On the probability distribution of power fluctuations in turbulence

0.23 MB·English

by  M. M. Bandi

On the probability distribution of power fluctuations in turbulence.
M. M. Bandi,1,∗ Sergei G. Chumakov,2 and Colm Connaughton3
1Center for Nonlinear Studies / MPA10, LANL, Los Alamos, NM 87545, USA
2Center for Nonlinear Studies / T3, LANL, Los Alamos, NM 87545, USA
3Mathematics Institute and Centre for Complexity Science,
University of Warwick, Coventry CV4 7AL, UK
(Dated: January 9, 2009)
9 Westudylocal powerfluctuationsinnumericalsimulations ofstationary,homogeneous, isotropic
0 turbulencein two and three dimensions with Gaussian forcing. Dueto the near-Gaussianity of the
0 one-pointvelocitydistribution,theprobabilitydistributionfunction(pdf)ofthelocal poweriswell
2 modelledbythepdfoftheproductoftwojointnormallydistributedvariables. Inappropriateunits,
thisdistributionisparameterisedonlybythemeandissipationrate,ǫ. Thelargedeviationfunction
n
for this distribution is calculated exactly and shown to satisfy a Fluctuation Relation (FR) with a
a
coefficient which depends on ǫ. This FR is entirely statistical in origin. The deviations from the
J
modelpdfaremostpronouncedforpositivefluctuationsofthepowerandcanbetracedtoaslightly
9
faster than Gaussian decay of thetails of theone-point velocity pdf. Theresulting deviations from

theFR are consistent with several recent experimental studies.
]
h
PACSnumbers: 47.27.Gs
c
e
m
We study the pdf of local power fluctuations in two– qualitative features of the pdf in both cases. We calcu-
-
t dimensional (2D) and three–dimensional (3D) turbu- late the large deviation function (Kramer function) and
a
lence, important practical examples of strongly non– find an exact FR with a rate depending on ρ, the cor-
t
s
equilibrium stationary states. Stationary turbulence re- relation coefficient of the two variables. For turbulence,
.
t quiresexternalforcingtocounterviscousdissipationpro- ρ is proportional to the mean dissipation rate. This is
a
m ducing a balance of the averagerates of energy injection entirely a consequence of statistics and has no relation
(power) and dissipation. The power is locally a scalar to the dynamicalarguments underlying some theoretical
-
d product of the force and velocity. The latter always has results. Thismaypartiallyexplaintheubiquityofexper-
n
intrinsic stochasticity. The power thus has non-trivial imental FRs in the literature and the lack of agreement
o
statisticsofitsown. Interestinthestatisticsofthepower onthevalueormeaningofthemeasuredrate(see[7]and
c
[ comes from two principal directions. From an engineer- the references therein for discussionof FR experiments).

ing perspective,the averagepowerrelatesdirectly to the Appliedtoturbulence,thismodel,whilequalitativelyap-
2
drag on a body in a turbulent flow. From a theoreti- pealing,doesnotcorrectlycapturethefarpositivetailof
v
3 cal perspective, interest focuses primarily on the power thepowerpdf. ThisistracedtoslightlyfasterthanGaus-
4 fluctuations. Such non-equilibrium fluctuations get to sian decay of the one-point velocity distribution. This is
7 theheartofthedifferencesbetweenequilibriumandnon- in accordance with theoretical expectations and results
0
equilibrium statistical mechanics as they relate directly in a deviation of the FR from the linear scaling which is
.
1 to the lack of detailed balance in turbulence. consistent with the results of severalexperiments [5, 7].
