Pattern fluctuations in transitional plane Couette flow

1 MB·

by  Joran Rolland

Pattern fluctuations in transitional plane Couette Flow
Joran Rolland, Paul Manneville
Laboratoire d’Hydrodynamiquede l’E´cole Polytechnique,91128 Palaiseau, France
1
1
0 January 14, 2011
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n
Abstract
a
J In wide enough systems, plane Couette flow, the flow established between two parallel plates translating in

3 oppositedirections,displaysalternativelyturbulentandlaminarobliquebandsinagivenrangeofReynoldsnumbers
1 R. Weshow that in periodic domains that contain a few bands,for given values of R and size, theorientation and
the wavelength of this pattern can fluctuate in time. A procedure is defined to detect well-oriented episodes and
]
to determine the statistics of their lifetimes. The latter turn out to be distributed according to exponentially
n
y decreasing laws. This statistics is interpreted in terms of an activated process described by a Langevin equation
d whose deterministic part is a standard Landau model for two interacting complex amplitudes whereas the noise
- arises from the turbulentbackground.
u
l
f
. 1 Introduction
s
c
si The main features of the transition to turbulence are well understood in systems prone to a linear instability like
y convection where chaos emerges at the end of an instability cascade. A much wilder transition is observed in wall-
h
bounded shear flows for which the laminar and turbulent regimes are both possible states at intermediate values
p
of the Reynolds number R, the natural control parameter, whereas no linear instability mechanism is effective. A
[
direct transition can take then place via the coexistence of laminar and turbulent domains in physical space. Two

1 emblematic cases are the pipe flow and plane Couette flow (PCF), the simple shear flow developing between two
v
parallel plates translating in opposite directions. Both of them are stable against infinitesimal perturbations for all
3
values of R and become turbulent only provided sufficiently strong perturbations are present. In both cases, strong
1
5 hysteresis is observed and, upon decreasing R, the turbulent state can be maintained down to a value Rg. Above Rg,
2 turbulence remains localised in space, in the form of turbulent puffs in pipe flow and turbulent patches in PCF. A
1. striking property of PCF or counter-rotating cylindrical Couette flow (CCF) is the spatial organisation of turbulence
0 in alternatively turbulent and laminar oblique bands that takes place in large enough systems in a specific range of
1 Reynolds numbers [1, Ch.7]. This regime was studied in depth at Saclay by Prigent et al. [2]. It can be obtained
1
by decreasing the Reynolds number continuously from featureless turbulence below R , the Reynolds number above
t
:
v which the flow is uniformly turbulent, or triggeredfrom laminar flow by finite amplitude perturbations above R , the
g
i Reynoldsnumberbelowwhichlaminarflowisexpectedtoprevailinthelongtimelimit. Asimilarsituationisobserved
X
in pipe flow but things are complicated by the global downstream advection so that the existence of a threshold R
r t
a above which turbulence is uniform is still a debated matter. In contrast for PCF, the pattern is essentially time-
independent and can be characterised by two wavelengths λ and λ in the streamwise and spanwise direction, x and
x z
z respectively,1 or equivalently by a wavevector k = (k ,k ) with k = 2π/λ . From symmetry considerations,
x z x,z x,z
two orientations are possible, corresponding to two possible combinations (k , k ). Whereas a single orientation is
x z
±
presentsufficiently farfromR sothateither mode(k ,+k )ormode (k , k )isselected,patchesofoneorthe other
t x z x z
−
orientationhave been reported to fluctuate in space and time when R approachesR from below [2, Figs. 2 & 3]. The
t
main features of the bifurcation diagram could then be accounted for at a phenomenological level by an approach in
terms of Ginzburg–Landau equations subjected to random noise featuring the small-scale turbulent background.
This patterning was reproduced by Duguet et al. [3] using fully resolved numerical simulations in an extended
systemofsizecomparablewiththatoftheSaclayapparatusbutthecomputationalloadwassoheavythatastatistical
1In the case of CCF, the pattern is time-independent in a frame that rotates at the mean angular velocity and the axial (azimuthal)
directioncorrespondstothespanwise(streamwise)direction.
