PARTICLE-LADEN FLOW - ERCOFTAC SERIES Phần 7 potx

Measurement volume length is always several times num-larger than the Eulerian integral length scale L s /L E, and the ratio of theaverage time-of-flight to the Lagrangian integral time s

250 M Bourgoin, P Gervais, A Cartellier, Y Gagne, C Baudet

Fig 4 Time-frequency representation.

Two bubble signals are visible

Table 1 Experimental

paramet-ers at different distances D from

the nozzle and the center of the

measurement volume D is

meas-ured in multiples of the nozzle

diameter N is the number of

in diameter) compared to the transducers diameter As a consequence, only

re-cordings were made at four distances (D) from the nozzle Every measurement

corresponds to the same Reynolds number, as it is constant along the jet, but

to different integral length scales [9, 8, 10] The measurement volume wascentered on the jet axis, to preserve cylindrical symmetry as much as pos-sible

Table 1 lists the main parameters of the different measurements The ber of velocity segments exceeds 1000 for all measurements, ensuring goodstatistical convergence Measurement volume length is always several times

num-larger than the Eulerian integral length scale (L s /L E), and the ratio of theaverage time-of-flight to the Lagrangian integral time scale (T s  /T L) is every-where above one

4.2 Velocity probability density function

The normalized velocity probability density functions (PDF) for the udinal velocity component measured at different distance from the nozzle arerepresented on figure 5 No significant change in shape can be seen between

longit-the four curves, indicating that longit-the variation of L s /L E does not break similarity The same remark is true for transverse components (figure 6) Allcurves are Gaussian, but small departures exist For the longitudinal com-ponent (figure 5), PDF edges are largely sub-Gaussian due to limitations ofthe velocity extraction algorithm and has no physical meaning For transversecomponents (figure 6), edges are over-Gaussian because of noise introduced

self-by the velocity extraction algorithm

Fig 5 Longitudinal velocity PDF with

zero mean and unity variance

Corres-ponding Gaussian curve is plotted in

Fig 6 Transverse velocity PDF with

zero mean and unity variance ponding Gaussian curve is plotted indashed line

Corres-Figure 7 shows the comparison of the Lagrangian PDF (P (u)) with the

corresponding Eulerian one (the hot-wire was located near the center of theLagrangian measurement zone) A reasonable agreement is found A slightlyhigher mean velocity is found in the Eulerian case (5 % higher), and thestandard deviation is higher for the Lagrangian velocity These effects resultfrom the inhomogeneity of the flow inside the acoustic measurement volume,which tend to under estimate the Lagrangian mean velocity on the axis andover estimate its fluctuations but is not visible on the Eulerian measurementwhich is carried out at a fixed point

Figure 8 shows isocontours of the joint PDF P (u, v) of longitudinal u and transverse v Lagrangian velocity A slightly elliptical shape is visible, indicat-

ing that no large-scale isotropy exists (horizontal and vertical coordinates areidentical) Standard deviation of the longitudinal component is higher thanthe corresponding one for the transverse component, by a factor ranging from1.1 to 1.25, depending on the position along the jet (resp farthest and nearestfrom the nozzle) A similar behavior exists for Eulerian velocity components

252 M Bourgoin, P Gervais, A Cartellier, Y Gagne, C Baudet

Fig 7 Lagrangian (solid line) and

Eu-lerian (dashed line) velocity PDF (80

dia-meters from the nozzle)

1 2 3 4 5 6

Velocity (v) [m/s]

Fig 8 Isocontours of joint Lagrangian

velocity PDF Contour values, from thecenter outward, are 10−3.5, 10−4, 10−4.5,

We have measured the velocity autocorrelation function for the Lagrangian

and the Eulerian signals The Lagrangian velocity correlation time T Lplays animportant role in modeling turbulent diffusion of passive tracers[14] Moreover

frame of numerical models such as RANS calculations [15] where this ratio is

a parameter to be calibrated

The statistical estimation of the Lagrangian autocorrelation function hasbeen obtained with an unbiased estimator, which also compensates to secondorder the axial inhomogeneity of the flow, inherent to open flows situation

[16] For the Eulerian autocorrelation, the integral scale L E is estimated fromthe hot wire measurement using a Taylor hypothesis based on the local mean

velocity and the integral time T E is then defined as T E = L& E /σ E , where σ E

is the Eulerian velocity standard deviation

Figure 9 shows the autocorrelation function of the Lagrangian velocitycomponents and the Eulerian longitudinal velocity for a measurement per-formed at 80 diameters from the nozzle The two curves for Lagrangian trans-verse components are almost identical, in accordance with the cylindrical sym-metry of the flow The longitudinal component exhibits a slightly longer timescale

