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Recovering coefficients of the complex Ginzburg-Landau
equation from experimental spatio-temporal data: two
examples from hydrodynamics
Patrice Le Gal, Jean-Francis Ravoux, Elena Floriani, Thierry Dudok de Wit
To cite this version:
Patrice Le Gal, Jean-Francis Ravoux, Elena Floriani, Thierry Dudok de Wit. Recovering coefficients of
the complex Ginzburg-Landau equation from experimental spatio-temporal data: two examples from
hydrodynamics. Physica D: Nonlinear Phenomena, Elsevier, 2003, 174, pp.114-133. �hal-00823553�
equation from experimental spatio-temporal data:
Two examples from Hydrodynami s
P. LeGal (1) ,J.F. Ravoux (1) , E. Floriani (2)
and T. Dudok de Wit
(3)
(1)
IRPHE,49Av. F.Joliot-Curie,B.P.146,Te hnop^oledeChateau-Gombert,13384 Marseille,Fran e
(2)
CentredePhysiqueTheorique,CNRSLuminyCase907, 13288Marseille edex 9,Fran e
(3)
LPCE-CNRS,3A,av. delaRe her heS ienti que,45071Orleans edex2,Fran e
April26,2002
Abstra t
Thereare manyexampleswherethedes riptionof the omplexityof ows anonlybe a hievedbythe
useofsimplemodels. Thesemodels,obtainedusuallyfromphenomenologi alarguments,needingeneral
theknowledgeof some parameters. The hallenge is then to determine thevalues ofthese parameters
from experiments. We will givetwo examples where we have been able to evaluate the oeÆ ients of
the omplex Ginzburg-Landauequation from spa e-time haoti data applied to rst arowof oupled
ylinderwakesandthentowavepropagationintheEkmanlayerofarotatingdisk. Inthe rst ase,our
analysisisbasedonaproperde ompositionofexperimental haoti ow elds,followedbyaproje tion
of theCGLE onto the proper dire tions. We show that our method is ableto re overthe parameters
ofthe model whi h permits to re onstru t thespatio-temporal haos observedin the experiment. The
se ondphysi alsystemunder onsiderationisthe owabovearotatingdiskandits ross- owinstability.
Ouraim is to study the properties of the wave eldthrough aVolterraseries equation. Thekernelsof
the Volterra expansion, whi h ontain relevant physi al information about the system, are estimated
by tting two-pointmeasurements viaanonlinear parametri model. We then onsider des ribingthe
wave eldwiththe omplexGinzburg-Landauequation,andderiveanalyti alrelationswhi hexpressthe
Despite the large Reynolds numbers of the ows that o ur in natural or industrial situations, their
dynami albehaviourisveryoftendominatedbythepresen eoflarges ale oherentstru tures. Classi al
examples an be found in atmospheri oro eani vortex ows and there is some hope that simpli ed
models might des ribe some of their omplex or even haoti spatio-temporal dynami s. Contrary to
the situation en ountered in many laboratory ows, one annot study the responses of these ows to
some ontrolperturbations. Inparti ular,whenfa ingtheproblemofre overingthe oeÆ ientsofsome
models, there is no way to ex ite these owsvia periodi for ingas it is often the ase to re overthe
dispersionrelationofthewaveswhi hareinvolvedinthesolutionsofthemodellingequations. Therefore,
itisne essarytodevelopsomete hniqueswhi hareabletoadjustanonlineardynami almodeltosome
measured data. This system identi ation thus leads to a problem of tting a nonlinear di erential
equationtoexperimental dataandisusuallyaddressedinthe eldofnonlinearsystemidenti ation[1℄.
Thisproblem anbedividedin threeparts:
model sele tion: determinewhattypeofmodelshould be ttedtothedata,
inferen e: estimatethemodel oeÆ ientsfrom thedata,
model validation: determineifthemodelindeed des ribestheobservationsadequately.
Unfortunately,theredoesnotexistaunifyingframeworkfordes ribingnonlinearsystems(liketheFourier
transformforlinearsystems). Oneshould therefore omparedi erentapproa heswheneverpossible. It
shouldalsobestressedthatthe riteriaforobtainingagoodmodeldi erdependingonwhetheronewants
tomakepredi tions, t data,reprodu etopologi alpropertiesin phasespa e, et . Inthefollowing,we
presenttwote hniqueswhi henableustoestimatewithagooda ura ythe oeÆ ientsoftheComplex
GinzburgLandau Equationinorder todes ribethespa e-timedynami so uringintwoinstabilitiesof
uid ows. The rst analysis is devoted to the oupled wakes downstream a rowof ylinders and the
se onddealswiththethree-dimensionnalinstabilityofarotatingdiskboundarylayer.
2 Part 1: Spatio-temporal haos generated by oupled wakes
2.1 Introdu tion
Itisknownfrom experimentalstudiesandnumeri alsimulations(seethereviewsbyZdrakovi h[2℄and
Changetal. [3℄),thatthewakesofblu bodiespla edsidebyside, anintera tand reatealargevariety
of phenomena. In the aseof interesthere, weanalyze thespa e-time haos reatedbythewakesof a
rowof16 ir ular ylinderspla edinawatertunnelperpendi ulartoanin oming ow. Figure1presents
asnapshotofthese16wakesmadevisible bydyeinje tionthroughasmallholedrilledattherearfront
ofea h ylinder.