1
study ofthe upper transitionalrangewasinconceivable. Earlier,Barkley& Tuckerman[4] alsosucceeded in obtaining
the bands by means of fully resolvedsimulations with less computational burden but using narrowelongated domains
alignedwiththe pattern’swavevector. Byconstruction,the fluctuatingdomainregimecouldnotbeobtained,whereas
a re-entrant featureless turbulence regime, called ‘intermittent’ was obtained closer to R .
t
In our previous work on this problem, we first showed that full numerical resolution was not necessary to obtain
realistic patterning but that a good account of the long range streamwise correlation of velocity fluctuations was
essential[5]. This next incited us to consider reduced-resolutionsimulations in systems of sizes sufficient to containat
least an elementary cell (λ ,λ ) of the pattern [6], thus avoiding the orientation constraint inherent in the Barkley–
x z
Tuckerman approach. Here, we expand our previous work to focus on pattern fluctuations in the upper part of the
PCF’sbifurcationdiagramwhenRapproachesR frombelow,takingthebestpossibleuseoftheinescapableresolution
t
loweringtoperformlongdurationsimulations,soasto obtainmeaningfulstatisticsaboutthe dynamicsofthis regime.
Systemsconsideredinournumericalexperiment,tobedescribedin 2,producepatternswithafewwavelengths. In
§
theneighbourhoodofR ,fluctuationsmanifestthemselvesasorientationchangesintimeinsteadofthespatiotemporal
t
evolution of well-ordered patches. It turns out that episodes of well-formed pattern between two orientation changes
can be identified reliably, so that the lifetimes of such episodes can be measured and their average determined as a
function ofR. The Langevinapproachinitiated by Prigentet al. in[2]was resumedin [6]as providinganappropriate
framework to interpret our numerical results. Orientation fluctuations were taken into account but their detailed
statistical properties left aside, which are the subject of the present paper.
In the context of pattern formation, the Langevin/Fokker–Planckapproach has a long history, dating back to the
1970’s when it was applied to convecting systems [7]. Noise of thermal origin is however extremely weak so that the
regionofparameterspacewherethesystemissensitivetothisnoiseisexceedinglynarrow[8]andnontrivialeffectscan
be observedonlyinveryspecific conditions[9]. Whenapplyingthe approachtothedescriptionofthe bifurcationfrom
featureless turbulence to pattern in shear flows, Prigent et al. [2] implicitly took for granted that the noise intensity
wasanadjustable parameterlinkedto the turbulentbackgroundatR>R . Herewe extendthe analysisstartedin[6]
t
withinthisconceptualframework,thesubjectof 3,andanalysesimulationresultspresentedin 2.4inthelightofthis
§ §
theory. Weconcludein 4bydiscussinghowwellthisapproachissuitedtodescribemodecompetitionandintermittent
§
re-entrance of featureless turbulence [4,6] and, more generally, how the noisy temporal dynamics of coherent modes
can hint at the spatio-temporal nature of transitional wall-bounded flows and explain the exponentially decreasing
probability distributions of residence times or decay times often observed in this field [12].
2 Conditions of the numerical experiment
2.1 Numerical procedure
Directnumericalsimulation(DNS)oftheincompressibleNavier–StokesequationsinthegeometryofPCFareperformed
using Gibson’s open source code ChannelFlow [10] that assumes no-slip boundary conditions at the plates driving
theflowandin-planeperiodicboundaryconditions. Theparallelplatesproducingtheshearareplacedatadistance2h
from each other in the wall-normaldirection y, they move at speeds U in the streamwise direction x, z labelling the
±
spanwisedirection. The lengthunitis h,the velocityunitU,the time unith/U,andthe ReynoldsnumberR=Uh/ν,
where ν is the kinematic viscosity of the fluid. The problem is completely specified when the in-plane dimensions
L and L of the set-up are chosen. The perturbation to the laminar flow U = yxˆ is noted u, so that u2 is the
x z
local Euclidian distance to the base flow squared. Periodic in-plane boundary conditions allow the definition of the
wave-vectorsk =2πn /L , where the wavenumbers n are integers. Without loss of generality, we can assume
x,z x,z x,z x,z
n 0.
x
≥
The resolution of the simulation is fixed by the number N of Chebyshev polynomials used to represent the wall-
y
normaldependence,andthenumbersN ofcollocationpointsusedtoevaluatethenonlineartermsinpseudo-spectral
x,z
scheme ofintegrationof the Navier–Stokesequations. The number ofFourier modes involvedin the simulationis then
2N /3, owing to the 3/2-rule applied to de-aliase the velocity field. The computational load necessary to obtain
x,z
meaningfulresultsinsufficiently widedomainswithfully resolvedsimulationsisunrealisticallyheavy. Accordingly,we
takeadvantageofourpreviousworkdevotedtothevalidationofsystematicunder-resolution asamodelling strategy[5].
