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Falling Paper: Navier-Stokes Solutions, Model of Fluid Forces, and Center of Mass Elevation

Umberto Pesavento 1, * and Z. Jane Wang 2,1 Depar tment of Physics, Cornell University, Ithaca, New York 14853, USA

2 Theoretical and Applied Mechanics, Cornell University, Ithaca, New York 14853, USA(Received 28 January 2004; published 27 September 2004)

We investigate the problem of falling paper by solving the two dimensional Navier-Stokes equations

subject to the motion of a free-falling body at Reynolds numbers around 103

. The aerodynamic lift on atumbling plate is found to be dominated by the product of linear and angular velocities rather thanvelocity squared, as appropriate for an ai rfoil. This coupling between translation and rotation provides amechanism for a brief elevation of center of mass near the cusplike turning points. The Navier-Stokessolutions further provide the missing quantity in the classical theory of lift, the instantaneouscirculation, and suggest a revised model for the uid forces.

DOI: 10.1103/PhysRevLett.93.144501 PACS numbers: 47.32.Ff , 47.85.Gj, 47.11.+j

A piece of paper or a leaf utters and tumbles down in aseemingly unpredictable manner (see Fig. 1). This richdynamical behavior has inspired many experimental andmodeling works on falling plates [14] and disks [5,6], aswell as observations of dispersing seeds [7]. Even beforethe establishment of aerodynamic theory, Maxwell of-fered a qualitative explanation of the correlation betweenthe sense of rotation and the drift direction of a tumblingcard [8]. More recently, Willmarth et al. classied dy-namics at different Reynolds numbers and dimensionlessmoment of inertia [5]. Belmonte et al. further identied adimensionless number governing the transition from ut-tering to tumbling [2]. Mahadevan et al. observed arelation between tumbling speed and the thickness of afalling card [3]. Models based on inviscid theory [914]also exhibit dynamics qualitatively simila r to those seenin experiments.

What appears to be lacking is a model of the uid forceand torque that is constructed and tested against experi-ments or computations. The force predicted by inviscidtheory includes added mass and a lift proportional to theproduct of velocity and circulation [10]. An unresolvedissue is the choice of the circulation around the fallingobject, which cannot be determined from inviscid theory.Previous models either assumed the circulation to be aconstant [14] or to be linearly proportional to the trans-lational velocity [2,9,11]. The latter is appropriate for asteady translating airfoil at a small angle of attack, asgiven by the celebrated Kutta-Joukowski condition whichrequires ow velocity to be nite at the singular trailingedge [15]. The resulting lif t is quadratic in velocity. Whilethe Kutta-Joukowski condition works remarkably well inthis special case, there is no direct evidence that it holdsfor an object uttering or tumbling in a uid.

Theoretical progress is in part hindered by the lack of simultaneous measurements of instantaneous forces andows around a falling object. Here we solve the Navier-Stokes equations for the uid subject to the dynamics of afalling body in two dimensions. Solving the falling paper

problem in the most general case, i.e., free fall of a t hreedimensional exible sheet, is daunting and unrealistic atReynolds numbers of the order of 10 3 . To simplify theproblem, we note that bodies of relatively large span-to-chord ratios fall essentially along a two dimensionalplane [4]. In the case of a business card, the span-to-chordratio is about 1.8, and the motion in the spanwise directionis negligible. Taking advantage of these observations wefocus on a falling rigid plate in a two dimensional uidgoverned by the incompressible Navier-Stokes equations.The choice of a rigid plate is convenient for numerics andfor comparison against existing theories. Obtaining ac-curate numerical solutions in this regime turns out to benontrivial due to the thin tip of the geometry, the movingboundary coupled to the uid forces, and the small uidtorque. Our numerical method is based on a vorticity-stream function formulation in a conformal grid tted tothe plate [16,17]. The conformal grid concentrates pointsat the edges of the plate and can be mapped onto aCartesian grid in which we discretize and solve theNavier-Stokes equations efciently via fast Fourier trans-form [18]. In the mapped Car tesian grid the Navier-Stokes equations take the following form:

FIG. 1. Rise of a falling journal cover under windless con-ditions, selected frames from a footage lmed at 300 frames=s .