These ylinders possess alength of 200mm and adiameter of 2mm. They are rigidlymaintained
in the wall of awater tunnel. The distan e separatingea h ylinder axes is equalto four times their
diameterandtheReynoldsnumberReof the owisequalto 80. Notethat theseparatingdistan ehas
been hosenina ordan ewithpreviousresultsobtainedonapairofwakesbyPes hardandLeGal[4℄
andin su hawaythatthe oupledwakesexperien easpatio-temporal haoti regime. Inordertobuild
spa e-timediagrams (512 time steps16 spa epositions) whi h representthe dynami sof thefamily
ofwakes,were ordauniquevideolineatthevideofrequen y(25Hz)andgathertheselinestogetheras
presentedin Figure 2. Thea quisition line issituated 12mm downstream therowof ylindersandthe
displa ementsof thedyestreaksarere ordedasafun tion oftime(timeunit is0:04s). We anseeon
gure2theerrati appearan ein spa eandtimeofamplitudeholes[5℄.
It is known that the Benard-von Karman wake of a ylinder pla ed in a ow appears via a Hopf
bifur ation. Thus theos illating ow anbemodeledbyaStuart-Landauequationasithasbeendone
and15.
Figure 2: Spa e-time diagram, time is running downwards (10 se onds). The errati appearan e of
amplitudeholesisvisible.
d t A(t)=(a r +ja i )A(t) (l r +jl i )jA(t)j 2 A(t) (1)
where A representsan order parameter(for instan e thetransverse velo ity at oneposition behind
the ylinder). The omplex oeÆ ientsa=(a
r +ja i )andl=(l r +jl i
)dependonthe hara teristi sof
thewake(aspe tratioor ross-se tionshapeofthe ylinder)andmustbedeterminedfromexperiments.
Therefore, the oupled os illators model that an be used to study the ow downstream the row of
ylinders is a dis rete versionof the Complex Ginzburg-Landau equation (CGLE) (see Cardoso et al.
[8℄): d t A i (t)=(a r +ja i )A i (t)+(g r +jg i )(A i+1 (t)+A i 1 (t) 2A i (t)) (l r +jl i )jA i (t)j 2 A i (t) (2)
withtheasso iatedboundary onditionsareA
0
(t)=A
17
(t)=0,whereA
i
(t)isthe omplexamplitude
ofthewakeof indexiandg=(g
r
+jg
i
)isthelinear oupling oeÆ ient.
Sin eE khaus[9℄, mostofthestabilityanalysis oftheGinzburg-Landauequationhavebeen arried
out for the ontinuous ase. The instability arises from a resonan eme hanismbetween wave trains,
and is alled the Benjamin-Feirinstability(or the sideband instabilitywith modulations at k0). In
parti ular,thewell-knowNewell's riterion[10℄isrelatedtotheinstabilityofanyplanewaveperturbed
bersperturbationsandnotonlythroughhomogeneousperturbationsasitwas lassi allystudied. Fora
dis retesystem,thepossiblewavenumbersofthewavesaregivenbythenumberofos illatorsandalso
bythe boundary onditionsapplied on thearray. The rstknown stabilitystudy forthe dis rete ase
hasbeenperformedbyWillaime et al. in1991[12℄with thewavenumbersq oftheperturbationsequal
to0or. These investigationshavethenbeenextendedtoallwavenumbersforthebasi solutionsand
toallwavenumbersfortheperturbationsbyRavouxetal. [13℄.
time
wakes
Figure3: Numeri alsimulationof 16 oupledLandau os illators(a=1+20j,g=1+3j,l=1-1j)with10 %
noiseaddedaposteriori.
2.2 Re overing the oeÆ ients of the model
Contrarytowhathasbeendoneforinstan ebyCroquetteetal. [14℄inthestudyofnonlinearwavesin
Rayleigh-Benard onve tion(wherethedispersionrelationwasdeterminedforea hfrequen ybyapplying
anexternalfor ing),herewedonotex iteourhydrodynami alsystembutletitevolvingin itsnatural
spatio-temporal haoti state. Therefore, ourgoal isto invert somemeasurementsof spa e-time haos
in order to re overthe oeÆ ientsof the model. Thus weneed to solve linearsystemsof equations of
the type M 1 = M 2 x, where M 1 and M 2
are matri es and x an unknown ve tor. These matri es are
madeupfrom data andthe ve torx onsists in the oeÆ ientsa, g andl oftheCGLE equation. The
resolutionis a hievedvia aleastsquare method wherethe linearsystem isover-determined. Thus, the
oeÆ ientve toris su h that the distan e betweenM
1
and M
2
x is minimized in the asso iated phase
spa e. Theinitialspatio-temporal eld isa51216 matrixA
ti
that anbegeneratedsyntheti allyby
numeri alintegrationof CGLEasshownon Figure3. However,asexperimental dataare orruptedby
noise,weadded10%Gaussiannoiseaposteriori(attheendofthewholesimulation)inordertotestthe
robustnessofourmethod [15℄. Se tion 1.3is thusdevoted totest ourmethods ontheresults obtained
fromnumeri alsimulations,where the oeÆ ientsaregivenapriori.