0 Experimental studies of the input power in turbu- We solve the 2D and 3D incompressible Navier–
9
lence initially focused on the mean and its scaling with Stokes equations for the velocity, v(x,t), with a time–
0
: Reynolds number [1]. The subsequent realisation that independent force, f(x) and bulk drag term, αv:
v
certain types of non-equilibrium fluctuations exhibit an
i
X exact symmetry known as a Fluctuation Relation (FR) ∂tv+(v )v = p+ν∆v αv+f (1)
·∇ −∇ −
r (see [2] andthe references therein) has focused attention v = 0
a on fluctuations about the mean in non–equilibrium sys- ∇·
tems. [3]. Turbulence has been harnessed as a source of Stationarity requires finite α for 2D flows where dissipa-
such fluctuations in various contexts [4]. Specific stud- tion of energy transferred to large scales by the inverse
ies of the power pdf have recently been undertaken for cascade is needed. α = 0 for 3D flows since there is no
waveturbulence [5] and 2D turbulence [6]. It was shown inverse cascade. Our simulations were done in biperi-
that the pdf of power fluctuations in different turbulent odic domains using standard pseudo-spectral methods.
systems can be qualitatively modelled by the pdf of the For numerical details, see [6] (2D) and [8] (3D). The
productoftwojointnormallydistributedvariables,vand forcing is central in what follows so let us clarify the
f, the velocity and force respectively. detail. Unlike the temporally-decorrelated forcing often
In this Letter, we consider the statistics of the power used to drive simulations of isotropic turbulence, our
in2Dand3DturbulencewithGaussianexternalforcing. forcing has no time-dependence. It does have spatial
Weshowthataproductofnormalvariablescapturesthe disorder. It is generated by selecting modes in a shell,
2
k < k < k , in the space of wave-vectors, k. These (A) Force pdfs (B) Velocity pdfs
1 2
are as|sig|ned an ampitude, A(k) and a random phase 100 100
uniformly distributed on [0,2π|).| We took A(k) to be 10-1 10-1
| | 10-2 10-2
the indicator function on [k ,k ]. We project out the
1 2 10-3 10-3
non-solenoidalcomponenttoassureincompressibilty. An
idnovmersfeorFcoinugriefireltdr.anIsnfotrhme 2thDensimpruoldauticoenssa, fsphaatsiaallysinragnle- σσ )]vf10-1 -4 -2 0 2 4 -4 -2 0 2 4
pcolimedpoinnetnhte:xf2dDir(exc)tio=n(a0n,df2t(hxe)m).agTnheeticcufirerlednitswpearspaenp-- Π[P/((2D)10-2
dicular to the fluid layer so that the Lorentz force acts 10-3
purely in the y direction (see [6]). This simplifies things IC (Lagrangian)
but is not an essential point. Indeed, in the 3D simu- 10-4 IC (Eulerian)
DC (Lagrangian)
lations, all three components of the force were present: Eq.(3) (ρ=0.12)
10-5
f3D(x)=(f1(x),f2(x),f3(x)). -10 -5 0 5 10
P /(σ σ)
The rationale for this forcing is two-fold. Firstly, our (2D) v f
2D forcing exactly mimics that used to generate turbu-
lence in electromagnetically driven fluid layers [9, 10]. FIG. 1: (Color online) Pdfs of the 2D power normalized by
It is thus of direct relevance to 2D experiments. Sec- σvσf for the inverse (◦) and direct (∗) cascades in the La-
ondly, since we are interested in power fluctuations, it is grangianframeandtheinversecascadeintheEulerianframe
((cid:3)) . The solid line is Eq. (3). The insets show the corre-
attractive to limit the sources of stochasticity to the in-
spondingpdfsoff (A)andv(B)normalizedbytheirstandard
trinsic randomness of the turbulent fluctuations. By the
deviations.
Central Limit Theorem, our forcing protocol produces a
Gaussian distribution for the single-point pdf of f pro-
vided that enough modes participate. This is shown in variables with mean zero, variances σ2 and σ2 and cor-
1 2
the inset of Fig. 1 for the 2D case and of Fig. 3 for the relation coefficient ρ, then their joint pdf is
3D case. We should be clear that we are not attempting
tfoorcminagkeisanoyfteunniuvseerdsailnstnautemmeerinctasl. siAmltuhlaotuigohnsGaanudsshiaans P(x1,x2)= 2πσ σ 1 1 ρ2 e−2(1−1ρ2)„σx1212−2ρσx11xσ22+σx2222«.
