On the scaling properties of 2d randomly stirred Navier--Stokes equation

0.21 MB·English

by  Andrea Mazzino

On the scaling properties of 2d randomly stirred Navier–Stokes equation
Andrea Mazzino∗
Department of Physics, University of Genova, and INFN, CNISM,
Genova Section, Via Dodecaneso 33, I–16146 Genova, Italy,
Paolo Muratore-Ginanneschi†
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 4, 00014 Helsinki, Finland,
Stefano Musacchio‡
Department of Physics of Complex Systems Weizmann Institute of Science Rehovot 76100, Israel
(Dated: February 8, 2008)
7
Weinquirethescalingpropertiesofthe2dNavier-Stokesequationsustainedbyaforcingfieldwith
0
Gaussianstatistics,white-noiseintimeandwithpower-lawcorrelationinmomentumspaceofdegree
0
2−2ε. This is at variance with the setting usually assumed to derive Kraichan’s classical theory.
2
Wecontrast accurate numerical experimentswith thedifferent predictions provided for thesmall ε
n regimebyKraichnan’sdoublecascadetheoryandbyrenormalization group(RG)analysis. Wegive
a clearevidencethatforallεKraichnan’stheoryisconsistentwiththeobservedphenomenology. Our
J
results call for a revision in theRG analysis of (2d) fully developed turbulence.

9
PACSnumbers: PACSnumber(s): 47.27.Te,47.27.−i,05.10.Cc
]
D
C In two dimensions, the joint conservation of energy nan’s predictions [3, 5, 21]. Very strong laboratory (see
. andenstrophy hasrelevantconsequencesfor the Navier– [17, 26] and reference therein) and numerical evidences
n
i Stokes equation. In 2d it is possible to prove that in (see e.g. [11] and references therein) support Kraichnan’
l
n the deterministic case the solution of the Cauchy prob- s theory. However, a first principle derivation of the sta-
[ lem exists and is unique [20] and, very recently, that in tistical properties of two dimensional turbulence is still

the stochastic case [8, 9, 10, 12, 19, 23] the solution is a missing. An attempt in this direction has recently been
1
v Markovprocessexponentiallymixingintimeandergodic undertaken [14, 15] (see also [24] and [16]) by applying
7 withauniqueinvariant(steadystate)measureevenwhen a renormalization group improved perturbation theory
1 the forcingactsonlyontwoFouriermodes[22]. Atlarge (RG) ([27] and [1] for review of applications to fluid tur-
0
Reynolds number, the 2d scaling properties also differ bulence)tothe randomlystirredNavier–Stokesequation
1
from the 3d-case. A long standing hypothesis [25], very withpowerlawforcing. Howeverhowitwillbediscussed
0
7 recently corroborated by numerical experiments [6, 7], in details in the sequel, RG analysis leads to a scenario
0 surmises the existence of a conformal invariance in 2d. notaprioriconsistentwithKraichnan’stheory. Thegoal
/ A phenomenological theory due to Kraichnan [18] and of the present work is to shed light on this issue.
