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Oscillating pendulum decay by emission of vortex rings

Diogo Bolster,1Robert E. Hershberger,2and Russell J. Donnelly2,*

1Department of Civil Engineering an Geological Sciences, University of Notre Dame, Notre Dame, Indiana, 46556 USA

2Department of Physics, University of Oregon, Eugene, Oregon, 97403 USA

?Received 23 September 2009; revised manuscript received 14 December 2009; published 26 April 2010?

We have studied oscillation of a pendulum in water using spherical bobs. By measuring the loss in potential

energy, we estimate the drag coefficient on the sphere and compare to data from liquid-helium experiments.

The drag coefficients compare very favorably illustrating the true scaling behavior of this phenomenon. We

also studied the decay of amplitude of the pendulum over time. As observed previously, at small amplitudes,

the drag on the bob is given by the linear Stokes drag and the decay is exponential. For larger amplitudes, the

pendulum bob sheds vortex rings as it reverses direction. The momentum imparted to these vortex rings results

in an additional discrete drag on the bob. We present experiments and a theoretical estimate of this vortex-

ring-induced drag. We analytically derive an estimate for a critical amplitude beyond which vortex ring

shedding will occur as well as an estimate of the radius of the ring as a function of amplitude.

DOI: 10.1103/PhysRevE.81.046317 PACS number?s?: 47.10.?g, 47.37.?q

I. INTRODUCTION

The study of pendulum motion is a classical problem in

physics and understanding the influence of fluid drag on its

decay dates back to Stokes ?1?. He derived a simple expres-

sion for drag on a sphere at low Reynolds numbers, which

was later expanded on to include the effects of added mass

and other phenomena ?e.g., Landau and Lifshitz ?2??.

At low Reynolds numbers, this drag, FD

as

s= 6??RsV?1 +Rs

s, can be expressed

FD

??.

?1?

At larger Reynolds numbers, it is observed that the drag has

an additional component, which is proportional to velocity of

the pendulum squared ?2?. This drag, FD

l, can be expressed as

FD

l= CDV2+ FD

s.

?2?

Here, ? is the viscosity, ? the density, and ?=?/? is the

kinematic viscosity of the fluid surrounding the sphere

whose radius is Rs. V=?A is the velocity, A is the amplitude

of the oscillation at frequency f, ?=2?f and ?=?2?/? is the

thickness of the boundary layer surrounding the sphere. CDis

a drag coefficient, which is typically empirically fit ?e.g.,

Gonzalez and Bol ?3? and Alexander and Indelicato ?4??.

More complex expressions for the drag on a sphere that in-

clude acceleration effects can be found ?e.g., Mordant and

Pinton ?5? and Lyotard et al. ?6??.

We define the Reynolds number as

Re=2RsV

?

,

?3?

The influence of fluid drag has become a topic of great

interest to the quantum fluids community, where studies of

oscillating objects in superfluid environments have been con-

ducted ?for a recent review, see Skrbek and Vinen ?7??. In

many of these cases, a transition from a drag that is linearly

proportional to the velocity to quadratically proportional is

also observed. In many cases, the quadratic drag is associ-

ated with vortical and turbulent structures behind the sphere.

Additionally, this drag and the interaction of a body with

vortical structures plays a crucial role in swimming dynam-

ics where animals oscillate or flap their bodies in such a way

as to generate vortices that propel them ?see Linden and

Turner ?8?, Dabiri and Gharib ?9?, von Ellenrieder et al. ?10?,

Blondeaux et al. ?11?, and Afanasyev ?12??.

Over the past few decades, significant experimental and

theoretical researches have been performed on unsteady flow

past a sphere. The generation of a vortex ring during the

impulsive flow of a sphere at low to moderate Reynolds

number was observed experimentally by Taneda ?13?. Later

Bentwich and Milow ?14?, Sano ?15?, and Felderhof ?16?

provided a theoretical solution to show the birth of such a

vortex ring. Various numerical studies ?e.g., Yun et al. ?17?,

Blackburn ?18?, and Constantinescu and Squires ?19?? at

small and large Reynolds numbers have observed vortex

rings and other vortical structures behind a sphere. Specifi-

cally, Yun et al. ?17? illustrated that a numerical model,

which does not capture the vortex rings, will underestimate

the actual drag on a body.

