VOLUME 81, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R S 13 JULY 1998

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From Flutter to Tumble: Inertial Drag and Froude Similarity in Falling Paper

Andrew Belmonte,1 Hagai Eisenberg,2 and Elisha Moses21Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 1526

2Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Is(Received 28 October 1997)

In an experiment on thin flat strips falling through a fluid in a vertical cell, two fundamentmotions are observed: side-to-side oscillation (flutter) and end-over-end rotation (tumble). At hReynolds number, the dimensionless similarity variable describing the dynamics is the Froude numFr, being the ratio of characteristic times for downward motion and pendular oscillations. Ttransition from flutter to tumble occurs at Frc ­ 0.67 6 0.05. We propose a phenomenological modelincluding inertial drag and lift which reproduces this motion, and directly yields the Froude similari[S0031-9007(98)06387-X]

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Not all falling objects travel straight downwards: a piecof paper dropped from the table or a leaf as it flutteto the ground are two common examples. This is dprimarily to the coupling of forward motion to lateraoscillations by the surrounding fluid [1–3]. The resultindynamics are relevant to systems ranging from aircrstability [3,4] to rising bubbles [5]. The problem of fallingpaper is rich in hydrodynamic effects, including lift, dragvortex shedding, and stall, and has a history dating bato Maxwell, who noticed in 1854 that the combination ogravity and lift results in a torque [6]. Despite subsequetheoretical work by Helmholtz, Kelvin, and others [1,3]including a formal irrotational analysis due to Kirchhof[1], a solution of the complete problem is impracticaand the correct description remains undetermined. Textensive theoretical background has not been matchexperimentally. Previous falling object experiments habeen fully three dimensional [7–11], reporting flutteringgyrating, or tumbling, but few quantitative specifics of thdynamics were given. While it is clear that vortices ashed simultaneous to the oscillations, it is unknown if thvortices result from or cause the oscillations [12,13].

More recently, phenomenological models have succefully elicited qualitative aspects of the motion [14–18], athough with some debate on the role of viscosity and texistence of chaotic tumbling. However, these modelsther ignore the vortices by treating the flow as irrotationaor assume a viscous drag, or omit drag altogether. Sprisingly, in almost 150 years, there is no experimentaverified description for this rather basic phenomenon.

We present here a simple laboratory experiment on flstrips dropped in a quasi-2D geometry [19]. Our expemental cell is a narrow fluid-filled glass aquarium, 60 cin height and length and 0.8 cm wide, into which thistrips made of various materials are dropped. The strare constructed of plastic, brass, or steel, each with abthe same thickness (0.1–0.2 cm) and width (0.75 cm), bwith a range of lengthsL. Using different materials al-lows us to varyL and the massM independently, from

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1 to 32 cm and 1 to 7 g, respectively. We use three dferent fluids: water, a glycerolywater mixture (40:60 byweight), and petroleum ether, with densitiesr of 1.00,1.10, and 0.67 gycm3, and viscosities of 0.010, 0.036, an0.003 P, respectively. Two thin stabilizing rings attacheto each side (typically of 1 mm diameter steel wire; sinset of Fig. 2) keep the strip from turning over in ththird (narrow) dimension. The ring radiusR is chosenso that its main effect is to increase the total mass [2Grazing or sticking to the wall is rarely seen due to lubrication layer effects. The motion of the strip is imagewith a video camera, and recorded for later analysis.

We observe two fundamental motions: side-to-sioscillation (flutter) and end-over-end rotation (tumbleshown in Fig. 1. By digitizing every frame, we obtainthe strip position vs time; the stabilizing rings are blacand therefore not visible. In Fig. 2a we show the timvariation of the angleQ that the strip makes with thehorizontal (see inset); it ranges between6Qmax. Thevertical velocityVy reaches its maximal downward valuas Q approachesQmax (Fig. 2b), corresponding to the

FIG. 1. Collage of consecutive video fields (Dt ­ 0.02 sec)for strips falling in water: (a)L ­ 5.1 cm, M ­ 2.9 g (Fr ­0.37); (b) L ­ 4.1 cm, M ­ 2.7 g (Fr ­ 0.45); (c) L ­2.0 cm, M ­ 1.4 g (Fr ­ 0.65); (d) L ­ 1.0 cm, M ­ 0.7 g(Fr ­ 0.89).