In that work,we showedthatqualitatively excellentand quantitativelyacceptable results couldbe obtainedby taking
N = 15 and N = 8L /3. The price to be paid for the resolution lowering was apparently just a downward shift
y x,z x,z
2
Figure 1: Snapshots of u2 in the plane y = 0.57 for R = 315 in a system of size L L = 128 84. From left to
x z
− × ×
right: one band pure state with each of the two possible orientations (n = 1, n = +1) or (n = 1, n = 1), two
x z x z
−
band pure state (n =1, n =+2), and mixed or defective pattern. Deep blue corresponds to laminar flow.
x z
Figure 2: Snapshots of u2 in the plane y = 0.57 for R = 290 in a system of size L L = 170 48. From left to
x z
− × ×
right: (n =1, n = 1), (n =1, n =+1), and (n =2, n =+1).
x z x z x z
−
of the range [R , R ] in which the bands are obtained, but everything else was preserved, including wavelengths. Of
g t
course, as far as resolution is concerned, the finest is the best on a strictly quantitative basis but we do not expect
that the observed trends and our conclusions be sensitive to our rules to fix N and N .
y x,z
2.2 Orientation fluctuations.
Inthisarticleweconsiderdomainsabletocontainpatternwithoneortwoelementarycells,i.e. L = n λ where
x,z x,z x,z
| |
n = 1 or 2 and n = 1 or 2. According to [2], in PCF wavelength λ is found to be approximately equal to 110
x z x
± ±
over the whole range [R , R ], while wavelength λ varies as a function of R in the range [40, 85] These observations
g t z
serve us to fix the size of the systems that we are going to consider below. As shown in [6], the specificity of such
systems is to convert the spatio-temporal evolution of fluctuating domains observed in the neighbourhood of R into
t
the temporalevolutionofcoherentpatterns characterisedby the amplitudes ofthe correspondingfundamentalFourier
modes; possible orientation changes are associated with changes of sign of the spanwise wavenumbers. Close enough
to R , there is also some probability that featureless turbulence, the state that prevails for R > R , be observed
t t
transiently, which is akin to the intermittent regime identified in [4]. In contrast,in the lowestpart of the transitional
range, close to R , the orientation remain frozen as expected for well-formed steady oblique bands. We first illustrate
g
this phenomenon using snapshots of u2 in Figures 1 and 2. The left and centre panels of Fig. 1 display well-oriented
patterns or ‘pure states’ showing the organised cohabitation of laminar and turbulent flow; an example of defective
pattern or ‘mixed state’ without much spatial organisation is shown in the right panel. (Orientation defects between
well-oriented domains require wider systems to be clearly identified as such.) Figure 2 similarly displays snapshots of
u2 obtained in a narrower but longer system.
Typically,duringlong-lastingsimulationsatgivenL andR,the flowdisplaysapurepatternforsometime,then
x,z
experienceabriefdefectivestage,andnextrecoversapurestate,possiblywithdifferentorientationor/andwavelength,
and so on. The spatial organisation of the pattern is detected via the Fourier transform of the perturbation velocity
field uˆ. It turns out that most of the information about the modulation is encoded in the amplitude of the dominant
wavenumber [2,6,11]. We consider time series of
1 +1
m2(n ,n ,t)= uˆ (n ,y,n ,t)2dy, (1)
x z x x z
2Z | |
−1
whichthuscharacterisesaflowpatternwithwavelengths(λ ,λ )=(L /n ,L /n )andorientationgivenbythesign
x z x x z z
| |
3
x 10−3
3
2.5
n=+1
2 z
n=−1
2 z
m1.5 n=+2
z
1 n=−2
z
0.5
0
0 1 2 3 4 5 6 7
time x 104
Figure3: Timeseriesofm2(t)forseveralwavenumbersn = 1, 2forR=315inasystemofsizeL L =128 84.
z x z
± ± × ×
of n . In the present study, we focus on the amplitude of the turbulence modulation in the flow and not on its phase,
z
i.e. on the position of the pattern in the system, which was shown to be a random function of time [6].
