Analysis of the free-fall of a disc in viscous fluid using thetechniques of nonlinear dynamics

Zoltan Pandula ∗ Laszlo Kullmann † Csaba Hos ‡

December 11, 2002

Abstract

The behaviour of a thin disc falling freely in viscous fluid in two dimensions is investigated. The equation ofmotion includes the lift and drag forces, their moment and the viscous breaking moment. An investigation of theseequations is undertaken in terms of bifurcation theory. For small oscillations around the equilibrium, analytical sta-bility analysis is performed revealing that the an unstable oscillation occurs as the angular damping ratio decreasesbelow a critical value. A numerical continuation analysis shows the route from stable autorotation (tumbling) to rock-ing motion (flutter), via a chaotic region with highly complex structure. The the fluid-to-body density and thickness-to-diamater ratio is found to govern the dynamics and a two-parameter continuation in terms of these parameters iscarried out to compute the borders between autorotation, rocking and chaotic motion.

1 Introduction

The dance of a leaf’s fall is not only a beautiful natural phenomenon but also leads to several questions arising in awide range of scientific areas and industrial applications. The complex behaviour of a body falling freely in a fluidhas already been described by Newton in Principia (1726), Maxwell (see [12] and [13]) and Kirchoff. Altough theproblem has been labeled as ‘toy aerodynamics’ by some [11], it yields a surprisingly diverse scientific backgroundsuch as bubble dynamics [7], wind engineering [17] and also biological studies in terms of gait of animals [1] – besidethe traditional ones like aircraft aerodynamics and stability [9].

A falling body travelling straight downwards is quite rare; e.g. a stripe of paper of (say) 3 × 5 cm would continuouslyrotate. The term ‘autorotation’ (introduced by Raibouchinsky in 1935) has been given to this phenomenon as the bodyrotates without any external force. (We use the term ‘pseudo-autorotation’ if the motion is induced by special geometrysuch as e.g. the Savenius rotor [11] or the meaple leaf.) The theoretical and experimental results before 1979 for boththe free-fall and wind-tunnel case of rectangular plates have been summarized by Iversen [6]. It has been found thata minimum moment of inertia (I) exists, below which autorotation is not possible. For values of moment-of-inertiaparameter greater than one, the difference between free-flight and wind-tunnel autorotation becomes indistinguishable(the moment-of-inertia parameter is defined as I/ρbc2, where ρ is the density of the fluid, b and c are the span and chordlength, respectively). Explanation for the hydrodynamical mechanisms responsible for autorotation has been given byLugt [10], assuming laminar flow of an incompressible fluid around an infitely long plate in two space dimensions.He presented a numerical solution of the two-dimensional Navier-Stokes equations for an ellyptic cylinder rotating

∗pandula@vizgep.bme.hu†kullmann@vizgep.bme.hu‡hoscsaba@vizgep.bme.hu

1

with constant angular velocity in the two-dimensional flow. The vortex structure around the body has been analysedin-depth and was found to be the main driving force. The same author gave a comprehensive overview of the fieldin 1983 [11]. More recently, Skews conducted a series of tests on the autorotation for rectangular plates [16] withinfinite aspect ratio (the test body spanned the full width of the wind tunnel). He found an excellent agreement withthe numerical results of Lugt mentioned earlier. He found that the tipspeed-to-airspeed ratio (i.e. Rω/U) is constant(approx. 0.46) for a thickness-to-chord ratio of 0.1. He also found that the drag coefficient cD remains constant whilevarying the thickness ratio, unlike the lift coefficient cL, which decreases about 30%. The observation of Reynoldsnumber independence of both Iversen and Lugt was confirmed.

In 1994, Tanabe and Kaneko gave a phenomenological model for free-fall of paper [18], assuming zero thickness andincompressible ideal fluid. Since the inviscous fluid does not yield friction, they included an ad hoc friction termproportional to the angular velocity. Based on numerical integration of the resulting system of ordianary differentialequations, they identified five falling patterns: autorotation (periodic rotation), chaotic rotation, chaotic fluttering,periodic flutterng and simple perpendilcular fall. Their model, despite of its simplicity, captured the essential of theunderlying mechanisms. Field et al. in [4] present a systematic experimental study of the motion of disks in fluid andalso address the application of modern tools of dynamical systems.

