Table Of ContentTransitions in the wake of a (cid:29)apping foil
∗
Ramiro Godoy-Diana, Jean-Lu
Aider, and José Eduardo Wesfreid
Physique et Mé
anique des Milieux Hétérogènes
(PMMH) UMR 7636 CNRS, ESPCI, Paris 6, Paris 7
10, rue Vauquelin, F-75231 Paris Cedex 5, Fran
e
8 Westudyexperimentallythevortexstreetsprodu
edbya(cid:29)appingfoilinahydrodynami
tunnel,
0 using2DParti
leImageVelo
imetry(PIV).Ananalysisintermsofa(cid:29)appingfrequen
y-amplitude
0 phase spa
e allows to identify: 1) the transition from the well-known Bénard-von Kármán (BvK)
2 wake to the reverse BvK vortex street that
hara
terizes propulsive wakes, and 2) the symmetry
n breaking of this reverse BvK pattern giving rise to an asymmetri
wake. We also show that the
a transition from a BvK wake to a reverse BvK wake pre
edes the a
tual drag-thrust transition and
J we dis
uss thesigni(cid:28)
an
e of thepresent results in the analysis of (cid:29)apping systemsin nature.
5
2
I. INTRODUCTION in most (cid:29)apping systems, a quasi-two-dimensional pi
-
] ture of the wake often
ontains key dynami
al elements,
n
su
h as the
reation and organization of vorti
ity, that
y Control of vorti
es produ
ed by (cid:29)apping wings or (cid:28)ns
are
ru
ial in every system involving (cid:29)apping foils as a
d togeneratepropulsivefor
esistheeverydaytaskofbirds,
means of produ
ing propulsive for
es [16, 24, 25, 26℄.
- inse
ts and swimming animals. Many studies of a
tual
u Thegoalofthepresentworkistoexplorethesequasi-2D
fl (cid:29)apping extremities have been driven by the need for me
hanisms using a simple but very well
ontrolled ex-
a better understanding of this form of propulsion with
s. periment, in order to provide a de(cid:28)nitive framework for
the ultimate goal of enhan
ing man-made propulsivede-
c the analysis of the transitions observed in the wake of
i vi
es [see extensive reviews in 1, 2, 3, 4, 5℄. A wake
s (cid:29)apping foils.
vortex street with the sign of vorti
ity of ea
h vortex
y
h reversed with respe
t to the typi
al Bénard-von Kár-
p mán(BvK)vortexstreetisan ubiquitousfeaturerelated
II. EXPERIMENTAL SETUP AND
[ to propulsive (cid:29)apping motion. Su
h reverse BvK vor- PARAMETERS
tex streets have not only been observed in the wakes
2
of swimming (cid:28)sh [see e.g. 6, 7℄ but also studied in de-
v The setup
onsists of a pit
hing foil of aspe
t ratio
9 tail through laboratory experiments with (cid:29)apping foils
4 pla
ed in a hydrodynami
tunnel (see Fig. 1). The
1 [8, 9, 10, 11, 12, 13, 14, 15℄ and numeri
al simulations
experimental
ontrol parameters are the (cid:29)ow velo
ity in
2 [11, 16, 17, 18, 19, 20℄. The mean velo
ity pro(cid:28)le asso-
2
iated with these thrust-produ
ing reverse BvK streets
0
has the form of a jet and the stability properties of this U
7 m
0 average jet (cid:29)ow seem
losely related to the propulsive 5 m A
s/ e(cid:30)
ien
y of the (cid:29)apping foil [21℄. D =
c Flapping systems, either natural orman-made, areof-
i tendis
ussedintermsofasingleparameter,theStrouhal c = 23 mm
s
Xiv:phy fSshrnhouarwaAovmwnebid=snsehttareoh,fmrwadvAptnea/l(cid:28)loitUtn0bhue..sad2deter<EapvAsxreSodptdprheA
iurevriluim<psdiirvseeo0eednd.t4eSubs(cid:30).t
yrwtD
otioiuaehtfhnhteata
h
yl(cid:29)
reanoupp(cid:29)umiespamaippinknbipgslgeeiwrndsspfgiobtefihfoyelrsidren[q2a[U8au1,we,,nni22ia.d
32erye-.℄℄ m flush−leveled reservoirthrough honeycomb (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) stfelappppeinr gm footiolr measurement plane
ar rangeof(cid:29)ying and swimming animals also lie on this in- fro
terval,suggestingthat,notsurprisingly,naturalsele
tion
has tuned animals for high propulsive e(cid:30)
ien
y. How- z y
x through flow meter
ever, less
lear are the physi
al reasons that determine to flush−leveled reservoir
ertain(cid:29)apping
on(cid:28)gurations,withtheirasso
iatedvor-
tex signature in the wake, to be the optimal
hoi
es for
FIG. 1: S
hemati
views of the foil pro(cid:28)le and the hydro-
c
e(cid:30)
ient thrust generation.