© 1998 The American Physical Society 345

VOLUME 81, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R S 13 JULY 1998

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FIG. 2. Measured dynamics of a falling strip (M ­ 2.0 g,

L ­ 3.0 cm, Fr ­ 0.53). (a) Angle Q vs time. The insetshows the strip, the stabilizing rings and the angleQ. (b) Ver-tical velocity Vy vs Q. (c) Horizontal velocityVx vs Q.

minimal drag of a small “angle of attack” [4]. AsQdecreases so doesVy, which reaches its minimum notat Q ­ 0, when the strip faces broadside into the flowbut at Q . 20±. A similar effect was observed in thepioneering work of Eiffel [21]. The butterfly shape inFig. 2b indicates thatVy oscillates at half the period ofQ.The horizontal velocityVx oscillates around zero with thesame period asQ (Fig. 2c); atQ . 20± Vx is maximum.

This description applies to all strips withQmax ,

90±. Beyond this value the strip tumbles end over en(Fig. 1d). We therefore useQmax as the order parameterfor this transition. We then find that the control parametis given by the ratio of characteristic times for thdownward fall and the rotational flutter. For large enougvelocities, the terminal downward velocity is determineby gravity and the “form” or inertial dragFd ­ CrV 2S[21], whereS is the cross-sectional area of the object,V isthe velocity, andC is a constant of the order of 1. For oustrips the terminal velocity scale isU0 ­ sM 0gyrLwd1y2,where M0 is the buoyancy corrected mass andw isthe width of the strip [22]. The downward motion ischaracterized by the time scalety ­ LyU0. The pendularmotion of fluttering is characterized by the time scale ofbuoyant pendulum:tp ­ sMLyM 0gd1y2. The ratio of thetwo time scales is the Froude number

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In our experiments Fr varies from 0.02 to 1.0. A plot osinsQmaxd vs Fr is shown in Fig. 3a; all data collapse onta single curve given by sinsQmaxd ­ s1.50 6 0.03dFr.The three fluids used differ in viscosity (factor of 10) andensity (factor of 0.6), but only the density as includein Eq. (1) could account for the changes in sinsQmaxd.The tumbling transition [sinsQmaxd . 1] occurs at a valueof Frc ­ 0.67 6 0.05. This corresponds to a pendulumdriven at resonance:ty . tp (Fr . 1) [23].

In general, the motion of the strips should also depeon the Reynolds number Re­ rULyh, which for ourexperiments ranges from about3 3 103 to 4 3 104. Ourscaling in terms of Fr, however, shows that the flutterinis independent of viscosity. This result is in agreemewith previous observations that, for high enough Re, tmotion is Re independent [7,13].

The role that Fr plays in our analysis is common tsystems which are controlled by the competition of a speof propagation and a dissipative speed [24], and in our cait provides a physically relevant equivalent to the scalinfirst implied in [7]. It was first defined by Froude in 1874as the similarity variable for surface ships [1,25,26], beinthe ratio of ship to wave speed. A more surprising examp

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VOLUME 81, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R S 13 JULY 1998

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of Fr similarity is the pendulum model of walking, whichaccounts for the exchange between gravitational potenand swinging kinetic energy of the leg during a forwarstride [24]. The comparative analysis of gait for a widvariety of animals [26,27] shows that the transition fromwalking to running in bipeds, or to trotting in quadrupedoccurs at Frc . 0.8 [27].

To verify our Fr scaling, we measure the average vetical downwards velocity:U ; 2kVyl. Plotting UyU0vs Fr in Fig. 3b, we find that all points fall onto a single curve, given byUyU0 ­ s1.25 6 0.09dFr 1 s0.45 6

0.04d. Note thatU . U0 over the range of Fr in ourexperiment.