An example of such time series is displayed in Fig. 3. A pure pattern stage corresponds to a single m(n ,n )
x z
fluctuating around a non zero value, the other m(n′,n′) remaining negligible. For instance the pattern keeps wave-
x z
number n = +2 from t = 3103 to t = 104. The defective stage corresponds to m(n ,n ) decaying to zero while
z x z
anotherone m(n′,n′) grows. Wavenumbers(n′,n′)may be different from(n ,n ), inwhichcase there is aneffective
x z x z x z
change of the orientation if n′ = n or a change of wavelength (sometimes combined with orientation changes) if
| z| | z|
n′ = n . InFig.3,achangeoforientationtakesplaceattime t=4104 (n = 1 +1),achangeofwavelengthat
| z|6 | z| z − →
time t = 1.7104 (n = 2 +1), the pattern with n =+1 growing back from a defective stage at time t = 4.3104.
z z
− →
Mostof the time there is no ambiguity aboutthe value of n involvedso that we shalluse simplified notations,i.e. just
m or m instead of m(n , n ), as often as possible.
± x z
±
ExceptveryclosetoR ,purestateintermissionslastlonganddefective episodesareshort,sothatseriesoflifetime
t
T of well-oriented lapses can be defined from recording simulations of duration sufficient to make reliable statistics.
i
2.3 Lifetime computations
Orientation and wavelength fluctuations are best characterised by lifetimes distributions. Beforehand, we have to
defineasystematicmethodtodetectthebeginningandthe endofpurepatternepisodesfromthem2 timeseries. This
is done by using two thresholds: one, s , for the start of a pure pattern episode and the other, s , for its termination,
1 2
see Fig. 4 (top). The fast growth of m2 makes it easy to choose s and the results are not much sensitive to its exact
1
value. In contrast, detecting the decay is more problematic. This will be discussed in detail after the presentation of
a typical result obtained by assuming that the difficulty has been properly resolved.
For practical reasons, we use a byproduct of the cumulated probability density function (PDF) Q:
# T T # T T
i i
Q(T)= { ≥ } =1 { ≤ }.
# T − # T
i i
{ } { }
Empirical distributions obtained in an experiment with L = 110 and L = 32 for R = 330 are displayed in Fig. 5.
x z
Theyareobtainedfromthetimeseries,asmallpartofwhichisshowninFig.4,distinguishingn = 1fromn = 2.
z z
± ±
Since, for symmetry reasons, the two orientations are supposed to have identical distributions, we sum over the
±
in each case. The semi-logarithmic coordinates used to represent Q(T) suggest exponentially decreasing variations,
whichmakesorientationchangeslook likederiving froma Poissonprocess. Assuming thatthey areindeed inthe form
exp( T/ T ),wecanobtainthemeanlifetime T fromtheplainarithmeticaverageoftheliftetimeseries. T canalso
− h i h i h i
be obtainedbyfitting theempiricalcumulateddistributionagainstanexponentiallaworitslogarithmagainstalinear
law. In addition to raw data, Fig. 5 displays the second kind of fits for n =1 and 2. These three different estimates
z
| |
are close to each other provided that the lifetime series comprise sufficiently large numbers of events. An average of
4
x 10−3
3.0
n =1
x
2.5 n =−1
z
2.0
2
m1.5
1.0 s1
0.5 s2
0
0 2 4 6 8 10 12 14
time x 104
x 10−3

1 n =+1
z
n =−1
0.8 z
2 0.6
m
0.4
0.2
0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
time
Figure 4: Time series of m2 for n = 1 at R = 330 (top) and R = 345 (bottom). The two horizontal lines in the
z
±
top panel locate threshold s (full line) and s (dashed line). Close to R , bottom panel, orientation fluctuations are
1 2 t
short-lived and much smaller, rendering the detection of well-oriented episodes more difficult.
0.0
|n|=2
z
−0.5 |nz|=1
linear fit of |n|=1
z
−1.0
linear fit of |n|=2
z
−1.5
)
Q
−2.0
(
n
l
−2.5
−3.0
−3.5
−4.0
0 2000 4000 6000 8000 10000 12000 14000 16000
lifetime
Figure 5: Logarithm of Q (right) for L L = 128 84, R = 315, computed with s = 0.001, s = 0.005, for both
x z 1 2
× ×
wave numbers n =1 and n =2. We have about 40 events for n =1 and about 20 events for n =2.
z z z z
| | | | | | | |
5
7000 s =0.000 3 5500 s =0.001
2 1
s =0.0005 s =0.0012
6000 2 5000 1
s2=0.0007 s1=0.0015
4500
5000 e s =0.0016
m 1
n lifetime34000000 ean lifeti34500000 s1=0.0017
a m
e 3000
m
2000
2500
1000
2000
00. 5 1 s 1.5 2 3 4 5 s 6 7
1 x 10−3 2 x 10−4
Figure 6: Mean lifetimes functions of s given s (left) and of s given s (right). R=315 and system size L L =
1 2 2 1 x z
×
128 84, n =1.
z
× | |
these threevalueswillbe usedto definethe meanlifetime andthe correspondingunbiasedstandarddeviationwillgive
an estimate of the “error”for each lifetime series.