dynami
tunnel. The foil
hord is 23mm and i×ts span is
Althoughthree-dimensionalityplaysanimportantrole 100mm whi
h
overs the whole height of the 100 150mm
se
tionofthetunnel. Thefoilpro(cid:28)leissymmetri
,openingat
D=5
theleading edge as a semi
ir
le of diameter mmwhi
h
is also the maximum foil width. The pit
hing axis is driven
∗
bya stepper motor.
Ele
troni
address: ramiropmmh.esp
i.fr
2
U f
thetunnel ,thefoilos
illationfrequen
y andpeak-to- with shedding frequen
y lo
ked-in to the (cid:29)apping fre-
A Re = UD/ν
peak amplitude . The Reynolds number , quen
y [see also 27, 28, 29℄. In parti
ular, the vorti
ity
D ν
where is the foil width (or thi
kness) and is the engendered at the boundary layers on ea
h side of the
kinemati
vis
osity, was set to 255 in the experiments foilisrolledupinvorti
esthat,intheirdownstreamevo-
reported here. This
orresponds to 1173 for the
hord- lution, stay on the sameside ofthe symmetryline of the
Rec = Uc/ν
based Reynolds number , whi
h is an inter- wake, de(cid:28)ned by the axis of the foil when parallel to the
100< Rec < 10000
mediate value in the range of `inter- free stream velo
ity. A
ordingly, the mean (cid:29)ow in this
mediate Reynolds numbers' that
hara
terize (cid:29)apping-
ase has a typi
al wake pro(cid:28)le with slower speed behind
based propulsion in nature [5℄. In addition, we de(cid:28)ne the obsta
le. In
reasing the (cid:29)apping amplitude allows
AD
a dimensionless (cid:29)apping amplitude and a Strouhal us to get a very
lear view of the me
hanism of rever-
Sr AD =0.71
number as sal of vortex position: for amplitude vorti
es
of alternating signs align on the symmetry line of the
wake (a 2S-type wake, using the nomen
lature of [30℄),
AD =A/D and Sr=fD/U . AD = 1.07
(1) whereas for , the vorti
es shed on one side of
the(cid:29)aporganizethemselvesinthewakeontheotherside
Although natural vortex shedding exists for the steady ofthesymmetryline
onstitutingthereverseBvKvortex
(cid:29)ap at this Reynolds number even at zero-angle of in- street. The mean (cid:29)ow pi
ture for this transition shows
iden
e [35℄, no mode
ompetition is observed in the thatthevelo
ityde(cid:28)
itinthewakepro(cid:28)leisalmost
om-
strongly for
ed (cid:29)apping regimes studied here. The (cid:29)ap- pletelyerasedwhenvorti
esarealignedinthesymmetry
Sr AD = 0.71 AD = 1.07
pingfrequen
yusedto de(cid:28)ne isthus equivalenttothe line ( ) prior to the apparition (at
AD = 1.77
main vortex shedding frequen
y. In addition, we have and more
learly at ) of the jet pro(cid:28)le that is
D
hosentousea(cid:28)xedlengths
ale( )insteadoftheusual typi
ally asso
iated to the reverse BvK street [see also
A Sr
peak-to-peakamplitude( )inthede(cid:28)nitionof sothat re
ent numeri
al simulations by 18℄.
ea
h degree of freedom of the (cid:29)apping motion (isSrr,eAprDe)- Further in
rease in the (cid:29)apping amplitude triggers a
sented in one non-dimensional parameter. The symmetry breaking in the reverse BvK wake (a feature
phase spa
ethus de(cid:28)ned allowsto give a
learpi
ture of previously observed for foils performing heaving motion
the di(cid:27)erent observed regimes. Measurements were per- [11, 13, 31℄. A strong dipolar stru
ture that propagates
formed using 2D Parti
le Image Velo
imetry (PIV) on obliquely to one side of the symmetry line of the wake
the horizontalmid-plane of the (cid:29)ap. PIV was16p0e0rf×or1m2e0d0 is formed in ea
h (cid:29)apping
y
le, while a mu
h weaker
using aLaVision system with an ImagerPro single vortex is shed on the other side. In these asym-
12-bit
harge-
oup∼le1d5devi
e (CCD)
amera re
ording metri
wakes, the quasi-two-dimensional nature of the
pairsofimagesat HzandatworodNd:YAG(15mJ) (cid:29)ow is rapidly lost and the
oheren
e of vorti
al stru
-
pulsed laser. L×aser sheet width was about 1mm in the turesin the horizontalmid-planewhere the observations
whole 100mm 80mm imtaging region. The time lapse areperformeddoes not prevail formorethan two
y
les.
