (1D), normalto the TNTI projected into the (x, y) plane (2D),or normal to the TNTI (3D). In the resulting con-ditional mean profile the TNTI is by definition lo-cated at yI = 0, while the irrotational and turbulentregions are defined by yI < 0 and yI > 0, respec-tively, where yI is normalised by the Kolmogorovmicro-scale in the turbulent region η = η(yI 0)[12].

Figure 6: Local coordinate systems used to computethe conditional mean profiles.

3.2. Wavelet LeadersInstead of building the structure function as inequation 19 with the discrete wavelet coefficients,those are replaced by a quantity derived from themcalled the Wavelet Leaders.The Wavelet Leaders, L(j, k), are defined as themaximum of the wavelet coefficients, d(j, k), in acertain modified dyadic interval I3 = [(k−1)2ji , (k+2)2ji ], where all the scales below the one in studyare considered (i.e for every ji ≤ j). The scalingfunction is determine by the slope of the curves

5

log(ZL(j, q)) versus log(2j) and the multifractalspectrum follows from it’s definition.

4. Results4.1. Flow Topology near a Turbulent/Non-

Turbulent Interface

Figure [7] details the enstrophy build up mecha-nisms across the TNTI by plotting conditional meanprofiles of vorticity magnitude ω, enstrophy viscousdiffusion ν∂2 (ωiωi/2) /∂xj∂xj , and enstrophy pro-duction ωiωjsij , for the present DNS.

A

B

C

D

EF

G

yI/η

ω=

(ωiω

i)1/2

-5 0 5 10 15 20 25 300

10

20

30

40

50

A

BCD

EF

F

G

G

0 10 20 30

ω2 Prod.ω2 Diff.

Figure 7: Mean conditional profiles (as function ofthe distance from the TNTI) of vorticity magnitudeω, enstrophy production (ω2 Prod.) and enstrophyviscous diffusion (ω2 Diff.).

Several letters (A-G) are assigned to specificlocations within the TNTI, where A denotes thestart or the ’outer edge’ of the TNTI (i.e. theorigin of the local reference frame yI = 0). Theviscous diffusion exhibits a characteristic shapewith positive/negative maxima at yI/η = 2.4 (B)and yI/η = 5.5 (E) associated with gain/loss ofenstrophy, respectively, as previously reportedby several authors[1], being clear that this is themechanism triggering the enstrophy rise observedinside the TNTI [12]. The diffusive transportswitches signal between the two extrema crossingzero at yI/η = 3.9 (D). On the other hand theenstrophy production becomes important afteryI/η ≈ 2 but by yI/η ≥ 3.1 (C) is already themain responsible for the enstrophy amplificationsurpassing the viscous diffusion. The conditionalmean enstrophy exhibits a sharp rise between0 ≤ yI/η ≤ 6.3 (A-F) until by yI/η = 6.3 (F) themaximum enstrophy is attained. By yI/η = 23.5(G) the flow exhibits all the characteristics of fullydeveloped turbulence, with no sign of the presenceof the TNTI.

Therefore, in the present case the VSL, as-sociated with the viscous diffusion of vorticitytowards the irrotational flow region [12], extendsfrom 0 ≤ yI/η ≤ 3.1 (A to C), i.e. with a meanthickness (defined by the region in Fig. 7 wherediffusion exceeds production), equal to 〈δν〉 ≈ 3η,while the TSL (associated with the rapid vor-ticity rise by vorticity production) lays between3.1 ≤ yI/η ≤ 6.3 (D to F), with an estimatedmean thickness (region where production exceedsdiffusion culminating in the maximum vorticity -figure 7) equal to 〈δω〉 ≈ 3η. Thus in the presentflow both 〈δω〉 ∼ 〈δν〉 ∼ η in agreement withreference [12]. The mean thickness of the VSL〈δν〉 was estimated (Corrsin and Kistler) to be ofthe order of the Kolmogorov micro-scale since thephysical process within this layer is the viscousdiffusion of vorticity from the turbulent core intothe irrotational region this process should besolely controlled by the amount of vorticity in theturbulent region ω′ and by the molecular viscosityν. On dimensional grounds it follows that thecharacteristic length scale for this process, definedas the thickness of the VSL, is δν = δν (ν, ω′),

leading to δν ∼ (ν/ω′)1/2 ∼

(ν3/ε

)1/4 ∼ η, where εis the mean rate of viscous dissipation in the coreof the turbulent region.The development of the ’teardrop’ shape acrossthe TNTI can be appreciated by plotting the jointpdfs of R and Q at several fixed distances fromthe TNTI layer and describes how an initiallyNon-Turbulent fluid element becomes Turbulent.Fig. 8 shows the pdfs in the VSL (at yI/η = 0, 2.4and 3.1 - A-B-C) while Fig. 9 shows the pdfs inthe TSL (at yI/η = 3.9, 5.5, and 6.3 - D-E-F).

