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PHYSICAL REVIEW E 77, 065302共R兲 共2008兲
Taylor’s frozen-flow hypothesis in Burgers turbulence
A. Bahraminasab,1,2 M. D. Niry,3 J. Davoudi,4 M. Reza Rahimi Tabar,5,6,* A. A. Masoudi,7 and K. R. Sreenivasan2
1
Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
2
ICTP, Starda Costiera 11, 34014 Trieste, Italy
3
Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran
4
Department of Physics, University of Toronto, 60 St. George Street, Toronto ON, Canada M5S 1A7
5
CNRS UMR 6529, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice Cedex 4, France
6
Carl von Ossietzky University, Institute of Physics, D-26111 Oldenburg, Germany
7
Department of Physics, Alzahra University, Tehran, 19834, Iran
共Received 26 November 2007; revised manuscript received 4 June 2008; published 30 June 2008兲
By detailed analytical treatment of the shock dynamics in the Burgers turbulence with large scale forcing we
calculate the velocity structure functions between pairs of points displaced both in time and space. Our
analytical treatment verifies the so-called Taylor’s frozen-flow hypothesis without relying on any closure and
under very general assumptions. We discuss the limitation of the hypothesis and show that it is valid up to time
scales smaller than the correlation time scale of temporal velocity correlation function. We support the analytical calculation by performing numerical simulation of the periodically kicked Burgers equation.
DOI: 10.1103/PhysRevE.77.065302
PACS number共s兲: 47.27.Gs, 05.45.⫺a
In a 1938 paper G. I. Taylor introduced an assumption by
which he deduced the spatial fluctuations of a turbulent velocity from the corresponding measurements of temporal
fluctuations at a single point 关1兴. This hypothesis, know as
Taylor’s frozen-flow hypothesis, relies on the existence of a
large mean flow which translates the spatial structures past a
stationary probe in a time smaller than the inherent evolution
time of the fluctuations 关2,3兴. As remarked by Tennekes 关4兴,
the Eulerian temporal scaling is dominated 关5,6兴 by large
scale energy containing structures which sweep inertialrange information past an Eulerian observer 关7,8兴.
The importance of this hypothesis relies on the fact that
most turbulence theories center on the scaling behavior of
spatial structure functions of the velocity field 关2,3兴. In experimental assessments of spatial fluctuations in turbulent
flows 关6–8兴 this hypothesis has been a general guideline for
extracting the information from one-point measurements
with hot wires. Original measurements treated by Taylor
were made in turbulence generated behind a stationary grid
in a wind tunnel, and his hypothesis has become a standard
technique employed in similar experiments which build our
current view on turbulence.
Although Taylor’s hypothesis is significantly clear when
there is a large scale average flow, its application in homogeneous isotropic flows has been a much more debated issue.
The latter always is meant to reflect the qualitative picture
that larger eddies randomly sweep the smaller eddies with
their root-mean-square velocities.
Apart from a few theoretical attempts in verifying the
hypothesis, a general quantitative framework for deriving it
does not exist 关9,10兴. Such a quantitative understanding calls
for characterization of the spatiotemporal fluctuations of
small scale eddies responsible for random sweeping 关10–13兴.
In retrospect the spatiotemporal structure functions are
central objects in providing the required information. Such
*[email protected]
1539-3755/2008/77共6兲/065302共4兲
structure functions of arbitrary order q are defined as
ជ
Sq共x , t兲 = 具兩关uជ 共xជ 2 , t2兲 − uជ 共uជ 1 , t1兲兴 · xx 兩q典, where the uជ 共xជ i , ti兲 are
Eulerian velocities at two spatially distinct points that are
measured at two different times. In a statistically stationary
state and in the inertial subrange, multiscaling assumption
implies
Sq共x,t兲 ⬀ x␰qFq
冉 冊
t
,
x zq
共1兲
where x = 兩xជ 2 − xជ 1兩 and t = 兩t2 − t1兩 关2,3兴. Fq’s are homogeneous
functions of their arguments and zq are dynamic exponents.
Two sets of exponents ␰q and ␨q are defined by casting two
asymptotic
limits
limt→0 Sq共x , t兲 ⯝ 兩x2 − x1兩␰q
and
limx→0 Sq共x , t兲 ⯝ 兩t2 − t1兩␨q, respectively.