between the two frames (d ) was set to 12ms. However, the symmetry features of these wakes do seem
tobede(cid:28)nedbyaquasi-two-dimensionalme
hanismthat
depends on the initial
onditions: the (cid:28)rst dipole that is
III. THE WAKE OF THE FLAPPING FOIL formed entrains (cid:29)uid behind it, de(cid:29)e
ting the mean (cid:29)ow
in the wake and for
ing the subsequent dipoles to fol-
(Sr,AD)
For a (cid:28)xed Reynolds number, a point in the low the same path. This me
hanism of dipole formation
parameter spa
e de(cid:28)nes thus the (cid:29)apping
on(cid:28)guration resulting in a de(cid:29)e
ted wake has also been observed in
and
orresponds to a given vorti
ity signature in the soap (cid:28)lm experiments studying the wake of os
illating
wake of the foil. A parametri
study as a fun
tion of
ylinders [32℄, supporting the idea of a mainly 2D phe-
the Strouhal number and the non-dimensional (cid:29)apping nomenon. In the mean (cid:29)ow pi
ture, the asymmetry of
amplitude allows to
hara
terize the di(cid:27)erent regimes in the wake
an be
learly seen as a de(cid:29)e
tion of the peak
(Sr,AD)
theaforementioned phasespa
eandtopindown in the jet pro(cid:28)le.
the transition zones.
(Sr,AD)
B. The phase spa
e
A. Symmetry properties
The experimental phase diagram is shown in Fig. 3,
In Fig. 2, we show snapshots of spanwise vorti
ity wheredi(cid:27)erentsymbolsrepresentdi(cid:27)erenttypesofwak(cid:3)e.
(left
olumn) as well as averagehorizontalvelo
ity (cid:28)elds Experiments where a BvK-type wake was observed (
AD
(right
olumn), where the (cid:29)apping amplitude is var- symbols) o
upy primarily the lower left region of the
iedfora(cid:28)xedvalueoftheStrouhalnumber. The(cid:28)rstrow plotbutextendtohigherStrouhalnumbersforthelowest
AD =0.36
( )isthetypi
al
aseoflow-amplitude(cid:29)apping amplitude tested. The transition from the BvK regime
whi
h produ
es a for
ed wake with features resembling to the reverse BvK regime (blue line in Fig. 3)
an be
(Sr,AD)
a natural Bénard-von Kármán (BvK) vortex street, but univo
ally de(cid:28)ned in the spa
e when the posi-
3
AD =0.36 50 0.11
40 0.10
30 0.09
20 0.08
10 0.07
0 0.06
0.05
-10
0.04
-20
0.03
-30
0.02
-40
0.01
-50
AD =0.71 50 0.11
40 0.10
30 0.09
20 0.08
10 0.07
0 0.06
0.05
-10
0.04
-20
0.03
-30
0.02
-40
0.01
-50
AD =1.07 50 0.11
40 0.10
30 0.09
20 0.08
10 0.07
0 0.06
0.05
-10
0.04
-20
0.03
-30
0.02
-40
0.01
-50
A =1.77 50 0.11
D
40 0.10
30 0.09
20 0.08
10 0.07
0 0.06
0.05
-10
0.04
-20
0.03
-30
0.02
-40
0.01
-50
A =2.80 50 0.11
D
40 0.10
30 0.09
20 0.08
10 0.07
0 0.06
0.05
-10
0.04
-20
0.03
-30
0.02
-40
0.01
-50
s−1
FIG. 2: (Colorm) Isn−s1tantaneous spanwise vorti
ity (cid:28)elds (left
olumnS,r = )0.a2n2d meaRne(cid:29)o=w2(5t5ime averaged horizontal velo
ity,
right
olumn, ) for (cid:28)xed Strouhal and Reynolds numbers ( and ) and from top to bottom, for
AD = 0.36,0.71,1.07,1.77, D D
and 2.8. The (cid:28)eld of view (pla
ed at mid-height of the foil)
overs from -2 to 20 on the
x D D y
horizontal (streamwise dire
tion ) and -8 to 8 in the verti
al (
rossstream dire
tion ) where the origin is de(cid:28)ned at the
trailing edge of the (cid:29)ap at zero angle of atta
k.