2.3 Data from numeri al simulations
2.3.1 Dire t inversion
Inthis ase,theCGLEsimplywrites:
D ti =aA ti +g ti lN ti ; with D ti = At+1 i At i t ; ti = A ti+1 +A ti 1 2A ti ; N ti = jA ti j 2 A ti ;
0
5
10
15
20
0.75
1
1.25
1.5
1.75
2
2.25
g
r true
= 1
Noise Amplitude ( %)
g
r
0
5
10
15
20
2.75
3
3.25
3.5
3.75
4
4.25
g
i true
= 3
Noise Amplitude ( %)
g
i
0
5
10
15
20
0.5
1
1.5
2
2.5
a
r true
= 1
Noise Amplitude ( %)
a
r
0
5
10
15
20
19.5
20
20.5
21
21.5
a
i true
= 20
Noise Amplitude ( %)
a
i
0
5
10
15
20
0.75
0.875
1
1.125
1.25
l
r true
= 1
Noise Amplitude ( %)
l
r
0
5
10
15
20
−1.25
−1.125
−1
−0.875
−0.75
l
i true
= −1
Noise Amplitude ( %)
l
i
Figure4: Dire tMethod,la kofa ura yoftheinversionasafun tion ofnoiseintensity.
where tis thetimeunit givenbythevideoa quisitionrate. Thegoalofourworkisthustoinvert
thealgebrai system and to obtainthevalues of the oeÆ ients a, g and l. Similardire t inversionof
haoti spatio-temporaldataserieswasdoneonthesameproblembyFullanaetal. [16℄,inasurfa ewave
studyby Gollubet al. [17℄, by Vosset al. forspatio-temporalmeasurementsofbinary- uid onve tion
[18℄,andin area tiondi usion partialdi erentialequationbyBaret al. [19℄.
As it anbeseen onFigure 4, thepresen e of noiseprohibits there overyof the oeÆ ients using
thisdire tinversionmethod. The al ulatedvaluesdependdrasti allyonthenoiseintensityanddepart
stronglyfromthetruevalueswhi hhavebeen hosento omputethesyntheti spatio-temporaldata.
Even averaging the al ulated oeÆ ients ona great numberof observation windows(typi ally 50,
orresponding to the total duration of the experimental data) does not an el the in uen e of noise.
Figure5showsinsolidlinesthisdeparturefromthetruevaluesandarapid onvergen etofalsevalues.
2.3.2 Dispersionrelation method
Inorderto lteroutthenoisepollutionwhi hisreminis entofanyexperiments,itistraditionaltousethe
Fourierre ipro alspa e. Unfortunately,asit anbeseenontheexampleofFigure6,theFourierpower
spe tra omputedonourexperimental haoti spa e-timediagramsdonotpermita leardetermination
ofthedispersionrelationofthewaveswhi hareinvolvedinthe haoti dynami s. Although,amoreor
lessparaboli shape an be observed(note the negative urvature linkedto thesigns of g and l), it is
0
10
20
30
40
50
0
0.5
1
1.5
2
a
r true
= 1
Number of windows
a
r
0
10
20
30
40
50
19
19.5
20
20.5
21
a
i true
= 20
Number of windows
a
i
0
10
20
30
40
50
0.5
0.75
1
1.25
1.5
l
r true
= 1
Number of windows
l
r
0
10
20
30
40
50
−1.5
−1.25
−1
−0.75
−0.5
l
i true
= −1
Number of windows
l
i
0
10
20
30
40
50
0.5
0.75
1
1.25
1.5
1.75
2
g
r true
= 1
Number of windows
g
r
0
10
20
30
40
50
2.5
2.75
3
3.25
3.5
3.75
4
g
i true
= 3
Number of windows
g
i
Figure 5: Simulated data: in solid line ({), la k of a ura y of the dire t inversion as a fun tion of
observationwindows. The rosses(x)showbetterresultsobtainedwiththerelationdispersionmethod.
Errorbarsshowthea ura yoftheinversion.
theCGLEwouldleadtoina urate oeÆ ients. ThereasonwhyFouriertransformsareoflittlehelpfor
makingaGalerkin proje tion,is be ause Fouriermodesare notanadequatebasis forrepresentingthe
spatio-temporal dynami s of thewave eld. It would bemoreappropriate to seek abasis that exploits
thepropertiesof thedata, giving what ouldbe onsidered as"eigenmodes". Su h abasis is provided
bytheBi-OrthogonalDe omposition,alsoknownastheProperOrthogonalDe omposition[20℄.
Chauve and Le Gal proposed in 1992 [21℄ a method where we an lter the data and get a good
hara terization of the dispersion relation obtained by a Galerkin proje tion onto the proper modes
of the Bi-Orthogonal De omposition (BOD) [22℄. Moreover, this method optimizes the total number
of modes whi h are needed for the series re onstru tion. The N = 16 proper modes of the omplex
eld A
ti
= A
i
(t) are rst al ulated by diagonalization of the temporal orrelation (1616) matrix.
Note that astheCGLE isa omplexmodel, the rststepof the method onsistsin omplexifyingthe
experimentaldatabytheuseoftheHilbertTransform. Then,theN=16propermodesofthe omplex
eld A
ti
=A
i
(t)are al ulatedbydiagonalizationof thetemporal orrelation(1616)matrixandby
theuseoftheproje tionrelation:
A ti = N P k =1 k k (t) k (i); (3)
wheretheoverbarreferstothe omplex onjugation,
k
thetemporalmodeand
k
0
5
10
15
20
25
30
35
40
45
50
Wave number
Frequency
0
−π
π
Figure6: (k,!)FourierTransformplane.
asso iatedwiththeeigenvalue
k .
1
4
8
12
16
1
4
8
12
16
0
2
4
6
8
x 10
−5
l
k
||
Γ
kl
||
1
4
8
12
16
1
4
8
12
16
l
k
1
4
8
12
16
1
4
8
12
16
0
0.1
0.2
0.3
l
k
||
κ
kl
||
1
4
8
12
16
1
4
8
12
16
l
k
1
4
8
12
16
1
4
8
12
16
0
0.1
0.2
0.3
0.4
0.5
l
k
||L
kl
||
1
4
8
12
16
1
4
8
12
16
l
k
1
4
8
12
16
1
4
8
12
16
0
2
4
6
8
10
12
l
k
||
Ω
kl
||
1
4
8
12
16
1
4
8
12
16
l
k
Figure7: Simulateddata: Matri es ,,Land.
