Reynolds number effect on the velocity increment skewness in isotropic turbulence

0.27 MB·English

by  Wouter J. T. Bos

Reynolds number effect on the velocity increment skewness in isotropic turbulence
Wouter J.T. Bos1, Laurent Chevillard2, Julian F. Scott1 and Robert Rubinstein3
1 LMFA, CNRS, Ecole Centrale de Lyon,
Universit´e de Lyon, 69134 Ecully, France
2 Laboratoire de Physique, ENS Lyon,
CNRS, Universit´e de Lyon, Lyon, France
3 Newport News, VA, USA
Second and third order longitudinal structure functions and wavenumber spectra of isotropic
turbulence are computed using the EDQNM model and compared to results of the multifractal
1 formalism. At thehighest Reynoldsnumberavailable in windtunnelexperiments, R =2500, both
λ
1 the multifractal model and EDQNM give power-law corrections to the inertial range scaling of
0 the velocity increment skewness. For EDQNM, this correction is a finite Reynolds number effect,
2 whereas for the multifractal formalism it is an intermittency correction that persists at any high

n Reynoldsnumber. Furthermore,thetwoapproachesyieldrealisticbehaviorofsecondandthirdorder
a statistics ofthevelocity fluctuationsinthedissipative andnear-dissipativeranges. Similarities and
J differences are highlighted, in particular theReynolds numberdependence.

5
2
I. INTRODUCTION non-zero intermittency corrections to the inertial range

] scalingoftheenergyspectrumandofhigherorderquan-
n
The nonlinearity in the Navier-Stokes equations gives tities.
y
d rise to aninteractionbetweendifferent length-scalesin a A valuable theoretical tool to study the statistical
- turbulent flow. These interactions are the basic mecha- properties of homogeneous turbulence is two-point clo-
u nism behind the celebrated Kolmogorov-Richardson en- sure theory. The first theoretical approach of this kind,
l
f ergy cascade [1, 2]. This phenomenological picture of derived from the Navier-Stokes equations, is the Direct
.
s energy cascading from scale to scale towards the scales Interaction Approximation (DIA) [10]. Subsequent im-
c in which dissipation becomes appreciable is the corner- provements [11] of this theory allowed to show that the
si stone of a large number of turbulence models (e.g. ref- k−5/3 dependence of the energy spectrum canbe related
y erence [3]). If locality in scale-space is assumed, energy- directly to the Navier-Stokes equations. Simplifications
h
conservation and local-isotropy will lead to a wavenum- led to different related closures such as the test-field
p
[ ber dependence of the energy spectrum of the form model [12], the Lagrangian renormalized approximation
[13] and the Eddy-Damped Quasi-Normal Approxima-
1 E(k)∼ǫ2/3k−5/3 (1) tion (EDQNM) [14]. EDQNM is of the closures named
v
herethe simplest. It is obtainedbyassuming inthe DIA
6 with ǫ the energy flux, which, using the assumption of
formulation that the two-time correlations decay expo-
2 statistical stationarity, equals the energy dissipation. A
nentially with a typical time-scale modeled phenomeno-
7
physical space equivalent of this scaling law is the scale
4 logically. We note that this time-scale can also be de-
dependence of the second-order longitudinal structure
. termined self-consistently within the EDQNM approach
1 function,
[15].
0
1 D (r)∼ǫ2/3r2/3. (2) Theseclosures,althoughdirectlyrelatedtotheNavier-
LL
1 Stokes equations, do not yield any intermittency correc-
: The definitions of D (r) and E(k) will be given below. tionstothescaling(1). However,thepredictedresultsfor
v LL
i Thepossibilityofcorrectionstotheinertialrangescal- scalingexponentsoftheenergyspectrumcomparerather
X
ing of structure functions, due to the intermittent char- well to experimentally observed values [16]. Indeed, at
r acter of the energy dissipation [4], was taken into ac- low Reynolds numbers, corrections to the scaling expo-
a
countin amoregeneraltheoryadvancedby Kolmogorov nentsduetothefiniteReynoldsnumberareusuallylarger
and Oboukhov [5, 6]. Experiments aiming at the mea- than the expected intermittency corrections and these
surement of the intermittency corrections(e.g. reference finite Reynolds number effects vanish very slowly [16–
[7, 8]) indeed showed small corrections to the scaling 18]. The factthat two-pointclosureandthe multifractal
which could be due to intermittency, in particular for formalism can treat both low and very high Reynolds
higher-order structure functions. Subsequently a large numbersusinglimited computationaleffort,makesthese
numberofphenomenologicalmodelswasproposedtode- approachesveryattractivetostudyReynolds-numberef-
scribe the intermittent character of turbulence. Refer- fects.