R/<sijsij>3/2

Q/<s ijsij>

-1 1

-1

1

-1 1

-1

1

C

R/<sijsij>3/2

Q/<s ijsij>

-1 1

-1

1

-1 1

-1

1

B

R/<sijsij>3/2

Q/<s ijsij>

-1 1

-1

1

-1 1

-1

1

A

Figure 8: Joint probability density function of R,Qacross the viscous-superlayer (VSL) region in theflow: start of the viscous superlayer (A), point ofmaximum mean enstrophy diffusion (B), end of theviscous superlayer (C).

6

R/<sijsij>3/2

Q/<s ijsij>

-1 1

-1

1

-1 1

-1

1

F

R/<sijsij>3/2

Q/<s ijsij>

-1 1

-1

1

-1 1

-1

1

E

R/<sijsij>3/2

Q/<s ijsij>

-1 1

-1

1

-1 1

-1

1

D

Figure 9: Joint probability density function of R,Qacross the turbulent sublayer (TSL) region in theflow: point of zero diffusion (D), point of minimumdiffusion (E), point of maximum enstrophy (F).

The VSL is predominantly linked with the for-mation of the ’teardrop’ shape in the 4th quadrant,associated with a predominance of sheet structures.Since the VSL is at the edge of the eddies fromthe T region[12], strain dominates over enstrophy,and R > 0 (sheet structures) is more frequent thanR < 0 (tube structures). In contrast in the TSLthe formation of the ’teardrop’ shape occurs in the2nd quadrant with the enstrophy now overcomingstrain, thus Q ≈ ωiωi/2 > 0 and R ≈ −ωiωjsij/4.The predominance of the non-linear over the viscousmechanisms implies that the only way the enstro-phy can grow is when ωi and sij are correlated sothat ωiωjsij > 0, i.e. more frequent events of R < 0(vortex stretching) than R > 0 (vortex compres-sion). The increasing intensity of R and Q as onemoves from A to F, naturally reflects the increasingintensity of the fluctuating fields of enstrophy andstrain. The results are in agreement with the condi-tional profiles displayed in Fig. 7. It is noteworthythat in contrast with the jet (where the ’teardrop’forms in 〈δω〉 ∼ λ), here, the ’teardrop’ shape de-velops in a much shorter distance, requiring only≈ 5.5η to form completely. Indeed, the joint pdf of(R,Q) at E is virtually identical to F and also G(not shown).

A very interesting perspective of the formationof the teardrop is provided by the trajectory of the(conditional average of the) invariants of the veloc-ity gradient tensor in the (R,Q) map, obtained fromthe conditional mean profiles using the three differ-ent orientations (1D, 2D and 3D), presented in Fig.10.

AD

E

F

G

R / < sijsij >3/2

Q/

<s ijs

ij>

-0.02 0 0.02 0.04

-0.1

-0.05

0

0.05

0.1

0.15

1D2D3DDA=0

A

BC

D

0 0.004 0.008

-0.1

2-0

.1-0

.08

Figure 10: Trajectory of the mean values of R andQ across the TNTI. DA = 27/4R2 + Q3 is the dis-criminant of the eigenvalues of Aij .

Only in the 2nd quadrant do we observe somedifferences in the statistics which shows the robust-ness of the conditional statistics. The T region ismainly in the 2nd but also in the 1st quadrants.There is an increasing tendency for finding a sheettopology as the fluid particles enter the VSL (A toC), followed by a sharply increased predominanceof vortex stretching and formation of tube struc-tures as the flow evolves inside the TSL (C to F),in agreement with the model for the VSL proposedin reference [12]. Interestingly, the trajectory con-necting the VSL and TSL regions (B to E) consistsin a straight line with a (constant) slope which isthe same for all the three orientations used in theconditional statistics (1D, 2D and 3D).