WhentheCGLEis proje tedontotheBODmodes,thefollowing(1616)matri esappear:
kl =((Dti il ) tk ) ; L kl =((Ati il ) tk ) ; k l =((ti il) t k ) ; kl =((Nti il) t k ) ;
andtheDispersionRelationlinkingmodeslandk anthenbewrittenas(seepages411-412of[21℄):
k l =aL k l +g k l l k l : (4)
These matri esaretherefore omputedandtheamplitudesoftheirentriesaregivenin Figure7. As
it anbeseen,theyareessentially onstitutedbydiagonalelements. Thereasonofthisparti ularshape
omesfromthefa tthattheseriesarenearlypuresinusoidalfun tionswhi harethemselvesproportional
we anplottheproje tionsofthedispersionrelation(equation(4))onthedi erentdire tions. Figure8
representssu haproje tioninthe3D-spa e
r ;
i
;
r
. Thelinearrelationbetweenthedi erentmatri es
isin ompletea ordan ewith equation(4). Therefore,determining the omplexdire torve torofthe
dispersionrelationplaneinthespa e(; ;) allowsthe al ulationofthe oeÆ ientsoftheCGLE.
−80
−60
−40
−20
−100
−50
0
50
100
−700
−600
−500
−400
−300
−200
−100
0
κ
r
2
1
7
12
1516
5
14
13
3
6
4
10
9
11
8
Γ
i
Ω
r
Figure 8: SimulatedData: Dispersion Relation in 3D spa e:
r ;
i
;
r
. In grey olor is represented a
portionof thedispersionrelationplane.
Figure5presentstheresults(withthe rosses(x))ofthe al ulationsofthe oeÆ ientswhenaveraging
theobtainedvaluesonobservationwindows. We anobservethat,althoughthe onvergen etowardsthe
truevalueislessrapidthanthe onvergen eofthedire tmethod,the nalresultissatisfa torywithan
a ura ybetterthan7%. Forallthese oeÆ ients,theresultisbetterthantheoneobtainedusingthe
dire tmethod wheresome oeÆ ients oulddepartfromtheirtruevalueofmorethan50%.
2.4 Re overing the oeÆ ients of the model from experimental data
As our inversion method was su essfully tested by our simulated data, we apply it to experimental
spa e-time diagrams asthe onepresentedin gure2. Figure 9showsthe amplitudes of the entries of
the four matri es , , L and . As it waspreviouslyobserved on thesimulateddata study, most of
the information is ontained in the diagonal elements of the matri es. Therefore, only these diagonal
elements, ordered by their index l will be onsidered in the following. Then it an easily be seen on
Figure10 thatmostof themodesline upandvalidatethe linearityofthedispersionrelation(equation
(4)).
Toin reasethea ura yofthelinearregression,wethen keeponlythe rsttwelvemodes. Thelast
fourmodesoftheBOD,wherethesignaltonoiseratioisobservedtobelessthan50%arethusnegle ted
in the inversionpro ess. The least square method then leads to the determination on the oeÆ ients
(averageon50temporalwindows):
a r = 0:0534[(t) 1 ℄;a i =0:4747[(t) 1 ℄, g r = 0:2396[(t) 1 ℄;g i = 2:7018[(t) 1 ℄, l r =0:0567[(t) 1 (A) 2 ℄;l i = 0:0795[(t) 1 (A) 2 ℄:
Infa t,the relevantparametersof(2) arenormalizedand dedu edfrom thelatter, =
g r a r , 1 = g i g r and 2 = l i l r : =4:48; 1 =11:27; 2 = 1:40.
1
4
8
12
16
1
4
8
12
16
0
0.2
0.4
0.6
0.8
1
x 10
−4
l
k
||
Γ
kl
||
1
4
8
12
16
1
4
8
12
16
l
k
1
4
8
12
16
1
4
8
12
16
0
0.5
1
1.5
l
k
||
κ
kl
||
1
4
8
12
16
1
4
8
12
16
l
k
1
4
8
12
16
1
4
8
12
16
0
0.1
0.2
0.3
0.4
0.5
l
k
||L
kl
||
1
4
8
12
16
1
4
8
12
16
l
k
1
4
8
12
16
1
4
8
12
16
0
2
4
6
8
l
k
||
Ω
kl
||
1
4
8
12
16
1
4
8
12
16
l
k
Figure9: Experimentaldata: Matri es ,,Land.
0
0.2
0.4
0.6
0.8
−2.5
−2
−1.5
−1
−0.5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
L
r
Ω
r
−2
−1.5
−1
−0.5
0
−2.5
−2
−1.5
−1
−0.5
1
2
3
4
5
6
7 8
9
10
11
12
13
14
15
16
κ
r
Ω
r
−1.5
−1
−0.5
0
x 10
−4
−2.5
−2
−1.5
−1
−0.5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Γ
r
Ω
r
−4
−2
0
2
4
x 10
−6
−2.5
−2
−1.5
−1
−0.5
1
2
3
4
5
6
7
8
9
10
1112
13
14
15
16
Γ
i
Ω
r
0
0.2
0.4
0.6
0.8
0
2
4
6
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
L
r
Ω
i
−2
−1.5
−1
−0.5
0
0
2
4
6
8
1
2
3
4
5
6
7
8
910
11
12
13
14
15
16
κ
r
Ω
i
−1.5
−1
−0.5
0
x 10
−4
0
2
4
6
8
1
2
3
4
5
6
7
8
9
10
11
12
13
15
14
16
Γ
r
Ω
i
−4
−2
0
2
4
x 10
−6
0
2
4
6
8
1
2
3
4
5
6
7
8
9
10
1112
13
14 15
16
Γ
i
Ω
i
Figure10: ExperimentalData: Proje tionofonsomeotherdire tions. Ea hsymbolislabeledbythe
indexloftheBODmodes.