1 2
experimental relevance, it has no a-priori justification. p − (2)
Turningtothevelocity,v,itssinglepointpdfisknown x1 and x2 should be thought of as components of v and
tobeclosetoGaussianforhomogeneous,isotropicturbu- f respectively. The pdf of the product, z =x1x2, is
lence since the early days of turbulence theory [11]. On
ρz
the other hand, the Navier–Stokes equation, Eq. (1) is P(z)= e(1−ρ2)σ1σ2 K |z| (3)
ntoonelxinpeeacrt.tEhavtenthweitphdGfaoufsvsiasnhofourldcinbge,tehxearcetliysnGoaruesassiaonn πσ1σ2p1−ρ2 0(cid:18)(1−ρ2)σ1σ2(cid:19)
and indeed it is not. While most investigations have fo- where K0(z) is the modified Bessel function of the sec-
cused on the relatively large non–Gaussianityof velocity ond kind of order zero. We take . to denote averaging
hi
differences, careful measurements show that the single- withrespecttothepdf, Eq.(3). Themomentgenerating
point pdf of v decays slightly faster than Gaussian in function, χ(θ)= eθz can be calculated explicitly:
h i
both the 2D [12] and 3D [13] cases.
1
We now consider the local power, denoted by p. The χ(θ)= (4)
1 2ρσ σ θ (1 ρ2)σ2σ2θ2
2D power is a simple product, p(2D) = vyfy. The 3D p − 1 2 − − 1 2
npoorwinerghfoarsntohwreeancyonsutrbi-bGutaiuosnssi:anpt(3aDil)s =of tPhe3i=p1dvfifoif.v,Igit- owbhtearienθmo∈me(−ntσs.1σT21(h1−eρm),eσa1nσ,21(v1a+rρia)n).ceIatndisstkheewnneesassyareto:
isclearthatmodelingpusingproductsofGaussianforce
and velocity components should capture the qualitative z = ρσ σ (5)
1 2
h i
features of the single point pdf. This has already been z2 z 2 = (1+ρ2)σ2σ2 (6)
proposedinaLagrangiansettingin2Dturbulence[6]and h i−h i 1 2
z3 3 z z2 +2 z 3 2ρ(3+ρ2)
in the context of wave turbulence [5] and shown to work h i− h ih i h i = . (7)
( z2 z 2)3/2 (1+ρ2)3/2
very well. In the present Letter, we extend the descrip- h i−h i
tion to 3D flows,calculate the large deviation properties Normalisingvandf bytheirstandarddeviations,σ and
v
of the model andaddress the meaning ofthe FR for tur- σ , the mean of the pdf Eq. (3) gives the correlationco-
f
bulent power fluctuations effient,ρ. For stationaryturbulence, this relatesρ to the
We needsomeresultsonproductsofnormalvariables. average dissipation rate, ε. Only a single component of
If x and x are two joint normally distributed random f contributes to the 2D power so ε=ρ. All components
1 2
3
(A) Force pdf (B) Velocity pdf 0.13. The (Taylor microscale) Reynolds number of the
100 100 3D simulation was 35 and ρ was 0.35. As expected, the
10-1 10-1
agreement is good. In detail, the 3D simulations show a
10-2 10-2
systematic deviation for large positive fluctuations. We
10-3 10-3
will return to this later.