It is possible to explain the result present infigure 10 by recalling that the small scale eddies(’worms’) determine the TNTI characteristics. Re-cently, using the Burgers vortex which is a goodmodel for the small scale eddies in turbulent flows,it has been shown that the periphery and core ofthe eddies existing near the TNTI determines thethickness of both the VSL and TSL [1],[12]. In-terestingly, resorting to the analytical radial pro-files of Q(r) and R(r) for a Burgers vortex onecan derive a linear relationship between the invari-ants: Q(r) = −R(r)/γB − γ2B , where γB is theaxissymmetric strain created by the axial veloc-ity uz(z) = γBz imposing the vortex core radius

RB = 2 (ν/γB)1/2

. Recalling that a typical smallscale eddy from inside the turbulent region has avortex core radius equal to Rivs/η ≈ 5 [13], thestrain rate imposed on this eddy can be estimated asγivs = 4ν/R2

ivs = 5.9. On the other hand, by mea-suring the slope of the straight line in the (R,Q)trajectory from B to E in Fig. 10 we arrive ata stretching rate of γ = 5.5, which is remarkably

7

close to the value obtained assuming the flow inthe VSL and TSL is described by a Burgers vor-tex. Thus, the (R,Q) trajectory shows the imprintof the small scale eddies existing near the TNTI,and unifies the existing models for the VSL andTSL within a TNTI. The small differences obtainedwith the three orientations in the 2nd quadrant cannow be explained: the 1D orientation will less likelyalign with the radial direction of the eddies form-ing the TNTI than the 3D orientation (because thevorticity is larger in the core of the eddy), thus ex-plaining why the straight line linking the VSL andTSL is longer for the 3D than for the 1D orien-tation. Both the ’tear-drop’ formation and meantrajectory of (R,Q) are expected to hold in otherflows e.g. fully developed wakes and jets, since theVSL forms at the edge of similar small scale eddiesin these flows[12].

4.2. Multifractal Analysis of Turbulence4.2.1 Dissipation in Fully-Developed Turbulence

The dissipation was analysed using DNS data fromfully-developed turbulence for several 2D planes ofthe simulation.In figure 11 is possible to observe that the scalingexponents are very similar in all planes. This is anindication of statistical convergence.

q

τ q+

αq

-6 -4 -2 0 2 4 6-10

-8

-6

-4

-2

0

2

4

6

plane 0245plane 0250plane 0255plane 0260

Figure 11: Scaling exponents obtained in differentplanes.

The obtained multifractal spectrum is shown infigure 12. In the same figure several models pro-posed are represented for comparison: the model inreference [8] - binomial cascade with m0 = 0.3 - isrepresented by a dash-dotted red line and the modelin reference [14] - binomial cascade with m0 = 0.36and m1 = 0.78 - is represented by a dashed blueline. A proposed model of our own to fit the ob-tained data is represented by a solid black line -binomial cascade with m0 = 0.4 and m1 = 0.75.

h - α + d

D(h

)

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1plane 0245plane 0250plane 0255plane 0260p=0.35 (H*=0.2)p=0.32 (H*=0.19)p=0.3

Figure 12: Multifractal spectrum obtained in differ-ent planes in comparison with several models pro-posed.

It can be observed, in the first place, that forall the planes in the turbulent zone analysed, thedata collapses with each other. Another impor-tant aspect is that the cancellation exponent (see[14]) is different from zero. This is an indicationthat the main conclusion of [14] is observed again:if the dissipation is to be modelled as a multi-plicative cascade, this cascade is not conservative- (m0 +m1 6= 1). Nevertheless it can be observedthat the model proposed in [14] does not agree withour results for the negative exponents (right part ofD(h)) because the negative q values are associatedwith the weakest singularities and in [14] the datais filtered we can conjecture that this is the reasonfor the differences in the negative exponents (andthey need to filter the data in order to make theirmethod work. We obtain this same behaviour witha method different from the WTMM that is usedin [14]). Since there is also some ambiguity in thechoice of the scaling range, we can say that only bybiasing the results one could match the results in[14]. Nevertheless the results obtained are in rela-tively good agreement.Another important conclusion about non conserva-tiveness of the cascade is that if one considers aconservative binomial cascade and fractionally in-tegrates it by a factor of , say, H∗ one can obtaina function that is also a multiplicative process butwith some degree of regularity [14]. In our case,a conservative cascade with m0 = 0.4

