j. fluid mech. (2012), . 693, pp. doi:10.1017/jfm.2011.515 ... · pdf fileexperiments on the...

J. Fluid Mech. (2012), vol. 693, pp. 216–242. c© Cambridge University Press 2012 216doi:10.1017/jfm.2011.515

Experiments on the periodic oscillation of freecontainers driven by liquid sloshing

Andrzej Herczyński1 and Patrick D. Weidman2†1 Department of Physics, Boston College, Chestnut Hill, MA 02467-3811, USA

2 Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309-0427, USA

(Received 15 April 2011; revised 12 November 2011; accepted 16 November 2011;first published online 6 January 2012)

Experiments on the time-periodic liquid sloshing-induced sideways motion ofcontainers are presented. The measurements are compared with finite-depth potentialtheory developed from standard normal mode representations for rectangular boxes,upright cylinders, wedges and cones of 90 apex angles, and cylindrical annuli. Itis assumed that the rectilinear horizontal motion of the containers is frictionless.The study focuses on measurements of the horizontal oscillations of these containersarising solely from the liquid waves excited within. While the wedge and cone exhibitonly one mode of oscillation, the boxes, cylinders and annuli have an infinite numberof modes. For the boxes, cylinders and one of the annuli, we have been able to excitemotion and record data for both the first and second modes of oscillation. Frequenciesω were acquired as the average of three experimental determinations for every fillingof mass m in the dry containers of mass m0. Measurements of the dimensionlessfrequencies ω/ωR over a range of dimensionless liquid masses M = m/m0 are foundto be in essential agreement with theoretical predictions. The frequencies ωR used fornormalization arise naturally in the mathematical analysis, different for each geometryconsidered. Free surface waveforms for a box, a cylinder, the wedge and the cone arecompared at a fixed value of M.

Key words: surface gravity waves, wave-structure interactions

1. IntroductionThe hydrodynamic coupling of liquid sloshing in containers moving in some

constrained manner has been the subject of scientific investigation for more thanseven decades. Much of the original work was concerned with the effect of liquidpropellant sloshing on the stability of ballistics and space vehicles in the 1960s (seeMoiseev 1964; Abramson 1966). Other studies have dealt with disturbances of trucksor ships transporting large partially filled liquid containers to external forcings inducedby a corrugated road on a moving truck or by periodic surface waves on a movingship – see Dodge (2000), Ibrahim (2005) and Faltinsen & Timokha (2009) for anextensive summary of these parametrically forced liquid transport problems.

The motion of a partially filled container supported as a bifilar pendulum is distinctfrom containers supported as a classic pendulum as studied by Moiseev (1953) and

† Email address for correspondence: [email protected]

mailto:[email protected]

Periodic oscillation of free containers driven by liquid sloshing 217

Abramson, Chu & Ransleben (1961). For small excursions of the pendulum fromvertical, containers partially filled with liquid suspended as a bifilar pendulum executenearly horizontal motion with negligible vertical deflection. This problem, studiedanalytically and experimentally by Cooker (1994) in the shallow-water limit, hasrecently generated attention on both sides of the Atlantic. Cooker (1996) provideda model for the dissipation of bores travelling along a suspended cylinder observedwhen the cylinder was released with moderate horizontal extension from its restposition. Weidman (1994, 2005) extended Cooker’s theory to multi-compartmentrectangular boxes and right circular cylinders suspended as bifilar pendula; he alsoobtained considerable experimental data at different pendulum lengths and differentliquid fillings for those geometries. Recently, Ardakani & Bridges (2010) providedan alternative derivation and a Lagrangian representation of Cooker’s problem, stillin the shallow-water limit. Yu (2010) reported finite-depth potential theory forCooker’s problem, for box and cylinder geometries, with two aims: (i) to eliminatethe shallow-water restriction, which assumes the pressure acting on the sidewalls ishydrostatic; and (ii) to display explicitly the effect of evanescent waves in the system.Moreover, Yu (2010), having taken an interest in our experiments on freely movingcontainers driven by liquid sloshing (Herczynski & Weidman 2009), also provided thefinite-depth infinite-pendulum-length limit eigenvalue equations for both the box andcylinder configurations.

In the present investigation we are interested in measuring the motion of a rigidcontainer, free to move horizontally without friction and subject only to pressureforces of the liquid sloshing inside. Motions of the container and the liquid arecoupled and resonate at the same frequency. Their relative amplitudes, however, aswill be shown, depend on the container geometry and the amount of liquid carriedby the vessel. The problem may be regarded as a limiting case of motion of a vesselsuspended as a bifilar pendulum, where the length of the support wires becomesinfinite. The physics of the sloshing pendulum, however, is substantially different fromthat of a free container. In the free-sloshing case, gravity does not directly affect themotion of the container, but only indirectly: the hydrodynamic pressure of the liquidon its walls provides the only restoring force. The sloshing liquid must therefore beout of phase with the oscillating vessel, accumulating always in the direction oppositeto that of the container’s motion. In the case of a container suspended as a bifilarpendulum, gravity provides the restoring force directly, working in concert with or inopposition to the hydrodynamic force, so that both in-phase and anti-phase oscillationscan occur (cf. Cooker 1994; Yu 2010). Note that the amplitude of the fundamental, in-phase mode of oscillations for the suspended container must vanish as the suspensionlength tends to infinity and, consequently, the next anti-phase mode takes on the roleof the fundamental in that limit; see the discussion in Yu (2010).

Our presentation begins with the problem formulation in § 2, followed by finite-depth potential theory solutions for the various geometries in § 3. The experimentalmeasurements given in § 4 are compared with theory, and the paper ends with adiscussion and concluding remarks in § 5.

2. Problem formulation

Consider a container of mass m0 partially filled with liquid of mass m free tomove in frictionless horizontal motion. If X(t) denotes the horizontal position of thecontainer in the stationary laboratory reference frame, then Newton’s second law for

218 A. Herczyński and P. D. Weidman

the container takes the form

m0X = Fp, (2.1)

where

Fp =∫

Sp(n · i) dS (2.2)

is the X-component of the pressure force acting on the container walls. Here p isthe hydrodynamic pressure acting over the wetted surface S of the container, n is theunit normal to S pointing out of the fluid domain and i is the unit vector directedalong the X-axis. Surface tension is neglected, though capillary effects could easily beincluded in the calculation of the sloshing waveforms. It is convenient to calculate thepressure, velocity potential φ and free surface displacement ζ in a coordinate system(x, y, z) attached to the container, with z pointing upwards and z= 0 the position of thequiescent free surface. Linearized potential motion is assumed so that the pressure inthe frame of reference moving with the container is given by

p=−ρ(φt + gz+ xX). (2.3)

Here ρ is the liquid density, g is gravity and −ρxX is the body force due to theacceleration of the container.