σ)]f -4 -2 0 2 4 -4 -2 0 2 4 We now calculate the large deviation properties of
σ f / ( 11v1100--21 tthheis mmoedaenlspadnfd, Ewqh.y(i3t).is Luesetfuuls. bSriuepflpyoseexpwlaeintawkehant
Π[v independent samples from the distribution Eq. (3), de-
10-3 noting them by z , i = 1...n. The large deviation
i
principle for Eq. (3) concerns the pdf of their average,
10-4 Numerical data M = 1 n z . It states that there exists a function,
10-5 Eq.(3) (ρ=0.35) I(xn), thnePrait=e1fuinction or Kramer function, such that
-10 -5 0 5 10
v f / (σ σ)
1 1 v f P(M >x) e−nI(x). (8)
n
≍
FIG. 2: (Color online) Comparison of the pdf of the v f
1 1
This is useful for several reasons. Firstly, the 3D power
contribution to the 3D power with Eq. (3). Insets show the
pdfs of f1 (A) and v1 (B) normalized by σv and σf. is a sum of 3 random variables with distribution Eq. (3)
so Eq.(8) providespartialinformationabout the tails of
thedistributionofthetotalpowerin3D.Secondlyexper-
contribute to the 3D power so ε = 3ρ. Eqs. (5) – (7) iments often measure global - or at least coarse–grained
thus link the statistics of p to the dissipation rate. - power rather than local power. Eq. (8) provides a link
The value of ρ can be measured. A comparison be- between the local and global power which may be more
tween Eq. (3) for the measured value of ρ and the con- accessible experimentally. Finally, a FR expresses a par-
tributionofa single componentofthe forceto the power ticular symmetry of the rate function for a stochastic
(normalised by the product of σ and σ ) is shown for process, so knowing I(x) allows us to address the ques-
v f
the2DpowerinFig.1andforthe3DpowerinFig.2. 2D tion of a FR for Eq. (3) directly. In this case, it is
results are presented for both direct and inverse cascade possible to obtain I(x) in closed form from the Chernoff
regimes with nominal (integral scale) Reynolds numbers formula [14]: I(x) = max θx lnχ(θ) . Lengthy but
θ
{ − }
of 1100and 7000respectively. The ρ values are 0.11 and straightforwardcalculations yield:
(ρ2 1)σ σ 2ρx+ 4x2+(1 ρ2)2σ2σ2 2(1 ρ2)σ σ ln 2x2
I(x)= − 1 2− p − 1 2 − − 1 2 (cid:20)σ1σ2((ρ2−1)σ1σ2+√4x2+(1−ρ2)2σ12σ22)(cid:21). (9)
2σ σ (1 ρ2)
1 2
−
This unwieldy expression is plotted in the inset of a quantity, X , derived from the entropy production or
τ
Fig. 3 for ρ = 0.35 and σ = σ = 1. The main part of energy dissipation in a non–equilibrium system. X is
1 2 τ
Fig. 3 illustrates how the asymptotic expression Eq. (8) obtainedbyaveragingaphysicalquantity,x(t), typically
capturestheessentialfeaturesofthepdfofthe3Dpower. the entropy produced or energy dissipated over a time
One cannot expect exact correspondence for severalrea- interval[t,t+τ]: X =τ−1 t+τx(t′)dt′. X ispositive
τ t τ
sons. Firstly,wehaveseenthatEq.(3)over-estimatesthe on average but, may fluctuaRte sufficiently that negative
probabilityoflargepositivevaluesofeachindividualcon- fluctuations are observable. A FR quantifies the relative
tribution to the total 3D power, an effect which remains probability of a negative fluctuation over a time interval
evident when these contributions are summed. Secondly compared to the probability of a positive fluctuation of
the components v are not strictly independent owing to the same magnitude. The ratioofprobabilitiestakesthe
the incompressibility condition. Finally, one should re- form:
member that Eq. (8) is an asymptotic statement. These Π(X )
objections notwithstanding, the correspondence is good. τ =eΣτXτ, (10)
Π( X )
τ
−
We now turn to the question of a FR for turbulent where Σ is a constant, independent of the averaging in-
power fluctuations. A FR is a symmetry of the pdf of terval,τ. Clearly this equates to the rate function of the
4
100 10
Kramer function 1 3-D
I(x) 2-D
12 0.8
10-1 8 8 0.6
ρ
4 0.4
Πσσ[P/()]vf1100--32 0 -8-4 x0 4 8 ΠΠ(P)/((-P)] 46 0 .02 2 lo 3g10(R e4) 5
n[
l
10-4
e-I[P/(σvσf)] 2
Numerical data
10-5
-10 -5 0 5 10 0
P/(σσ) 0 2 4 6 8 10
v f
P
FIG.3: (Color online) Comparison ofthepdfofthelocal 3D FIG. 4: (Color online) Asymmetryof thepdfsof local power
power with e−I(x). Inset shows the Kramer function, I(x), in2-Dand3-D.SolidlinesindicatethepredictionofEq.(12).