whi hare oeÆ ientswithvaluesin a ordan ewith thevaluesgenerallymeasuredonwakes. Using
these oeÆ ients,itiseasynowtotestthisvaliditybyanumeri alsimulationoftheCGLE.Asit anbe
seenon gure11,there onstru tion ofthespatio-temporaldynami s showsagoodagreementwiththe
observedspatio-temporal haoti behavior. In parti ular, theintermittent extin tionsof os illators (or
amplitudeholes)are re overed. Notethat themain limitationoftheBODis thatthis method exploits
se ond order momentsof the data only. Be ause of this, it annot properly apture deviationsfrom a
gaussiandistribution,whi harepre iselyahallmarkofnonlinearsystems.
Anotherwaytotestourestimationofthe oeÆ ientsofthemodelequation,isto omparesimulated
predi tionswithvariousexperimental on gurationsandverifythattheregimereallyexperien edbyour
system anbeindeedpredi ted. Inour ase,we hangedthedistan ebetweenthe ylinders,sothatthe
oupling oeÆ ientbetweenthewakesisvaried. Fordistant ylinders,thewakesarepoorly oupledand
thephasebetweensu essivewakesis. Onthe ontrary,forshortdistan ebetween ylinders,thewakes
arestrongly oupledandos illateinana ousti modewithaphasedi eren e loseto0(there anbea
longwavespatialmodulationalongtherow). Aspresentedin gure12,ournumeri alsimulationsre over
time
wakes
Figure11: Spa e-timediagramofthe16simulated oupledwakeswiththe oeÆ ientsobtainedfromthe
experiment.
Wakes
Time
Wakes
Time
Figure 12: left: Visualizations of \anti-phase" (top) and \in-phase" modes (bottom, with long waves
modulations) behind a row of ylinders. right: numeri al simulations for weak and strong oupling,
(=0:035(top)and5(bottom)),withtheother oeÆ ientsobtainedfromexperiments(
1
=11:27;
2 =
1:40).
arealsore overed)andthe\anti-phase"modeof theweak oupling ase[5℄.
3 Part 2: Rotating disk ow instabilities
3.1 Introdu tion:
Ekmanwasthe rstin 1905[23℄, toformulate inageophysi al ontext, themathemati alexpressionof
thevelo ity eld of arotatingboundarylayer. His analysis wasbasedon thelinearization ofthe uid
diskhaveangularvelo itiesvery loseonetotheother. Theself-similarsolutionhewrotetakestheform
ofaspiral,now alled the"EkmanSpiral",and ismainly lo alizedin athin boundarylayerofdepth Æ
nearby therotatingboundary (Æ =
p
=). In1921, Karman [24℄this sear h for self-similarsolutions
tothefullnonlinear ase. Twotypesofinstabilities (typeI andtypeII) instabilities anappearinthe
rotatingboundarylayer. TypeIIinstability orrespondstoadestabilizationbythe ombinede e ts of
thefor es due tothe Coriolisand vis ous e e ts. It produ eswaveswhi hare rolledupin spirals in a
ontrarydire tion to thedisk rotation. Stuartin 1955 [25℄showsthat typeI instability isinvis id and
omesfromthepresen eofunstablein e tionpointsintheradialvelo itypro les. Thisinstabilityalso
produ esspiralwavesbutwhi harerolledupinthedire tionofrotationofthedisk. Ourexperimental
devi e is fully presented in Jarreet al. [26℄ and mainly onsists of a 50 m diameter horizontal disk,
whi hisimmersedinawatertank. VisualizationofthetypeIwavesismadepossiblewhenusingawhite
dyeasitispresentedon gure13. Around30wavelengths anbe ountedallaroundthedisk.
Figure14: Wavepa ketshapewidensasitsamplitudegrows.
Anemometri measurementsofthewavesgeneratedby asmallroughness elementgluedon thedisk
surfa ejustunderthelinearthresholdhavebeenperformedbytheasso iationoftwoanemometri probes,
lo atedatadistan eof8mmonefromtheother,onadire tionmakinganangleof45degreesfromthe
areobtainedfrom Fourieranalysis aregivenin gure14. These measurementsallowedin parti ular to
estimatetheazimuthal oheren elength
0
from the urvatureofthemarginalstability urve: avalue
around1.2mmwasfound[26℄attheonset oftheinstabilitywhi ho urs aroundR e=280. Thenand
before fullydevelopedturbulen etakesoverforR e510itwasshownin Jarreetal. [27℄that thenon
linearwavespropagatewithawellde nedpatternandgroupvelo itywhi hjustifyanamplitudeequation
approa h. However,thefulldetermination ofthevaluestakenbythe oeÆ ientsoftheCGLE ouldnot
beobtainedbytheFourieranalysisdevelopedin[27℄. Nextse tionwillpresentournewte hniquebased
onVolterraserieswhi h allowssu h adetermination [28℄.
Figure 15: Growth of the wavesalong the radius and marginal stability urve. Note the se ond lobe
orrespondingtothegrowthofharmoni sdrivenbynonlineare e ts.
3.2 De nition of the Volterra model
We use here adi erent approa h for inferring the oeÆ ients of CGLE model. Instead of tting this
equation dire tlyto the data (whi h would be an indu tive approa h), we rst des ribethe data with
ageneral lass of models, basedon Volterraseries. If the underlying physi sis indeed des ribed by a
CGLEmodel,then adire tmappingexists betweenthemodel oeÆ ientsand theVolterrakernels. In
thiswaywe annotonlyestimatetheCGLEmodel oeÆ ients,butalsoandmoreimportantly,dedu e
whetherthismodelisindeed orre t.