ence [9] gives anoverviewof workon intermittency upto The present work will compare the predictions of clo-
1995. One of the more successful models, in the sense sure for second and third-order quantities with results
of reproducing the different features of isotropic turbu- of the multifractal description. This will allow to show
lence,is the multifractalmodel[9]. This phenomenologi- to what extent intermittency corrections can be distin-
caldescriptioncompareswelltomeasurementsandgives guished from Reynolds number effects at low, moder-
2
ate and high Reynolds numbers. We choose to compare with
quantitiesinphysicalspace,sincemostexperimentaland
theoretical efforts aiming at the understanding and de- Pijm(k)=kjPim(k)+kmPij(k), (7)
scription of intermittency focus on these quantities (we k k
P (k)=δ − i j. (8)
note howeverthat in principle intermittency corrections, ij ij k2
if any, should also be observed in wavenumber spectra).
Therefore we need to convert the Fourier-space quanti- To derive (6), the incompressibility condition was used
ties into physical space quantities. It is described in the to eliminate the pressure term. In isotropic non-helical
next section how this is done. The relations to convert turbulence, the energy spectrum is related to u (k) by
i
physical space quantities into their Fourier-space coun-
terparts is also given. In section III we will present the Pij(k)E(k)=u (k)u∗(k) (9)
EDQNM model and we will give anoutline of the multi- 4πk2 i j
fractal description. In section IV we present the results
of the EDQNM model for these quantities and compare and since ui(x) is real, this gives
withthemultifractalresultsinbothFourierandphysical
space. Section V concludes this article. E(k)=2πk2ui(k)ui(−k). (10)
In order to derive the equation for E(k), we multiply
(6) by u (−k). Then we write a similar equation for
i
II. EXACT RELATIONS BETWEEN SECOND
u (−k) and multiply by u (k) Summing both equations
AND THIRD ORDER QUANTITIES IN i i
and averaging yields,
FOURIER SPACE AND IN PHYSICAL SPACE
∂
+2νk2 E(k) = iπk2P (k) T (k)−T∗ (k)
In this section we will give the relation between the ∂t ijm ijm ijm
energy spectrum and the second order structure func- (cid:20) (cid:21)
= T(k) (cid:2) (11(cid:3))
tion D (r), and between the nonlinear transfer and
LL
the third-order longitudinal velocity structure function,
with
D (r). Even though the relations given here are not
LLL
new (e.g. [19–21]), the details of the derivation are dis-
T (k)= u (k)u (p)u (q)δ(k+p+q)dpdq
persedornotwelldocumentedinliteratureandwethink ijm i j m
that it is therefore worth to write down in detail this ZZ
(12)
derivation, which can be found in the appendix.
∗
T (k)= u (−k)u (−p)u (−q)δ(k+p+q)dpdq.
ijm i j m
ZZ
(13)
A. Derivation of the Lin-equation
By isotropy it can be shown that T∗ (k) = −T (k).
ijm ijm
The starting point is the Navier-Stokes equations for TheRHSofthe Lin-equation(11)isthe nonlineartrans-
incompressible flow, ferT(k),whichwewillrelatetothethird-orderlongitudi-
nalstructure function. But first we will give the relation
∂ui(x) +u (x)∂ui(x) =−1∂p(x) +ν∂2ui(x) (3) between the energy spectrum and the second order lon-
∂t j ∂x ρ ∂x ∂x2 gitudinal structure function.
j i j
∂u (x)
i
=0, (4)
∂x
i B. Relation between E(k) and D (r).
LL
with ρ the density and p the pressure. Time arguments
are omitted for brevity. The three-dimensional Fourier Thesecondorderlongitudinalstructurefunctionisde-
transfer is defined as fined as
u (k)= 1 u (x)e−ik·rdk. (5) DLL(r)=δu2L (14)
i (2π)3 i
Z
with
InFourierspacetheNavier-Stokesequationscanbewrit-
r r
ten δuL =uL−u′L = riui(x)− riui(x+r). (15)
∂u (k)
i +νk2u (k)= its relation to the energy spectrum is
i
∂t
i
− P (k) u (p)u (q)δ(k−p−q)dpdq (6) D (r) = E(k)f(kr)dk (16)
ijm j m LL
2
ZZ Z
3
with f(x) given by Using this in (16) and (20), we find for very small r,
2
1 sin(x)−(x)cos(x) D (r) = r2 k2E(k)dk
f(x) = 4 − . (17) LL
3 (x)3 15
(cid:20) (cid:21) Z
ǫr2
= (27)
Thederivationofthisexpressionisgivenintheappendix. 15ν
Aconvenientexpressiontocomputetheenergyspectrum 4 2
from the second order structure function is DLLL(r) = r T(k)dk− r3 k2T(k)dk
5 35
Z Z
2
u2 DLL(r) = − r3 k2T(k)dk (28)
E(k)= 1− kr[sin(kr)−krcos(kr)]dr 35
π 2u2 Z
Z (cid:18) (cid:19)
(18) in which we used that
andwerefertoMathieuandScott[21]forthederivation. 2ν k2E(k)dk = ǫ (29)
Z
T(k)dk = 0, (30)
C. Relation between T(k) and D (r).
LLL Z
with ǫ the energy dissipation. So we find that the struc-
The third order longitudinal structure function in ho- ture functions oforder2 and3 scale as r2 and r3 respec-
mogeneous turbulence can be expressed as tively for very small r, which is expected since at small
enough scales the flow can be considered as smooth.