In this paper, we restrict our attention to the case when the axis of rotation is perpendicular to the flow and is not fixed(i.e. free-fall). We also assume disc geometry, which results in a two-dimensional model and small (< 0.1) thickness-to-chord ratio. The model defined in §2 is rather simple but hopefully captures the essential of dynamics. We usetrigonometrical functions to describe the change of lift, drag and torque coefficients as the angle of attack varies andinclude them in the equation of motion. By introducing dimensionless variables and a polar coordiante system, we endup with a set of four nonlinear ordinary differential equations. Two important parameters are identified; the fluid-to-body density ratio and the thickness-to-chordlength ratio. In §3, we assume small perturbations around the equilibrium(which is falling horizontally with constant velocity and without rotation) and show that it becomes unstable belowa critical value of the angular damping. It is also shown that the stability loss corresponds to a Hopf bifurcation andthat the appearing oscillation is unstable. Having completed the analytical calculations, in §4 we turn to numericaltechniques. First, a bifurcation diagram computed by a series of numerical intergration is presented, showing severalinteresting features of the problem: rocking motion (side-to-side oscillation), autorotation and chaotic regions arefound in the parameter space. With the help of the numerical continuation package AUTO [3], we give a qualitativeexplanation of the dynamical features of the problem. Note that we are not aware of any previous work presentinglinear or nonlinear stability analysis, neither analytical nor numerical. The authorts would also like to emphasise thataltough the model presented here might be found too simple to yield quantitative results, the techniques used here -particularly the AUTO computations - are powerful, efficient and can be redone on much more realistic models.

2 Governing Equations

The physical model of the problem is sketched in Fig.1. The body has three degrees of freedom; two velocity com-ponents (Ux and Uy) and the inclination angle (compared to the horizontal axis) φ. The angle of attack (the angleenclosed by the velocity vector U and the plate) represented by α will be also needed. The plate with thickness h hasa total cord length of l (for visibility reasons, not shown in Fig.1) and a size of b in the z dierction (‘depth’, also notindicated), which is a functions of z. The forces acting on the plate are (a) resistance force E, (b) lift force due to theflow F, (c) gravity and lifting force G due to the density difference. There is a moment M 0 originating from the factthat the forces E and F are acting not at the center of gravity (law of Kutta-Joukowski), for details, see e.g. [15] or[14]. Another moment M f is present due to friction forces.

For a thin plate in case of perpendicular angle of attack (i.e. α = π/2), the resistance force can be expressed as

E⊥ = ce⊥ρ

2U2 A, (2.1)

2

PSfrag replacementsM0,M f

FE

GU

x

y

α

α

α

γ

φ

h

S P

Figure 1: Physical model of the problem.

where ce⊥ denotes the resistance coefficient, ρ the density of the fluid and A the surface of the plate. The value of c e⊥is constant for Re = U D/ν > 104 and can be chosen as 1.11 ∼ 1.17 (see [5]). If the fluid flow is parallel to the plate,the resistant coefficient can be expressed as a function of the Reynolds number as given in [5]

ce‖ = 2

(

0.455lg

(

Re2.58) − 1700

Re

)

. (2.2)

For Re = 105, (2.2) gives ce‖ = 0.0365. As the plate is not infinitesimally thin, this value has to be modified as givenin [14] (p.42):

ce = ce‖

(

1 + 2hl+ 60

h4

l4

)

, (2.3)

which gives ce = 1.026 ce‖ for hl = 0.1. Between the two extrema (ce⊥ and ce‖) the coefficient will be approximated

by the analytical function ce(α) = ce⊥ sin2 α. This approximation is qualitativly acceptable although it misses severaladditional features, such as hysteresis effects and the linearity around alpha = 0 and π (for measurement results, see[14]). In what follows, we assume that Re = 105 and h

l < 0.1. In this case, the relative error caused be neglecting ce‖ is1.11−0.05684

1.11 = 5.12% and 0.6% for Re = 106, assuming hl = 0.1. Thus, the resistance coefficient ce can be approximated

using an analytical function as

E = ce(α)ρ

2U2 A = ce⊥ sin2 α

ρ

2U2 A. (2.4)