Theoretical arguments resorting to the multifractal phenomenology support the existence of a hierarchy of dynamic
exponents zq if one compensates the sweeping effects by
choosing a quasi-Lagrangian frame 关14兴. In an Eulerian
frame for which the sweeping dominates the temporal fluc␰
tuations, zq = ␨qq = 1, at least up to leading order 关2,14兴.
Here we calculate ␨q for the one-dimensional Burgers
equation stirred a forcing with large scale correlation in
space and with a Wiener scaling in time. The exponents are
derived from equations of motion without relying on any
closure by which the dynamic exponents zq are inferred. The
crucial role of shocks in establishing the result is spelled out
and the scaling solutions of the dynamic structure functions,
i.e., limx→0 Sq共x , t兲, are obtained. Numerical simulations on
the periodically kicked Burgers equation 关15兴 are performed
to support the analytical results for ␨q.
Our numerical and analytical calculations both indicate
that for incremental time t = t2 − t1 less than or at the order of
the correlation time scale, i.e., t ⱗ tcorr, the scaling of
limx→0 Sq共x , t兲 are dominated by the shock dynamics and
saturate, i.e., ␨q = 1 for q 艌 1, while ␨q = q for q ⬍ 1. As a
by-product the dynamical exponents are equal to unity, i.e.,
zq = 1, consistent with Taylor’s hypothesis. However when t
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BAHRAMINASAB et al.
⬎ tcorr the dynamics of limx→0 Sq共x , t兲 are controlled by the
scaling limit of the forcing. Hence we show that the Taylor
frozen hypothesis does not hold for t ⬎ tcorr when the forcing
scales as a Wiener process in time. The crossover from
shock-dominated to forcing-dominated regime is also verified numerically.
We consider the Burgers equation in one dimension, i.e.,
ut + uux = ␯uxx + f共x,t兲,
共2兲
where f共x , t兲 is a zero-mean Gaussian statistically homogeneous, and white in time random process with covariance,
具f共x2 , t2兲f共x1 , t1兲典 = C共x2 − x1兲␦共t2 − t1兲. The spatial correlation
C共x2 − x1兲, as a function of 兩x2 − x1兩, is assumed to have a
finite support ␴ ⯝ L, where L is the system’s domain
关16–18兴.
A convenient way to derive the dynamical evolution of
Sq共x , t兲 is via the standard generating function method
关17,18兴. Defining ⌰ = exp共−i␭1u1 − i␭2u2兲 the two-point generating function is given by 具⌰典, where 具¯典 is an average
over the forcing statistics. The Fourier transform of the generating function is the two point probability density function
共PDF兲 P共u1 , u2 , x1 , t1 , x2 , t2兲 defined at the points x1, t1 and
x2, t2 with their related velocities u1 and u2. From Eq. 共2兲 the
dynamical equations of PDF P at distinct times, say t1 and t2,
are sought. Using the following change of variables t = t2
t +t
u +u
− t1, T = 1 2 2 , and transforming u1 and u2 with u = 1 2 2 , ␻
= u2 − u1, one obtains the following equation for the PDF of
velocity field difference as
Pt = − uPx −
−
1
2
1
2
冕冉
冕冉
K u+
K u−
冊冉
冊
G2 − G1 = − ␭2具⑀2共␭兲e−␭u1 − ⑀1共− ␭兲e␭u2典,
␻
␻
− u⬘ Px u − ,u⬘ du⬘
2
2
冊冉
冊
1
␻
␻
− u⬘ Px u⬘,u +
du⬘ + C共0兲Pu␻
2
2
2
1
G2 G1
− C共x兲关 41 Puu − P␻␻兴 +
−
,
2
2
2
In principle the solutions of the above equation in the
inviscid limit should have scaling forms of the type introduced in Eq. 共1兲. However, Eq. 共4兲 is not obviously tractable
because the first and last set of terms on the right-hand side
are not expressed in terms of Sq共x , t兲 and the equation is not
closed. In what follows we show that it is possible to treat
the unclosed terms by means of shock representation and
without resorting to any closure model.
In the inviscid limit both types of the terms are expressible in terms of operators localized on shocks when x → 0.