Let v(x;t;R e) bethe azimuthal uid velo ity re ordedat time t,position x and Reynolds number
R e. Ageneraldynami almodelforthewave eldamplitudeis
v(x;t;R e)
x
=F v(x;t;R e)
;
whereF isa ontinuous,nonlinearandtime-invariantoperator. Weassumewe anwriteFasaVolterra
series[29℄,[30℄: v i (x;R e) x = 1 X k =0 g k (R e)v i k (x;R e) (1) + 1 X k =0 1 X l=0 g k ;l (R e)v i k (x;R e)v i l (x;R e) + 1 X k =0 1 X l=0 1 X m=0 g k ;l;m (R e)v i k (x;R e)v i l (x;R e)v i m (x;R e)+
here,with the notationv i
(x;R e) =v(x;t =t
i
;R e). The oeÆ ients g
k ;g k ;l and g k ;l;m are respe tively
alled rst,se ond andthird orderVolterrakernels. Furthermore,sin eweare dealingwithnonlinearly
intera ting waves, it is appropriate to onsider Fourier modes of the wave eld. The dis rete Fourier
transformin timegives:
^v(x;!) x = (!)^v(x;!) (2) + X ! 1 +! 2 =! (! 1 ;! 2 )^v(x;! 1 )^v(x;! 2 ) + X !1+!2+!3=! (! 1 ;! 2 ;! 3 )v(x;^ ! 1 )^v(x;! 2 )^v(x;! 3 ) +
where^v(!)standsfortheFouriertransformofv
i
atfrequen y !. ThelinkbetweentheVolterrakernels
inFourierspa eandtheirtemporal ounterpartsisobviously:
(!) = 1 X k =0 g k e i!k ; (3) (! 1 ;! 2 ) = 1 X k =0 1 X l=0 g k ;l e i(!1k +!2l) ; (! 1 ;! 2 ;! 3 ) = 1 X k =0 1 X l=0 1 X m=0 g k ;l;m e i(!1k +!2l+!3m) ;
andsoonforhigherorder kernels.
3.3 Inversion of the Volterra model: determination of the Kernels
We an give an estimation of the spatial derivative using the two-point measurements: as the probe
separationxissuÆ ientlysmall omparedtothewave-length,we anwrite:
v i (x;R e) x v i (x+x;R e) v i (x;R e) x :
TheVolterramodelmaynowbefullyexpressedintothemore onvenientframeworkoftransferfun tions:
u i = v i (x;R e) (the input) (4) y i = v i (x+x;R e) (theoutput) = n X k =0 g k u i k + n X k =0 n X l=0 g k ;l u i k u i l + n X k =0 n X l=0 n X m=0 g k ;l;m u i k u i l u i m + +" i ; where" i
istheresidualerrorthathastobeminimised andwheregis relatedto gby
g=
g
x
Thedis retetransferfun tion(refequation4)isalsoknownasaNX(NonlinearwitheXogeneousinput)
sin eevenfor loworder polynomials, thenumberofunknown oeÆ ients anbe huge. Many solutions
have been developed for that purpose, see for example [1℄. The key problem here is the sele tion of
the model stru ture, i.e. the determination of kernels g that signi antly ontribute to the wave eld
dynami s. Thepro edure wehavefollowedisdetailedin [28℄.
Forexample,forR e=387,aleastsquares tyieldsthemodel oeÆ ients,whi hpermittore onstru t
thesignalsasitispresentedin gure16(Notethattheresidualsbetweentheoutputandthepredi tions
arebarelyvisibleon16-b):
y i = 0:7610u i 8 +0:0032u i 8 u 2 i 1 0:0105u 2 i 24 u i 3 +0:1022u i +0:0017u i 23 u i 20 u i 8 0:0115u i 14 u i 3 u i 1 +0:0103u i 23 u i 11 u i +" i
Themostsigni antkernelsare hosenamongallpossible ombinationsoflinear,quadrati and ubi
terms, with amemory (i.e. anumberof delaysn) equaling upto three wave eld periods. For allthe
Reynoldsnumbersofinterest(R egoingfrom 250to 505),nohigherordertermsareneededtoproperly
modelthewave elddynami s. Thisisanimportantresult,sin eitjusti esthetrun ationoftheVolterra
seriesat ubi terms,justifyingtheuseoftheCGLEasamodelof wavepropagation. Moreexpli itely,
letusemphazisethat thetrun ation oftheVolterraequation(2) at ubi termsisnotmade \apriori"
but omesfromtheexaminationoftheparti ularexperimentaldataunder onsideration: forthesedata,
higherorderkernelsintheseriesarenegligible. Asa onsequen e,thetrun atedVolterraequtionhasto
berelatedtothe ubi CGLE.
3.4 Relation to the omplex Ginzburg-Landau equation
Thehydrodynami eldde ningthewavepa ket,v(x;t), anbewrittenas:
v(x;t)=A(x;t)e
ik x i! t
+ : :; (5)
whereA(x;t)isa omplexfun tion,slowlyvaryingin spa eandtime, andwhi h obeystheCGLE:
0 A(x;t) t +V g A(x;t) x =A+ 2 0 (1+i 1 ) 2 A(x;t) x 2 l r (1+i 2 )jA(x;t)j 2 A(x;t); (6) where = R e R e R e ; V g
is thegroupvelo ity,
0 =a r 1 and 2 0
= givethe hara teristi timeandlengthoftheinstability.