given byEq (9), for ρ=0.35. Inset shows the decrease of the correlation coefficient for the
2-D case, ρ, as the notional Reynolds number increases. We
expect a similar trend in 3-D.
pdf of x(t) asymptotically possessing the symmetry:
I(x) I( x)= Σx. (11) nonlinearityandlargescaledissipation. Nonetheless,itis
− − −
widely used sowe adoptit here to parameteriseoursim-
It is easy to show that Eq. (9), satisfies this symmetry
ulations. We observe that ρ decreases as Re increases so
exactly with a rate, Σ, given by
thatthepdfofthepowerbecomesmoresymmetricasthe
flow becomes more turbulent. This make physical sense
2ρ
Σ= . (12) as the greaterthe turbulent fluctuations, the less the ve-
(1 ρ2)σ σ
− 1 2 locity can correlate with the forcing. As the pdf of the
From this, we conclude that when it is reasonable to powerbecomesmoresymmetric,Eq.5demonstratesthat
model non-equilibrium fluctuations using a product of the decrease in the correlation must be compensated for
correlated normal variables a FR will result. The value byanincreaseinthevarianceofthevelocityfieldifoneis
of the entropy rate, Σ, depends on the dissipation rate. to maintaina fixed mean rate of energy injection. There
This observation may partially explain the proliferation are clearly some important questions to address here in
of empirical Fluctuation Relations in the literature and understandingthe relationshipbetweenρ andReaswell
the lack of consensus on the value and meaning of the as investigating the corresponding issues in 3-D. These
entropy rate measured for different experimental situa- are, however, beyond the scope of the present work.
tions. This result is entirely statistical and does not re- We have already discussed how the Kramer function,
quire any restrictions on the microscopic dynamics such Eq. 9, encodes information about the behaviour of sums
as time–reversibility. Indeed it tells us very little about ofsamplesfromthepdf. Ifwethinkoflocalaveragingas
the physics of the system under study. such a summation procedure, the fact that the Kramer
Let us now reconsider the specific case of turbulent functionexhibitsaFRwitharategivenbyEq.12,means
powerfluctuations. Fig.4 showsthe degreetowhichour thatwemightexpectthe coarse-grainedpowerto satisfy
numerical data satisfies the symmetry of Eq. (11) with thisFRprovidedthatwecoarse-grainthedataoverinter-
the appropriate values of Σ from Eq. (12). As in many vals longer than the correlation length. This latter con-
cases in the literature, a good agreement is found for dition is important since the Kramer function describes
relatively small fluctuations but a systematic deviation the asymptotics of sums of independent samples. This
appearsforverylargefluctuations. Unusually,weunder- providesa way to link our discussionof localpower fluc-
stand completely the observed values of Σ. It is deter- tuations to “global”fluctuations (in the sense offluctua-
minedsolelyfromthethecorrelationcoefficient,ρ,which tionsatscalesofmanycorrelationlengths). Thiscoarse-
isnotknowna-priori. TheinsetofFig.4showsnumerical graining could be done either in space or in time. In
measurements of how ρ varies as the notional Reynolds our numerical simulations, the spatial correlation length
number,Re,isincreased. In2-D,inthepresenceofanin- was too long to allow us to perform the coarse-graining
versecascade,theusualdefinitionofReisofquestionable convincingly and will require further effort. This is un-
usefulness, since the principal energy balance is between fortunate, this being most relevant to experiments. The
5
statistical model proposed in [6] and [5] but rather are
a signature of some underlying dynamics. For the spe-
1.4 n=0
n=5 cificcaseofturbulence,theworkof[15]identifiedspecific
n=10
1.2 flow configurations (“instantons”) which are responsible
n=15
)]n n=20 for the faster–than–Gaussian decay of the single point
P 1
Π(- Theory velocity distribution in forced turbulence. This theory
P)/n 0.8 may provide a starting point for analysis of the devia-
Π(+ tions from the FR observed in our data but given the
n[ 0.6 non-universal nature of the force, it seems unlikely that
1/n) l 0.4 there is anything universal about these deviations.