n
i Batchelor [2] predicts the presence of a double cascade RG starting point is the randomly stirred Navier–
l
n mechanism governing the transfer of energy and enstro- Stokes equation
: phy in the limit of infinite inertial range. Accordingly,
v (∂ +v·∂)vα−ν ∂2vα =−∂αP +fα−ξ vα (1)
i an inverse energy cascade with spectrum characterized t 0 0
X
by a scaling exponent dE = −5/3 appears for values of ∂·v =0
r the wave-number k smaller than the typical scale k of
a F where α = 1,2, P is the pressure enforcing incompress-
the forcing scale. For wave-numberslargerthan k a di-
F
ibility and ξ is an Eckman type coupling providing for
rect enstrophy cascade should occur. The corresponding 0
large scale dissipation. The forcing f is a Gaussian field
energy spectrum scales as dE =−3+... where the dots
with zero average and correlation
here stand for possible logarithmic corrections. As em-
phasized in [3, 5] Kraichnan’s theory is encoded in three ≺fα(x,t)fβ(y,s)≻=δ(t−s)Fαβ(x−y) (2)
hypothesis, (i) velocity correlations are smooth at finite
ddp
viscosityandexistintheinviscidlimitevenatcoinciding Fαβ(x):= eıp·xFˇ(p)Tαβ(pˆ) (3)
points, (ii)Galileaninvariantfunctions andin particular Z (2π)d
structure functions reach a steady state and (iii) no dis- F χ p, p
Fˇ(p):= 0 m M (4)
sipative anomalies occur for the energy cascade. Under pd−(cid:0)4+2ε (cid:1)
these hypotheses,if the forcing field is homogeneousand
isotropic Gaussianand time δ-correlatedit is possible to F0 is a constant specifying the amplitude of forcing fluc-
derive asymptotic expressions of the three point struc- tuations and Tαβ(p) is the transversal projector. The
turefunctionsofthevelocityfieldconsistentwithKraich- function χ in (4) is slowly varying for m ≪ p ≪ M and
set infra-red m and ultra-violet cut-offs M scales for the
2
forcing. Itsdetailedshapedoesnotaffectthescalingpre- ToattainthisgoalweintegratedtheNavier–Stokesequa-
dictionsoftheRG.Irrespectivelyofthespatialdimension tion(1)forthevorticityfield(ω =ǫ ∂βvα)withastan-
αβ
d, the cumulative spectrum Fα (0) of the forcing (Ein- dard, fully-dealiased pseudospectral method in a doubly
α
steinconvectionforrepeatedvectorindexes)divergesfor periodic square domain of resolution 10242. Time evo-
0 ≤ ε < 2 asthe ultra-violetcut-offM tends to infinity
hence providing for stirring at small spatial scales. For
10-1
ε > 2, Fα (0) is dominated by small wave-numbers and
α
thusdescribesinfra-redstirring. Atε=0,thescalingdi-
10-2
mensionsofthematerialderivative,dissipationandforc-
ing in (1) coincide for a scalingdimensionof the velocity
10-3
field dv = 1 in momentum units. Thus ε = 0 provides a k)
marginal limit around which scaling dimensions can be E(
10-4
perturbatively determined with the help of ultra-violet
RG. The main result [14, 15, 16] is the existence of a
10-5
non-Gaussianinfra-red stable fixed point of the RG flow
yielding for the energy spectrum the prediction
10-6
1 101 102
E(k)=ε1/3F2/3k1−43ε R ε,m,kb (5) k
0 (cid:18) k k (cid:19)
FIG.1: Energyspectrumfor ε=0.5andstandard molecular
The adimensional function R depends upon infra-red dissipation. Aninertialrangewithinverseenergycascadesets
scales m and k = (εξ3/F )1/(6−2ε) and admits a reg- inforsufficientlysmall viscosityν =O(10−4)(solid line). At
b 0 0
largerviscositytheinertialrangeissuppressedanddissipative
ular expansionin powers of ε at the RG fixed point [14].
spectrum E∼k1−2ε is observed (dashed line).