We denote in the following the longitudinal and transverse Lagrangian

integral time scales by T l

by fitting an exponential curve on the autocorrelation Corresponding values

for Eulerian components are denoted by T l

E Only T l

obtained from measurements, because of the necessity of a Taylor hypothesis

reasons may be responsible for that On the one hand, the jet self-similaritycan be broken Wygnansky and Fiedler [8] have shown that self-similaritycan be violated for distances as large as 100 nozzle diameters, depending onthe quantity considered, in which case actual measurements of Eulerian timescales would lead to similar results On the other hand the velocity profilevaries linearly with the distance to the nozzle, while the measurement volumesize is constant, so that the flow homogeneity in the measurement volume

depends on the position in the jet As T l

L /T t

L increases when D increases, this

indicates that large-scale isotropy either does not exist whatever the distance,

or is recovered very slowly Lagrangian times T Lcan be considered as a rough

These results show that whatever the component considered, both times arevery close, the turnover time being slightly longer Obtained ratios are com-

phenomen-ological analysis leads to T L  T E [18] A larger Eulerian time scale can beexplained by sweeping effects The advection of the internal scales by theenergy-containing scale leads to broadening of the Eulerian autocorrelation incomparison with the Lagrangian one [19]

L T t L

T L l

t L

T E l

T L l

T E t

T t L

254 M Bourgoin, P Gervais, A Cartellier, Y Gagne, C Baudet

0.7 0.8 0.9 1 1.1

Fig 9 Lagrangian velocity autocorrelation (solid line) for longitudinal and

trans-verse components Eulerian velocity autocorrelation (dashed) An exponential fithas been superimposed to Lagrangian correlations (dot-dashed)

5 Conclusion

Lagrangian measurements in a free turbulent air jet were performed usingacoustical Doppler effect This method is adapted to collecting large data setswithout tremendous memory requirement, contrary to visualization methods

A single tracer at a time can be detected, with the time- and space- dynamics

of the measurements comprising a large part of the inertial scales, comparable

to previously-obtained results [6] Simultaneous Eulerian measurements wereperformed

We show that the Eulerian integral time is larger than the Lagrangian one.This might be a consequence of the Eulerian statistics sensitivity to sweepingeffects, which instead do not affect Lagrangian statistics This result holdsfor distances in the jet ranging from 60 nozzle diameters up to 110 nozzle

E /T l

on the distance from the jet nozzle

The acoustic technique is now being adapted to study two phase flowsladen with inertial particles The first experiments aim to explore Stokesnumber dependence of individual particles dynamics, with a particular fo-cus on the effect of particles finite size and of the particle to fluid density ratio

[1] Virant, M., and Dracos, T., 1997 3D PTV and its application on rangian motion Measurement science and technology, 8, pp 1539-1552[2] Ott, S., and Mann, J., 2000 An experimental investigation of the relativediffusion of particle pairs in three-dimensional turbulent flow Journal ofFluid Mechanics, 422, pp 207-223

Lag-[3] LaPorta, A., Voth, G A., Crawford, A M., Alexander, J., andBodenschatz, E., 2002 Fluid particle accelerations in fuly developpedturbulence Nature, 409, February, p 1017

[4] Bourgoin, M., Ouellette, N T., Xu, H., Berg, J., and Bodenschatz, E.,

2006 The role of pair dispersion in turbulent flow Science, 311, February,

p 835

[5] Xu, H., Bourgoin, M., Ouellette, N T., and Bodenschatz, E., 2006 Highorder Lagrangian velocity statistics in turbulence Physical Review Let-ters, 96, January, p 024503

[6] Mordant, N.,Metz, P.,Michel, O., and Pinton, J.-F., 2001 Measurement

of Lagrangian velocity in fully developed turbulence Physical ReviewLetters, 87(21), p 214501

