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Transitions 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 reasing (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 produ 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)apping motion: (cid:28)rst, the transition from the well-known A knowledgments Bénard-von Kármán (BvK) wake to the reversed vortex 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. 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