The lifting force can be expressed in a similar way as

F = c f (α)ρ

2U2A = c f∞max sinα cosα

ρ

2U2A (2.5)

where c f∞max represents the maximum value of the lifting coefficient of an infinitely ‘deep’ (i.e. b→ ∞) plate. Again,we used the diagrams given in [14] to fit the analytical function. The value of c f∞max is in the range of 0.8 ∼ 0.96 for

3

NACA profiles, 0.8 ∼ 0.96 for Gottingen 459 − 460 ones and 1.23 ∼ 1.34 for Gottingen 766 profile. Choosing c f∞max

as

cdef= ce⊥ = c f∞max = 1.11 (2.6)

unburdens the calculations and is also an acceptable simplification.

The gravitational force is reduced by the mass of the fluid equal to the volume of the body. It is convenient to define a’virtual’ gravity as

G = Vρb(ody)g − Vρg = Vρb × g

(

1 − ρρb

)

def= mg∗. (2.7)

ahol g = 9.81[

ms2

]

.

The moment of the flow forces can be calculated as M(α) = k(α) N(α), where N(α) is the component of the resistantand lifting force normal to the surface of the disc. The location of the point where these forces act is highly non-trivial. For small α values, analytical calculations show that it is located at at one fourth cord length (i.e. l

4 ) from thestagnation point (see e.g. [15]), which has also been validated by several experiments (e.g. [14]). It is also clear thatfor symmertic profiles with α = π/2, this moment has to vanish while F(α = π/2) = 0 and E(α = π/2) , 0, hence, inthis case k has to vanish. Taking these facts into account, we have chosen k(α) = l/4 cosα. Using this approximations,for narrow, symmetric profiles, the resistance and lift force induce a moment of dM0 on the surface of the plate withlength l and ‘depth’ dz:

dM0 = k(α) dN =l4

cosα (dF cosα + dE sinα)

=l(z)4

cosα(

c f∞max sinα cos2 α + ce⊥ sin3 α) ρ

2U2 l(z)dz

=l(z)2 dz

4c

sin 2α2ρ

2U2 (2.8)

In the case of a disc, the cord length can be expressed as l(z) = 2√

R2 − z2 and by integrating (2.8) one gets

M0 =

∫ z=R

z=−RdM0 =

2c3π

D sinα cosαρ

2U2 A = cM D sinα cosα

ρ

2U2 A. (2.9)

Based on the studies concerning to friction force acting on a flat disc rotating with a constant angular velocity [8], [2]and [19], the viscous breaking force in turbulent flow can be written as

M f =D5ρ

60ω|ω|. (2.10)

For small Reynolds numbers, the flow around the body is laminar and the breaking moment is a linear function of theangular velocity. Thus, the friction force will be approximated by

4

M f (ω) =

{

kω for laminarε ρD5 ω |ω| for turbulent

flows. (2.11)

with friction factor ε = 0.0167, which has been validated for thin, flat, disc-shaped bodies in [19]. It is to be noted herethat the critical (rotational) Reynolds number corresponding to the laminar/turbulent transition is rather uncertain. Inthis article, we will use the laminar term for analyzing ’small’ perturbations around the equilibrium (linear stabilityanalysis) and the turbulent model for global, large-amplitude oscillations.

Applying Newton’s second law gives the equations of motion:

mx = F sin γ − E cos γ (2.12)

my = −F cos γ − E sin γ +G (2.13)

Θφ = −M0 − M f . (2.14)

We introduce the re-scaling parameters l0 and t0 and define new, dimensionless variables as

ζ =xl0, η =

yl0, v =

Uv0

and τ =tt0, where

v0 =

√

2mg∗

cAρ, t0 =

l0v0

and l0 = D. (2.15)

Notice that v0 represents the steady-state sinking velocity of a horizontal oriented plate. We also introduce the polarcoordinate system via dimensionless velocity v and angle of attack γ (ζ = v cos γ and η = v sin γ. In this new system,the governing equations yield:

v′ = −a0v2 sin2(γ − φ) + a1 sin γ

γ′ = −a0v sin(γ − φ) cos(γ − φ) + a1

vcos γ

φ′ = ω

ω′ = −a2v2 sin(γ − φ) cos(γ − φ) −M f (ω)