In this study, we show that, as expected, at sufficient am-

plitude, the drag on a spherical pendulum is greater than that

predicted by Stokes ?1?. We demonstrate experimentally the

existence of a regime where a vortex ring is shed at the end

of each swing and show that the additional decay on ampli-

tude beyond Stokes ?1? can be estimated analytically as the

impulse given to these rings. The pendulum system can be

characterized by two dimensionless numbers. These are

KC=?A

RS

and St=4fRS

2

?

,

?4?

which are the Keulegan-Carpenter and Stokes numbers, re-

spectively.

*rjd@uoregon.edu

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II. APPARATUS

We operated the pendulum in two ways: with a fixed sus-

pension and with a motor-driven suspension. The fixed sus-

pension was used to observe the decay of the pendulum’s

amplitude. The driven suspension permits observation of the

pendulum motion at constant amplitude near the resonant

frequency of the pendulum.

A glass aquarium tank is filled with the fluid in which the

pendulum bob oscillates. For the cases described here, the

working fluid is de-ionized water, with a small amount of

thymol blue dissolved therein. The pendulum is constructed

using various spherical metal balls suspended on either light-

weight magnet wire or fishing line. A brushless linear motor,

driven sinusoidally, was used for the driven suspension.

A digital video camera was used to measure the amplitude

of oscillation at any point in time. The ring radii were mea-

sured with a digital camera. The camera setups were cali-

brated by imaging a steel rule located on the focal plane.

In the decaying oscillation experiments, two pendulum

lengths were used ?315 and 155 cm?. Assuming a maximum

amplitude of 10 cm, then the angle cosines would be cos ?

=0.9995 and 0.9979, thus preserving the small-angle ap-

proximation. For most of the observations, the camera loca-

tion was such to limit the parallax error to less than 0.15%.

The Baker electrolytic technique is used to visualize the

vortex rings from the spherical bobs. The electrochemistry

and other physical details are described in Mazo et al. ?20?.

The pendulum and resultant rings are photographed in sil-

houette.

The oscillation amplitude is extracted from the recorded

images with the use of an in-house-written software. Vortex

ring sizes were measured using standard software packages

?such as PHOTOSHOP?.

III. DRAG COEFFICIENT IN WATER

For small amplitude motion of the driven pendulum, no

vortex rings are shed as illustrated in Fig. 1?a?. As the am-

plitude of oscillation increases, the pendulum bob begins to

shed vortex rings as it reverses direction at the top of each

swing. This is shown in Fig. 1?b?, where for a continuously

driven pendulum, vortex rings stack up as they migrate to-

ward the tank boundaries. Figure 2 shows a time sequence of

images depicting the shedding of the boundary layer from

the pendulum bob during a directional reversal.

A commonly measured parameter used to quantify the

drag force experienced by an object is the dimensionless

drag coefficient CD, defined as

CD=

F

1

2?V2A

,

?5?

where V=?A is taken as the characteristic velocity for a

given swing, A is the projected area of the sphere, ?Rs

F is the average force over one period. Our photographs

allowed us to determine the height y of the sphere above the

lowest point at the center of the arc as well as the amplitude

A. The change in potential energy ?U can be easily related to

2, and

the work done if the change in height ?y is small, ?U

=Mg?y, where M is the physical mass corrected for buoy-

ancy and g is acceleration due to gravity. The distance trav-

eled during one period is approximately 4A, so the average

work done is 4AF and by conservation of energy

F =?U

4A.

?6?

The resulting drag coefficient is shown in Fig. 3. Pixel reso-

lution limits this technique below Re=300.

IV. DRAG COEFFICIENT IN LIQUID HELIUM

Schoepe’s group at Regensburg produced several pioneer-

ing papers on the motion of a small sphere of magnetic ma-

terial ?100 ?m in radius, suspended between the supercon-

ducting plates of a capacitor, and carrying an electric charge

?e.g., ?21??. The velocity amplitude and resonance frequency

are measured as a function of driving force and temperature

in liquid helium at temperatures between 0.35 and 2.2 K.

Liquid helium is a Navier-Stokes fluid above 2.176 K and we

show their results at 2.2 K in Fig. 3. We also show results at

2.1 K, which can be considered a mixture of normal and

superfluid, with the normal-fluid density about 75% of the

total density and behaves not far from being a classical fluid.