The velocity potential, pressure and free surface displacement are determined fromthe solution of the linearized potential flow boundary-value problem:

∇2φ = 0 (in D), (2.4a)

φt + gζ + xX = 0 (z= 0), (2.4b)φz = ζt (z= 0), (2.4c)

n ·∇φ = 0 (on S), (2.4d)

in which D is the domain of the quiescent liquid. For future reference we note thatζ(x, t) may be eliminated from (2.4b,c), yielding the combined kinematic and dynamicfree surface condition

φtt + gφz + x...X = 0 (z= 0). (2.5)

We are interested in container shapes amenable to analytic solution, which usuallyrequire some form of symmetry about the vertical axis or plane. Considered below arerectangular geometries, upright cylinders, cones and wedges with 90 apex angles, andcylindrical annuli. Experiments show that complicated rectilinear motions can arisedepending on how the system is put into motion. With some practice, we have learnedhow to manually excite the containers to yield damped periodic motion with little drift.With ‘improper’ excitation, the container was observed to oscillate while translating.Although these latter cases are certainly interesting from a dynamical systems point ofview, we analyse here only periodic motions of the simplest form

X(t)= X0 cosωt. (2.6)

The important dimensionless parameter for this study is the ratio of liquid mass m todry container mass m0, denoted by

M = m

m0. (2.7)


The goal is to calculate the dimensionless finite-depth frequency of periodic motionω/ωR as a function of M and compare with laboratory experiments. Natural choicesfor the reference frequency ωR will avail themselves for each geometry considered.

The theoretical results, presented here for comparison with experiment, are simplyan implementation of well-known linear modal methods. The theory is essential,however, for distinguishing between the shallow-water and deep-water limits of thefinite-depth theory. The adopted classical modal method inherently includes the effectof evanescent waves but does not explicitly separate out their contribution to thecomposite travelling and evanescent wave solution as in Yu (2010). While oursolutions can be cast in a more compact form compared to those presented in Yu(2010), this is at the expense of slower convergence of the series expressions obtained.Nevertheless, there is no hindrance for determining accurate solutions with the aid ofMathematica (Wolfram 1991).

A comment about a resonance for suspended containers reported by Cooker (1994)is in order. Based on his linear model, Cooker shows that, for a rectangular containerof length L = 2D and width W filled with liquid to depth H, the governing shallow-water equation will be secular when the length of the pendulum satisfies the relation

l= (1+M)D2

n2π2H

(n > 2). (2.8)

Yu (2010) contends that this is not a resonance because ‘the mathematical formulationdoes not include a mechanism for continued energy input to the system’. Indeed, thereis no external forcing of the system independent of the hydrodynamic motion andso there is no mechanism for energy input. While the pressure force due to sloshingcan, like gravity, provide a restoring force allowing for the possibility of the resonantcondition within Cooker’s linear model, the system’s behaviour when condition (2.8) issatisfied has yet to be elucidated mathematically or observed experimentally. However,since the two mechanisms are coupled, suspended containers are not expected toexhibit exponential growth of oscillation amplitude. The point to stress is that, forcontainers moving freely in the horizontal, as considered here, there is only therestoring force of the sloshing liquid, and no other body force to provide resonance.This is clear from the solutions provided in the sequel.

3. Analytical solutions3.1. Rectangular containers

3.1.1. Finite-depth solutionConsider a rectangular box of length L = 2D, width W and mass m0 filled to

depth H with liquid of density ρ. The origin for x is located midway between theendwalls. A modal solution of (2.4a) satisfying the impermeability condition (2.4d) onthe vertical walls at x=±D and on the horizontal bed at z=−H is given by

φ(x, z, t)=∞∑

n=1

Ancosh kn(z+ H)

cosh knHsin knx sinωt, (3.1)

where kn = (2n − 1)π/2D and An are coefficients to be determined. Inspection of (2.5)and (3.1) shows that the coordinate x should be expanded in the Fourier sine series

x=∞∑

n=1

2 (−1)n+1

k2nD

sin knx. (3.2)


Inserting (3.1) and (3.2) into (2.5) then furnishes the coefficients

An =− 2 (−1)n+1 X0ω3

k2nD (gkn tanh knH − ω2)

. (3.3)

Having determined φ(x, z, t) and using X(t) in (2.6), we apply (2.4b) to obtain the freesurface displacement,

ζ(x, t)= 1g

(xX0ω

2 − ω∞∑

n=1

An sin knx

)cosωt. (3.4)

The eigenvalue equation for the oscillation frequency ω is obtained from (2.1), whichrequires computation of the sideways pressure force (2.2) wherein p calculated from(2.3) is given by

p(x, z, t)=−ρ[ω

∞∑n=1

Ancosh kn(z+ H)

cosh knHsin knx+ gz− ω2X0 x

]cosωt. (3.5)

The contributions to (2.2) at the left wall x = −D for which n = −i and at the rightwall x= D for which n= i give the resultant sideways hydrodynamic force

Fp = 2ρW

[DHX0 ω

2 − ω4∞∑

n=1

(−1)n+1 An tanh knH

]cosωt. (3.6)

We note here, and in the sequel, that the hydrostatic pressure component −ρgz giveszero net sideways force. Inserting (3.6) into (2.1) yields, on simplification,

− m0 = 2ρDWH

[1+ 2ω2

∞∑n=1

tanh knH

(knH) (knD)2(gkn tanh knH − ω2)

]. (3.7)

Since the mass of liquid in the container is m = 2ρDWH, we introduce the mass ratiofrom (2.7) to obtain the desired result

1+M

(1+ 2ω2

∞∑n=1

tanh knH

(knH) (knD)2(gkn tanh knH − ω2)

)= 0. (3.8)

This is the finite-depth eigenvalue equation for ω to be determined at each value ofM for given box dimensions. Solutions with waveforms antisymmetric about x = 0appear with increasing frequency: these will be denoted as successive modes for thesloshing-induced motion of the container. The effect of box width W appears implicitlythrough M. In solving the equation it must be remembered that H = H(M). Equation(3.8) includes the effect of evanescent waves.