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@S!@t

Sp u r ! 1

Re r 2 !; (1)

r Sp u 0 ; (2)where ! and u are, respectively, the vorticity and velocityelds, and S is the scaling factor associated with the

conformal mapping from the exterior of the plate to asemi-innite strip. The use of body-xed coordinateseliminates spatial interpolation as the plate moves withrespect to the uid. T his turns out to be crucial forobtaining the almost periodic trajectories seen in Fig. 2.The forces on the plate are calculated by integrating thestress tensor along the body [19]. The updated velocity isthen fed back to the Navier-Stokes solver through theboundary conditions. The method is applicable to arbi-trary shapes of the cross section, and for simplicity, wechoose an ellipse. To gauge the grid dependence of theresults, we repeat simulations using grid systems of 5121024 and 256 512 and nd that the results presented

below hold for both resolutions.A falling ellipse is characterized by six parameters: the

major and minor semiaxes, a and b, the densities of theellipse, b , and of the uid, f , the kinematic viscosity of the uid , and the gravitational acceleration g [seeFig. 2(a)]. From these parameters, three dimensionlessquantities can be dened: the Reynolds number, Re 2 uta= , using the terminal velocity estimated by balanc-ing gravity against the uid force on a plate of size 2a anddrag coefcient 1, ut bg b= f 1q ; the dimension-less moment of inertia, I b a

2 b2 b2 a 3 f

; and the aspect ratioof the ellipse e

b=a . Note that the dimensionless iner-

tia is related to the Froude number Fr dened in previouswork [2], I / Fr 2 .In Fig. 2 we show the Navier-Stokes solution of atumbling ellipse for Re 1100 , I 0 :17 , and e 0 :125 , released from rest with an initial angle of 0:2radwith respect to the horizontal. After uttering for a shorttransient, the ellipse tumbles with an almost periodicmotion [ve periods are shown in Fig. 2(b)], which con-sists of gliding and quick 180 rotations. During theglide, the ellipse pitches up due to the uid torque. Theincreased angle of attack results in an increased drag;thus the ellipse slows down while pitching up. The smalltranslational velocity results in a cusplike shape near theturning points. As the plate initiates a turn, the wakebecomes unstable and breaks up into vortices due toKelvin-Helmholtz instability. The vortices have a char-acteristic size of half of a chord.

A casual observer of falling leaves or paper mightnotice that while falling downward on average, they canrise momentarily as if picked up by wind. In Fig. 1 weverify this observation by lming a falling journal coverat h igh speed. The case shown in Fig. 2 is an example of

center of mass elevation for a rigid plate without ambientwind. The swinging up motion may be familiar to thosewho have seen sailplane stunts performing dead loops. Atsufciently high Reynolds numbers, Joukowskis theorypredicts phugoid motion which swings up periodically inthe special case where the angle of attack is constant anddrag is negligible [9]. The situation here is different. TheReynolds number is relatively low, about 10 3 , and drag isnon-negligible. A straightforward modication of Joukowskis model to incorporate the lift-drag polar atReynolds number about 10 3 predicts no center of masselevation [20].

To investigate what might be missing in the Joukowski-like model of a falling plate, we turn to the instantaneousforces (Fig. 3). It is instructive to decompose the pressureforce into the contribution of the added mass and the li ftproportional to circulation, as described in inviscid the-ory [10,14]. The viscous force and torque F x , F y , and are relatively small (see Fig. 3) and can be treated asperturbations. The force components, F x and F y, andthe torque are dened with respect to body-xed axes

-1 0 1 2 3 4 5 6 7 86

5

4

3

2

1

0

x' [chords]

y ' [ c h

o r

d s

]

(b)

342

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(c)

1 2

mg

-1.8 -0.67 0 0.67 1.8

(d)

(a)