LetusnowcometotheproblemofthesensitivityofQ(T)tothevalueofthethresholdss ands usedtodetermine
1 2
thelifetimesofthepurepatternepisodes. InFigure6(left)themeanlifetime T displaysaclearplateauasafunctions
h i
ofs . The width ofthis plateaudoes notdepend ons thoughits value depends onit. The existence ofthis plateauis
1 2
easily seen to be related to the fast growth of m when the pattern sets in: m always goes through most of the values
corresponding to the plateau in a very short time. In practice, for 10−3 s 1.510−3 the very same episodes are
1
≤ ≤
detected whatever the precise value of s . That the plateau value still depends on s just expresses that the duration
1 2
of the detected episodes are modified in the same way due to changes in the detection of their termination. Of course,
when s is taken too large, some less-well orderedepisodes escape detection or are detected too late, which artificially
1
decreases the mean. On the other hand, if s is taken too small, the “signal” gets lost in the “noise”: a large number
1
of brief noisy excursions are detected as relevant ordered episodes, again decreasing the mean.
The variationofthe meanlifetime with s is completely different asseeninFig.6 (right). Here, T variesroughly
2
h i
linearly with s in a wide interval above the noise level ( 310−4, see Fig. 4):
2
∼
T (s ) a(1 bs ).
2 2
h i ≃ −
Coefficient b = 1000 100 does not vary significantly over the cases that we have considered. This dependence fully
±
explainsthechangeofplateauvalueinplotsof T asafunctionofs . Coefficienta,correspondingto T extrapolated
1
h i h i
towards =0howeverstilldependonRandthegeometry. Henceforth,wedefinethisextrapolatedvalueastherelevant
2
average lifetime T , which will be supported by the theoretical considerations to be developed in the next section.
h i
The observeddependence of T on s can be explained by the fact that the decay of a pure pattern is much more
2
h i
gradualthan its growth,whichcausessignificantdifferences when the durationof anepisode is measured,leading to a
decrease of T as s increases since the termination of the episode is detected earlier. A second reasonwhy the mean
2
h i
lifetime increasesass decreasesarisesfromthefactthatsomeexcursionsarenotcountedasdecayevents. Inphysical
2
space, this corresponds to an irregular and slow disorganisation of turbulence, contrasting with the fast installation
of the pattern. In fact T cannot be obtained otherwise than by extrapolation of threshold s to zero, as will be
2
h i
discussed in 3.3.
§
6
5.0
4.5
)
τ
(
g
o
l4.0
3.5
325 330 335 340
R
Figure 7: Mean lifetime τ as a function of R for L L =110 32 (log scale).
x z
× ×
2.4 DNS results
The twosystemssizes,L L =128 64and110 32,alreadyconsideredinourpreviouswork[5,6]arestudiedhere
x z
× × ×
overthe wholerangeofReynoldsnumberswherethepatternexistsatthe chosennumericalresolution,R [R , R ]=
g t
∈
[275,345]. Orientation fluctuations are systematically found close enough to R , see Cases 1 & 2 below.
t
In addition, wavelength fluctuations can take place when the size of the system is too far away from resonating
with the pattern’s elementary cell λopt λopt, where ‘opt’ means ‘optimal’, in a sense to be defined below in 3.1.
x × z §
Orientation and wavelength fluctuations are observed at R = 315 for L = 128, L = 84 and 90, and for L = 110,
x z x
L = 84, meaning that both n = 1 and n = 2 are competitive for L = 84 or L = 90. In contrast, lifetimes
z z z z z
| | | |
of single mode patterns are extremely long for L < 84 and L > 90, meaning that L < 84 is optimal for n = 1
z z z z
| |
and L > 90 is optimal pour n = 2. Orientation and wavelength fluctuations are similarly present in several other
z z
| |
circumstances, at lower Reynolds number R = 272 and R = 275 for L L = 110 32, as well as at R = 290 for
x z
× ×
L =48 and L =80 or at R=330 for L =90, 140, and 150.