Thegeneralsolutionofalinearstability hydrodynami problem anbeexpressedusingafrequen y
andawavenumberthatverifya omplexdispersionrelation!=!(k;R e). The oeÆ ients
0 ,V g , 0 , 1
arerelatedto theTaylorexpansionofthefrequen y!(k;R e)nearthe riti althresholdinthefollowing
way[31℄: 1 0 = iR e ! R e ; (7) V g = ! k ; (8) 2 0 (1+i 1 )= i 0 2 2 ! k 2 ; (9)
−5
0
5
u(t), y(t)
(a)
−5
0
5
model fit
(b)
−5
0
5
residuals
(c)
−5
0
5
linear term
(d)
−5
0
5
quadratic term
(e)
0
0.1
0.2
0.3
0.4
0.5
−5
0
5
cubic term
time [sec]
(f)
Figure16: Ex erpt of the wave eldamplitude atR e=387showing fromtop to bottom (with the
down-streamprobealwaysinbold): (a)themeasuredin-andoutput,(b)themeasuredoutputanditspredi tion,
( )the measuredoutput andthe residuals, (d)themeasuredoutputandthe linear onstituent ofthe
pre-di tion,(e) themeasuredoutput andthequadrati onstituentofthe predi tion, (f)the measuredoutput
andthe ubi onstituent ofthe predi tion. The measuredsignals are enteredandredu ed.
wherej
meansthatthepartial derivativesare al ulatedatthe riti alpointR e=R e
, k=k
. Ifthe
solutionA(x;t) isdevelopedin atemporalFourierseries
A(x;t)= X ^ A(x;)e it ;
itiseasytoverifythat theCGLEisequivalentto
0 = i ^ A(x;) ! k ^ A(x;) x + i 2 2 ! k 2 2 ^ A(x;) x 2 i ! R e (R e R e ) ^ A (x;) q X 1+2+3= ^ A(x; 1 ) ^ A (x; 2 ) ^ A (x; 3 ); (10) whereqisde ned by q= l r (1+i 2 ) 0 ; (11)
3.3): ^v(x;!) x = 1 (!)^v(x;!) (12) + X !1+!2=! 2 (! 1 ;! 2 )^v(x;! 1 )^v(x;! 2 ) + X !1+!2+!3=! 3 (! 1 ;! 2 ;! 3 )^v(x;! 1 )^v(x;! 2 )^v(x;! 3 ):
Nearthe riti althresholdR e=R e
, wewritev(x;t) intheform:
v(x;t)= A(x;t)e ik x i! t +B(x;t)e i2k x i2! t + : :; (13)
withA(x;t),B(x;t) slowlyvaryinginspa eandtime. Equation(12)givesthen,for! loseto !
: ^ A(x;! ! ) x = [ 1 (!) ik ℄ ^ A(x;! ! ) (14) + 2 X !1+!2=!; !1'2! ;!2' ! 2 (! 1 ;! 2 ) ^ B(x;! 1 2! ) ^ A (x; ! 2 ! ) + 3 X !1+!2+!3=!; ! 1 '! 2 ' ! 3 '! 3 (! 1 ;! 2 ;! 3 ) ^ A(x;! 1 ! ) ^ A(x;! 2 ! ) ^ A (x; ! 3 ! ); andfor! 1 '2! : ^ B(x;! 1 2! ) x = [ 1 (! 1 ) 2ik ℄ ^ B(x;! 1 2! ) (15) + X !3+!4=!1; !3'!4'! 2 (! 3 ;! 4 ) ^ A(x;! 3 ! ) ^ A (x;! 4 ! ):
Theadiabati approximationleadsthento:
^ B(x;! 1 2! )' 1 <( 1 (! 1 )) X ! 3 +! 4 =! 1 ; ! 3 '! 4 '! 2 (! 3 ;! 4 ) ^ A (x;! 3 ! ) ^ A (x;! 4 ! ): (16)
Substitutingthis expressionin(14),weget
^ A(x;! ! ) x = [ 1 (!) ik ℄ ^ A (x;! ! ) (17) + X !1+!2+!3=!; !1'!2' !3'! (! 1 ;! 2 ;! 3 ) ^ A (x;! 1 ! ) ^ A (x;! 2 ! ) ^ A (x; ! 3 ! );
whereisde ned by:
(! 1 ;! 2 ;! 3 )= 2 2 (! 1 ;! 2 ) 2 (! ! 3 ;! 3 ) <( 1 (! ! 3 )) +3 3 (! 1 ;! 2 ;! 3 ): (18)
2 ^ A(x;! ! ) x 2 = [ 1 (!) ik ℄ 2 ^ A (x;! ! ) + X ! 1 +! 2 +! 3 =!; !1'!2' !3'! ^ A (x;! 1 ! ) ^ A (x;! 2 ! ) ^ A (x; ! 3 ! )(! 1 ;! 2 ;! 3 ) [( 1 (!) ik )+( 1 (! 1 ) ik )+( 1 (! 2 ) ik )+( 1 (! 3 )+ik )℄: (19)
Again,wekeeponlytermsproportionaltoA
n
withn3. Wewillnowrepla etheexpressions(17)
and(19)for ^ A(x;) x and 2 ^ A(x;) x 2 intheCGLE: 0 = i ^ A(x;! ! ) (! ! )+i ! k [ 1 (!) ik ℄ ! R e (R e R e )+ 1 2 2 ! k 2 [ 1 (!) ik ℄ 2 + X !1+!2+!3=!; !1'!2' !3'! ^ A (x;! 1 ! ) ^ A (x;! 2 ! ) ^ A (x; ! 3 ! ) ! k (! 1 ;! 2 ;! 3 ) (20) + i 2 2 ! k 2 (! 1 ;! 2 ;! 3 )[ 1 (!)+ 1 (! 1 )+ 1 (! 2 )+ 1 (! 3 ) 2ik )℄ q :
Andidentifyingtheterms,it omes:
V g =i 1 ! 1 ; (21) 0 = R e V g 1 R e 1 ; (22) 2 0 (1+i 1 )= 0 2 V 3 g 2 1 ! 2 ; (23) l r (1+i 2 )=V g 0 2 2 (! ;! ) 2 (2! ; ! ) <( 1 (2! )) 3 3 (! ;! ; ! ) : (24)
The imaginarypartof thelinearkernel (!)is dire tly relatedto the wave elddispersionrelation.