D (r)=δu3 =3 u u′ 2−u′ u2 . (19) The velocity-increment skewness is defined as
LLL L L L L L
(cid:16) (cid:17) D (r)
LLL
which is related to the transfer spectrum by S(r)= . (31)
D (r)3/2
LL
∞
D (r)=r T(k)g(kr)dk, (20) Since at very small scales
LLL
Z0 ∂u
δu ≈r , (32)
with L ∂x
3(sinx−xcosx)−x2sinx one finds that
g(x)=12 , (21)
x5 (∂u/∂x)3
limS(r)= . (33)
with details given in the appendix. The equivalent ex- r→0 3/2
(∂u/∂x)2
pressiontocomputethetransferspectrumfromD (r)
LLL
is [21], (cid:16) (cid:17)
Using expressions (27) and (28), we find [19],
T(k)= 6kπ sin(rkr)∂∂r 1r∂∂r r4DLLL(r) dr. (22) limS(r)= [(∂xu)3] =− 153/2 ∞0∞k2T(k)dk (.34)
Z (cid:20) (cid:0) (cid:1)(cid:21) r→0 [(∂xu)2]3/2 35(2)1/2[ 0R k2E(k)dk]3/2
In the case of high-Reynolds numRber, if the non-
D. Small scale behavior of DLL(r) and DLLL(r) stationarity can be neglected at high k, or if the tur-
bulence is kept stationary by a forcing term acting only
Before continuing, let us have a look at the behavior at small k, we have
of the functions (17) and (21),
k2T(k)dk≈ 2νk4E(k)dk, (35)
1 sinx−xcosx Z Z
f(x)=4 − (23)
3 x3 so that [19]
(cid:20) (cid:21)
3(sinx−xcosx)−x2sinx ∞
g(x)=12 . (24) 153/221/2ν k4E(k)dk
x5 limS(r)≈− ∞0 . (36)
r→0 35 [ k2E(k)dk]3/2
0R
Taylorexpansionsofthesineandcosinetermsshowthat
for x↓0, The velocity-derivative skewneRss is then completely de-
termined by moments of the energy spectrum. Using
2 (29), andassuminganinertialrangespectrumextending
f(x)= x2+O(x3) (25)
15 upto kf of the form
4 2
g(x)= − x2+O(x4). (26) E(k)∼ǫ2/3ka (37)
5 35
4
we obtain for α>−3 DIA is a self-consistent two-point two-time theory with-
out adjustable parameters. Simplifications are needed
1/(α+3)
k ∼ ǫ1/3/ν . (38) to obtaina single-time (or Markovian)descriptionintro-
f
ducting assumptions and adjustable parameters. In the
(cid:16) (cid:17)
Substituting this in (36) we obtain, case of EDQNM the simplifying assumption is that all
time-correlations decay exponentially, with a time-scale
limS(r) ∼ k−21(5+3α) Θkpq modeled phenomenologically by
r→0 f
−3α+5 1−exp(−(η +η +η )×t)
∼ R α+3 , (39) Θ = k p q (42)
λ kpq
η +η +η
k p q
inwhichweusedthattheTaylor-scaleReynoldsnumber,
to be defined later, is proportional to ν−1/2. It follows in which η is the eddy damping, expressed as
fromthisexpressionthatthevelocityderivativeskewness
is independent of the Reynolds number if α = −5/3. If k
corrections to the Kolmogorov 1941 (K41) scaling are ηk =λ s2E(s)ds+νk2. (43)
s
present,as is the casein the multi-fractalformalism,the Z0
skewness becomes a function of the Reynolds number.
related to the timescale associated with an eddy at
This dependence is by (39) directly related to the inter-
wavenumber k, parameterised by the EDQNM parame-
mittency correction to the K41 scaling.
ter,λ,whichischosenequalto0.49[15]. Theexponential
time-dependence in(42)appearsbythe assumptionthat
the initial conditions have vanishing triple correlationas
E. Large scale behavior of D (r) and D (r)
LL LLL wouldbe the casefor a Gaussianfield. Its influence van-
ishes at long times. For an extensive discussion of the
At large r we find EDQNM model see [22, 23], but we want to stress that
oneofthekeyfeaturesofEDQNMisthatitisapplicable
4
D (r) = E(k)dk =2u2 (40) atallReynolds numbers (itis notanasymptotic theory)
LL
3
Z and at all scales of a turbulent flow. In other words, its
whichisexpectedfrom(A1)sinceatlargeseparationdis- results go beyond mere scaling and can give insights on
tances the correlationbetweenthe velocityattwopoints the Reynolds number dependency of different quantities
is supposed to vanish. D (r) tends for the same rea- related to turbulence.