The eigenvalue equation (3.8) can also be obtained using the formalism proposedby Faltinsen & Timokha (2009) developed for applications to ships carrying liquidloads in their hulls (their equations (5.70), (5.72) and (5.73)). In that approach, thegoverning equations of motion are written in tensorial form, equivalent to Newton’ssecond law in both translational and rotational forms, wherein the net mass is given asthe sum of the vessel’s mass and the frequency-dependent ‘added mass’ including thehydrodynamic effect of sloshing. However, for our one-dimensional problem – withoutroll, pitch, yaw, sway and heave motions of the container – the direct derivationpresented here, for all shapes considered, is simpler and more intuitive.


3.1.2. Shallow- and deep-water limitsThe long-wave limit knH→ 0 gives the shallow-water behaviour for the box. In this

case we make the approximation tanh knH ∼ knH in (3.8) and insert the definition forH = H(M) to obtain

1+M

(1+ 2

∞∑n=1

Z2

α2n(α

2n − Z2)

)= 0, (3.9a)

where αn = (2n− 1)π/2 and

Z = 1M1/2

(ω

ωR

), ωR =

√m0g

2ρD3W. (3.9b)

Using partial fractions and summing the independent terms we find

∞∑n=1

Z2

α2n(α

2n − Z2)

=∞∑

n=1

[1

α2n − Z2

− 1α2

n

]= tan Z

2Z− 1

2. (3.10)

Inserting this into (3.9a) yields the shallow-water eigenvalue equation

Z +M tan Z = 0. (3.11)

The reference frequency ωR provides the natural non-dimensionalization for oscillationfrequency ω, since boxes of dimensions L = 2D and W with dry masses m0 must allexhibit the same ω/ωR behaviour in the shallow-water limit.

The deep-water frequencies are found by the simple expedient of setting tanh knH =1 in the numerator and denominator of (3.8). This yields

1+M + 2σ 2β2∞∑1

1α3

n(αn − β2)= 0, σ 2 = 2ρD2W

m0, (3.12)

where β = ω/ω0 and ω0 = √g/D. Summing this series with the aid of Mathematicawe obtain the implicit solution for the deep-water behaviour,

M = σ 2

π3β2 − 2π2ψ

(12

)+ 2π2ψ

(12− β

2

π

)− β4ψ (2)

(12

)π3β4

− 1, (3.13)

where ψ(z) and ψ (2)(z) are the digamma and tetragamma functions defined inAbramowitz & Stegun (1972).

3.1.3. Sample solutionsSolution curves obtained from the finite-depth result along with the limiting shallow-

and deep-water representations for two box geometries are shown in figure 1. Forthe box, cylinder and annular geometries, we tested convergence with respect to thenumber of spatial modes and found that including more than 10 modes producedno change in results to three or four decimal places; for insurance on numericalaccuracy we used 15 modes in all our calculations using Mathematica. For this boxexample we take the nominal values ρ = 1.0 g cm−3 and g = 1000 cm s−2. The boxof square planform has L = W = 20 cm, m0 = 1000 g with shallow-water referencefrequency ωR = 5.0 rad s−1, while the box of rectangular planform has L = 40 cm,W = 20 cm and m0 = 2000 g for which ωR = 2.5 rad s−1. The dashed line at low M


0 2 4 6 8 10

1

2

3

4

5

Rectangular box

Square box

M

FIGURE 1. Normalized oscillation frequencies for two box geometries computed from (3.8):a box of square planform (L = 20 cm, W = 20 cm, m0 = 1000 g) and a box of rectangularplanform (L = 40 cm, W = 20 cm, m0 = 2000 g). The dashed lines exhibit the commonshallow-water asymptotes at small M computed from (3.11) and the distinct deep-waterasymptotes at high M computed from (3.13) with tanh knH = 1. The shallow-water referencefrequencies for the square and rectangular boxes computed from (3.9b) are ωR = 5.0 rad s−1

and ωR = 2.5 rad s−1, respectively.

is the common shallow-water limit given as solution of (3.11) and the two dashedlines at high M are the deep-water behaviours computed from (3.13). Note that bothresponse curves exhibit a value of M at which the frequency is maximum, andthis feature is more evident for the square box. For the square box, the maximumω/ωR = 3.031 47 occurs at M = 3.84 and this corresponds to a dimensional frequencymaximum ω = 15.16 rad s−1. For the rectangular box, the maximum ω/ωR = 4.458 84occurs at M = 6.65 and this corresponds to a dimensional frequency maximumω = 11.15 rad s−1, smaller than that for the square box. Note that both frequencycurves merge smoothly to the same shallow-water limit even though the two boxeshave different masses and dimensions.

3.2. Cylindrical containers3.2.1. Finite-depth solution

Now consider a circular cylinder of radius R and dry mass m0 filled with liquid todepth H. Cylindrical coordinates (r, θ, z) in the moving frame are now incorporatedand the linearized problem is still that given by (2.4) but with x replaced by r cos θ .A finite-depth solution form for the velocity potential satisfying Laplace’s equation(2.4a) and the impermeability condition (2.4d) on the vertical wall at r = R and on thehorizontal bed at z=−H is given by

φ(r, θ, z, t)=∞∑

n=1

AnJ1(knr)cosh kn(z+ H)

cosh knHcos θ sinωt, (3.14)

where J′1(knR)= 0 fixes the radial wavenumbers kn and J1 is the Bessel function of thefirst kind. Expanding r in the Fourier–Bessel series

r =∞∑

n=1

2knR2

(k2nR2 − 1)

J2(knR)

J21(knR)

J1(knr), (3.15)


and inserting this expression and (3.14) into (2.5) determines the coefficients as

An =− 2αnRω3X0

(α2n − 1)J2

1(αn)

J2(αn)

(gkn tanh knH − ω2), (3.16)

where αn = knR. The free surface displacement determined from (2.4b) is

ζ(r, θ, t)= 1g

[ω2X0 r − ω

∞∑n=1

AnJ1(knr)

]cos θ cosωt. (3.17)

For calculation of the sideways pressure force, the outward normal to the verticalsidewall is n= er, where er is the radial unit vector. Since er · i= cos θ , the expressionfor the pressure force in (2.2) in the cylindrical geometry is