z x x
Case 1: L = 128, L = 84, R = 315, wavelength fluctuations. Several experiments under the same protocol
x z
havebeenperformed,usingdifferentinitialconditions. Integrationtimesrangedfrom5104to105h/U. Alargeenough
ensemble of lifetimes has been sampled, both for n = 1 and n = 2, allowing us to compute the corresponding
z z
| | | |
order parameters M – the conditional time averages of m(t) as defined in (1) – with sufficient accuracy. Snapshots
corresponding to this aspect ratio are displayed in Fig. 1, a typical part of the corresponding time series is shown in
Fig. 3. For n = 1 and n = 2, we obtain M = 0.033 0.001 and M = 0.038 0.001, respectively. From the
z z 1 2
| | | | ± ±
lifetime distributions in Fig 5, we get τ =8100 200 and τ =3800 100. The fact that M <M is not surprising
1 2 1 2
± ±
and is understood in term of optimal wavelength ( 3.1, λopt 39 at R=315 [6]). The reason why one has τ >τ is
§ z ≃ 1 2
however not clear.
Case 2: L = 110, L = 32, variable R, orientation fluctuations. A thorough account of the behaviour of M
x z
and the re-entrance featureless turbulence has been given in [6]. Here, lifetimes are computed for Reynolds number
ranging from R=325 to R=340. Below R=325,the lifetimes are so long that a small number of events is observed
despitethelengthoftimeseriesused(>2105),whichforbidsthedeterminationofτ asameaningfulaverage(Fig. 10).
Above R=340, a clear separation of time scales is lacking, which now forbids the definition of lifetimes of individual
events, compare the two panels in Fig. 4.
Figure 7 displays the variation of the average lifetime τ with R, showing that it increases by a factor of 10 as
R decreases from R = 340, which is somewhat below R = 355, down to R = 325 below which it is too long to be
t
measured reliably. “Error bars” suggested by up and down triangles in Fig. 7 correspond to the unbiased standard
deviation of the three estimates for τ mentioned earlier.
7
3 Conceptual framework and application to DNS results
3.1 The Landau–Langevin model
Prigent et al. proposed to consider the turbulent bands as resulting from a conventional pattern formation problem
described at lowest order, from symmetry arguments, by two coupled cubic Ginzburg–Landauequations, one for each
band orientation, further subjected to noise featuring the turbulent backgroundabove R . The slowly varying part of
t
the velocity field component away from the laminar profile can be written as
ux =A+(x˜,z˜,t˜)eikxcx+ikzcz+A−(x˜,z˜,t˜)eikxcx−kzcz +cc,
where A C are the amplitude fields accounting for the two modulationwaves,and x˜, z˜and t˜are slow variables[1].
±
∈
Then, following this approach, we assume
τ ∂ A =(ǫ+ξ2∂2 +ξ2∂2 )A g A 2A g A 2A +αζ , (2)
0 t˜ ± x x˜x˜ z z˜z˜ ±− 1| ±| ±− 2| ∓| ± ±
the quantity ǫ = (R R)/R measures the relative distance to the threshold2 R , τ is the ‘natural’ time scale for
t t t 0
−
pattern formation, ξ are streamwise and spanwise correlation lengths, g and g are the self-coupling and cross-
x,z 1 2
couplingnonlinearcoefficients,andαthe strengthofthe noiseζ supposedtobe acentredGaussianprocesswithunit
±
variance. The strength α of the noise is expected to grow smoothly with R, regardless of the existence of the pattern
since the local intensity of the turbulence is empirically not directly correlated to the amplitude and phase of the
modulation A . The tilde variables describe the long-wave modulations to an ideal pattern with critical wavelengths
±
λc to which correspond critical wavevectors kc = 2π/λc , the term critical referring to the most unstable wave
x,z x,z x,z
vectornearR=R . ThesystemsthatweconsiderhaveperiodicboundaryconditionsplacedatdistancesL . Fourier
t x,z
analysis then leads to characterise the pattern by wavevectors k = (k ,k ), with k = 2πn /L . It is assumed
x z x,z x,z x,z
that the wave numbers obtained during a given experiment will be the integers that will be as close as possible of
nc = L /λc . Furthermore, our systems can accommodate a small number of cells of size (λ ,λ ) so their modes
x,z x,z x,z x z
are well isolated [1, Ch.4]. Assuming that a single pair (n , n ) is involved, the partial differential equation (2) is
x z
±
turned into an ordinary differential equation for A(n , n ) simply denoted A Ar +iAi , close enough to R [6]:
x ± z ± ≡ ± ± t
τ dA =˜ǫA g A 2A g A 2A +αζ , (3)
0dt ± ±− 1| ±| ±− 2| ∓| ± ±
where ˜ǫ = ǫ ξ2δk2 ξ2δk2 controlling the linear stability of these modes, is evaluated for δk = k kc with
the relevant−k x=x2π−nz /Lz , as well as the nonlinear coefficients g ( R because the pattxe,rzn doexs,zn−ot xd,rzift, at
x,z x,z x,z 1,2
∈
least in the absence of noise). Coefficient α is the effective strength of the noise affecting the mode that we consider.