Therealpartof (!)isrelatedto thelineargrowthrateandis displayedin gure17togetherwiththe
frequen yf
ofthefundamentalmode. It anbe he kedthatthezerogrowthrate urveisequivalentto
themarginalstability urvedisplayedin gure14. Positivevaluesofthegrowthrate on rmtheonset
oftheinstabilityaroundR e=300,ingoodagreementwiththeFourieranalysis.
UsingthevaluesofVolterrakernelsweget,nearthelinearthreshold
V g = 0:160:03ms 1 ; 0 = 15:13:2ms; 0 = 2:10:5mm; 1 = 0:470:28:
Atthis stage,wewere notableto obtainstatisti ally signi antvaluesforthe oeÆ ientsl
r
and
2
thatareasso iatedwiththenonlineartermoftheCGLEmodel. ExtrapolatingExtrapolatingformulas
(21)and(23)forR e6=R e
,we angetanestimationofthedependen eofthegroupvelo ityV
g
andthe
di usionlength
0
fromReynoldsnumber. Theresultingbehaviouris onsistentwiththeonepresented
−300
−200
−100
0
100
300
350
400
450
500
20
30
40
50
Re
f [Hz]
Figure 17: Real part of the linear Volterra kernel
1
for various frequen ies and Reynolds numbers.
Superimposedonitisthe frequen y f
ofthe fundamental.
between the probes dire tion and the radial one, we nd
0
= 1:4 mm whi h is lose to the 1.2 mm
obtainedpreviously. Moreover,thetransitionfrom onve tivetoabsoluteinstability analsobe he ked,
usingthe riterionobtainedontheCGLEbyMoonet al. in 1983[31℄:
abs = R e R e R e V 2 g 2 0 4 2 0 (1+ 2 1 ) : (25) abs
getspositiveforR e=380,whi hisinagreementLeGal[32℄whereatransitiontowardsabetter
spatio-temporalorganisationofwaveswasdete tedforthisvalueoftheReynoldsnumber. Notethatthis
thresholdislowertowhatwaspredi tedbyLingwood[33℄andthusdoesnot orrespondtothetransition
towardsturbulen ewhi hisobservedforaReynoldsnumberaround510.
Clearly,theseresultsarestillopentoimprovements. TheNXnonlinearmodelwehaveusedisstati in
thesensethatit annotgeneratesustainedos illationsiftheinputde aystozero. Asigni antredu tion
of theresiduals and probablya better des riptionof the topologi al properties of the system ould be
a hievedbyusingamoregeneral lassofmodels, alledNonlinearAuto-RegressiveMovingAveragewith
eXogeneousinput(NARMAX).Thiswillbetheobje tofaforth omingpubli ation.
4 Con lusion
Inthese studies, wehave shown that measurements oming from image analysis orfrom anemometri
signalsand des ribingspatio-temporal haoti propagation ofwaves, anbe des ribed bytheComplex
Ginzburg-LandauEquation.
In the rst experiment whi h is devoted to oupledwakes,our on ern wasessentially to lterout
the noise that pollutes the video images. Our method is based on the Bi-Orthogonal De omposition
(orProperOrthogonalDe omposition)and leadsto ageneralizedform ofthedispersionrelationofthe
waves. Thenoiseistheneasilyremovedasitis on entratedonmodeshavingahighindex: thesemodes
arein fa t poorly orrelated. Therefore, theinversionproblemis solvedand themodel oeÆ ients an
beextra tedfromtheexperimentaldata. Theirvaluesarein agreementwithknownpropertiesofwakes
andthe re onstru tionof haoti spa e-timesignalsis thenpossible. Predi tionsof thedynami s have
alsobeenmade by theuse ofthe CGLE but fordi erent oupling oeÆ ients. These predi tions have
beenfavourably omparedwithexperiments.
Ourse ondanalysisisdevotedtothepropagationofdestabilizingwavesinarotatingboundarylayer.
200
300
400
500
0
0.2
0.4
0.6
0.8
Re
amplitude
V
g
[m/s]
ξ
0
[cm]
Figure18: Group velo ity V
g
proje ted along the probeseparation ve tor, and di usionlength
0
. Error
barsrepresentone standarddeviation.
200
300
400
500
−0.5
0
0.5
Re
η
Re
c
Figure 19: The owbe omesabsolutelyunstablewhen be omespositive.
OuranalysisisbasedonVolterraseriesthatallowthe al ulationofthelinearpropertiesofthewavetrains
(their non linear ones are urrentlyunder study). An analyti al al ulation that uses the elimination
of fast harmoni modes made the onne tion between the Volterrakernels and the Ginzburg-Landau
model. It isthen possibleto dedu ethe main properties of the wavepropagation: growth rate, group
velo ityand oheren elength. Moreoveratransitionfroma onve tiveinstabilitytoanabsoluteonehas
been dis overedandexplainsin fa t thegrowing oheren e of thewavepatternthat hasbeen observed
previouslybeforeits naltransitionto turbulen e.
To on lude,letusremarkthatasbothmethodsarebasedongeneralprin iples,theyarenotrestri ted
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