LLL
son to zero for large separation distances r. At large We performed simulations of the EDQNM model in
r the velocity increment skewness should therefore go the unforced case by integrating numerically Eq. (41),
smoothlytozero,sincethevelocitycorrelationshouldde- starting from an initial spectrum,
caysmoothlyatlarger. TheexactwayinwhichD (r)
LLL E (0)=Bk4exp −(k/k )2 , (44)
tendstozerodependsonthebehavioroftheenergyspec- k L
trum at the very low wavenumbers.
(cid:2) (cid:3)
withB chosentonormalizetheenergytounityandk =
L
8k ,k beingthesmallestwavenumber. Theresolutionis
0 0
III. THE EDQNM MODEL AND THE chosen 12 gridpoints per decade, logarithmically spaced.
MULTIFRACTAL DESCRIPTION In the decaying simulations results are evaluated in the
self-similar stage of decay, in which ǫ/e , with e the
kin kin
kineticenergy,isproportionaltot−1. Forcedsimulations
A. The EDQNM model
areevaluatedwhenasteadystateisreached. Theforcing
in these cases corresponds to a region k < k , in which
The EDQNM model is a closure of the Lin-equation L
the energy spectrum is kept constant.
in which the nonlinear transfer T(k) is expressed as a
function of the energy spectrum. The transfer T(k) is
given by
B. The multifractal description
T(k)= Θ (xy+z3) k2p E(p)E(q)
kpq
In this section some key concepts of the multi-fractal
ZZ∆
(cid:2) dpdq description will be presented. A more detailed presenta-
−p3E(q)E(k) . (41)
tion and references can be found in appendix C.
pq
In the multifractal description the velocity-increments
(cid:3)
In equation (41), ∆ is a band in p,q-space so that the δu (x,r)=u (x+r)−u (x) at scales r in the inertial
L L L
three wave-vectorsk,p,q form a triangle. x,y,z are the range are modeled by the product of two independent
cosines of the angles opposite to k,p,q in this triangle. random variables
This particular structure is common to all closures de-
rived from the Direct Interaction Approximation [10]. δu (x,r)=β ξ. (45)
L r
5
In this expression ξ is a zero average Gaussian random 1010
380
variableofvarianceσ2,whereσ2istwicethemean-square 2500
108 25000
of the velocity fluctuations. The quantity β introduces
r
the scale dependence in the statistics of the velocity in- 106
crements. It is defined as
r h 51/4ν) 104
βr = L . (46) εk)/( 102
(cid:18) 0(cid:19) E(
100
withL theintegrallengthscale. Theparticularityofthe
0
approach lies in the fact that the exponent h is a fluc- 10-2
tuating quantity. If a constant value h = 1/3 is chosen,
10-4
K41behaviorisrecovered. Inthemultifractalframework
h is determined by the probability density function 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102
kη
r 1−D(h) 2.5
P (h)∝ . (47) 380
r L 2500
(cid:18) 0(cid:19) 25000
2
If the unknown function D(h) is given (for example by
comparison with experimental results), a complete de-
scription of the inertial range statistics of the velocity -5/3k) 1.5
iancccroeumnetntthsecdanissbipeaotibvteaienffeedc.tsAwnasexptreonpsoiosnedtobytaPkaelaindtino 2/3εk)/( 1
andVulpiani [24]and Nelkin [25]. Further details onthe E(
multi-fractaldescription,includingtheexpressionsforβ
r
and P (h), are given in the appendix. 0.5
r
0
IV. RESULTS FOR SECOND AND THIRD 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102
ORDER QUANTITIES kη
0.7
380
A. Results in Fourier space 0.6 2500
25000
0.5
In the following we will consider three different values 0.4
of the Reynolds number 0.3
εk)/( 0.2
u22 kT( 0.1
Rλ =s15 νǫ . (48) 0
-0.1
These values are R = 380, 2500, 25000. The low-
λ -0.2
est corresponds to a typical Reynolds number for lab-
-0.3
oratory experiments in jets or wind-tunnels, the second
-0.4
one to the highest Reynolds number obtained in wind- 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102
tunnel turbulence, i.e. in the Modane windtunnel [26], kη
and R =25000 corresponds to the Reynolds number of
λ
FIG. 1: Results for the energy spectrum computed by the
large scale atmospheric flows and no controlled experi-
EDQNMmodel. Inthecenterplotweshowthecompensated
mental results of isotropic turbulence are available. In
spectrum. Inthebottomfigurethenonlineartransferisplot-
the following we will present results for these Reynolds
ted. All quantitiesare normalized by Kolmogorov scales.