Fp =∫ 2π

0

∫ 0

−Hp(R, θ, z, t) cos θ R dθ dz. (3.18)

Using (2.6) and (3.18), the linearized pressure (2.3) evaluated at the cylindrical wall is

p(R, θ, z, t)=−ρ[ ∞∑

n=1

AnωJ1(knR)cosh kn(z+ H)

cosh knH+ gz− Rω2X0

]cos θ cosωt, (3.19)

and carrying out the integral in (3.18) gives

Fp = ρπR

[Rω2HX0 −

∞∑n=1

Anω J1(αn) tanh knH

kn

]cosωt. (3.20)

Inserting (3.20) into the equation of motion (2.1) yields, on simplification andidentifying m= ρπR2H as the fluid mass, the finite-depth eigenvalue equation

1+M

(1+ 2ω2 R

H

∞∑n=1

J2(αn)

(α2n − 1)J1(αn)

tanh knH

(gkn tanh knH − ω2)

)= 0, (3.21)

where again it must be kept in mind that H = H(M). As in the case of the rectangularcontainer, eigenvalue equation (3.21) for the cylinder can, alternatively, be derivedusing the ‘added mass coefficients’ in the method of Faltinsen & Timokha (2009).

3.2.2. Shallow- and deep-water limitsThe shallow-water limit is obtained from (3.21) by replacing tanh knH with knH and

incorporating the definition for H = H(M). This yields

1+M

(1+ 2

∞∑n=1

αnJ2(αn)

(α2n − 1)J1(αn)

Z2

(α2n − Z2)

)= 0, (3.22a)

where J′1(αn)= 0 and

Z = 1M1/2

(ω

ωR

), ωR =

√m0g

2ρπR4. (3.22b)

Though we did not analytically sum this series, we find that the term in parenthesesin (3.22a) is numerically equal to J1(Z)/ZJ′1(Z). Inserting this into (3.22a) we find theshallow-water eigenvalue equation for the cylinder

ZJ′1(Z)+MJ1(Z)= 0. (3.23)


This is in agreement with the infinite-pendulum-length limit Ω → 0 of equation (35)in Yu (2010).

As with the rectangular box, the deep-water eigenvalue equation for the cylinder isobtained by replacing tanh knH in (3.21) with unity. We do not attempt to sum theresultant series for this case, but simply compute the deep-water behaviour using thedeep-water series representation.

3.3. Wedge geometry

Cooker (1994) presented a potential theory solution for a suspended planar hyperboliccontainer. The asymptotes of these hyperbolae form a wedge with apex angle β = π/2.The streamlines for this geometry coincide with those depicted by Lamb (1932,§ 258). For this limiting geometry, Cooker presented a formula for the frequencyof suspended container motion that exhibits two frequencies, the lower (higher) ofwhich corresponds to wave oscillations in phase (anti-phase) with the motion of theoscillating tank. Anxious to confirm his result, one of the present authors (P.D.W.)performed several experiments at different pendulum lengths, only to find that thetheory consistently over-predicted the experimental measurements for all values of M.Communication with Cooker led to a correction of the theory (M. J. Cooker, 2009,personal communication) for this case, wherein application of the hydrostatic pressureat the sidewalls is replaced by the potential pressure as given in (2.3). The revisedtheory then gave predicted frequencies in accord with the experiments. In hindsight, itis clear that all frequencies for the motion of the wedge with π/2 apex angle must begoverned by just one formulation because the streamlines for each liquid mass placedin the wedge are self-similar: for the 90 wedge studied here the formulation must befinite-depth potential theory. It is possible that the motion in suspended wedges withmuch larger apex angles π/2 β < π will be adequately described by shallow-watertheory, but no antisymmetric solutions are known for values of β > π/2. A case inpoint is a wedge with included angle β = 2π/3, for which only symmetric waveformsolutions are known (see Haberman, Jarski & John 1974). Nevertheless, accurateestimates of the sloshing frequencies for wedges of any angle β < π may be foundusing conformal transformation techniques (see Davis & Weidman 2000).

The following result for the frequency of free motion of the π/2 wedge coincideswith the infinite-pendulum-length limit of the corrected theory due to M. J. Cooker,2009 (personal communication) mentioned above. Consider a wedge of mass m0,width W and apex angle bisected by the vertical axis aligned with gravity. In themoving reference frame with z = 0 located at the mid-point of the quiescent liquidsurface, the wetted container shape z=−h(x) is given by

h(x)= H + x (−H 6 x 6 0), h(x)= H − x (0 6 x 6 H). (3.24)

Posited solution forms for φ satisfying Laplace’s equation (2.4a) and free surfacedisplacement ζ satisfying (2.4c) are given by

φ(x, z, t)=−ζ0

Hωxz sinωt + C(t)x, (3.25a)

ζ(x, t)= ζ0

Hx cosωt, (3.25b)

where C(t) is an arbitrary function. It is clear from (3.25b) that the posited solutionrepresents a periodically oscillating free surface that is always planar. Inserting (3.25a)


into free surface condition (2.5) yields, upon integration,

C(t)=− gζ0

ωHsinωt − X + at + b, (3.26)

where a and b are constants. For the assumed periodic motion (2.6), we can takea= b= 0 without loss of generality. Computing X using (2.6) and inserting (3.26) into(3.25a) gives the potential function

φ(x, z, t)= x

[X0ω − gζ0

ωH− ζ0ω

Hz

]sinωt. (3.27)

The outward normals to the left and right walls are

n=− (i+ k)√2

(−H 6 x 6 0), n= (i− k)√2

(0 6 x 6 H), (3.28)

where i and k are unit vectors aligned with the x- and z-coordinates, respectively. Theimpermeability condition (2.4d) then yields

ζ0

X0= 1

g

ω2H− 1

. (3.29)

We refer to ζ0/X0 as the amplification ratio: for a given sideways amplitude X0 of thewedge, a maximum deflection ζ0 of the liquid in the container is realized.