(
Acknowledgements
0.2 Thisworkwaspartiallycarriedoutunderthe auspices
of the National Nuclear Security Administration of the
0
0 0.5 1 1.5 2 2.5 3 3.5 4 U.S.DepartmentofEnergyatLosAlamosNationalLab-
Pn oratory under Contract No. DE-AC52-06NA25396.
FIG.5: (Coloronline)FluctuationrelationfortheLagrangian
power averaged over time (τ) intervals in multiples of the
correlation time (τc) of the power signal for τ/τc ≡ n = 0, ∗
5, 10, 15, and 20. The solid line is the theoretical prediction Corresponding Author: [email protected]
from Eq. 12. [1] D. Lathrop, J. Fineberg, and H. Swinney, Phys. Rev. A
46, 6390 (1992).
[2] V. Chernyak, M. Chertkov, and C. Jarzynski, J. Stat.
Phys. 8, 08001 (2006).
Lagrangian correlation time is relatively much shorter.
[3] R.Labb´e,J.-F. Pinton,andS.Fauve,J.Phys.IIFrance
Therefore we can illustrate the point by coarse-graining
6, 1099 (1996).
temporally using data gathered from measurements of
[4] S.Aumaˆıtre,S.Fauve,S.McNamara,andP.Poggi,Eur.
the force and velocity in the Lagrangian frame. Full de- Phys. J. B 19, 449 (2001).
tails of the Lagrangian measurements are already avail- [5] E. Falcon, S. Aumaˆıtre, C. Falc´on, C. Laroche, and
ablein[6]. Wedefineatemporallycoarse-grainedpower, S. Fauve,Phys. Rev.Lett. 100 (2008).
P (t),againnormalisedbythestandarddeviationsofthe [6] M. M. Bandi and C. Connaughton, Phys. Rev. E 77,
n
036318 (2008), arXiv:0710.1133 [cond-mat.stat-mech].
force and velocity:
[7] M. M. Bandi, J. R. Cressman, and W. I. Goldburg, J.
1 1 t+nτ Stat. Phys. 130, 27 (2008), arXiv:nlin/0607037v2.
P (t)= P(t′)dt′. (13) [8] S. G. Chumakov,J. Fluid Mech. 562, 405 (2006).
n σvσf nτ Zt [9] J. Paret and P. Tabeling, Phys. Rev. Lett. 79, 4162
(1997).
Hereτ isthe Lagrangiancorrelationtime(τ 0.1inour
[10] M. Rivera, W. Daniel, and R. Ecke (2005), eprint:
≈
simulationscomparedwithalargeeddyturnovertimeof arXiv:cond-mat/0512214, arXiv:cond-mat/0512214.
about 10) and P(t) is the local power in the Lagrangian [11] A. Monin and A. Yaglom, Statistical fluid mechanics
frame. The results forthe pdfs ofP areshownin Fig.5 (MIT press, Cambridge, 1975).
n
[12] Y. Jun, X. Wu, and J. Zhang, Phys. Rev. Lett. 96,
forcoarse-grainingtimesrangingfrom5to20correlation
164502 (2006).
lengths. It is clear that the symmetry of the Kramer
[13] T. Gotoh, D. Fukayama, and T. Nakano, Phys. Fluids
function demonstrated in Eq. 11 produces a FR for the 14, 1065 (2002).
coarse-grainedpower.
[14] J. Lewis and R. Russell (1997), Dublin Institutefor Ad-
It has been rightly argued [4, 5] that the deviations vanced Studies.
from Eq. (11) evident in Fig. 4 are typical. Here we un- [15] G. Falkovich and V.Lebedev,Phys.Rev.Lett. 79, 4159
derstand that these deviations do not follow from the (1997).