numbers. All quantities are normalized by Kolmogorov
scales,whichmeansthattheyarenon-dimensionalizedby
usingthevariablesν andǫ. Forexample,alllengthscales
are normalized by η = (ν3/ǫ)1/4. This normalization InFigure1,theenergyspectrumisshownforthreedis-
allows to collapse the dissipation range of the different tinctTaylor-scaleReynoldsnumbers. We observeaclear
quantities if this range becomes independent of the vis- k−5/3 power-law in the log-log representation. However,
cosity. ThisisthecaseintheK41phenomenology. Inthe when showing the compensated spectra in log-lin repre-
presence of intermittency this is not the case anymore. sentationitisobservedthatonlyatthehighestReynolds
It will however been shown in the following that also in numberaclearplateaucanbe discerned. Atsmallk this
thatcasethedissipationrangesofthedifferentquantities plateaudropstozero,andatlargekaviscousbottleneck
nearlycollapseinthepresentrangeofReynoldsnumbers. is observed.
6
105 105
200 EDQNM 380
300 MF 380
104 400 104 EDQNMMF 22550000
EDQNM 25000
103
103
51/4ενE(k)/() 102 1/2νεD(r)/()LL 110012
101
100
100 10-1
10-1 10-2
10-3 10-2 10-1 100 100 101 102 103 104 105 106 107
kη r/η
3 2.5
200 EDQNM 380
300 MF 380
400 EDQNM 2500
2.5 2 MF 2500
EDQNM 25000
2
2/3-5/3εE(k)/(k) 1.5 2/3εD(r)/(r)LL 1. 51
1
0.5
0.5
0 0
10-3 10-2 10-1 100 100 101 102 103 104 105 106 107
kη r/η
0.6 200 FIG. 3: The second order londitudinal structure function
300
0.4 400 computed by EDQNM and the multifractal model. Straight
blacklinesindicatepowerlawbehaviorproportionaltor2 and
0.2 r2/3 respectively. Inthebottomfigurethefunctionsarecom-
0 pensated according to K41 scaling.
εk)/( -0.2
kT( -0.4
-0.6
-0.8
-1
-1.2
10-3 10-2 10-1 100 are shown for relatively low Reynolds numbers (upto
kη R = 400), since the numerical integration for higher
λ
values yielded extremely noisy results in the dissipation
FIG. 2: Results for the energy spectrum computed from the
range. A bumpy large-scale behavior is observed in the
multi-fractaldescription. Inthecenterplotweshowthecom-
compensatedenergyspectra,correspondingtothead-hoc
pensatedspectrum. Inthebottomfigurethenonlineartrans-
modeling of the large scales, as explained in appendix
fer is plotted.
C. This modeling also causes the relatively narrow neg-
ative peak in the transfer spectrum. In the following,
when presenting the structure functions, we do not need
Sinceweareinterestedinsecondandthirdorderquan- a smooth behaviorfor the large-scalesandwe will there-
tities in the present work, we also show the nonlinear fore not use the ad-hoc modification of the large scales.
transfer. Again we observe that the asymptotic case, In the dissipative and near-dissipative ranges, the spec-
here indicated by a plateau around zero in between the tral quantities (such as power spectrum and nonlinear
negative and the positive lobe of the transfer spectrum, transfer) obtained from EDQNM closures and the mul-
is only observed at the highest Reynolds number. tifractal formalism are very similar. In the dissipation
The energy and transfer spectra computed from the range a viscous bottleneck is observed in both descrip-
multifractal descriptionare shownin Fig. 2. The results tions.
7
106 ism. Note that we prefer to show here the results with-
EDQNM 380
105 MF 380 out the ad-hoc modificationproposedin the last section.
EDQNM 2500
MF 2500 Wefurtherobservethatthestructurefunctioncomputed
104 EDQNM 25000
from EDQNM, as for the energy spectra, does not dis-
103 play a clear plateau in the compensated representation
3/4νε) 102 for Rλ <25000.
D(r)/(LLL 101 shoTwhne irnesfiugltusrfeor4.thAegtahinirdw-eorcdleerarsltyruocbtsuerrevefutnhcetisomnsooatrhe
- 100 small-scalebehaviorproportionaltor3. Inthisrangethe
10-1 multifractal model closely follows the EDQNM results.
For larger r a close to linear dependence and at large
10-2
scalesadecreasetowardszero. Alsoherethemultifractal
10-3 formalism does not take into account the large scales.