The results are plotted in Fig. 3 and fit with our data remark-

ably well, especially at higher Reynolds numbers. The effec-

tive kinematic viscosity of liquid helium at 2.1 K is about

1.67?10−4cm2/s ?Stalp et al. ?22?? with a Stokes number

St=642. For our pendulum in water, St=653.

(b)

(a)

FIG. 1. Photographs of ?a? laminar flow at small amplitude os-

cillation and ?b? a street of vortex rings at larger amplitude

oscillation.

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V. DRAG COEFFICIENT FROM VORTEX RING

EMISSION

There is another interesting and useful way to look at the

decay data. Figure 4?a? shows a representative sample plot

for the amplitude against time for one of the pendulum de-

cays. The plots for the other experimental runs are similar. A

list of the experiments conducted is given in Table I. For

smaller amplitudes ?approximately given by KC??

times, the decay is largely exponential and is well approxi-

mated by the Stokes drag given in Eq. ?1?.

At higher amplitudes ?early times?, the decay exceeds this

purely exponential rate. This behavior has previously been

observed by many researchers ?e.g., Gonzales and Bol ?3??.

Often nonlinear damping functions are fit to the curve with-

out much physical insight into this additional drag. Figure

4?b? shows a typical log plot of the difference between the

Stokes decay and the measured decay of Fig. 4?a? ?i.e., the

difference in amplitude between the solid and dashed

curves?. What is noteworthy here is that this difference also

appears to decay exponentially. At early times ?large ampli-

tudes?, a vortex ring is shed from the bob each time it

changes direction ?at the local position extremes?. At later

times, no ring is emitted and the boundary layer ?largely the

inked region? remains attached to the sphere. The point at

which this ejection of vortex rings ceases coincides with the

point where the amplitude decay begins to follow the Stokes

drag law. Thus, we postulate that the excess loss in ampli-

tude, as illustrated in Fig. 4?b?, can be accounted for by the

impulse lost to each shed vortex ring.

We can estimate the excess drag owing to vortex ring

emission in an elementary way. The characteristic momen-

tum of the pendulum is M?A, where M is the hydrodynamic

mass of the bob. This means that the loss of momentum at

each half period is M??A. A certain amount of momentum

is lost owing to Stokes drag.Assuming the excess beyond the

2? at later

FIG. 2. Time sequence showing the shedding of a vortex ring.

Image ?a? shows the bob accelerating toward the right.

10-2

10-1

100

101

102

101

102

103

104

CD

Re

H2O, 293K

Schoepe’s data, 2.1K

Schoepe’s data, 2.2K

Laminar drag

FIG. 3. ?Color online? Drag coefficient as a function of Rey-

nolds number, Re=2RS?A/?. The solid line is the result of Eqs. ?1?

and ?5?

10-2

10-1

100

101

0 50 100150 200250300

Amplitude (cm)

Time (s)

10-3

10-2

10-1

100

101

0 20 4060 80 100

Amplitude difference (cm)

Time (s)

(b)

(a)

FIG. 4. ?a? Amplitude decay of a 5.08 cm pendulum bob and ?b?

its deviation from Stokes drag ?case 6?.

OSCILLATING PENDULUM DECAY OF VORTEX RINGS…

PHYSICAL REVIEW E 81, 046317 ?2010?

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Stokes drag is imparted to the fluid with a characteristic ve-

locity equal to that of the pendulum ?i.e., ?A?, then the mo-

mentum imparted to the ejected fluid is given by M??A

=m?A, where m is the mass of the fluid to which momentum

is imparted each half cycle. This conservation of momentum

principle also works very well for vortex ring guns ?Sullivan

et al. ?23?? This gives

?A

A

= 2m

M,

?7?

where the value of m is the mass of the boundary layer of

thickness ? during oscillation. Using the flat plate analogy,

?=?2?/?, and

m = 4?Rs

2??,

?8?

the hydrodynamic mass of the bob is

M =4

3?Rs

3??s+?

2?.

?9?

We investigated six different cases. The physical proper-

ties are contained in Table I and the corresponding decay

constants are contained in Table II.

For sufficiently large amplitudes, the excess drag beyond

the simple Stokes drag for classical fluids is due to a loss of

impulse to the shed vortex rings. This is a discrete, rather

than continuous, phenomenon. A good analogy would be that

of a bouncing inelastic ball. While the ball will continuously

lose momentum due to drag exerted by the surrounding fluid,

it will also lose momentum during each inelastic collision

with the surface on which it is bouncing.