θ(2.16)

To simplify the analysis, we introduce the density ratio κ = ρ

ρband the ’slenderness’ λ = h

D as new, dimensionlessparameters. The coefficients in (2.16) in terms of these new parameters yield:

a0def= p0 = c

ρAl02m=

c2κ

λ, a1 =

G l0v2

0 m= a0 and a2

def= p1 = cM

ρAl202Θ= 4c

κ

λ

134 + λ

2, (2.17)

where we also took into account that l0 = D and Θ = m12

(

34 D2 + h2

)

.

Let us offer a few remarks here. First, the dimensionless variables defined by (2.15) are meaningless for κ = 1 (i.e.ρ = ρb) although system (2.16) remains non-singular. Hence, for physical relevance, we restrict our attention to the

5

parameter range 0 < κ < 1 and λ < 0.1 (the latter assumption ensures that ce‖ can be neglected). Secondly, note thatsystem (2.16) is invariant under the transformation

(v, γ, φ, ω)→ (v, π − γ,−φ,−ω) . (2.18)

As known from the literature [XXX], there are two important kinds periodic motions; swaying (swinging) motion andautorotation. In the case of swaying motion, the angle coordinates γ and φ remain bounded and are a periodic withsome T < ∞, hence, transformation (2.18) does not give a new solution, it represents only a phase shift. In the caseof autorotation, the angular coordinates are periodic, with modulo an even multiple of π, i.e. φ(0) = mod(φ(T ), k 2π)with some k. Hence, transformation (2.18) represents two independent autorotating solutions with opposite directionof motion.

The equilibria are given by

ve

γe

φe

ωe

=

1π2 + 2kπ

2lπ0

where l, k = �. (2.19)

Physically all these equilibria represent the sinking of a horizontally orientated disc with constant U = v0 velocity.Thus, in what follows, it is sufficient to analyze the equilibrium given by

(

1, 12π, 0, 0

)

.

3 Analytical Stability Analsysis

In this section, we present analytical calculations under the assumption of small perturbation around the equilibrium,which lets us include the linear friction model M f (ω) = K ω. We start with the conventional linear stability analysis,i.e. qualitative behaviour of small-amplitude motions around the fixed point. After performing the linear stabilityanalysis and detecting a Hopf bifurcation, we carry out the Hopf normal form transformation to determine the stabilityof the appearing limit cycle. The presence of the Hopf bifurcation gives the idea that the swaying motion observed inexperiments [XXX] is born via this stability loss and as the ampitude of the swinging grows and reaches φmax = ± π2 ,it turns into autorotation. However, the normal form transformation reveals that this is not the case as the appearinglimit cycle is unstable, hence, one can never come across it in experiments. Also, latter nonlinear analysis shows thatthe branch corresponding to swinging motion is a completely seperate and closed branch; it is independent from theautorotation. The results of this section are rather theoretical and contribute to the qualitative understanding of thebehaviour of the solutions.

The change of variables x =(

v − 1, γ − π2 , φ, ω)

turns system (2.16) into

x′ = A x + F(x). (3.20)

with the linear coefficient matrix

6

A =

−2 p0 0 0 00 0 −p0 00 0 0 10 p1 −p1 −k

, (3.21)

where k = KΘ

. The characteristic polinomial is given by

D(λ) =4

∑

i=0

aiλi = λ4 + (k + 2p0) λ3 + (2kp0 + p1) λ2 + 3p0 p1 λ + 2p2

0 p1. (3.22)

Applying the Routh-Hurwitz criteria, it turns out that all the eigenvalues are with negative real part if and only ifk > p0. Subsituting k = p0 into the characteristic polinom, one finds that

λ1 = −2p0, λ2 = −p0 and λ3,4 = ± i√

p1. (3.23)

As

d<λd k

∣

∣

∣

∣

∣

∣

k=p0, λ=i√

p1

= − p1

2(p20 + p1)