Theresummationleadingtok1−43ε in(5)isderivedfrom
the solutionof the Callan–Symanzikequation [27]which
lution was computed by means of a standard second-
in the present case requires scaling at finite ε to stem
order Runge–Kutta scheme. We repeated our numeri-
fromthebalanceofthetwotermsinthematerialderiva-
cal experiments for different values of the hyperviscos-
tivewiththeforcingi.e. fromtherequirementofGalilean
ity (−1)p+1ν ∂2pv including p = 1, standard viscosity
invariancealone. Accordingto[1,14,15]compositeoper- 0
as in (1), and p = 4,6 obtaining qualitatively identical
atorsatsmallεdoesnotinduceanyself-similaritybreak-
results. The outcomes evince that for all 0 ≤ ε ≤ 2
ing by the infra-red scales m and k . The conclusion is
b
and sufficiently small viscosity, an inverse energy cas-
thatthespectrumshouldscaleforsmallεwithexponent
dE = 1 − 4ε/3 as in the 3d-case [1]. Such conclusion cade with exponent consistent with dE = −5/3 is ob-
served (see Fig. 1). If the viscosity increases, finite
seemsatvariancewithKraichnan’stheorywhichinstead
resolution effects prevent the observation of an inertial
suggests the occurrence of an inverse cascade at small
range as the Kolmogorov scale becomes of the order of
ε. Namely under the assumptions (i),(ii),(iii) [3, 4] the
the infra-red dissipation scale. In such a case a dis-
energy balance equation
sipative spectrum appears with scaling exponent con-
∂ +ξ −ν ∂2 ≺vα(x,t)v (0,t)≻ sistent with the prediction dE = 1 − 2ε dictated by
t 0 0 α
the balance between forcing and dissipation (see again
(cid:0) 1 (cid:1)
− ∂ ≺||δv||2(x,t)δvµ(x,t)≻=F(x) (6) Fig. 1). In agreement with (7), local balance scaling
µ
2
becomes dominant in the range 2 < ε < 3 (see Fig.
withδvα(x,t)=vα(x,t)−vα(0,t)yieldsforξ =0,ε<2 2). There, energy spectra scale with an exponent com-
0
and r2 :=||x||2 patible with dE = 1 − 4ε/3. Finally, for ε > 3 a di-
rect enstrophy cascade invades the whole inertial range.
S (r)≃c F M4−2εr+ c2F0 , m−1 ≫ r ≫M−1 (7) The phenomenology just described is confirmed by the
3 1 0 r3−2ε inspection of the energy and enstrophy fluxes in the in-
ertial range (see Fig. 3). They are respectively defined
S3 denotesthethreepointvelocitylongitudinalstructure by Π (k)∝ k d2q ℜ ≺F{v}α(−q)F{(v·∂v )}(q) ≻
rfuelnecvtaionntfaonrdthceipir=ese1n,t2atrwgoumadeinmt.enPsoiwonearlccoouenffiticnigenbtassierd- and ΠEZ(k) ∝Rkokko(2(π2d)2π2q)2ℜ ≺ F{ω}(−q)F{(v·∂ωα)}(q) ≻
on (7) then predicts a dE = −5/3 inverse cascade spec- where F denoRtes the Fourier transform and ko is cho-
trumatsmallεwithat most sub-leading correctionscon- sen such that Π (k) and Π (k) are positive quantities
E Z
sistent with the RG prediction. inthe inertialrange. Asspectraandstructurefunctions,
The currently available numerical resources permit to fluxes highlight the existence versus ε of three distinct
contrast the two apparently discordant predictions (5) scaling regimes the origin of which can be understood
and (7) with the actual Navier–Stokes phenomenology. by contrasting (6) with the energy and enstrophy inputs
3
11
11 (a)
10-1
10-2
(r)3 10-3 Π(k)E 10-1
S 10-4
10-2
10-5
10-6 11 101 102
k
10-710-2 10-1 11 103
(b)
r 102
FIG.2: Thirdorderstructurefunctionoflongitudinalvelocity 101
(intrciraenmgelenst)s.SW3(rh)enforǫǫ<=21th(sequparreesse)n,cǫe=of2a.5n(icnivrecrlesse),enǫe=rgy4 (k)E,Z 1
Π
cascade with constant flux results in the scaling S3(r) ∼ r 10-1