Recall the velocity u satisfying the Burgers equation develops shock solutions in the limit ␯ → 0. One may represent a
shock locally as u共x , t兲 = u+H关x − x0共t兲兴 + u−H关x0共t兲 − x兴, where
H is a Heaviside function. The position x0 is identified by
two quantities, namely the velocity u in positions x0+, x0− 共set
it, for example, to 0兲. In other words the velocity gradient
共corresponding Burgers velocity兲 is not continuous at points
x0. At these singular points u⫾ is defined as u⫾共x0 , t兲
= u共x0⫾ , t兲 keeping in mind that u− ⬎ u+, while the shock
strength s and the shock velocity ū are defined as s = u+ − u−
and ū = 21 共u+ + u−兲.
Here we argue that the last term in the right-hand side of
Eq. 共4兲 vanishes in a particular space-time window. Indeed in
the inviscid limit only small intervals around the shocks will
contribute to the Gi terms. Each of Gi’s at space-time 共xi , ti兲
are nonvanishing only on the shocks 关17,18兴. As a neat way
of demonstrating the cancellation of Gi terms we return to
the Fourier space and represent the related terms with Gi. The
anomaly terms G2 − G1 in this space can then be written as
共3兲
x +x
where x = x2 − x1, y = 1 2 2 . The C共x兲 is the spatial correlation
function of forcing, K共u兲 = H共u兲−H共−u兲
, and H共u兲 is the
2
Heaviside function. The terms Gi’s are defined as Gi =
where
具uixixi 兩 u1 ,
−␯⳵ / ⳵ui兵具uixixi 兩 u1 , u2 , x1 , x2 , t1 , t2典P其,
u2 , x1 , x2 , t1 , t2典 is the average of uixixi with the conditions that
the velocity field has the values of ui’s at points xi and in
times ti. Indeed the term Gi is the only term preventing Eq.
共3兲 to be closed, which can be referred to a sort of dissipative
anomaly 关16兴.
Multiplying both sides of Eq. 共3兲 with ␻q and integrating
it over u and ␻ one obtains
⳵Sq
⳵Sq,1 1
=−
+ q共q − 1兲C共x兲Sq−2 + 具␻q共G2 − G1兲典. 共4兲
2
⳵t
⳵x
The first term in the right-hand side of Eq. 共4兲 is of the
type of mixed structure functions of center of mass and incremental velocities Sq,r = 具␻qur典. The second term is the contribution of forcing and the last term is the combination of
dissipative terms 具␻q共G2 − G1兲典 in Eq. 共4兲 that are at variance
with the usual anomaly terms in the equations of spatial
structure functions at one time 具␻q共G2 + G1兲典 关17,18兴.
共5兲
where xi is the position of the ith shock and ␭ = ␭2 − ␭1 is the
conjugate of ␻ and ⑀共␭ ; x , t兲 = 兺iF共ūi , si兲␦关x − xi共t兲兴. The form
factors F are read F共ū , s兲 = −2e␭ū␭−1⳵␭␭−1 sinh共 s2␭ 兲 关19兴. This
representation shows that the anomaly contribution is generally very complicated for arbitrary separation distance x and
time difference t. However it is possible to identify a spacetime regime in which the following operator vanishes. In fact
⳵u
by Taylor expansion one easily sees that for 兩t2 − t1兩兩 ⳵s2 兩s=t1
⳵u1
⯝ −兩t2 − t1兩ū兩 ⳵x 兩x=xi共s兲, given x → 0, G2 and G1 cancel out in
the leading order. This condition physically means that the
anomaly due to a shock at point x2 and in time t2 is the same
in statistical sense at a point x1 but with a time delay approxi兩x −x 兩
mately equal to 兩t2 − t1兩 ⯝ 2 ū 1 , because the shocks move
with their local velocity ū. In this spatiotemporal window the
time variations of velocities are mostly dominated by the
random shocks which sweep the spatial fluctuations past a
point 关1,4兴.