4
CD/CD0
surfa
e obtained by interpolation of the mea-
(Sr,AD)
sured points in the spa
e are plotted as labeled
2
iso-
ontoursin Fig. 4. The drag
oe(cid:30)
ient
Sr =0.3
A
FD
1.5 CD = 12ρU02D , (3)
Sr =0.4
A
normalizedbyitsvalueforthenon-(cid:29)appingfoilatzero
D CD0 (Sr,AD)
A 1 Sr =0.2 angle of attaC
Dk , allows to pin down the
A
urvewhere
hangessign,markingthetransitionbe-
tween drag and propulsive regimes (heavy bla
k line in
Fig. 4). A
omparison of this zero-drag
urve with the
0.5 transition from a BvK vortex street to a reverse BvK
pattern in the wake (gray line) shows that, for in
reas-
ing (cid:29)apping amplitude or frequen
y, the reversal of the
vortexstreethappensbeforethea
tualdrag-thrusttran-
0
0 0.1 0.2 0.3 0.4 0.5 sition in almost all the parameter range studied here.
Sr (Sr,AD)
This shift shows that a region in the plane ex-
istswhereareversedBvKpattern
anbeobservedinthe
AD Sr Re=255 wake of the (cid:29)apping foil, but the relative thrust engen-
FIG. 3: (Color) v(cid:3)s. map for (cid:4) . Experimental
dered by the (cid:29)apping motion is not yet enough to over-
points a+re labeled as : BvK△wake; : aligned vorti
es (2S Sr > 0.4
ome the total mean drag. For , the two lines
wake); : reverseBvKwake; : de(cid:29)e
tedreverseBvKstreet
resulting in an asymmetri
wake. Blue line: transition be-
rossandtheestimateofthedrag-thrusttransitionstays
tween BvK and reverse BvK. Green line: transition between a
tually slightly below the observed BvK-reverse BvK
reverse BvK and thSerAas=ym0.m3±etr0i.
1regime. The shSadrAed area transition threshold. A possible explanation for this is
orresponds to the SinrAte=rvaSlr,×whAeDre is the that the quasi-2D hypothesis underlying the drag
oe(cid:30)-
amplitude based Strouhal number (see text).
ient
al
ulation degrades faster the higher the Strouhal
number, so that 3D e(cid:27)e
ts might be
ooperating to pro-
du
e this result that is unexpe
ted from a 2D point of
tion of vorti
itymaxima, at any given time and horizon- view.
tal position,
ross the symmetry axis of the wake. We Itshouldbenotedthattheapproximationsunderlying
remark tAhDat≈th0e.6transSirti>on0l.i4ne tends to an asymptoti
the momentumbalan
e usedto obtainEq. (2)makeit a
valueof for sothatathresholdampli-
rude estimate forthe mean dragon the(cid:29)appingfoil. In
tude value exists for the produ
tion of a reverse street. parti
ular, as soon as the (cid:29)ow
eases to be parallel the
+
The regionwherethe reverseBvKregimeisobserved( momentumbalan
e
hangesandEq. (2)nolongerholds.
symbols) is bounded on t△he other side by the transition The mean drag in the regimes where the average wake
to asymmetri
regimes ( symbols) represented by the is asymmetri
and a mean lift for
e is likely to be non-
green line in Fig. 3. negligible, will thus be poorly estimated by Eq. (2). We
nonetheless verify a posteriori that the expe
ted drag-
thrust transition does not overlap the region of the pa-
C. The drag(cid:21)thrust transition rameter spa
e where asymmetri
wakes develop, so that
the predi
tion should be reasonable. The sensitivity of
x
the drag
oe(cid:30)
ient
al
ulation to the -position of the
Using the mean velo
ity (cid:28)elds to perform a standard
wake pro(cid:28)le, whi
h de(cid:28)nes the downstream boundary of
momentumbalan
eina
ontrolvolumeen
losingthefoil
the
ontrol volume, was tested performing the momen-
(see e.g. [33℄, pp. 349-351) we
an obtain an indire
t
tum balan
e at di(cid:27)erent horizontal stations in the wake.