, 0, (3.24)

the equilibrium looses its stability via a non-degenerate a Hopf bifurcation. To gain information about the stabilityof the appearing cycle, we shall proceed as described in [Kuznetsov]. First, let q ∈ �4 be a complex eigenvectorcorresponding to λ1 = iω (i.e. A q = iω q) and p ∈ �4 the adjoint (left) eigenvector (i.e. AT p = −iω p) satisfyingthe normalization condition 〈p, q〉 = 1, where the scalar product for complex numbers is defined as 〈x, y〉 = ∑n

i=1 xi yi.The proper vectors in our case are

q =1ω2

0p0

− iωω2

and p =12

ω

ω + i p0

0iω

p0 − iω1

. (3.25)

The next step is to decompose the phase space with the help of these vectors and project the system onto the centermanifold. This can be achieved by virtue of the Center Manifold Theorem [see e.g.Kuz or Holmes], which states thatthe bifurcations occur essentially on a two-dimensional hypersurface (in case of Hopf bifurcation), while the behaviour’off’ tha plane (in the directions of all the non-critical eigenvectors) is trivial: the trajectories are either convergingor diverging. In our case, the critical real eigenspace T c is spanned by

{<(q),=(q)}

while the stable/unstable (non-critical) eigenspaceT su is spanned by the non-critical eigenvectors. An arbitrary vector y ∈ T su if and only if 〈p, y〉 = 0(for details, see [Kuz] or the Fredholm Alternative Theorem). Hence, we can decompose any x ∈ �4 as

x = z q + z q + y, (3.26)

where z is a complex number, z q + z q is a coordinate on the two-dimensional centre manifold and y represents thecoordinate along the stable/unstable manifold. We have an explicit formula for z:

7

〈p, x〉 = z 〈p, q〉 + z 〈p, q〉 + 〈p, y〉 = z, (3.27)

where we have used the facts that 〈p, q〉 = 1, 〈p, q〉 = 0 (see Kuznetsov) and 〈p, y〉 = 0. Using these variables, theoriginal system

x = A x + F(x) (3.28)

is projected onto the critical (center) manifold

z = 〈p, x〉 = 〈p, Ax〉 + 〈p, F(x)〉 = iω 〈p, x〉 + 〈p, F(x)〉= iω z + 〈p, F(z q + z q + y)〉 (3.29)

while along all the other directions, the dynamics is ’trivial’ as these are the directions concerning to the ‘non-critical’eigenvectors:

y = A y + F(z q + z q + y)

− 〈p, F(z q + z q + y)〉 q− 〈p, F(z q + z q + y)〉 q. (3.30)

As now the system is restricted to the centre manifold, we can proceed with the standard Hopf normal form transfor-mation (see also in Kuz). Let us define the multilinear functions B(x, y) and C(x, y, z) as

Bi(x, y) =n

∑

j,k=1

∂2 Fi(ζ)∂ζ j ∂ζk

∣

∣

∣

∣

∣

∣

ζ=0

x j yk (3.31)

and

Ci(x, y, z) =n

∑

j,k,l=1

∂3 Fi(ζ)∂ζ j ∂ζk ∂ζl

∣

∣

∣

∣

∣

∣

ζ=0

x j yk zl. (3.32)

With the help of these functions, the nonlinear term of the original system can be written as

F(x) =12

B(x, x) +16

C(x, x, x) + O(‖x‖4) (3.33)

and we also gain an invariant expression for the first Lypunov coefficient (taken from Kuz):

δ =1

2ω<

[

〈p,C (q, q, q)〉 − 2⟨

p, B(

p, A−1B (q, q))⟩

+⟨

p, B(

q, (2iωE − A)−1 B(q, q))⟩]

. (3.34)

After carrying out these lengthy computations, δ turns out to be

8

δ = p30

p20 + 2 p1

4 p32

1

(

p20 + p1

)2> 0. (3.35)

4 Numerical nonlinear bifurcation analysis

In this section we present the global, nonlinear behaviour of the system. We have used direct numerical integration(Runge-Kutta algorithm) and the numerical continuation package AUTO [3]. The analysis of small perturbationspresented in §3 showed that there is a branch corresponding to unstable flutter originating at the Hopf bifurcationpoint. This branch has also been computed with AUTO and it was found that it terminates at p0 = 0 without furtherbifurcations. Hence, stable periodic motions such as flutter or tumble cannot bifurcate from this branch. Thus, weturn to nonlinear analysis of large-amplitude motions. However, in this case, one cannot use the laminar friction termas the flow might be strongly turbulent around the body. Thorough this section, we use the turbulent friction modeldefined by (2.11). We are aware that Tanabe and Kaneko used a laminar friction term in [18] but based on the researchcorresponding to dynamical behaviour of check valves [Chemnicky], we believe that the turbulent modelling is muchmore realistic. The actual term included in the equations are:

M f (ω)

θ=

ε ρD5

148ρb D2 πh

(

34 D2 + h2

)ω|ω| = 48 επ

κ

λ

134 + λ

2ω|ω|. (4.36)

A typical bifurcation diagram corresponding to the Poincar e section at φ = 0 is shown in Figure 2. Note that largevalues of dimensionless moment of inertia corresponds to small values of κ (with fixed λ). The transition from autoro-tation to rocking (from tumbling to flutter) appeares at κ ≈ 0.0147. The measurements of Field et al. [4] show thatthis transition is nearly independent of the Reynolds number and happens at I∗ ≈ 0.04. Taking into account that thedimensionless moment-of-inertia parameter is defined as I∗ = πρbh / 64ρD = 0.0491λ / κ, the transitio is expected tooccur at κcrit = 1.234 λ, which gives κcrit = 0.01234 for λ = 0.01. The difference between the measurement and theanalytical prediction is by all means promising. A zoom close the transition zone in panel (b) shows that the autorotat-ing solution undergoes several period-doubling bifurcations as κ increases and becomes chaotic. It is interesting thatthere are several transient chaotic solutions in this narrow parameter range, i.e. the solution passes in and out chaoticand periodic regions. The presence of chaos between autorotation and rocking motion is also clearly shown in themeasurements of Field et al. Unfortunately, their experimental ‘bifurcation diagram’ [4] is depicted in terms of I ∗ andthe Reynolds number, which makes the quantitative verification cumbersome.

We now try to explain the dynamics qualitatively with the help of the continuation software AUTO [3], which makesit possible to compute stationary and periodic solutions (both stable and unstable ones), locate the special points suchas e.g. Hopf bifurcation or period-doubling bifurcations and compute the two-parameter locus of such points. A fullperiod of an autorotation has been imported from direct numerical simulation, and AUTO used this curve to initializethe solution and start the continuation. A special and useful feature of AUTO is that it automatically adopts the solutionif rotations are found (solutions which are periodic in the sence x(t) = mod (x(t + T ), 2π k). The results are shown inFigure 3. Starting with small values of κ, the autorotation is stable until it loses its stability at κ ≈ 0.147. More detailsof this branch are presented in panel (b). The original solution becomes unstable at BP and another periodic orbit withdouble period is born, which then undergoes several further period-doubling bifurcations and finally becomes chaotic.The same technique was used the compute the flutter solution (rocking motion) depicted in Figure 4. Continuationreveals that this branch is closed and is independent of other solutions. The branch consists of a stable and an unstablepart, which are seperated by two fold points.

Combibing these two AUTO runs, one can explain the results of direct integration qualitatively (see Figure 2). As arule of thumb, bodies with small fluid-to-body density ratio (large moment of inertia) would autorotate independently

9

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

PSfrag replacements

κ

v

Rocking

(a)

0.0144 0.0148 0.0152 0.0156 0.016

2

2.4

2.8

3.2

3.6PSfrag replacements

κ

vRocking

(a)

κ

v

Rocking

Autorotation

Autorotation

(b)

Figure 2: Bifurcation diagram obtained by direct numerical integration of system (2.16). Parameter values used toplot: λ = 0.01, ε = 0.016 and c = 1.11. As shown in panel (a), for large moment of inertia autorotation exists untilκ ≈ 0.0147. At this point, rocking motion (flutter) begins, which is followed by another large chatic region. (b) Azoom close to the flutter-tumble transition revealing several series of period-doubling bifurcation.