This phenomenon also explains why for larger amplitude

decay models such as that presented by Digilov et al. ?24?

the amplitude decay rate depends on the initial condition.

This is because the geometric decay rate per oscillation ?A is

proportional to A as given by Eq. ?7?. Of course, at even

larger amplitudes one might expect the flow behind the

sphere to become more complex, requiring an empirical fit as

is done traditionally.

Referring to Fig. 4?b?, we see that

?A/A = ?sP,

?10?

where P is the period, and we can compare the calculated

and observed values of ?sfrom

?sP =2?m

M

,

?11?

where ? is a constant of order unity to be derived from

experiment. The value ? is needed to allow for such effects

as we see in Fig. 2, where the boundary layer is not actually

spherically symmetrical. Comparison to experiment ?see

Table III? yields ?=7.39?1.24.

The fact that ? is considerably larger than expected sug-

gests that considerably more fluid is being shed into the ring

than is given by the boundary layer. Perhaps the roll-up of

the boundary layer entrains external fluid, just as the roll-up

from a piston gun entrains external fluid ?Eq. 2.19 of ?23??.

Clearly, this problem merits further investigation.

TABLE I. Properties of the spherical bob pendulums used in this investigation.

Case

Rs

?cm?

?s

?g/cm3?

m

?g?

M

?g?

?

?s−1?

?

?cm?

1

2

3

4

5

6

1.27

1.91

2.54

1.27

1.91

2.54

8.45

7.65

7.63

8.45

7.65

7.67

2.29

5.13

9.08

1.90

4.33

7.66

76.8

238

561

76.8

238

560

1.58

1.60

1.59

2.27

2.24

2.24

0.113

0.113

0.112

0.0939

0.0945

0.0945

TABLE II. Results from decay experiments for the six cases of Table I. CDcomes from ?l. Since ?s

comes from a difference in amplitude, a drag coefficient has no simple meaning. KCcand Re are the

Keuligan-Carpenter and Reynolds numbers at critical amplitude.

Case

?l

?s−1?

?s

?s−1?

KCc

StRe

CD

1

2

3

4

5

6

0.0240?0.0002

0.0136?0.0005

0.0115?0.0003

0.0225?0.0012

0.0176?0.0005

0.0135?0.004

0.1153?0.0040

0.0665?0.0044

0.0630?0.0030

0.1020?0.0076

0.1001?0.0052

0.0928?0.0037

2.69

1.78

1.34

2.24

1.50

1.13

162

371

653

233

521

919

436

666

876

524

782

1040

0.847

0.654

0.738

0.670

0.714

0.731

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VI. VORTEX RING RADIUS AS A FUNCTION

OF AMPLITUDE

The radius of the ejected rings grows with the amplitude

of swing. We can account for this variation as follows. The

impulse of a ring of radius R is ??R2?. This must be equal to

the momentum in the boundary layer

m?A = ???R2.

?12?

In the spirit of the slug model, we assume the circulation

must have the form ??Rs

boundary layer is given by Eq. ?8?. Thus,

R2= ??A?,

2f, so the mass of fluid in the

?13?

where ? is a constant of order unity to be derived from

experiment. The results are shown in Table IV.

VII. CRITICAL AMPLITUDE FOR VORTEX RING

EMISSION

The critical amplitude Acmust be sufficiently large to

support a viable vortex ring. The core size a of such a ring in

this experiment must be of order ?. The smallest vortex ring

will be given by this largest value of the “slenderness ratio”

?=a/R. Direct numerical simulation by Archer et al. ?25?

suggested ?=0.37 is the largest value beyond which vorticity

from the core will begin to leak out of the bubble. Thus, we

estimate that the smallest value of R will be of order ?/0.37.

Then,

Ac=R2f

4??=

R2

8??.

?14?

Since there are uncertainties in all these estimates, we

simply write

Ac? ??

?15?

and determine ? from experiment. We attempt to determine

Acby using plots such as Fig. 4?b?. If we take the time at

which the amplitude difference falls to some arbitrary mini-

mum ?10−2cm in our analysis? and use that time to identify

Acfrom the plots such as Fig. 4?a?, we find from Table V,

?=7.53?2.46 ?26?. Thus,

Ac? 7.53?2?

?.

?16?