100 101 102 103 104 105 106 107
r/η In the inertial range, at very large Rλ, the third order
structure function should scale as
1
EDQNM 380 4
MF 380 D (r)=− ǫr. (49)
EDQNM 2500 LLL
5
0.8 MF 2500
EDQNM 25000
it is observed that this is only reached at the highest
R for EDQNM and only for a short range of scales.
λ
εD(r)/(r)LLL 00..46 TRhλe=m3u8l0t.i-fractal results collapse with (49) already at
-
C. Results for the velocity increment and
0.2
derivative skewness
0 In figure 5 top, we show the velocity increment skew-
100 101 102 103 104 105 106 107
r/η ness for different Reynolds numbers. In the K41 phe-
nomenology, this quantity should give a constant value
FIG.4: Thethirdorderlonditudinalstructurefunctioncom- in the inertial range. It is observed that the fact that
putedbyEDQNMandthemultifractalmodel. Straightblack D (r) tends to zero smoothly for large r results in a
LLL
lines indicate powerlaw behavior proportional to r3 and r1 gently decreasingfunction, rather thana constantvalue.
respectively. In the bottom figure the functions are compen- In the dissipation range all curves nearly collapse.
sated according to K41 scaling. The asymptoticresult 4/5 is Only the multifractal approach gives a slightly higher
indicated bya dashed line.
value than the rest, since the velocity derivative skew-
ness is a function of R , as will be shown later, in figure
λ
6. At large scales, the multifractal result closely follows
B. Results for structure functions
thehigh-ReynoldsEDQNMresultuptothecut-offofthe
multi-fractal result.
Weusedequations(16)and(20)tocomputethestruc- In the center and top graph of figure 5, we compare
turefunctionsfromtheenergyspectraandtransferspec- the results also with experimental results. At R =380,
λ
tra shown in the previous section. The results for the wecomparewithhot-wiremeasurementsinanair-jetex-
secondorderstructurefunctionareshowninfigure3. We periment [27]. At small scales the experimental value is
show the multifractal prediction in the same graph. In significantly larger than the theoretical results. At these
the log-log representationwe clearly observe the smooth scales the accuracy of the hot-wire probe decreases how-
r2 small scale behavior and the plateau proportional to ever. In the inertial range the multi-fractal approach is
thekineticenergyatlargescales. Inbetweenapower-law veryclosetotheexperimentalvalue. TheEDQNMcurve
dependence close to r2/3 is observed. dropsmuchfastertozero. Inhomogeneityandanisotropy
The multifractal prediction closely ressembles the of the experimental turbulent field could be behind this
EDQNM result in the dissipation range. The differences discrepancy.
between the two models are more clearly visible in the At R = 2500 a comparison is made with the veloc-
λ
compensatedplot, wherewe observethatforR =2500, ity increment skewness measured in the return-channel
λ
the power-law dependence is clearly steeper than r2/3. of the ONERA wind-tunnel in Modane. The Reynolds
The largestdifference is observedat large r. Indeed, the number obtained there is one of the highest measured
multifractal description does not take into account the in wind-tunnel turbulence. Unfortunately at large scales
shape of the velocity correlation at large r. This corre- the third-order statistics are not fully converged so that
lation should in a realistic flow smoothly tend to zero, no smooth curve is available there. However the gen-
but this effect is not taken into account in the formal- eral trend of the curves is quite similar at all scales. A
8
0.6 1
EDQNM 380
MF 380
EDQNM 2500
0.5 MF 2500
EDQNM 25000
0.4 0)
S(
-
S(r) 0.3
-
EDQNM
0.2 MF
0.1 DNS
101 102 103 104
0.1
Rλ
0
100 101 102 103 104 105 106 107 FIG.6: Thevelocity derivativeskewness as afunction of the
r/η ReynoldsnumbersforEDQNMandMF.AlsoshownareDNS
results from references [28,29].
EDQNM 380
MF 380
Jet 380
EDQNM 380 Forced
0.4 transient effect, due to the finite-Reynolds number. In
EDQNM this power-law vanishes thus at high Reynolds
number.
S(r) In figure 6 we show the Reynolds number dependency
- of the longitudinal velocity derivative skewness, as com-
0.2
puted by equation (34). We see that this quantity sat-
urates for R > 100 at a value around 0.4. In the mul-
λ
tifractal approach this quantity follows a power-law of
the Reynolds number (see Eqs. (C7) and (C9)) with
an exponent around 0.13. Also shown are the results of
0.1
100 101 102 103 104 105 Direct Numerical Simulations [28, 29]. Note that in [28]
r/η morenumericalandexperimentalresultsareavailablein-
cluding the experimental compilation by [30]. We chose
EDQNM 2500
MF 2500 however those which give the general trend. The DNS
Modane 2500
results show a slightly increasing trend from 0.4 to 0.6
0.4
for a Reynolds number going from 10 to 1000.