(solid line). When ǫ > 3 the presence of a direct enstrophy
cascade with constant flux results in the scaling S (r) ∼ r3 10-2
3
(dashed line). For 2 < ǫ < 3 the scaling is compatible with
10-3
theRG argument (dash-dotted line) 101 102
k
10-1
(c)
I (k) ∝ k dqqFˇ (q) and I (k) ∝ k dqqFˇ (q). For
E ko v Z ko v
0≤ε< 2Rboth IE andIZ arepeakedRinthe ultra-violet. 10-2
Since stirringoccursmainlyatsmallspatialscalesanin-
verse energy cascade sets in the whole inertial range. In Π(k)Z
therange2 < ε < 3theenergyinputbecomesinfra-red 10-3
divergent whilst I is still ultra-violet divergent in the
Z
absence of cut-offs. In this situation, the steady state is
attained when the energy and enstrophy fluxes balance 10-4
101 102
scale by scale the corresponding inputs (see Fig. 3). In k
thisε-rangetheenergyspectrumscalesinagreementwith
the exponent which can be extrapolated but no longer FIG.3: Theenergy fluxΠE (symbols) andtheenergy input
fully justified using the RG. Finally, for ε > 3 both the IE(k) (thin line) for different values of ǫ. (a) ǫ = 1; (b)
energyandenstrophyinputsbecomepeakedintheinfra- ǫ = 2.5; (c) ǫ = 4. The energy input is defined as IE(k) ∝
red. In such a case [3, 4] the right hand sides of Eq. R0kdqqFˇv(q)forǫ<2andasIE(k)∝Rk∞dqqFˇv(q)forǫ>2.
(6) and of the analogous equation for the vorticity cor- In (b) the enstrophy input, IZ(k) ∝ R0kdqq3Fˇ(q) with and
relation admit a regular Taylor expansion for mx ≪ 1. the enstrophy flux ΠZ(k) are also plotted in order to show
theirbalance at each inertial range scale.
The hypotheses (i), (ii), (iii) thus recover S (x)∝r3 i.e.
3
Kraichnan’s scaling for the direct enstrophy cascade in
agreement with our numerical observations.
As far as intermittency is concerned, our numerics to the convolution of the response field with the forc-
support normal scaling of velocity structure functions ingkernelintheSchwinger-Dysonequation[1,27]giving
(Fig. 4). Higher order vorticity structure function are rise to a non-local scaling field of dimension dO = −1/3
compatible with a weakly anomalous scaling (Fig. 5). independently of ε. This point however deserves more
These resultsprovideanindirectpositivetestfor the oc- theoretical inquiry.
currence of a vorticity dissipative anomaly [4]. We are pleased to thank J. Honkonen, A. Kupiainen
In conclusion, our numerics fully support the validity and P. Olla for numerous friendly and enlightening dis-
of Kraichnan’s theory for all values of the Ho¨lder expo- cussions. Numericalsimulationshavebeenperformedon
nent of power law forcing in 2d. RG scaling prediction the “Turbofarm” cluster at the INFN computing cen-
is not observed in the range where it was supposed to ter in Torino. A.M. was supported by COFIN 2006
appear. It seems to us that this in not trivially a conse- N.2005027808 and by CINFAI consortium, P.M.-G. by
quence of low Reynolds number (O(ε1/2) for F = O(ε) centreofexcellence “Geometric Analysis and Mathemat-
0
see discussion in section 9.6.4 of [13]) entailed by per- ical Physics” ofthe AcademyofFinlandandbyFP5EU
turbative expansion. Instead scaling seems to be related network contract HPRN-CT-2002-00300.
4
105 102
(a) 1.5
104 1 ζn
0.5
103
101 0
0 1 2 3 4
S(r)n 110012 ω()S(r)n n
1
1
10-1
10-2 10-1
10-1 1 101 10-2 10-1 11
S2(r) r
104 (b) FIG.5: StructurefunctionofvorticityincrementsSn(ω)(r)for
n=1,2,3,4(square,circles,triangles,poligons). Intheinset
102 we show thescaling exponentsζn. Here ǫ=4.