Equation (3) can be written as deriving from a potential:
∂
τ dAr,i = V +αζ ,
0dt ± −∂Ar,i ±
±
with
= 1ǫ˜ A 2+ A 2 + 1g A 4+ A 4 + 1g A 2 A 2. (4)
V −2 | +| | −| 4 1 | +| | −| 2 2| +| | −|
(cid:0) (cid:1) (cid:0) (cid:1)
Usually, when making use of phenomenological equations such as (2), one relies on values of critical wavevectors kc
thatarecomputedonceforallfromsomelinearstabilitytheoryandfurtherintroducedintheperturbationexpansions
solvingthenonlinearwavelengthselectionproblembeyondthethreshold[13]. Herethetheoryisnotdevelopedenough
tohavesuchadefinitionandsuchanevaluationofnonlinearlyselected‘optimal’wavevectorsfarenoughfromthreshold.
Accordingly,in(3)weintroducevaluesof˜ǫthatdonotmakereferencetosomeexplicitcomputationinvolvingmeasured
values ofǫ andξ but valuesthat arejust estimatesconsistentwith the empirically determinedoptimalwavelengths.
x,z
In the same way, we keep the cubic Landau expressions (3), neglecting higher order terms that would introduce too
many little-constrained parameters, without deeper insight into the problem.
2The existence of a well defined threshold in this system is attested by the behaviour of the turbulent fraction and spatially averaged
kineticenergywhichdisplayamarkedchangeofslopeatRt [6]
8
The stable fixed points of the deterministic part of (3) were shown to correspond to the permanent state of the
pattern and the additive noise term seen to account for fluctuations quite well by solving the corresponding Fokker–
Planck equation [6]. The stationary probability distribution for the moduli A =Am was obtained in the form:
| ±| ±
Π(Am,Am)=Z−1AmAmexp( 2 /α2), Z = AmAmexp( 2 /α2)dAmdAm. (5)
+ − + − − V Z + − − V + −
The time behaviour of Am is easily discussed by considering the shape of within the stochastic process framework.
± V
Two limiting cases can be identified, depending on whether ǫ˜is (1) or 1. In the first case, excursions from the
O ≪
neighbourhood of the minima of are rare; the lifetime of an ordered episode can be defined as the average time
V
necessary for the system to go from the neighbourhood of a minimum to the potential’s saddle. It is expected to
increase with the height of the potential barrier, i.e. as parameter ǫ˜grows, and to fall off as α increases. The lack of
symmetry between the growth and the decay of a pattern has then a clear explanation when ǫ˜is large: The growth
corresponds to the system falling from the neighbourhood of the saddle into one of the wells; even in the presence of
noise,this evolutionis fastandmostly deterministic. Incontrast,the decaycorrespondsto the systemslowlyclimbing
toward the saddle against the deterministic flow, driven by the sole effect of noise. In the opposite limit, when ǫ˜
approaches zero, the definition of a lifetime no longer makes sense since the characteristic times for growth and decay
become of the same order of magnitude.
3.2 Orientation lifetimes from the model
Orientation changes and associated lifetimes are analysed in terms of first passage time and escape from metastable
states [14]. The distribution of lifetimes is anticipated to be Poissonian as expected from a jump process controlled
by an activation “energy”. In a simplified one-dimensionalversionof potential [14, Ch.11, 2,6–7],if the well is deep
V §
enough, within a parabolic approximation the mean escape time, the average time necessary to go from a well to
another, is given by
2π
s w
τ/τ = exp 2V −V , (6)
0 ′′ ′′ (cid:18) α2 (cid:19)
Vw|Vs|
where‘w’standsfor‘well’and‘s’for‘saddle’; parethevaluesofthepotentialsatthecorrespondingpointsand ′′
Vw,s Vw,s
the values of the second order derivatives of the potential with respect to the variable at these points. The derivation
of this formula shows that τ is dominated by the time spent around the saddle. In our two dimensional system with
potential (4), at lowest order in α the coordinates of the well and saddle points are:
(Aw,Aw)= ǫ˜/g , 0 and (As ,As )= ǫ˜/(g +g ), ǫ/(g +g )
± ∓ 1 + − 1 2 1 2
(cid:16)p (cid:17) (cid:16)p p (cid:17)
and the corresponding values of the potential:
= ǫ˜2/4g and = ǫ˜2/2(g +g ).