[7] Sato, Y., and Yamamoto, K., 1987 Lagrangian measurement of particle motion in an isotropic turbulent field Journal of Fluid Mechan-ics, 175, pp 183-199

fluid-[8] Wygnanski, I., and Fiedler, H., 1969 Some measurements in the preserving jet Journal of Fluid Mechanics, 38(3), pp 577-612

self-[9] Pope, S B., 2000 Turbulent flows Cambridge University Press

[10] Tennekes, H., and Lumley, J L., 1992 A first course in turbulence MITpress

[11] Voth, G A., la Porta, A., Crawford, A M., Alexander, J., andBodenschatz, E., 2002 Measurement of particle accelerations in fullydeveloped turbulence Journal of Fluid Mechanics, 469, pp 121-160[12] Poulain, C., Mazellier, N., Gervais, P., Gagne, Y., and Baudet, C., 2004.Spectral vorticity and Lagrangian velocity measurements in turbulentjets Flow, Turbulence and Combustion, 72, pp 245-271

[13] Flandrin, P., 1993 Temps-frequence Hermes

[14] Taylor, 1921 Diffusion by continuous movements Proc London Math.Soc., 20, p 196

[15] Lipari, G., Apsley, D D., and Stansby, P K., 2006 Numerical particletracking studies in a turbulent jet in the present Proceedings of Eur-omech Colloquim 477

[16] Batchelor, 1957 Diffusion in free turbulent shear flows Journal of FluidMechanics, 3, pp 67-80

[17] Yeung, P K., 2002 Lagrangian investigations of turbulence Annual view of Fluid Mechanics, 34, pp 115-142

Re-256 M Bourgoin, P Gervais, A Cartellier, Y Gagne, C Baudet

[18] Corrsin, S., 1963 Estimates of the relations between Eulerian and rangian scales in large Reynolds number turbulence Journal of the At-mospheric Sciences, 20(2), pp 115-119

Lag-[19] Kraichnan, R H., 1964 Relation between Lagrangian and Eulerian relation times of a turbulent velocity field Physics of Fluids, 7(1), pp.142-143

cor-Beat L¨uthi, Jacob Berg, Søren Ott and Jakob Mann

Risø National Laboratory, Wind Energy Department, P.O Box 49,

Frederiksborgvej 399, DK-4000 Roskilde, Denmark beat.luthi@risoe.dk

Summary Combined measurements of the Lagrangian evolution of particle

con-stellations and the coarse grained velocity derivative tensor ∂ 'u i /∂x j are presented.The data is obtained from three dimensional particle tracking measurements in aquasi isotropic turbulent flow at intermediate Reynolds number Particle constella-tions are followed for as long as one integral time and for several Batchelor times

We suggest a method to obtain quantitatively accurate ∂ 'u i /∂x jfrom velocity

meas-urements at discrete points We obtain good scaling with t ∗ =

2r2/15S r (r) for

filtered strain and vorticity and present filtered R-Q invariant maps with the typical

’tear drop’ shape that is known from velocity gradients at viscous scales Lagrangianresult are given for the growth of particle pairs, triangles and tetrahedra We findthat their principal axes are preferentially oriented with the eigenframe of coarsegrained strain, just like constellations with infinitesimal separations are known to

do The compensated separation rate is found to be close to its viscous counterpart

is governed by the coarse grained velocity derivative field 'A ij = ∂ u'i /∂x j.Moreover, it has been recognized for a few years now that constellations withmore than two particles have a rich structure at scales smaller than the integralscaleL [4, 5, 6, 7, 8] Work that started with [5] and currently is being further

developed by [9] is relating the dynamics of 'A ij to the evolution of tetrahedraand a stochastical model has been developed for its simulation Experimentaland numerical studies have investigated some of the properties of 'A ij[10, 11].The most important finding is that coarse grained velocity derivatives exhibit

Bernard J Geurts et al (eds), Particle Laden Flow: From Geophysical to Kolmogorov Scales, 257–269.

© 2007 Springer Printed in the Netherlands.

258 Beat L¨uthi, Jacob Berg, Søren Ott and Jakob Mann

roughly the same properties like their small scale counterparts Probably themost important property is that

"

'

Λ2

#

> 0, where ' Λ i are the eigenvalues

of the rate of strain tensor 's ij = 1/2 (∂ 'u i /∂x j + ∂ 'u j /∂x i) It means thatalso for inertial range scales the field of velocity derivatives experiences self-amplification

In this contribution, we present for the first time experimental results that

of particle pairs, triangles, and tetrahedra The filter scale covers a good part

of the inertial range and the particle constellations are followed as long as theintegral time,T , and for several Batchelor times, τ B = R 2/3