estimate of the mean drag. It is given by CD =0
Thus,thegray`errorband'aroundthe transition
line in (cid:28)gure 4 represents the (cid:29)u
tuation of the transi-
CD
FD =ρU0Z [U0−u(y)]dy , (2) tion line when the
al
ulatiXon±of4D is performed using
a wake pro(cid:28)le measxured at ≈4(i0.%e. representing an
overall span in the dire
tion of of the length of
ρ U0
where is the water density, is the free stream ve- the PIV measurement window)[36℄.
lo
ity at the
enter of the tunnel upstream of the (cid:29)ap
u(y)
and is a velo
ity pro(cid:28)le measuredXin t≈he1w2Dake. The
position of this wake pro(cid:28)le is set to down- IV. CONCLUSIONS
stream from the trailing edge, where the wake
an still
bereasonably
onsideredquasi-2Danda
ross-wakepres- The experimentalstudyofthe wakevorti
esprodu
ed
sure (cid:29)u
tuations
an be negle
ted. A drag
oe(cid:30)
ient by a high-aspe
t-ratio pit
hing foil presented in this pa-
5
dimensional nature of the wake dynami
s resulting from
the(cid:28)nitespanofthefoilinthehydrodynami
tunnel,the
2
quasi-two-dimensional pi
ture studied here by fo
using
on the near wake
aptures the essential of the vorti
ity
−0.6
dynami
s be
ause the main vortex shedding is indu
ed
1.5 −0.3 when the foil motion
hanges dire
tion [see also 24, 34,
where 2D theory and simulations show good agreement
0.3 −0.6
with experimental studies on (cid:29)apping wings℄.
−0.3
D
A In
on
lusion, we
an
omeba
k to the (cid:29)apping-based
1
0.6 0.3 propulsiveregimesen
ounteredin natureS,rwAh=er0e.(cid:28)3s±h-0li.k1e
swimming and (cid:29)apping (cid:29)ight o
ur for .
SrA =
ISnr×terAmDs of the parameters used in this paper,
0.5 , so that the previous interval represents a re-
(Sr,AD)
0.3 gion bounded by hyperboles in the phase spa
e
0.6
0.6 (shadedareainFig. 3). Our(cid:28)rstremarkisthatthedrag-
thrusttransitionshownin Fig. 4
ompareswell with the
0 SrA = 0.3
0 0.1 0.2 0.3 0.4 0.5
urve (dashed line) that
hara
terizes animal
Sr propulsion. Within the limits imposed by the quasi-2D
pi
ture analyzed in the present work, this suggests that
CD/CD0 the
on
lusions obtained here should be relevant as in-
FIG. 4: Contoursof a meandrag
oe(cid:30)
ient surfa
e terpretation tools in the study of natural systems. In
estimated using a momentum-balan
e approa
h (see text).
The bla
k line
orresponds to CD = 0 where the estimated aSdrAdit=io0n.,3F±ig0..13 also shows that the region de(cid:28)ned by
overlaps not only with the reverse BvK
drag-thrusttransitiono
urs. Theshadedarearepresentsthe
CD =0 regime, but also with the asymmetri
regime. We may
estimatederrorforthe
urveduetosensitivityonthe
onje
ture that animals using (cid:29)apping-based propulsion
hoi
e of the
ontrol volume. The gray line is the transition
must either exploit the
reation of asymmetri
wakes as
from BvK to reverse BvK (blue line in Fig. 3). The dashed
line
orresponds to SrA=0.3. partof theirmanoeuveringte
hniquesor, when
ruising,
avoid (cid:29)apping regimes where the symmetry breaking of
the reverseBvK street will o
ur.
per gives eviden
e
hara
terizingtwokey dynami
alfea-
turesrelevanttowakevortexsystemsengenderedby(cid:29)ap-
ping motion: (cid:28)rst, the transition from the well-known
A
knowledgments
Bénard-von Kármán (BvK) wake to the reversed vor-
tex street that signals propulsive wakes, and se
ond, the
symmetry breaking of this reverse BvK pattern giving We thank warmly D. Pradal for his invaluable help in
riseto an asymmetri
wake. Albeit the inherently-three- the design and
onstru
tion of the experimental setup.
[1℄ F. E. Fish and G. V. Lauder, Annu. Rev. Fluid Me
h. [11℄ K.D.Jones,C.M.Dohring,andM.F.Platzer,AIAAJ.
38, 193 (2006). 36, 1240 (1998).