S(r) D. Influence of large-scale forcing
-
0.2
To conclude this results section we address the influ-
ence ofa largescaleforcingonthe scalingofthe velocity
increment skewness. Indeed, in experiments of nearly
isotropic turbulence we often consider a turbulence gen-
0.1
100 101 102 103 104 105 erated by a grid, advected by a mean velocity. This cor-
r/η responds in the frame moving with the mean flow, to
freely decayingturbulence. Directnumericalsimulations
FIG. 5: Comparison of the longitudinal velocity increment
ofisotropicturbulenceareoftenforcedatthelargescales
skewness between EDQNM and the multifractal approach.
in order to obtain a as high as possible Reynolds num-
Inthecenterfigureresultsarecomparedtoair-jetresults. In
ber. The difference between the two types of turbulent
the bottom plot the results are compared to high-Reynolds
flows is important. For example in [31] it was shown
numberwind-tunnelexperiments. The straight black line in-
dicates a powerlaw proportional tor−0.04 that the normalized dissipation rate is nearly twice as
high in decaying turbulence as it is in forced turbulence.
Also for the appearance of scaling ranges this difference
can be important. The difference between decaying and
surprising fact is here the power-law that is observed in forcedturbulence inapproachingthe asymptotic formof
the inertial range of both the multifractal result and the the third-order structure function was reported in [32].
EDQNMresult. Indeed,inthemultifractalapproachthis In figure 7 we show how the inertial range scaling of the
power-law is a signature of inertial-range intermittency energy spectrum improves when considering statistically
and the model is developed to take this into account. stationary forcedturbulence at the same Reynolds num-
In the EDQNM approach, however, this power-law is a ber. We observe the appearance of a large peak in the
9
4.5
380 themultifractalmodelalmostcollapsedwiththescaling-
4 2500 correctioninducedbythefinitenessoftheReynoldsnum-
25000
ber in the EDQNM simulations. In particular at a
3.5
Reynolds of R = 2500 the two corrections almost co-
λ
3
-5/3k) 2.5 iancchiideev.abTlehiisnschoonwtrsoltlheadtexatpeRriemynenotldssanndumsibmeurslactuiornres,ntinly-
2/3εE(k)/( 2 tteinrgmuiitstheendcyfrcoomrrelocwtioRnesytnooltdhsensukmewbneersesffceacnts.noAtnbeintdeisr--
1.5
esting perspective is to investigate to what extent inter-
1 mittency corrections to higher-order quantities such as
the flatness can be distinguished from Reynolds number
0.5
effects(seee.g. Ref. [33]). Thistaskiswithinthe frame-
0
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 workofclosure-theoryfarfromtrivialandwillbeleftfor
kη future work.
EDQNM 380
EDQNM 2500
EDQNM 25000
Appendix A: Relation between second order
0.4
structure function and the energy spectrum
We will give here a detailed derivation of the relation
S(r) betweensecondorderstructurefunctionsandthe kinetic
-
0.2 energy spectrum. Parts of this derivation can be found
in different textbooks, but we think it is useful for the
interested reader to give all the details in this work.
Starting from (14), homogeneity allows to write
0.1
100 101 102 103 104 105 106 D (r) = 2 u2 −u u′ (A1)
LL L L L
r/η
= 2r(cid:16)irj u u −(cid:17)u u′ . (A2)
FIG.7: Theinfluenceofalargescaleforcingonscalingareil- r2 i j i j
lustrated fortheenergyspectrum andthevelocityincrement (cid:16) (cid:17)
skewness. The black-dotted lines correspond to forced tur- UsingtheinverseFouriertransform,and(9)wecanrelate
bulence at the same Reynolds number as the decaying cases this to the energy spectrum
considered.
r r
D (r) = 2 i j u u − u (k)u (−k)eik·rdk(A3)
LL r2 i j i j
spectrum, corresponding to the forcing. We also observe (cid:18) Z (cid:19)
r r P (k)
thatthecompensatedenergyspectradisplayaclearscal- = i j u u − ij E(k)eik·rdk (A4)
r2 i j 4πk2
ing range, already at a Reynolds number of Rλ = 380. (cid:18) Z (cid:19)
The velocity-increment skewness of these forced calcu-
Defining φ the angle between k and r, we find
lations shows however no clear plateau at this Reynolds
number,butitsinertialrangebehaviorfollowscloselythe r r
Rλ =2500 decaying turbulence result. ri2jPij(k)=(1−cos2φ). (A5)
Also, in isotropic turbulence, the Reynolds stress tensor
V. CONCLUSION takes the form,
2
Inthe presentworkwe computedsecondandthird or- uiuj =u2δij, u2 = E(k)dk, (A6)
3
der structure functions from EDQNM results. We com- Z
paredthesestructurefunctionswithresultsfromthemul-
so that, introducing conveniently oriented spherical co-
tifractal formalism. It was shown that in the near dis-
ordinates, we write,
sipation range the different approaches give very similar
results. Itwasshownthattheappearanceofclearscaling (1−cos2φ)
rangesisveryslowforthestructurefunctionsaswasalso DLL(r) = 2u2−2 4πk2 E(k)eik·r2πk2sinφdφdk
observed in previous work [16, 17]. Z
4
The results for the velocity increment skewness were = E(k) − (1−cos2φ)eik·rsinφdφ dk.