0 /ε 1/3 , where R0

is the scale of the constellation at t = 0 Since Batchelor [12] it is known that

changes from r2− r2

∝ t2, known as the ballistic regime, to r2(t)

= gεt3,which is known as the Richardson law The importance of having observation

times t > τ Bcan also be expressed in terms of kinetic energy of relative motion

in a particle swarm of size R with N points, E = 1/2"

uN − U 2#

R

: Only if

tracking times are long enough a transition from the regime where dE/dt < 0

to a regime with dE/dt > 0 can be observed [13] The former regime is

essentially governed by Eulerian dynamics while the latter is governed by theLagrangian evolution of particle constellations

One can define the tensor 'A ij coarse grained at scale ∆ as

approxima-tion to the velocity field as

∂x j directly but can insteaddifferentiate the filtered velocity field to obtain

'

The left hand side of eqn 2 can be approximated by least square fitting

∞ this operation becomes equivalent to top-hat filtering the spatial velocity

derivative field Different to [14] here spherical polynomials that by definition

are incompressible and orthogonal are used Since for ∆ > η the velocity field

is not smooth n > 4 is necessary to obtain convergence As we will demonstrate below in the result section we have found that n > 12 is sufficiently high.

2 3D-PTV Experiment

constellations we have performed a Particle Tracking Velocimetry (PTV) periment in an intermediate Reynolds number turbulent flow PTV is by now

ex-a well estex-ablished non-intrusive flow meex-asuring technique [15, 16, 17, 18, 19,

14, 20, 2] which naturally allows to probe a flow’s Lagrangian properties Tomeet the competing goals of high tracer seeding density to allow for coarse

graining, and high trackability of particle constellations to reach t > τ B sometrade off’s in the experimental design had to be made: Typically 900 particles

average particle distance of d p ≈ 50η and tracking lengths longer than

integ-ral scales t T > T and t T > 10τ B For the sake of ’good’ statistics the total

recording time is t R ≈ 500T

The flow is forced with eight rotating propellers placed in the corner of

recorded with four synchronized, 50Hz CCD cameras To suppress the opment of a mean flow the propellers change their rotational direction after

devel-0.5s of stirring and after an additional devel-0.5s of pausing A typical propeller tip velocity is 50cm/s Further details of the experiment are described in [2] The

characteristic flow properties are summarized in table 1 A recent

modifica-Table 1 Flow properties of the turbulent flow as already reported in [2].

0.25mm 48mm 0.07s 2.45s 168mm2/s3 23mm/s 190 172

tion of tracking 3d particle positions through consecutive time frames allows

to connect tracked particle trajectories that are only interrupted by one ing’ point The main impact of this feature is a drastic increase of the number

’miss-of long trajectories The number ’miss-of tracks with length t T > T has more than

doubled while the number of tracks with t T > 2 T is one order of magnitude

the number of particles is n > 12 In fig 1(a) we plot the averages of 's2and'ω2

as a function of filtering scale ∆/η The comparison with the straight dashed

260 Beat L¨uthi, Jacob Berg, Søren Ott and Jakob Mann

Based on the longitudinal second order structure function S2(r) we construct

a time, t ∗ (r) with which a better compensation of ' A ij can be obtained Wedefine

r 2/3 With the parameterized form of S2(r) employed by [21]

the second order structure function is expressed as a function of separation

r, the viscosity ν, the flow properties L and ε and the Kolmogorov constant

C k Again in fig 1(a) we show 's2

· t2

∗ and 'ω2

· t2

Clearly the more general scaling with eqn 4 holds over our entire range of

100 < ∆/η < 300 as 's2· t2 ≈ 1 To see how far off the approximation of ' A ij

is with a too low number of points we show in fig 1(b) the same quantities

≈ 50, which is much too high if we keep

in mind that for ε = 168mm2/s3at the smallest scale s2

20 30 50 70 100

are plotted versus

filtering scale ∆/η along with the compensated values 's2

· t2

∗and 'ω2

· t2

∗ In (a)results are obtained from n > 12 points per least square fit to linear polynomials and in (b) only n = 4 points are used.

The qualitative difference between 'A ij as obtained from n = 4 or n > 12

It is known that in the viscous range Λ2 / Λ1 ≈ 0.15 over a wide range

of Reynolds numbers [22, 23, 14] and also for inertial scales it is reportedthat

Contrary, in fig 2(b) the same PDFs but obtained from only n = 4 points

peak at zero and

"

'

Λ2/ ' Λ1

#

≈ 0, i.e one of the most important turbulent

properties is lost completely

Fig 2 PDFs for the coarse grained strain shape 'Λ2/ ' Λ1 for filter scales 100 <

∆/η < 300 In (a) results are obtained from n > 12 points per least square fit to linear polynomials and in (b) only n = 4 points are used.