[2℄ K. V. Rozhdestvensky and V. A. Ryzhov, Progr. [12℄ M. M. Koo
hesfahani, AIAAJ. 27, 1200 (1989).
Aerospa
e S
i. 39, 585 (2003). [13℄ J.C.S.LaiandM.F.Platzer,AIAAJ.37,1529(1999).
[3℄ M.S.Triantafyllou,A.H.Te
het,andF.S.Hover,IEEE [14℄ K.Parker,K.D.vonEllenrieder,andJ.Soria,Exp.Flu-
J. O
ean. Eng. 29, 585 (2004). ids 39, 281 (2005).
[4℄ M. S. Triantafyllou, G. S. Triantafyllou, and D. K. P. [15℄ N. Vandenberghe, J. Zhang, and S. Childress, J. Fluid
Yue, Annu.Rev. Fluid Me
h. 32, 33 (2000). Me
h. 506, 147 (2004).
[5℄ Z. J. Wang, Annu.Rev. Fluid Me
h. 37, 183 (2005). [16℄ S. Alben and M. Shelley, Pro
. National A
. S
i. (USA)
[6℄ E.G.Dru
kerandG.V.Lauder,J.Exp.Biol.204,2943 102, 11163 (2005).
(2001). [17℄ P. Blondeaux, F. Fornarelli, L. Guglielmini, M. S. Tri-
[7℄ M. J. Wolfgang, J. M. Anderson, M. A. Grosenbaugh, antafyllou, and R. Verzi
o, Phys. Fluids 17, 113601
D.K.P.Yue,andM.S.Triantafyllou,J.Exp.Biol.202, (2005).
2302 (1999). [18℄ H. Dong, R. Mittal, and F. M. Najjar, J. Fluid Me
h.
[8℄ J. M. Anderson, K. Streitlien, D. S. Barret, and M. S. 566, 309 (2006).
Triantafyllou, J. Fluid Me
h. 360, 41 (1998). [19℄ L.Guglielmini,P.Blondeaux,andG.Vittori,O
eanEng.
[9℄ J. H. J. Bu
hholz and A. J. Smits, J. Fluid Me
h. 546, 31, 883 (2004).
433 (2006). [20℄ Q. Zhu, M. J. Wolfgang, D. K. P. Yue, and M. S. Tri-
[10℄ R. Gopalkrishnan, M. S. Triantafyllou, G. S. Triantafyl- antafyllou, J. Fluid Me
h. 468, 1 (2002).
lou, and D. Barrett, J. Fluid Me
h. 274, 1 (1994). [21℄ M.S.Triantafyllou,G.S.Triantafyllou,andR.Gopalkr-
6
ishnan, Phys.Fluids A 3, 2835 (1991). [30℄ C.H.K.WilliamsonandA.Roshko, J.FluidsStru
t.2,
[22℄ D. A. Read, F. S. Hover, and M. S. Triantafyllou, J. 355 (1988).
Fluids Stru
t.17, 163 (2003). [31℄ G. C. Lewinand H. Haj-Hariri, J. Fluid Me
h. 492, 339
[23℄ G.K.Taylor,R.L.Nudds,andA.L.R.Thomas,Nature (2003).
425, 707 (2003). [32℄ Y. Couder and C. Basdevant, J. Fluid Me
h. 173, 225
[24℄ F. O.Minotti, Phys.Rev. E66, 051907 (2002). (1986).
[25℄ B. Protas and J. E. Wesfreid, C.R. Mé
anique 331, 49 [33℄ G. K. Bat
helor, An introdu
tion to (cid:29)uid dynami
s
(2003). (Cambridge UniversityPress, 1967).
[26℄ Z. J. Wang, Phys.Rev. Lett. 85, 2216 (2000). [34℄ Z. J. Wang, J. M. Bir
h, and M. H. Di
kinson, J. Exp.
[27℄ B.Thiria,S.Goujon-Durand,andJ.E.Wesfreid,J.Fluid Biol. 207, 449 (2004).
Me
h. 560, 123 (2006). [35℄ The
riti
al Reynolds number for the (cid:29)ap is approxi-
[28℄ B. Thiria and J. E. Wesfreid, J. Fluid Me
h. 579, 137 mately 140.
CD
(2007). [36℄ A dis
ussion of the sensitivity of the
al
ulation to
[29℄ M. Vial, L. Bellon, and R. H. Hernández, Exp. Fluids the
hoi
eof
ontrolvolumeisavailableassupplementary
37, 168 (2004). material.