The other set of terms on the right hand side of Eq. 共4兲,
Sq,1 = 具␻qu典, are rooted in the uPx term in Eq. 共3兲. Using
stationarity, i.e., ⳵T P = 0, it is easy to reexpress limx→0 Px in
terms of the measures describing the statistics of shocks. In
the spatiotemporal regime where x ⱗ ūt it is possible to show
lim Px共u, ␻,x,t兲 = Nx共␻兲M共u兲 = 关␳具s典␦1共␻兲 + ␳S共␻,T兲兴M共u兲,
x,t→0
共6兲
␦ 1共 ␻ 兲 = d ␦ 共 ␻ 兲 / d ␻ ,
and
Nx共␻兲 = 兰limx→0 Px
where
共u , ␻ , x , t兲du. Here ␳ = 兺i␦关x − xi共T兲兴 and S共␻ , T兲 are the
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TAYLOR’S FROZEN-FLOW HYPOTHESIS IN BURGERS TURBULENCE
shock number density in space and the PDF of s共x1 , t1兲 conditioned to x1 being on a shock location, respectively.
To leading order the result is t independent and the averages on u and ␻ on the shocks separate. The functional
dependence of limx,t→0 Px共u , ␻ , x , t兲 on u appears as an unknown function M共u兲, where 兰M共u兲du = 1 共see also 关14兴兲.
Although the precise form of M共u兲 does not enter in later
arguments the separation of u and ␻ is central for obtaining
the later results.
In order to calculate ␨q we use the representations in Eqs.
共5兲 and 共6兲 at leading order and substitute them back into Eq.
共4兲. Therefore we obtain
q 艌 2,
共8兲
where ␻ = limx→0 关u共x1 + x , t1 + t兲 − u共x1 , t1兲兴 and ␳ is the average density of shocks in time. We remark that the leading
term in S2共t兲 has a forcing contribution proportional to C共0兲.
The higher order moments, i.e., q ⬎ 2 do not obtain contributions from forcing because they are always subleading in
time.
For the lower order moments the competition of the regular part of the velocity increment in time dominates the contribution of shocks. One writes 关u共x , t1 + t兲 − u共x , t1兲兴 ⯝
−兰tt1+tū共⳵xu兲ds + 兰tt1+tdW共s兲, where the first integral in the
1
1
right-hand side resembles the sweeping of the spatial increment and hence scales as t. The second integral however
scales as t1/2 because the integral of the forcing is a Wiener
process in time by definition. Consequently in leading order
one may easily write
lim Sq共x,t兲 ⯝ Aq兩t兩q/2 + Bq兩t兩,
x→0
-1
<|ω|q>
10-2
10
q = 0.1
q = 0.3
q = 0.5
q = 0.7
q = 0.9
q = 1.2
q = 1.4
q = 1.6
q = 1.8
q = 2.0
-3
10-4
10
-5
10-4
10-3
共7兲
Solving the analytical scaling solutions in Eq. 共7兲 we find the
moments q 艌 2 of the velocity increments in time behave as
具兩␻兩q典 = 兩t兩关− ␳具ū典具兩s兩q典 + C共0兲␦q,2兴,
10
共9兲
where Aq and Bq are unknown amplitudes. Therefore, the
low order moments scale as tq/2 and the saturation should
begin for orders q 艌 2.
To test the predictions of the analytical calculation, we
have carried out numerical simulations of the Burgers turbulence by means of the so-called particle method 关15兴. The
numerical experiments reported hereafter have been made
with the kicking force and a kicking period tkick = 0.001. The
number of collocation points chosen for our simulations is
generally Nx = 105. In order to perform temporal averages,
since we need a large sequence of velocity time series to
calculate intermittency in time, we run the algorithm for
about 103tkick. For time integration we adapt a time discretization of order td = 10−5. Between two subsequent kicks we
advance the minimizers in time for 100dt. We then construct
the Eulerian velocity by means of the particles velocities
after reaching to statistically stationary state. The results are
shown in Fig. 1, in which the log-log plot of 具兩u共x , t1 + t兲
− u共x , t1兲兩q典 is depicted as a function of the time increment t
for different q’s specified in the inset. The time span t in the
measurements of velocity increments lies in the extended
range of t 苸 关10−5 , 2 ⫻ 10−1兴. Therefore, we have the informa-
t
10-2
10-1
FIG. 1. 共Color online兲 The scaling of the time increments of
velocity 具␻q典 verses t2 − t1 = t is sketched for different moments q
苸 共0 , 2兴. The kicking time is 10−3 and the measured time span is
showing the scaling both below and above it.
tion about scaling both above and below the characteristic
correlation time. The typical correlation time is about tcorr
⯝ 3 ⫻ tkick.