For strips with Fr. Frc, we observe a continuous tumbling rotation; for values just above Frc we also some-times observe random reversals of the rotation directiosimilar to what is observed in simulations [15,16]. However, the length of the trajectories is insufficient to detemine if they are strictly chaotic [11].

We also directly measure the parallel (Fk) and per-pendicular (F') drag force by towing the strips in eachdirection with a known force, while restraining the fluttering. The drag force is indeed quadratic in velocity [21Fk ­ 2AkrwLV 2 and F' ­ 2A'rwLV 2, with A' ­4.1 6 0.1 andAk ­ 0.88 6 0.03. Our experimental con-clusion that the relevant drag force is quadratic and nlinear in velocity constitutes a fundamental departure froexistent models, in which drag terms originate from eithviscosity [16] or an irrotational representation [1,18].

We therefore modify the model in [16] to include onlylift, gravity, and inertial drag. Although added mass effects are surely present [1,15,18], our experiments indicthat they are not significant. The drag parallel and pependicular to the strip isFk ­ 2AkrwLV 2 cossg 2 Qdand F' ­ 2A'rwLV2 sinsg 2 Qd, whereg is the an-gle of the velocity vectorsVx , Vyd from the positivexaxis, andV 2 ­ V 2

x 1 V 2y . The rotational drag isFv ­

2AvrwL4v2, where v is the angular velocity. Thesereplace the viscous forces in [16]; the lift force is thsteady state value given by the Kutta-Joukowsky therem [1,4,16], where the circulation depends on the vlocity, and thus can vary. From our simulations wchooseAv ­ 0.0674, which fixes the tumbling transitionat Frc ­ 0.67.

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Note that the Fr scaling is clearly evident in the equationthis arises from the dependence of both lift and drag oV 2. The sign conventions ensure that the strip is alwayinside the right angle formed by the lift and velocityvectors, and that rotation byp does not change theequations. We emphasize that our equations are noparticular case of the potential flow solutions to Laplace’equation [1,18], but the result of including the effects ovortex shedding via inertial drag and variable circulation

Employing a standard Runge-Kutta integration schemwe obtain trajectories that are very close to the expermental ones, as shown in Fig. 4 for the parametersFig. 1b [28]. Just near Frc the model exhibits additionalfrequencies and apparent chaotic behavior, similar to eisting models [14–17]; these are not observed experimetally. The coefficientAv affects not only the value of Frc

but also the smoothness of the curves; small differencesAv may explain qualitative differences in the trajectorie(Fig. 1a–1c). A detailed comparison of the experimentaand numerical trajectories shows that the model reproducthe fluttering dynamics extremely well. A slight deviationdoes occur at the turning points, whereQ changes rapidly.This tendency towards apparently singular behavior growinversely withAv , and must also depend on the magnitudof the lift coefficients. This effect points to the necessityof explicitly treating the vortex shed at each turning poinfor example, by including another differential equation fothe lift [12].

What are the dynamics of the surrounding fluid? Thfalling strip creates a zigzag wake by shedding vorticesynchronized with its fluttering oscillations, as shown balumina particle visualization in Fig. 5. A small whirldevelops behind the side which is swinging upwards, anis shed when the angle reachesQmax. This attachedvortex may be a major factor in stabilizing the momendriving rotation [13], and may also modify the lift [29].An oscillating lift could drive the fluttering motion, since

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FIG. 4. Two collages of the simulation (see [28]), with Frcorresponding to the data in Figs. 1(b) and 1(d). The strips ashown everyDt ­ 0.4tp: (a) Fr ­ 0.45, L ­ 0.75 (flutter); (b)Fr ­ 0.89, L ­ 0.19 (tumble).

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VOLUME 81, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R S 13 JULY 1998

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FIG. 5. Aluminum particle visualization of the flow behind afluttering strip:L ­ 4.1 cm, M ­ 2.7 g (Fr ­ 0.45).

vortex shedding produces an oscillating side thrust [1A full treatment of these effects is lacking in ouanalysis.