0 0.05 0.1 0.15 0.21

2

3

4

5

6

7

8

9

10

κ0.0145 0.015 0.0155 0.016

2

2.5

3

3.5

4

κ

PSfrag replacements

κ

Ma

x(v

)

Ma

x(v

)

BP

PD

(a) (b)

Figure 3: (a) Bifurcation diagram of the autorotating solution computed with AUTO (λ = 0.01, ε = 0.016 andc = 1.11). (b) A zoom showing the branching point (BP) and the first two period-doubling bifurcations (PD). Filledcircles correspond to stable, empty ones to unstable solutions.

from other parameters such e.g. thickness-to-chord ratio or Reynolds number, as also indicated by the measurementsin [4]. As the density ratio is increased, the autorotating solution undergoes several period-doubling bifurcations andexhibit complex dynamics including chaotic attractors. If κ is increased further, rocking motion appeares suddenlyand chaos dies away. Another chaotic region was found after the rocking motion, but this has not been observedexperimentally yet.

Figure 5 depicts how the thickness parameter (λ) influences the dynamics. The two fold points in Figure 4 and thefirst period-doubling point in Figure 3 were chosen to compute the two-parameter bifurcation diagrams. As it can beseen, the two fold points are located on the same continuous, closed curve bordering the parameter region where flutteroccurs. (Let us not forget that the model is not physically realistic if κ < 1 or λ < 0.1 – the scaling of the κ and λ axis inFigure 5 is merely to depict the closedness of the fold curve.) This figure coincides qualitatively with the experimentalresults of Field et al.: large moment of inertia (small κ) induces autorotation and during the transition from flutter(rocking) to tumble (autorotation) the system crosses several chaotic regions, with a surprisingly fine structure (seeFigure 2 (b)).

An interesting feature of the problem is that while the body flutters, two time scales are competiting: one of them is

10

0 0.05 0.1 0.151

2

3

4

5

6

7

8

0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3PSfrag replacements

κ v

γMa

x(v

) Folds

(1)(2)(3)(4)

(a) (b)

Figure 4: (a) Bifurcation diagram of the flutter solution computed with AUTO (λ = 0.01, ε = 0.016 and c = 1.11).Filled circles correspond to stable, empty ones to unstable solutions. (b) Projection of the phase-space to the (v, γ)plane, showing the trajectories at λ = 0.015, 0.05, 0.1, 0.17 (from left to right).

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

PSfrag replacements

κ

λ

Autorotation

PD

Rocking motion

LP

Highly non-trivial orbits,

chaos-like motion

Figure 5: Two-parameter bifurcation diagram, computed with AUTO. Solid line corresponds to the fold points enclos-ing the flutter region, the dashed line indicates the first period-doubling bifurcation of the autorotating solution.

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the time scale of the vertical sinking (t0), the other one is the time scale of the pendulum-like motion. The ratio of thetwo numbers is the Froude number Fr, which was found to be constant at the flutter-tumble transition for geometryand density combinations [1]. When calculating the two-parameter bifurcation diagrams, AUTO also computed theperiod of the flutter T , which is the time scale of the pendulum-like motion. In our calculations the Froude numberturned out to be approximately constan as well with a value of

Fr =Tpendulum

Tdownward∼ T t0

t0= T = 1.51 ± 0.032 (4.37)

along the ‘left arm’ of the fold curve (the one closer to the period-doubling curve) in the region of 0 < κ < 1 (seeFigure 5). In [1], a somewhat lower value of the critical Froude number has been found: 0.67 ± 0.05. It is also to benoted that based on the AUTO computations, the average velocity ratio v = U/v0 was found to be approx. 1.31, whichis very close to the one given by Belmonte et al. [1]. As known from the literature ([10], [4], [1]), for large enoughReynolds-numbers, the motion is independent of viscousity. This can also be seen directly from the dimensionlessequation of motion, as there is now explicit Re-dependence as long as now laminar/turbulent transition occurs. Basedon experimental results in [4], the dimensionless moment-of inertia parameter I∗ is also approximately constant at theflutter/tumble transition. The reason for this can be summarized as follows. The Froude number can also be expressedin terms of parameters κ and λ by taking into account that the time scale of the bouyant pendulum can be written asTpen. ∼