[u /a]t

v

g

b

a

xy

u

FIG. 2 (color online). Navier-Stokes solution of a tumblingellipse at Re 1100 , I 0 :17 , and e 0 :125 . (a) Body- xedcoordinate system and kinematic variables. (b) Trajectory andorientation of the chord (major axis of the ellipse) over veperiods of motion. (c) The history of the chord and the forcevector, equally spaced in time for the rst period of thetrajectory in (b). The chords numbered from 1 to 4 correspondto the frames shown in (d) and to the times marked with dots onthe force h istory of Fig. 3. (d) Vorticity eld at four instantsduring a full rotation. The frames display an area of 5 2 :5chords and they are 4 a=u t ti me units apart. The vorticity eld iscolor coded on a logarithmic scale.

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[see Fig. 2(a)] and can be modeled as

F x f v m12 v m11 _u f b Agsin F x ;(3)

F y f u m21 u m22 _v f b Ag cos F y ;(4)

m11 m22 uv I a _ ; (5)where u t and v t are the components of the ellipsevelocities along its major and minor axes, t is itsangular velocity, t is the angle between the x axisand the direction of gravity [see Fig. 2(a)], is thecirculation around the body, A is the area of the ellipse,mij is the components of the added mass tensor, and I a isthe added moment of inertia [15]. In the expression forthe forces F x and F y, the rst term is the lift due to thecirculation around the ellipse, the second and the thirdterms correspond to the added mass, the fourth term is thebuoyancy corrected gravity, and the F s are the viscous

forces.The circulation is unknown and needs to be modeled

to complete the equations for the pressure force. Here, wet the pressure force from the Navier-Stokes solution byusing Eqs. (3) and (4), where the added mass coef cientsare left as free parameters and has two contributions,one proportional to the angular velocity of the ellipse

and the other given by the Kutta-Joukowski condition (seeFig. 3).

cRa 2 cLajv jsin 2 ; (6)where jv j is the speed of the ellipse and is its angle of attack, de ned as the angle between the major axis of theellipse and its velocity vector, 2 0 ; 2 .For the falling ellipse shown in Fig. 2, the rotationalcontribution is about 10 times larger than the translationalone. The lift corresponding to the circulation is about75% of the total lift, the remaining lift being generated bythe added mass terms with coef cients m12 a nd m21 . It isworth noting that values mij from the force t di ffer fromthose given by inviscid theory (see the caption of Fig. 3).Skin friction gives a contribution of about 25% of thetotal force and can be approximated with an expansion inthe kinematic variables the ellipse u, v, and (see thecaption of Fig. 3). The pressure torque is 2 orders of magnitude smaller than the torque on an ellipse steadilytranslating with speed ut . However, it can be modeled by

a term proportional to uv as predicted by inviscid theory(see Fig. 3).The circulation model of Eq. (6) can be validated by

integrating the velocity eld outside the ellipse. Figure 4shows the ci rculation obtained both by tting the pressureforce with Eqs. (3) and (4) and by integrating the velocityeld. The circulation displays a strong dependence on themotion of the ellipse and cannot be modeled by a constantvalue as in [14] or by the classical expression for a trans-lating airfoil as in [2,11]. Instead, the circulation is betterapproximated by Eq. (6). The negative peaks in the cir-culation correspond to vortices shed at the turning pointsof the ellipse trajectory.

In terms of the traditional decomposition of forces intolift and drag, the proportionality between the circulationaround the ellipse and its rotational velocity corresponds

5 10 15 20 25 30

-0.5

0

0.5

t [a/u t]

F x

[ m g

]

5 10 15 20 25 30

-1

0

1

t [a/u t]

F y

[ m g

]

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t [a/u t]

[ m g

c h o r d

]