In fig 3(a,b) we show how the large scale axis-symmetry that for this iment was already reported in [2] is reflected in 'A ij , especially for large ∆/η.

stretch-ing principal strain axis 'λ1 of 's ij x3 is the vertical tank axis, which with

32× 32 × 50cm3 is higher than wide Towards smaller scales a slow tion of this anisotropy can be observed, similar to reports of [9] For 'ω the

relaxa-situation is slightly different as is shown in fig 3(b) Consistent with the tankdimensions it seems that large scale vorticity is preferentially aligned with thelonger vertical tank axis but equally distributed over both directions of rota-tion Towards larger scales the symmetry is broken slightly as the horizontal

component of vorticity starts to align with +x1and−x2

Finally we show in fig 4 the topological property of measured 'A ijby means

of the two invariants R and Q [24] The normalized invariants are defined as

Q = −1

2T r

'

262 Beat L¨uthi, Jacob Berg, Søren Ott and Jakob Mann

0.4 0.6 0.8 1 1.2 1.4

cos( ω,x1,2,3)

x1, ∆/η=100 x

1 , ∆/η=200 x

1 , ∆/η=300 x

2 , ∆/η=100 x

2 , ∆/η=200

x2, ∆/η=300 x

3 , ∆/η=100 x

3 , ∆/η=200 x

3 , ∆/η=300

b) a)

∆/η=300

∆/η=300

Fig 3 (a) PDFs of the cosines between the most stretching principal coarse grained

eigenvector 'λ1and the coordinate directions x1,2,3 for three different scales ∆/η x 1,2

are the horizontal directions and x3 is the vertical direction (b) PDFs of the cosinesbetween coarse grained vorticity 'ω and the coordinate directions x 1,2,3 for three

different scales ∆/η.

For plots obtained from only n = 4 points we see that essentially for all scales 100 < ∆/η < 300 the RQ-shapes look like such obtained from Gaussian velocity fields [22] Very differently for n > 12 the well known ’tear drop’

shapes are recovered for all scales This is at first surprising since we would

most symmetric RQ-shape for the n > 12 figures The only explanation we

have is that the observed ’tear drop’ shapes at larger than integral scales arecaused by large scale mean strain This effect has already been observed instochastical model results [9]

4 Multi point statistics

In the previous section we established that the measured 'A ijis approximatingwell the actual coarse grained velocity derivative tensor In this section weshow how particle pairs, triangles and tetrahedra grow in time and how theirprincipal axes are oriented with respect to 'A ij In addition, we check to what

degree the kinematic relation for the growth of pairs r

12

−2 0 2 4

−4

−2 0 2

4

>12 points >12 points >12 points

Fig 4 Joint PDFs of the invariants R and Q as defined in eqn 6 Shown are

results from n > 12 points per least square fit to linear polynomials (bottom row) and n = 4 points (top row) for three different filtering scales ∆/η = 100, 200, 300.

The isoprobability contours are logarithmically spaced

For evolving triangles or tetrahedra we use R0= √g1, where g1 is the largesteigenvalue of the moment of inertia tensor

We attribute this to the too low scale separation of our experiment In fig 5(a)

it can be seen how shortly before r0=L (denoted by circles) the growth rate

starts weakening In the case of triangles and volumes fig 5(b) the statisticsbecome too sparse even before integral scale is reached

for triangles and tetrahedra respectively These shape factors are a measure

264 Beat L¨uthi, Jacob Berg, Søren Ott and Jakob Mann

objects Following [6] they are defined as

w = 2

do not reach Richardson scaling that is denoted by straight dotted t3 lines in

fig 6(a,b) For the tetrahedra it can be observed how at early times, where due

to small scales the velocity field is still quite smooth, the volumes are almost

conserved This is reflected in initially decreasing mean values of g3, which arethe most compressed directions of the tetrahedra Both shape factors reach astable plateau after a short transient time in which the initially regular shapesassume their intermediate state It is difficult to decide if these intermediatestates reflect self-similarity or just ’Gaussianity’ The noise level is relativelyhigh and our inertial range is very small In addition, as can be seen bythe straight dotted lines of fig 6(c), the values for self-similar and Gaussianshapes are fairly close together From slightly higher Reynolds number DNS

value is I2≈ 0.22 [6] Our data lies in-between for tetrahedra and, since three

points are ’easier’ to follow, probably also for triangles for which we don’tknow the corresponding values