Using (2.6) and (3.27), the pressure

p(x, z, t)= ρ[ζ0

H(g+ ω2z)x cosωt − gz

](3.30)

is obtained from (2.3). Note that the terms proportional to the amplitude X0 ofsideways motion cancel, but the pressure still depends on X0 through the amplificationratio (3.29). Again, the hydrostatic contribution to (2.2) is zero and the remainingterms give the sideways component of the pressure force,

Fp = ρWH2

(g

H− 1

3ω2

)ζ0 cosωt. (3.31)

Inserting (3.31) into the governing equation of motion (2.1) yields

ω2m0X0 = ρWH2

(ω2

3− g

H

)ζ0. (3.32)

Since the wedge carries fluid mass m = ρWH2, a second expression for themagnification ratio

ζ0

X0= 1

M

113− g

ω2H

(3.33)


is obtained. Equating (3.29) and (3.33) gives

ω2 = g

H

1+M

1+ M

3

. (3.34)

We eliminate H using H =√m0M/ρW to arrive at the exact solution

ω

ωR= 1

M1/4

√√√√ 1+M

1+ M

3

, ωR =(ρg2W

m0

)1/4

(3.35)

for the sloshing-induced oscillation frequency of the free wedge. It is pertinent toobserve that this ωR is not a shallow-water reference frequency, for there is no shallow-water limit for this wedge geometry; the solution for all M is a finite-depth solution.

Note that the singular behaviour in (3.29) appears only when the containers areempty so that no sloshing-induced motion can occur, namely when ω2 = g/H, whichmeans, according to (3.34), that M = 0. Equation (3.33) is never singular except atM = 0, since the case ω2 = 3g/H is excluded by (3.34).

3.4. Cone geometry

Now we consider a cone of mass m0 and apex angle π/2. Cylindrical coordinates(r, θ, z) in the moving reference frame are used with the conical z-axis antiparallel togravity. The analysis follows closely that for the 90 wedge. We find that the potentialfunction and the free surface displacement are given by the wedge solutions (3.27) and(3.25b) with x replaced by r cos θ. As with the 90 wedge, the free surface is alwaysflat. The wetted surface of the cone is given by z=−h(r), where h(r) and the outwardnormal to the surface n are given by

h(r)= H − r, n= (er − ez)√2

, (3.36)

in which er and ez are unit vectors pointing along positive r and z, respectively.Imposition of the impermeability condition (2.4d) on the conical wall yields amagnification ratio identical to (3.29). The linearized pressure field is identical to(3.30), when x is replaced by r cos θ . Evaluating this pressure on the boundary h(r),inserting it into (2.2) and performing the integration yields a sideways force differentfrom that for a wedge, viz.

Fp = πρζ0H3

[( g

3H− ω2

)+ ω

2

4

]cosωt. (3.37)

Inserting this into (2.1), and identifying the fluid mass in the cone as m= ρπH3/3, wearrive at an expression for the magnification ratio for the cone:

ζ0

X0= 1

M

114− g

ω2H

. (3.38)


Equating (3.29) and (3.38) and eliminating H in favour of M furnishes the exactsolution

ω

ωR= 1

M1/6

√√√√ 1+M

1+ M

4

, ωR =(ρπg3

3m0

)1/6

(3.39)

for the oscillation frequency of the freely moving cone with apex angle π/2 radians.The similarity with the wedge solution given in (3.35) is apparent.

3.5. Annular containersWe now consider the annular region between two vertical right concentric circularcylinders. The original work on this problem dates back to Sano (1913), who studiedthe seiching motion observed in a circular lake with central circular island (see also,Campbell 1953; Bauer 1960). The annulus of inner radius R1, outer radius R2 and drymass m0 is filled with liquid to depth H. We denote η = R1/R2 as the radius ratio. Aswith the cylinder, we use (r, θ, z) for the coordinate system attached to the sideways-moving container and align the z-coordinate with the axis of the concentric cylinders.It is clear that any description of the free surface deflection for sloshing between thecylinders must include higher-order azimuthal (e.g. cos nθ ) terms, particularly in thenarrow gap limit – the free surface cannot slosh back and forth in vertical planes asit does in the fundamental sloshing mode of a cylinder with a single nodal diameter.Nevertheless, one can compute the liquid–structure interaction for purely rectilinearmotion of the annulus owing to the orthogonality of trigonometric functions. We thusproceed to determine the frequency of motion of the system by retaining only thelowest azimuthal dependence, cos θ , realizing that computation of the time-dependentfree surface will not be available at this level of analysis. A finite-depth solutionform for the velocity potential that satisfies Laplace’s equation and the impermeabilitycondition on the bottom boundary is given by

φ(r, θ, z, t)=∞∑

n=1

[CnJ1(knr)+ DnY1(knr)]cosh kn(z+ H)

cosh knHcos θ sinωt, (3.40)

where J1 and Y1 are Bessel functions of the first and second kind. Satisfyingthe impermeability condition on the inner and outer walls gives two homogeneousequations for Cn and Dn, the determinant of coefficients of which provides theeigenvalue equation for the radial wavenumbers kn, namely

J′1(knR1)Y′1(knR2)− J′1(knR2)Y′1(knR1)= 0. (3.41)

Thus the potential (3.40) may be written in the form

φ(r, θ, z, t)=∞∑

n=1

AnP(r)cosh kn(z+ H)

cosh knHcos θ sinωt, (3.42a)

where

P(r)= Y′1(knR1)J1(knr)− J′1(knR1)Y1(knr). (3.42b)

We now expand r in the Fourier–Bessel series,

r =∞∑

n=1

BnP(r), (3.43)


which, after a lengthy but straightforward calculation, gives

Bn = 2knR21Y′1(knR1)[J2(knR2)− η2J2(knR1)] − J′1(knR1)[Y2(knR2)− η2Y2(knR1)]

η2P2(knR2)[(knR2)2−1] − P2(knR1)[(knR1)

2−1] .