(r)n 1 [4] D. Bernard, SPhT-00-039 and chao-dyn/9904034.
S
[5] D. Bernard, SPhT-00-093 and cond-mat/0007106.
[6] D. Bernard, G. Boffetta, A. Celani and G. Falkovich,
10-2 Nature Physics 2 124 (2006)
[7] D. Bernard, G. Boffetta, A. Celani and G. Falkovich,
nlin.CD/0609069
10-4
10-2 10-1 1 101 [8] J. Bricmont, A. Kupiainen and R. Lefevere, J. Stat.
Phys., 100 743 (2000) and math-ph/9912018.
S (r)
2
[9] J. Bricmont, A. Kupiainen and R. Lefe-
10-3 vere, Comm. Math. Phys. 224 65 (2001) and
(c)
www.ma.utexas.edu/mp arc/
10-5 [10] J. Bricmont, A. Kupiainen and R. Lefevere, Comm.
Math. Phys.230 87 (2002) and math-ph/0010028.
10-7 [11] G. Boffetta, submitted toPhys. Fluids (2006).
S(r)n [12] WPh.yEs.,2J2.4C8.3M(2a0tt0i1n)g.ly, and Y. G. Sinai. Comm. Math.
10-9
[13] U. Frisch, Turbulence, Cambridge Univ. Press, Cam-
bridge, (1995).
10-11 [14] J. Honkonen,Phys.Rev. E 58 4532 (1998).
[15] J. Honkonen, Yu.S. Kabrits and M.V. Kompaniets,
10-13 nlin/0201025.
10-5 10-4 10-3 10-2 [16] L.Ts.Adzhemyan,J.Honkonen,M.V.Kompanietsand
S2(r) A. N. Vasil´ev, Phys. Rev. E 71, 036305 (2005) and
nlin.CD/0407067.
FIG. 4: Structure function of longitudinal velocity incre- [17] H. Kellay and W.I. Goldburg, Rep. Prog. Phys. 65 845
mentsS4(r)(squares)andS6(r)(circles)vs. S2(r). Thelines (2002).
represent the non-intermittent scalings S4(r)∼S2(r)2 (solid [18] R. Kraichnan, Phys.Fluids 10 (1967) 1417.
line)andS6(r)∼S2(r)3 (dashedline). (a)ǫ=1;(b)ǫ=2.5; [19] S. Kuksin and A. Shirikyan. Comm. Math. Phys. 213
(c) ǫ=4 291 (2000).
[20] O. A. Ladyzhenskaya The Mathematical Theory of Vis-
cous Incompressible Flow, Gordon and Breach, New
York, 1969.
[21] E. Lindborg, Phys.Fluids 11 510 (1999).
[22] M. Hairer and J.C. Mattingly, Annals of Mathematics,
∗ Electronic address: mazzino@fisica.unige.it 164 (2006) 993 and math.PR/0406087.
† Electronicaddress: paolo.muratore-ginanneschi@helsinki.fi [23] J.C. Mattingly, Comm. Math. Phys. 230 421 (2002).
‡ Electronic address: [email protected] [24] P. Olla, Phys.Rev. Lett. 67 2465 (1991).
[25] A. Polyakov, Nucl.Phys. B 396 367 (1993) and
[1] L.Ts. Adzhemyan, A.V. Antonov and A.N. Vasil´ev The
hep-th/9212145
field theoretic renormalization group in fully developed
[26] P. Tabeling, Physics Reports 362 1 (2002)
turbulence Gordon and Breach, Amsterdam (1999).
[27] J. Zinn-Justin, Quantum field theory and critical phe-
[2] G.K. Batchelor, Phys. Fluids, Supp II 12 233 (1969).
nomena, 4th ed.Clarendon Press, Oxford (2002).
[3] D. Bernard, Phys. Rev E 60 6184 (1999) and
chao-dyn/9902010.