Note that the critical velocities observed by Schoepe’s

group in helium II are not accounted for by Eq. ?16?. The

resultsat2.2K arein

?10−4cm2/s and Ac=3.34?10−3cm with corresponding

velocity 6.1 cm/s. According to Eq. ?16?, the three highest

points in Reynolds number should start to show a break,

which they do. It is gratifying to find that Eq. ?16? holds over

such a large range of scales. The ratio of the radius of our

2-inch bob to the radius of Schoepe’s microsphere is a factor

of 276 and the ratio of the masses of the spheres is 37 mil-

lion.

helium I,where

?=1.80

VIII. DISCUSSION AND CONCLUSION

We conducted a series of controlled experiments where

we measured the decay of a pendulum in water ?subject to

fluid drag?. With these data, we measured the drag coefficient

over a range of Reynolds numbers. These measurements

compared very favorably to those of ?21?, who measured the

drag on a 100 ?m sphere in liquid helium. This illustrates

the true scaling behavior of such a system.

As expected, at smaller amplitudes, the classical Stokes

drag theory works well at describing the decay. For larger

amplitudes where this does not hold, we have identified a

discrete drag mechanism, where the pendulum loses momen-

tum by shedding vortex rings at the maximum amplitudes

while reversing its direction motion. We can estimate the

momentum lost to each of these rings by assuming that the

mass of fluid in the boundary layer surrounding the sphere,

of thickness ?, has the same characteristic velocity as the

bob. To the best of our knowledge, this discrete mechanism

has not previously been identified and only complicated non-

TABLE III. Geometric decay of excess drag.

Case

?A/A

2?m/M

1

2

3

4

5

6

0.458

0.261

0.249

0.283

0.280

0.260

0.440

0.319

0.239

0.365

0.269

0.202

TABLE IV. Comparison of Eq. ?2? to experiment. The value of

? is determined to be 2.96?0.44.

A

0.306?ARexp

?

0.945

1.219

1.440

1.619

1.787

2.115

2.392

2.607

2.638

0.297

0.338

0.367

0.389

0.409

0.445

0.473

0.494

2.58

0.660

0.805

0.985

1.09

1.32

1.41

1.69

1.67

1.65

2.22

2.38

2.68

2.80

3.23

3.17

3.57

3.38

3.20

TABLE

??=7.53?2.46?.

V.Experimentaldeterminationof

?

Case

Ac

?Ac/?

1

2

3

4

5

6

0.786

0.618

1.161

0.347

0.778

0.980

6.96

5.52

10.4

3.70

8.23

10.4

OSCILLATING PENDULUM DECAY OF VORTEX RINGS…

PHYSICAL REVIEW E 81, 046317 ?2010?

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linear fitting curves have been used to attempt to model it.

The relatively large scatter in the experimentally deter-

mined quantities ?, ?, and ? is, in part, the consequence of

using such a simple apparatus. For example, when vortex

rings are ejected, the pendulum is buffeted and only the

2-inch pendulum was massive enough to persist in its path to

measure the drag coefficient in decay. We know from expe-

rience that even vortex rings generated in a careful experi-

ment by a piston have substantial scatter in velocity and

decay owing to the growth of bending waves on the core,

which are very difficult to control ?see Fig. 12 in ?25??.

There are other limitations to our analysis. The boundary

layer on the sphere is not uniform. The vortex rings are not

isolated. The rings are close enough to interact with each

other and the bob. Further, the errors in the critical amplitude

we measure come about because the amplitude at which ring

shedding stops in an experiment does not necessarily coin-

cide exactly with the critical amplitude. The amplitudes

reached by the pendulum are discrete rather than continuous

numbers ?i.e., the amplitude at which ejection stops is the

first amplitude less than or equal to the critical one?. This

may also induce errors in the vortex ring radius predictions.

Nevertheless, we believe much insight is gained from this

simple experiment and we know of no results in the literature

for the onset of discrete vortex emission which have the pre-

dictive power of Eq. ?15?.

ACKNOWLEDGMENTS

We thank Wilfried Schoepe for reading the paper and sug-

gesting changes, as well as supplying his unpublished data at

2.2 K. Joe Vinen encouraged us to do this investigation and

helped clarify a number of points.

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?26? Case 4 of Table V looks like an outlier: it was the least massive

ball and the shortest pendulum and therefore the fewest

swings. If case 4 were omitted, we would have ?

=8.29?1.93, a considerably lower uncertainty.

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