(3.44)

Inserting (3.42) and (3.43) into the combined free surface condition (2.5) determinesthe coefficients

An =− ω3X0

(gkn tanh knH − ω2)Bn. (3.45)

For calculation of the sideways pressure force, the outward normal to the innersidewall is ni = −er and that to the outer wall is no = er, where er is the unit vectoralong r. Thus ni · i = − cos θ and no · i = cos θ , yielding the following expression forthe resultant sideways hydrodynamic force on the annulus:

Fp =∫ 2π

0

∫ 0

−H[p(R2, θ, z, t)R2 − p(R1, θ, z, t)R1] cos θ dθ dz. (3.46)

The linearized pressure is

p(r, θ, z, t)=−ρ[ ∞∑

n=1

AnωP(r)cosh kn(z+ H)

cosh knH+ gz− rω2X0

]cos θ cosωt. (3.47)

Evaluation of (3.46) gives

Fp = ρπR1

[X0R1H

(1− η2)

η2ω2 − ω

∞∑n=1

An[η−1P(R2)− P(R1)] tanh knH

kn

]cosωt, (3.48)

and insertion into Newton’s equation of motion (2.1) using (2.6) yields

−m0ω2 = ρπR1

[ω2HR1

(1− η2)

η2

+ω4∞∑

n=1

[η−1P(R2)− P(R1)]Bntanh knH


]. (3.49)

The fluid mass in the annular domain is m = ρπR21H(1 − η2)/η2 so using (2.7) we

obtain the desired eigenvalue equation for the frequency of tank motion as

1+M

(1+ ω2 η

(1− η2)

∞∑n=1

Bn

knR1H[P(R2)− ηP(R1)] tanh knH


)= 0, (3.50)

where the coefficients Bn are given in (3.44). A long, tedious calculation in the limitR1→ 0 shows that this result reduces to eigenvalue equation (3.21) for a cylinder, asexpected.

3.5.1. Sample solutionsThere now arises the manner in which we normalize the frequencies computed. We

have not determined the shallow-water behaviour for the annulus, and since it probablydepends on the radius ratio η, we choose to normalize all frequencies with the shallow-water value ωR for a cylinder of radius R2 with a dry container of mass m0. To exhibit


1

2

3

4

5

6

0 1 2 3 4 5 6M

0.2

0.40.60.8

FIGURE 2. Normalized oscillation frequencies computed from (3.50) for selected radiusratios of an annulus (R2 = 20.0 cm, R1 = 0, 4.0, 8.0, 12.0, 16.0 cm, and m0 = 2000 g). Theshallow-water reference frequency computed from (3.22b) is ωR = 1.995 rad s−1.

η M ω (rad s−1)

0.2 0.700 4.0860.4 3.56 3.8000.6 5.16 1.740

TABLE 1. Crossing points of the η = 0.8 frequency curve with those at η = 0.2, 0.4, 0.6for the annulus calculations displayed in figure 2.

sample results, we choose annuli with radius ratios η = 0, 0.2, 0.4, 0.6, 0.8, eachwith identical dry masses m0 = 2000 g. The selected radius is R2 = 20 cm, whichgives R1 = 0, 4.0, 8.0, 12.0, 16.0 cm. As with the sample box calculations given infigure 1, we choose the nominal values ρ = 1.0 g cm−3 and g = 1000 cm s−2, whichgive ωR = 1.9947 rad s−1 computed from (3.22b) for a cylinder (η = 0). The resultsdisplayed in figure 2 are somewhat surprising. None of the curves at low η cross anyother up to η = 0.6, but the curve for η = 0.8 crosses those for η = 0.2, 0.4, 0.6twice, yet it does not cross the frequency curve for the cylinder at η = 0. The uppercrossing points are listed in table 1. A more detailed investigation shows that, as ηdecreases from η = 0.8, the two crossing points move towards each other to form asingle point of tangency. Our estimate is that the point of tangency occurs at η ' 0.56when M ' 1.6. Thus the crossing phenomena exist only for η > 0.56. For η very small,the bulk of the fluid sloshes back and forth in vertical planes, but as η increases, largeazimuthal excursions of the fluid around the inner cylinder take place. The criticalcrossing point at η ' 0.56 evidently heralds this transition in sloshing behaviour.

4. ExperimentsInitial experiments for the cylindrical geometry were carried out by P.D.W. using

both flat and V-shaped air-bearing tables available at the University of Colorado. Thefrequencies obtained at one liquid filling measured using a stopwatch were found to be


L (cm) W (cm) Hb (cm) R (cm) R1 (cm) R2 (cm) m0 (g) %Mloss

Large box 24.74 8.255 7.5 — — — 886.5 0.76Tall box 14.67 9.59 14.7 — — — 922.4 0.33Large cylinder — — 7.5 13.02 — — 1129 0.76Tall cylinder — — 17.6 7.335 — — 1086 0.27Wedge — 24.75 12.3 — — — 1138 0.82Cone — — 10.5 — — — 1228 1.48Annulus(η = 0.364)

— — 7.5 — 4.73 12.98 1207 0.76

Annulus(η = 0.777)

— — 15.2 — 10.113 13.013 1644 0.32

TABLE 2. Container dimensions, dry masses m0 and maximum percentage change in Mdue to evaporation.

10–15 % lower than those predicted by finite-depth theory. Seubert & Schaub (2010)have shown that air-bearing tables do not provide frictionless motion of the containerowing to the fact that the container moves back and forth into the air stream. Seubert& Schaub (2010) have realized nearly frictionless motion by incorporating a feedbackcontrol that permits air support only from holes directly beneath the container, cuttingoff pressure to all other holes. However, a much simpler apparatus available at BostonCollege proved entirely adequate.

The experimental set-up consisted of a low-friction cart and a 1.2 m long aluminiumtrack commercially available from PASCO (specializing in physics apparatus forteaching laboratories). The cart has a mass of ∼0.5 kg and is outfitted with fourknife-edge wheels rotating freely on high-quality ball bearings, which are attachedto the chassis via a suspension system with springs above each wheel. The cart’swheels fit snugly into two parallel grooves along the track, which could be accuratelylevelled using four adjustment screws to assure one-dimensional, horizontal motionwith minimal mechanical resistance.

A small plastic insert was fastened with two screws to the cart inside the hollow onits upper surface (designed to carry extra masses). The insert provided a flat surfaceon which each container could be attached using strong, double-sided adhesive tape.This mounting system proved reliably rigid and allowed us to attach and detach thecontainers with ease. However, we were limited in the maximum weight on the cart’swheels to less than 40 N, since beyond this load the springs began to give in andbecame extremely soft, making the cart wobbly and subject to transverse oscillations.Since our typical dry mass m0 (cart plus insert plus dry container) was in the range900–1600 g, our containers could be filled with roughly 2–3 kg of water, dependingon the container being tested. We were also limited by the brimful heights Hb of thecontainers (see table 2) and could fill them only up to about 2 cm below the rim inorder to prevent spilling during the back-and-forth motion of the cart.