We now look at how evolving particle constellations are oriented withrespect to the strain eigenframe spanned by the coarse grained eigenvectors

'λ i For the following Lagrangian results we use as an evolving filtering scale

R0(t) = r (t) or R0(t) =

g1(t) For infinitesimal separations and also for

infinitesimal areas it is well known that after a transient time of t/τ η > 1

separation vectors are predominately aligned with the most stretching axis 'λ1

Fig 6 Temporal evolution of mean eigenvalues of the tensor g ab, solid lines for

g1, dashed lines for g2 and dotted lines for g3(a,b) Initial separations range from

14 < r0/η < 30 for triangles and from 22 < r0/η < 30 for tetrahedra with 4η

bins (c) Mean shape factors
w and
I2 for triangles and tetrahedra The straight dotted lines denote inertial range value I2≈ 0.16 and the Gaussian value I2≈ 0.22.

and that surface normals are predominately aligned with the most compressingaxis 'λ3 [25, 22, 14, 26] It is natural to expect the larger scale counterparts

r0, g1, and g3 of pairs, areas and volumes to behave similarly In fig 7 and 8

we present experimental evidence for this PDFs for all cases are shown for

t > τ B , i.e τ B is replacing τ η as the relevant time scale Initial scales range

from 6 < η < 30, 14 < η < 30, and 22 < η < 30, for pairs, triangles and

tetrahedra respectively This is reflecting that it is more difficult to find e.g.four points close by and to be able to track them for a long time than it

is to find and track ’just’ a pair In all cases we observe a clear alignment

of r and g1 with 'λ1 (fig.7), and moreover, the PDFs for pairs, triangles andtetrahedra are almost identical also on a quantitative level For the surface

normals of triangles and the smallest eigen-direction of tetrahedra, g3, we see

a clear preferential alignment with the compressing principal axis 'λ3 (fig 8).These alignments are one way to explain why in the inertial range flat andelongated structures can be observed as it is reported in [6, 8]

Such alignments to principal axes only affect separations if the ing strain field 's ij is strong enough, e.g as one would expect from K41 typearguments As we have seen above in fig 1 this seems to be the case Forparticle pairs we now directly check how much the coarse grained counterpart

is balanced In other words, we check to what degree particle separation

dr (t) /dt is governed by the strain field filtered at the local scale ∆ = r (t)

as it is assumed in [3] Fig 9(a) shows the temporal evolution of averages

of the l.h.s and r.h.s of eqn 11 for initial separations 6 < r0/η < 30 For

the r.h.s values are only given if separations are large enough to find n > 12

266 Beat L¨uthi, Jacob Berg, Søren Ott and Jakob Mann

1 2 3

|cos(g

1 , λi)|

0 0.2 0.4 0.6 0.8 1 0

1 2 3

Fig 7 PDFs of the cosines between the most elongated axis of particle

constel-lations and the eigenframe of the filtered strain tensor 's ij at t > τ B (a) Particle

pairs with initial separations r0 of 6 < r0/η < 30 (b) Particle triangles with initial

Fig 8 PDFs of the cosines between the shortest axis of particle constellations

and the eigenframe of the filtered strain tensor 's ij at t > τ B (a) Triangles with

/r2while the compensated r i r j 's ij /r2continues to decrease The

two straight dotted lines at 0.11 and 0.14 denote the interval of the seemingly

universal stretching rate for the viscous scales [25, 22, 14, 26, 2] Our datafor the compensated large scale separation rates fall into this range It thusappears, that the total separation rate indeed does behave like its viscouscounterpart, also on a quantitative level, as it was assumed in [3] However,

we infer that the total separation must be the sum of contributions that stem

Fig 9 (a) Temporal evolution of the means of l.h.s (solid lines) and r.h.s (dashed

lines) of eqn 11 Initial separations are 6 < r0/η < 30 with 4η spacing (b) Same data as in (a) but all quantities are compensated with t ∗ / √

2 of eqn 4 The two

straight dotted lines denote the interval between 0.11 and 0.14.

also from smaller scales ∆ < r The contribution from r i r j 's ij, filtered at scale

∆ = r, is in our case roughly 50%.

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