w 1 s 1 2
V − V −
The second derivatives have to be replaced by the eigenvalues of the Hessian matrix of computed at these points:
V
2ǫ˜ 0 2ǫ˜ g g
H = and H = 1 2 .
w (cid:18)0 ǫ˜(g2/g1−1)(cid:19) s g2+g1 (cid:18)g2 g1(cid:19)
At point ‘w’, H is diagonalandthe eigen-directionpointing to point ‘s’ has eigen-value ǫ˜(g /g 1). At point ‘s’, H
w 2 1 s
−
is diagonal in the basis (1,1),(1, 1) and has eigenvalues 2ǫ˜, 2ǫ˜(g g )/(g +g ) . The unstable eigen-direction
1 2 1 2
{ − } { − }
correspond to the second one which is negative (g >g ). Inserting these values in (6), we obtain:
2 1
2π√2 g (g +g ) ˜ǫ2 g g
1 1 2 2 1
τ/τ = exp − . (7)
0 ǫ˜ p g g (cid:18)2α2g g +g (cid:19)
2 1 1 1 2
−
In its exponential factor, this formula points out an “energy” scale ˜ǫ2/g to be compared to the characteristic noise
1
energy α2 which play the role of the Boltzmann energy in thermal problems. It also shows that, especially when g is
2
larger but comparable to g , the noise energy has to remain small enough because the parabolic approximationwhich
1
underlies the formula assumes sufficiently deep wells. The main difference between the one and two dimensions cases
are in the shape of the “energy landscape”, corrections are therefore expected to be multiplicative and not depend on
the value of ǫ˜.
9
x 10−3
2
mode +
mode −
1.5
2
A| 1
|
0.5
0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
time
Figure8: Typicaltimeseriesfrommodel(3),forasetofparameterscorrespondingtotheNavier–StokesDNS,ǫ˜=0.05,
g =60, g =120, α=0.002 [6].
1 2
300 600
250 500
200 400
τ150 τ 300
100 200
50 100
0 0
0.04 0.06 0.08 0.1 s 0.12 0.14 0.16 0 0.005 0.01s 0.015 0.02
1 2
Figure 9: Mean lifetime τ, extractedfrom the model as a function of s , for s =0.0163,0.0193,and 0.0225(left) and
1 2
as a function of s for several s ranging from 0.0784 to 0.1296 (right). ˜ǫ=0.075, g =1, g =2, α=0.002.
2 1 1 2
3.3 Simulation of the model
For the deterministic part of equation (3) we use a simple first-order implicit Euler algorithm, while the additive
noise αζ(t) is treated as a Gaussian random variable with standard deviation α√dt at each time step. The model is
integratedoverarangeof˜ǫ,giveng ,g (severalvalues),andα(assumedconstant). Thetimeseriesof A 2 displayed
1 2 ±
| |
in Fig. 8 are indeed reminiscent of those obtained by numerical integration of Navier–Stokes equations (Figs. 3 and
4): pure states at the bottom of the wells correspond to A 2 fluctuating around 0 and A 2 away from 0, or the
− +
| | | |
reverse. Large excursions can lead to a change of the dominant orientation. These excursions are more likely to occur
when ǫ˜is decreased.
LifetimeshavebeencomputedinthesamewayasforFig.6. Thedependenceofthemeanlifetimesonthresholdss
1,2
is displayedin Figure 9. A neat plateau is obtainedfor s [0.075,0.115]for different values of s (Fig. 9, left), which
1 2
∈
corresponds to the trajectory getting away from the saddle. Extremes values of s lead to bad estimates of τ for the
1
same reasons as stated before. As to threshold s , an orientation change has taken place when the trajectory goes
2
beyond the saddle, while a pure state corresponds to one amplitude large and the other at the noise level. We have
thus to detect the change from large to small for one or the other amplitude. It is extremely difficult to detect the
precise passage at the saddle, since it is dominated by the time spent in that region, contribution from the two sides
of the saddle point having the same weight. On the contrary the passage from one state to another leaves no doubt
10