With the system set in motion, its position was recorded using a PASCO motionsensor aligned with, and located 30–40 cm from, the end of the cart. The motionsensor works by repeatedly sending bursts of 49 kHz ultrasonic pulses and measuringthe time they take to reflect back from the moving cart. The sensor is connected toa computer via a PASCO universal interface (ScienceWorkshop 750) and the positionversus time data can be saved in tabular form and/or displayed graphically on acomputer screen. We used the sampling rate of 100 or 120 Hz and the position data


0.420

0.422

0.424

0.426

0.428

0.430

0.432

0.434

0.436

0.502

0.504

0.506

0.508

0.510

0.512

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9t (s)

X (m)

X (m)

(a)

(b)

FIGURE 3. Damping of horizontal motions initiated for (a) the tall box at M = 1.66 and(b) the large box at M = 1.52. Details for the tall and large boxes are given in table 2.

were obtained with the nominal accuracy of ±0.001 m; in practice, the measurementsare reliable at least to ±0.1 mm. To determine the frequency of the cart’s oscillations,we read the elapsed time over multiple cycles (peak-to-peak or trough-to-trough), andaveraged the resultant periods over three separate runs for each liquid filling. We madesure that there was a few seconds delay between releasing the cart and the start ofthe recording so that most transients would attenuate. We also ignored the first fewrecorded cycles, and the ringing at the end of each run with very small amplitudes; seefigure 3(b).

Each container was fabricated from transparent lucite to minimize the total weightm0 of the tank plus the cart. The disadvantage is that the damping of standing wavesin lucite containers is much greater compared to that in glass containers (Keulegan1959). Boxes and wedges had sidewalls composed of 1/8 inch plate, while thecylindrical and annular geometries were composed of 1/8 inch wall cylindrical stock.The bottom surfaces were fabricated from 3/16 inch or 1/4 inch plate. The wedgewas cemented to a base platform 5.1 cm × 7.6 cm. The cone and its 7.6 cm diametercylindrical base were machined as a single unit from solid cylindrical stock. Relevantdetails of the containers are given in table 2 in which L = 2D for the boxes, Hb is thebrimful height of a container and m0 is the dry mass of the system, i.e. the mass ofthe cart and insert, double-sided adhesive tape and Plexiglas container. To verify thatthe liquid volume remained reasonably constant, we have measured the evaporationrate of water from our containers. We found a very linear rate of evaporation perunit surface area with time, the value being Revap = 7.00 × 10−3 g h−1 cm2. For themaximum six-hour period over which frequency measurements were made, we canestimate the maximum percentage change in M for each container, %Mloss, and thesedata are also included in table 2.


Containers partially filled with water were set into motion by manually oscillatingthe system and setting it free. Two traces of recorded container motion are presentedin figure 3. Figure 3(a) for the tall cylinder at M = 1.66 shows a complicatedresponse of the system that can arise depending on the initial conditions, in thiscase a superposition of two distinct frequencies. More typical were traces resultingfrom the superposition of oscillations in the lowest mode with a nearly constantspeed translation of the centre of mass (drift). None of these more complicated traceswere deemed usable for our measurements. We relied on regular, single-frequencyoscillatory traces with minimal drift, such as that shown in figure 3(b), taken usingthe tall box at M = 1.52. We recognize that a sinusoidally driven linear actuator couldhave been devised to place the cart supporting the container into motion at preciselythe expected frequency for each liquid filling. Sophisticated control systems have beenused, for example, to prevent sloshing of liquid moved in an open container as it iscarried by a robotic arm (see Feddema et al. 1996). However, as figure 3(b) illustrates,the desired single-frequency oscillation mode can be obtained manually with somepractice. The drawback of this approach was that many runs had to be discarded sincethey had unacceptable drift. For any particular configuration, our batting average for agood, usable run was about one out of three attempts when exciting the fundamentalmode, and perhaps one out of 10 when trying to excite the second mode. The cart’soscillation frequency was determined by measuring the elapsed time for 3–10 cycles,always at low oscillation amplitude to stay within linear theory.

Below, we present measurements for the geometries tested. In some casesmeasurements could be made of the second mode of oscillation, but in no casewere we able to excite the third or higher modes, presumably because its frequencywould be too high, and its amplitude too low, to initiate manually. In the theoreticalcomputations we used the local value of the gravitational constant g= 980.366 cm s−2

provided to us by a colleague in the Department of Earth and Environmental Sciencesat Boston College. The density was taken to be that for pure water at the average roomtemperature for the experiments, namely ρ = 0.9977 g cm−3.

4.1. Rectangular containersWe have determined the number of terms necessary in eigenvalue equation (3.8) toachieve three-decimal-place accuracy for the large box and compared the results withthose using the formulation of Yu (2010). We find that 15 terms are needed in ourmodal expansion while only six terms are needed using equation (19) of Yu (2010) toattain this level of accuracy.

Results for the large box determined from (3.8) are shown in figure 4 in whichthe solid lines are the numerically computed frequencies for the first two modes overthe range 0 6 M 6 2.0. Visible in this figure is the mode 2 maximum ω/ωR = 3.7269at M = 1.900 corresponding to a maximum frequency ω = 19.678 rad s−1. Mode 1oscillations were recorded over a wide range of liquid fillings 0.5 < M < 1.5; belowM = 0.5 there was too little liquid mass to excite well-defined oscillations, and aboveM ≈ 1.5 liquid sloshed out of the container. We were also able to initiate mode 2oscillations, but only over a narrow range centred about M = 1. As will be seenwith the other geometries, the mode 1 frequency measurements agree better withtheory than the mode 2 measurements. It was always the case that the amplitudesof the mode 2 oscillations were considerably diminished compared with those ofthe fundamental mode and, as a result, a relatively small number of well-defineddamped oscillations were observed for the higher mode. Nevertheless, the agreement isconsidered to be very good for both modes.


0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 0.5 1.0 1.5 2.0M

Mode 2

Mode 1

FIGURE 4. Normalized oscillation frequencies for the first two modes of the large boxcomputed from (3.8) (solid lines) and corresponding experimental data (solid symbols). Theshallow-water reference frequency computed from (3.9b) is ωR = 5.280 rad s−1; details forthe large box are given in table 2.

0 0.5 1.0 1.5 2.0M

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

FIGURE 5. Normalized oscillation frequencies for the first mode of the tall box computedfrom (3.8) (solid line) and corresponding experimental data (solid symbols). The referencefrequency computed from (3.9b) is ωR = 10.943 rad s−1; details for the tall box are given intable 2.