√

g∗l0/g (see [1]). Thus, the Froude number in terms of κ and λ yields Fr = v0/√

g∗D/g ∼√λ/κ. Expressing

I∗ (defined by Field in [4]) also as a function of κ and λ yiedls I∗ = πρbh / 64ρD ∼ λ/κ. Comparing the definition ofFr and I∗ suggests that the parameter ratio λ/κ is responsible for the similarity. Indeed, computing this ratio alongthe corresponding section of the fold curve gives λ/κ = 1 ± 0.01. The same conclusion can be drawn by looking atthe definition of the parameters p0, p1 (2.17) and p2: λ/κ is the only parameter varying if density or geometry dataare changing (for λ < 0.1, the λ-dependence of term 0.75 + λ2 can be neglected). The time series and phase spaceprojection of a flutter/tumble transition is depicted in Figure 6.

5 Discussion

The above results demonstrate an analytical approach of the behaviour of a falling thin disc in viscous media. Thefidelity with which such a model represents the ‘real life’ is always of concern: however, it is also important to find the‘simplest’ model capturing the experimentally observed motions. Although the modelling equations presented hereinclude subtantial simplifications, they still describe most of the experienced phenomena such autorotation (tumble),stable rocking motion (flutter) and the chaotic region between them. The analysis based on bifurcation theory is clearlyof benefit in understanding the dynamics of falling bodies. Several features of the system have been explained, e.g.the route from autorotation to rocking motion, the independence from Reynolds number or the universal value of theFroude number and the dimensionless moment-of-inertia parameter at the transition from autorotation to flutter. Also,it has been shown that the ratio of density parameter κ and the thickness-to-diameter parameter λ governs essentiallythe dynamics.

There are several ways of improving the model. Obviously, reliable, experiment-based approximations of the lift,drag and moment coefficients are needed and also the viscous friction term is to be verified. Including the regionof transition from laminar to turbulent flow in a single function to compute each coefficients is a highly challangingtask. The importance of this study is that, whatever level of accuracy of these functions is incorporated in the mathe-matical model, numerical continuation techniques can be used to provide an in-depth understaning of the underlyingmechanisms.

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0 20 40 60 80 1001

2

3

1 2 31.5

2

2.5

0 20 40 60 80 1001

2

3

1 2 31.5

2

2.5

0 20 40 60 80 1001

2

3

1 2 3

1

2

3

0 20 40 60 80 1001

1.5

2

2.5

1 1.5 2 2.5

1

2

3

PSfrag replacements

(a) (b)

κ = 0.014

κ = 0.0148

κ = 0.0156

κ = 0.016

Figure 6: (a) Time histories showing the transition from autorotation to flutter: velocity v versus dimensionless timeτ. (b) Projection of the trajectories to the plane (v, γ).

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References

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[4] S.B. Field, M. Klaus, M.G. Moore, and F. Nori. Chaotic dynamics of falling discs. Nature, 388:252, July 1997.

[5] R.W. Fox and T. McDonald. Introduction to fluid dynamics. Wiley, 1994.

[6] J.D. Iversen. Autorotating flat-plate wings: the effect of moment of inertia, geometry and reynolds number.Journal of Fluid Mechanics, 92(2):327–348, 1979.

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[12] J.C. Maxwell. On a particalar case of the descent of a heavy body in a resisting medium. Scientific Papers, page115, 1890.

[13] J.C. Maxwell. Rotation of a falling card. Camb. Dublin Math. J., 9:145, 1894.

[14] F.W. Riegels. Aerodynamische Profile. R. Oldenburg, Munchen, 1958.

[15] A. Robinson and J.A. Laurmann. Wing Theory. Cambridge University Press, 1956.

[16] B.W. Skews. Autorotation of rectangular plates. Journal of Fluid Mechanics, 217:33–40, 1990.

[17] B.W. Skews. Autorotation of polygonal prisms with an upstream vane. Journal of Wind Engineering andIndustrial Aerodynamics, 73:145–158, 1998.

[18] Y. Tanabe and K. Kaneko. Behaviour of a falling paper. Physical Review Letters, 73(10):1372–1375, September1994.

[19] J. Szira Z. Pandula. Experimentelle untersuchung des bremsmomentes das auf eine in zahem medium gedrehtescheibe wirkt. In Proceedings of the MicroCad Conference, Miskolc, Hungary, 2001.

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