alsocomparedtoexperimentalresults. Itwasshownthat 3
Z (cid:20) Z (cid:21)
the intermittency correction to this quantity given by (A7)
10
The integralover φ can be performed analytically by in- after multiplication of both sides by P (k) one finds
ijm
troducing ζ =cosφ and x=kr:
P (k) T(k)
π(1−cos2φ) sinφ eikrcosφdφ T(k)= ij4mk2 Tijm(k)−Ti∗jm(k) = 4iπk4. (B10)
(cid:0) (cid:1)
Z0 We substitute this in (B8),
1
= (1−ζ2) eixζdζ
r r r T(k)
Z−1 ∂2 1 DLLL(r)=3 irj3m 4iπk4Pijm(k)eik·rdk. (B11)
= 1+ eixζdζ Z
(cid:18) ∂x2(cid:19)Z−1 Defining φ the angle between k and r, we find that
∂2 eix−e−ix
= (cid:18)1+ ∂x2(cid:19) ix rirrj3rmPijm(k)=2kcosφ(1−cos2φ) (B12)
∂2 sinx
= 2 1+ and thus
∂x2 x
(cid:18) (cid:19)
T(k)
sinx cosx D (r)=6 cosφ(1−cos2φ) eik·rdk. (B13)
= 4 x3 − x2 , (A8) LLL 4iπk3
(cid:18) (cid:19) Z
Introducingagainconvenientlyorientedsphericalcoordi-
yielding
nates, we write this as
DLL(r) = 4Z E(k)(cid:20)31 − sin(kr)−(k(rk)r3)cos(kr)(cid:21)(dAk9.) DLLL(r) = 6 ∞ πcosφ(1−cos2φ)4Tiπ(kk)3eik·r×
Z0 Z0
2πk2sinφdφdk (B14)
Appendix B: Relation between third order structure ∞ T(k) π
function and the energy transfer spectrum = −3i cosφ(1−cos2φ)eik·r ×
k
Z0 Z0
sinφdφdk. (B15)
The Fourier transform of u u′ 2 with respect to r is
L L
As for D (r), the integral over φ can be performed an-
LL
FTr uLu′L2 = (B1) alytically by introducing ζ =cosφ and x=kr:
h i π
= uL(k)u2L(−k) (B2) cosφ (1−cos2φ) sinφ eikrcosφdφ
Z0
= u (k) u (p)u (q)δ(−k−p−q)dpdq (B3) 1
L L L
= ζ(1−ζ2) eixζdζ
ZZ
Z−1
= u (k)u (p)u (q)δ(k+p+q)dpdq (B4)
L L L 2sinx 6cosx 6sinx
ZZ = −2i + − , (B16)
r r r x2 x3 x4
= i j m u (k)u (p)u (q)δ(k+p+q)dpdq (cid:18) (cid:19)
r3 i j m
ZZ yielding
(B5)
= rirrj3rmTijm(k) (B6) DLLL(r) = 12r ∞T(k)3(sinkr−krc(oksrk)r5)−(kr)2sinkrdk
Z0
Analogously we find (B17)
r r r
FTr u2Lu′L = irj3mTi∗jm(k) (B7) Appendix C: The multifractal description
h i
So that
The multifractal formalism can be seen as a proba-
r r r bilistic interpretation of the averaged behavior of ve-
D (r)=3 i j m T (k)−T∗ (k) eik·rdk.
LLL r3 ijm ijm locity structure functions. More precisely, for a scale
Z (cid:0) (cid:1) (B8) r in the inertial range, using both the standard argu-
It is clear from the definitions (12) and (13) that ments of the multifractal formalism [9] and the proba-
T (k)−T∗ (k is a third order tensor, symmetric bilistic formulation of Castaing [34], the velocity incre-
ijm ijm
in the indices j,m and solenoidal in the index i, so that ment δuL(x,r) = uL(x+r)−uL(x) can be represented
(cid:0)its general form is(cid:1) by the product of two independent random variables,
δu (x,r) = β ξ, with ξ a zero average Gaussian ran-
L r
T (k)−T∗ (k)=T(k)P (k) (B9) dom variable of variance σ2 = h[δu (x,L )]2i, where
ijm ijm ijm L 0