Our experiment has shown that the transition froflutter to tumble is determined by the Froude numbewith a velocity scale set by the inertial drag. The modmotivated by these observations reproduces this transitiand implies the Fr similarity. The vortices and the higRe flow around the strip are responsible for the inertidrag, although they do not appear dynamically in omodel. Nonetheless, Fr adequately describes the grfeatures of the motion. What remains experimentais to explorequantitatively the attached vortex, the liftdynamics during flutter, and the dynamics of tumble.

We thank J. Fineberg, L. Mahadevan, H. StonR. Zeitak, and O. Zik for discussions. A. B. acknowledges support from the NSF and a NASA Research Gra

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This work was supported by the Minerva FoundationMunich, and the Minerva Center for Nonlinear Physics.

[1] H. Lamb, Hydrodynamics (Dover, New York, 1945),6th ed.

[2] M. J. Lighthill, Annu. Rev. Fluid Mech.1, 413 (1969).[3] H. J. Lugt, Annu. Rev. Fluid Mech.15, 123 (1983).[4] R. von Mises,Theory of Flight(McGraw-Hill, New York,

1945).[5] E. Kelley and M. Wu, Phys. Rev. Lett.79, 1265

(1997).[6] J. C. Maxwell, in The Scientific Papers of James Clerk

Maxwell (Dover, New York, 1890), pp. 115–118.[7] W. Willmarth, N. Hawk, and R. Harvey, Phys. Fluids7,

197 (1964).[8] E. H. Smith, J. Fluid Mech.50, 513 (1971).[9] R. E. Stewart and R. List, Phys. Fluids26, 920 (1983).

[10] E. Tränkle and M. Riikonen, Appl. Opt.35, 4871(1996).

[11] S. Field, M. Klaus, M. Moore, and F. Nori, Nature(London)388, 252 (1997).

[12] T. Sarpkaya, J. Appl. Mech.46, 241 (1979).[13] H. J. Lugt, J. Fluid Mech.99, 817 (1980).[14] V. V. Kozlov, Izv. Akad. Nauk SSSR Mekh. Tverd. Tela

24, 10 (1989).[15] H. Aref and S. W. Jones, Phys. Fluids A5, 3026

(1993).[16] Y. Tanabe and K. Kaneko, Phys. Rev. Lett.73, 1372

(1994); Phys. Rev. Lett.75, 1421 (1995).[17] L. Mahadevan, H. Aref, and S. W. Jones, Phys. Rev. Le

75, 1420 (1995).[18] L. Mahadevan, C. R. Acad. Sci. (Paris)323, 729 (1996).[19] A. Belmonte, H. Eisenberg, and E. Moses, Bull. Am. Phys

Soc.41, 1731 (1996).[20] Using Istrip ­ ML2y12 and Iring ­ mR2, the radius is

calculated by settingIstripyM ­ Iringym, which yieldsR ­ Ly

p12. ThusItot ­

112 sM 1 mdL2.

[21] G. Eiffel, La Résistance de l’Air et l’Aviation(Dunod,Paris, 1911).

[22] Since the width of the stripsw is always close to the widthof the cell, we usew ­ 0.8 cm in our calculations.

[23] The geometric asymmetry condition derived via a lineastability analysis in [18] is an equivalent condition.

[24] D. W. Thompson, On Growth and Form (CambridgeUniversity Press, Cambridge, England, 1942), Chap. 2.

[25] W. Froude, Trans. Inst. Naval Arch.11, 80 (1874).[26] S. Vogel, Life’s Devices (Princeton University Press,

Princeton, NJ, 1988).[27] R. M. Alexander, Nature (London)261, 129 (1976); Am.

Sci. 72, 348 (1984).[28] We determine the lengthL using the experimentally

measured distance that a strip falls in one oscillationThen for the other strips we useL1yL2 ­ sFr2yFr1d2.

[29] P. G. Saffman and J. S. Sheffield, Stud. Appl. Math.57,107 (1977).