Although all high order moments q 艌 2 display a slope of
approximately 1 the moments of orders q 艋 2 exhibit a crossover in their scaling. For t ⱗ tcorr the scaling exponent is read
to be ␨q = q, while for t Ⰷ tcorr one observes ␨q = 2q . When t
ⱗ tcorr the regular part of the velocity increment is dominated
by ureg共t1 + t兲 − ureg共t1兲 ⯝ 共⳵tu兲t. Similar to the decaying probq
典 ⬃ tq. We have plotted
lem the latter gives the scaling 具␻reg
the behavior of ␨q in terms of order q in Fig. 2. The figure
shows the saturation for time scales t ⱗ tkick and t Ⰷ tkick, respectively.
We also checked the statistical independence of the center
of mass velocity u = 2ū = u1 + u2 and velocity increments ␻
= u1 − u2. As an instance we numerically measure the quantity
具共u1 + u2兲2 兩 ␻ ; x , t典 for the range of space-time where statistical convergence are accessible. Indeed Fig. 3 demonstrates
that for the time span of the order t 苸 关10−3 , 10−1兴 the conditional averages display a regime of independency for u and
␻. The measurements shown here are done for a fixed sepa-
1
ζq
1
⳵Sq
= − ␳具ū典具sq典 + q共q − 1兲C共x兲Sq−2 + o共t兲.
2
⳵t
100
0.5
0
Time spane smaller than kicking
Time spane bigger than kicking
0
1
2
q
3
4
5
FIG. 2. 共Color online兲 The saturation of ␨q for different q’s,
where ␨q is the scaling exponents 具兩␻兩q典 ⬃ 兩t兩␨q. The upper and lower
figures are the saturation of scaling exponents for the time span
smaller and bigger than kicking time, respectively. The correlation
time scale is about tcorr ⯝ 3 ⫻ tkick. The results shows that Taylor
frozen hypothesis is preserved just for time scales less than tcorr.
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3
<u 2|ω> / <u 2>
-5
t = 2×10
t = 6×10-5
t = 10-4
t = 10-3
t = 2×10-3
-3
t = 6×10
t = 0.01
t = 0.04
t = 0.1
2
1
-2
-1
0
ω / <ω 2>1/2
1
2
FIG. 3. 共Color online兲 The conditional moment 具u2 兩 ␻典 as a
function of ␻ at different time spans 兩t2 − t1兩 below and above the
kicking time. For any given time increment t the conditional average is normalized by the corresponding 具u2典.
ration x → 0, but at different time windows t. As time increment t grows the domain of velocity increments ␻ where the
independency is observed extends to larger values.
In summary, scaling exponents ␨q of the moments of velocity increment in time are derived from randomly forced
Burgers equation. When the incremental time t = t2 − t1
ⱗ tcorr, they are analytically shown to saturate to unity, i.e.,
␨q = 1 for q 艌 1, while ␨q = q for q ⬍ 1. Our numerical simu-
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support the results. At large times when t Ⰷ tcorr, our numerical results reveal a non-universal scaling regime in which the
higher order moments still saturate ␨q = 1, but for q ⬎ 2 while
the low order moments display a normal scaling ␨q = q / 2 for
q ⬍ 2. The later nonuniversal scaling of low order moments
␨q = q / 2 for q ⬍ 2 is a direct consequence of the forcing
which is Wiener in time at every point of space.
Although the advective terms in the Navier-Stokes and
the Burgers equations are similar the nonlocal pressure contribution prevents the formation of shocks in the NavierStokes turbulence. Closed analytic forms of the pressure and
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analytic treatment of the relevant small-scale singularities in
the Navier-Stokes turbulence is yet an open problem and,
moreover, the closures are typically not based on the geometry of the flow singularities. To conclude, our work highlights the importance of a detailed knowledge of singularity
dynamics in deriving Taylor’s hypothesis. Generalizations to
the Navier-Stokes turbulence requires further investigations.
We thank Uriel Frisch and Jeremie Bec for useful comments and discussions. J.D. thanks the ICTP and the Max
Planck Institute for Complex Systems for partial support.
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