1

2

34

1 2

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FIG. 3. The uid force and torque on the tumbling ellipseshown in Fig. 2. Solid lines are computed forces, with the totalforce shown as thin lines and the pressure force as thick lines.The corresponding dashed lines are the best ts of the databased on the quasisteady model of Eqs. (3) (5). The addedmass tensor m11 0 :53 m and m12 3 :1 m, m22 1 :5 m, m21 0 :56 m, and the circulation around the ellipse 2 :6 a 20 :49 ajv jsin 2 are obtained from the pressure force.These mijdiffer from those predicted by inviscid theory ( minv11 minv21 0 :0491 m and minv22 minv12 3 :14 m). The viscous correctionsare modeled with the expansions F x 11 u 12 u2 , F y

21 v 22 v 2 , and 31 uv 32 33 j j, with11 0 :18 , 12 0 :0075 , 21 0 :070 , 22 0 :054 , 31 0 :31 , 32 0 :05 , and 33 0 :16 .

30 40 50 60 70 80 90

-0.5

0

0.5

1

1.5

t [a/u t]

FIG. 4. Circulation as a function of time for the fallingellipse of Fig. 2. The thick line corresponds to the value of the circulation found by integrating the vorticity eld up to aradius of 3 =2 chords from the center of the ellipse. The th in linecorresponds to the circulation obtained from tting the pres-sure forces, the dashed line to the contribution of the rotationalterm of Eq. (6) alone. The peaks of negative circulation notcaptured by the t correspond to the vortices shed by t he ellipseat the turning points of its trajectory.

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to a lift proportional to jv j instead of jv j2 as in the caseof a translating plate. In the classical context, jv j is thepredicted lift for an airfoil translating and pitching atsmall amplitude [21]. This is par ticularly important at t heturning points of the trajectory, where the translationalvelocity is small. There, the increased angular velocity compensates for the decreasing velocity jv j and the liftgenerated is suf cient for the center of mass of the ellipseto elevate. This mechanism for lift augmentation has beenrecently emphasized in insect hovering [22].

To further verify the connection between center of mass elevation and rotational lift, we a rbitrarily varythe coef cients of the rotational and translational contri-butions in Eq. (6). The tumbling trajectories obtainedwith such a procedure are shown in Fig. 5. Models with-out rotational li ft display center of mass elevation only forunphysical values of the lift coef cient ( cL > 7 ). On theother hand, models including rotational lift show centerof mass elevation for the coef cients obtained fromNavier-Stokes solutions ( cL 0 :49 and cR 2 :6 in thecase shown in Fig. 5).

Finally the ow-induced coupling between translationand rotation can decrease the speed of descent. The tum-bling ellipse shown in Fig. 2 has an average descent speedof 0:4 u t . In contrast, an identical ellipse parachutingdown with its major axis perpendicular to the directionof motion would reach a terminal velocity of 0 :77 u t . Itwould be interesting to nd out whether the slow descentand the stable direction of tumbling motion are exploitedby nature, for example, in seed dispersion.

In our current work we are further validating the modelpresented in this Letter experimentally and using it to

address the nature of the transition between uttering andtumbling [23]. Understanding free-falling bodies mightalso have interesting applications to insect ight, an areaof research that partly motivated this study. Althoughinsects might take advantage of both active and passivemechanisms to control their apping wings, only pre-scribed motions have been considered so far [24,25].Falling paper is a beautiful example of a passive ight.

We hope that the model presented here will be alsorelevant to descriptions of forces in general appingmotion.

We wish to thank M. Salganick for discussion at theearly stage of this work and A. Andersen for discussion of the ODE model. This work is supported by NSF, by ONR,by AFSOR, and by Packard Foundation.

*Electronic address: up22@cornell.edu Electronic address: jane.wang@cornell.edu

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4 2 0 2 4 6 8

9

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6

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1

0

1

x [chords]

y [ c h o r d s

]

(a)(b)

mg

FIG. 5 (color online). Trajectories obtained from ordinarydifferential Eqs. (3) (5) for different values of the t ranslationaland rotational lift coef cients cL and cR: (a) cR 2 :6 , cL 0 :49 , from the t of Fig. 3; (b) cR 0 , cL 1 :5 , as in classicaltranslational lift. Center of mass elevation occurs in (a) but isabsent in (b).

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