0.4+0.75 ≈ 0.35will have a multifractal spectrum equal to the oneobtained with the non conservative model but it isshifted to the left by a factor of H∗ = log2(0.4 +0.75) ≈ 0.2.This result can have a very simple interpretation:since the Reynolds number is finite the function willhave some degree of regularity below a given scale

8

(in this case the Kolmogorov micro-scale) and so itmust be expected that, if the process is a pure mul-tiplicative one (in the limit of infinite Reynolds) itmust display some kind of behaviour of this type.This result cannot be achieved with box countingmethod because by construction this method alwaysyield a conservative cascade even if it is not the case[14].The generalized dimensions for our data (accordingto the model which agrees better with the results)are given by

Dq =− log2(0.4q + 0.75q)

q − 1(21)

4.2.2 Dissipation near the T/NT Interface

The dissipation was also analysed by using dataconditioned to the distance to the interface. In fig-ure 13 is represented the scaling functions obtained;it is also shown for comparison a scaling function forone the planes in the turbulent zone.

q

τ q+

αq

-6 -4 -2 0 2 4 6-10

-8

-6

-4

-2

0

2

4

6

distance 0000distance 0001distance -0001plane 0260

Figure 13: Scaling exponents obtained for differentdistances.

It can be observed that the scaling functions inthe interface are very similar to those in the turbu-lent zone.With respect to the multifractal spectrum the re-sults are present in figure 14. The comparativemodels shown are the same as in figure 12.

h - α + d

D(h

)

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1distance 0000distance 0001distance -0001plane 0260p=0.35 (H*=0.2)p=0.32 (H*=0.19)p=0.3

Figure 14: Multifractal spectrum obtained for dif-ferent distances.

It can be observed that, for this simulation, thereis evidence that the multifractal spectrum in the in-terface is similar to the spectrum in the turbulentzone and the same comments apply to this case.If one recalls equation 18 and establishes and anal-ogy, since we determined the multifractal spectrumfor one dimensional signals, we have

df =7

3+

2

3

(1−D1/3

)(22)

From equation 21 one can calculate that D1/3 ≈1.0776 and predict, with this results, df ≈ 2.28.This value is slightly lower that the value predictedfrom [8] where df ≈ 2.36. Remembering that thecascade is non conservative, the value of the fractaldimension should indeed be lower.This is a very interesting result since, as shown inreference [15], experimentally it was obtained thatthe fractal dimension of the interface was df = 2.37consistent with a value slightly higher than 7/3 - asit should be in the case of a conservative cascade.With this results on the multifractal spectrum ofthe dissipation (together with the one found in [14])the fractal dimension should be lower than 7/3.

5. ConclusionsThe invariants of the velocity gradient tensor showthe specific topology of the flow in the VSL andTSL within the TNTI and can be described by asimple model unifying the previous models for eachseparate (sub)layer. With this model knowledge ofthe eddy characteristics near the TNTI e.g. eddyradius Reddy allows the determination of the back-ground strain γeddy which permits estimating the(mean) values of the invariants across the VSL andTSL, therefore opening the door to the developmentof new models for the turbulent entrainment mech-anism based on the small scale eddies, associatedwith the ’nibbling’ mechanism [16], and to mixing

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models for free shear flows, which are often basedon the dissipation rate, which is simply ε ≈ −4νQin the VSL. The TNTI provides also an interest-ing example where the coherent features of the floware directly linked to the geometrical and dynami-cal features of the flow.The analysis of trial functions have shown that themethodology implemented for the analysis of mul-tifractal systems is able to recover the theoreticalpredictions. By investigating the multifractal spec-trum of dissipation in fully-developed turbulencethe main result present in [14] was recovered witha different method: the non conservativeness of themultiplicative process which describes energy dissi-pation. At the interface, evidences that the multi-fractal spectrum is similar to the turbulent regionwas found, which is in agreement with the experi-mental findings of [2] and these results yield a frac-tal dimension of the interface of about df ≈ 2.28.This issue needs further assessment. The expan-sion of the method to analyse 3D fields is sufficientto develop a tool to analyse without limitation themultifractal spectrum in the interface, being there-fore able to determine the fractal dimension of theinterface. The proposed analysis for several differ-ent flows will be able to clarify the universality ofthe interesting results obtained.

Acknowledgements

The author would like to thank to Prof. CarlosB. da Silva and Rodrigo R. Taveira for their greatorientation and daily availability to teach, help anddiscuss. They opened my horizons in this amazingfield of science.

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