In an effort to observe the frequency maximum, we fabricated the tall box.Theoretical and experimental results for the first oscillation mode of this containerare displayed in figure 5. The maximum ω/ωR = 1.4796 in the numerical calculationoccurs at M = 1.47; this corresponds to a maximum frequency ω = 16.191 rad s−1.Owing to the relatively high frequencies associated with the first mode in the tall box,the agreement with theory is not as good as with mode 1 in the large box (cf. figure 4).Nevertheless, the experimental results track the theoretical curve fairly well, even if themaximum frequency cannot be discerned in the measurements shown in figure 5.


M

Mode 2

Mode 3

Mode 1

1

2

3

4

5

6

7

0 0.5 1.0 1.5 2.0 2.5 3.0

FIGURE 6. Normalized oscillation frequencies for the first three modes of the large cylindercomputed from (3.21) (solid lines), the distinct shallow-water asymptotes computed from(3.23) (dashed lines) and the experimental data for modes 1 and 2 (solid symbols). Theshallow-water reference frequency computed from (3.22b) is ωR = 3.505 rad s−1; details forthe large cylinder are given in table 2.

4.2. Cylindrical containersWe have determined the number of terms necessary in eigenvalue equation (3.21) toachieve three-decimal-place accuracy for the large cylinder and compared the resultswith those using the formulation of Yu (2010). We find that 12 terms are needed in ourmodal expansion while only five terms are needed using equation (30) of Yu (2010) toattain this level of accuracy.

Numerical and experimental results computed from (3.21) for the first two modesof oscillation for the large cylinder are shown in figure 6. In this case we take theopportunity to display theoretical results for mode 3 oscillations. The dashed lines,representing the low-M asymptotic solutions, show that each successive mode has itsown shallow-water behaviour. The maxima for modes 1 and 2 occur beyond M = 3but the maxima for mode 3 occurs in the plotted region at M = 2.61 with the valueω/ωR = 7.2828, corresponding to ω = 25.53 rad s−1. In comparison to the large box,we note that experimental data may be gathered over wider ranges of M for bothmode 1 and mode 2 free oscillations. Again, the agreement with theory is better forthe larger-amplitude mode 1 oscillations compared to mode 2.

In order to try to capture a maximum in the frequency oscillation curve, wedesigned the tall cylinder. The numerical and experimental data for the mode 1response in this geometry are shown in figure 7. The maximum in the frequencycurve occurs at M = 1.31 with the value ω/ωR = 1.612 44, corresponding toω = 17.46 rad s−1. It is clear that the measurements track the theoretical curve veryclosely; however, as with the tall box, the maximum is not clearly defined in theexperimental data.

The sloshing-induced motion of a cylindrical container has been observed in anatural setting; see the Appendix.

4.3. Wedge geometryTheoretical and experimental results for the wedge with 90 apex angle are shown infigure 8. The theory for the single mode possible in this geometry is that given in


0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 0.5 1.0 1.5 2.0M

FIGURE 7. Normalized oscillation frequencies for the first mode of the tall cylindercomputed from (3.21) (solid line) and the corresponding experimental data (solid symbols).The shallow-water reference frequency computed from (3.22b) is ωR = 10.833 rad s−1; detailsfor the tall cylinder are given in table 2.

(3.35). For this container, data could be obtained up to M ≈ 2.8, above which fluidsloshed out of the wedge. The agreement between experiment and theory is consideredgood, but we now see a clear trend – theory and experiment are generally in betteragreement for rotationally symmetric geometries compared to planar geometries (boxand now the wedge). Note in this case that ω/ωR ∼ M−1/4 as M→ 0, in distinctcontrast to the box and cylinder geometries. But one must bear in mind that as M→ 0the liquid mass m in the container tends to zero, so there is no liquid to excite thesideways periodic container motion in this limit. For this experiment the referencefrequency calculated from (3.35) is ωR = 12.017 rad s−1.

4.4. Cone geometryTheoretical and experimental results for the cone with 90 apex angle are shownin figure 9. The theoretical frequencies for the only mode in this geometry is thatgiven in (3.39). For this container, data could be obtained only up to M ≈ 0.7, abovewhich fluid sloshed out of the cone. The agreement between experiment and theoryis considered excellent and supports the trend observed previously that theory andexperiment are generally in better agreement for axisymmetric geometries (cylinderand now the cone) compared to planar geometries (box and wedge). For the cone,ω/ωR ∼M−1/6 as M→ 0, but again in this limit there is no liquid in the cone to excitethe horizontal oscillations. For this experiment, the reference frequency calculatedfrom (3.39) is ωR = 9.638 rad s−1.

4.5. Annular containersWe now present results for the sloshing-induced motions of partially filled concentriccylindrical annuli. Experimental data and theoretical calculations for the first twomodes of sideways oscillation at η = 0.3644 are displayed in figure 10. While there isno maximum in the plotted range of M for mode 1, mode 2 displays a maximum atM = 2.20 with value ω/ωR = 6.1798 corresponding to ω = 21.66 rad s−1. Agreementbetween theory and experiment for both modes is considered excellent. As mentioned


0.5

1.0

1.5

2.0

0 0.5 1.0 1.5 2.0 2.5 3.0M

FIGURE 8. Normalized oscillation frequencies for the 90 wedge computed from (3.35)(solid line) and corresponding experimental data (solid symbols). The small-M asymptoteM−1/4 (dashed line) is also shown. The reference frequency computed from (3.35) isωR = 12.017 rad s−1; details for the wedge are given in table 2.

0.5

1.0

1.5

2.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8M

FIGURE 9. Normalized oscillation frequencies for the 90 cone computed from (3.39)(solid line) and corresponding experimental data (solid symbols). The small-M asymptoteM−1/6 (dashed line) is also shown. The reference frequency computed from (3.39) isωR = 9.638 rad s−1; details for the cone are given in table 2.

in § 3.5, we cannot determine the free surface deflection for the annulus using only thefirst azimuthal mode in the analysis. However, we could observe damped oscillationsin the annular region of wave sloshing and it was very interesting indeed. The freesurface signature of the motion revealed waves propagating around opposite sidesof the annulus that ultimately met in head-on collisions at θ = 0,π. Collisions withsplashing were observed only during the first couple of oscillations for which thewave amplitudes were relatively large. The splashing observed in our experiment is


1

2

3

4

5

6

7

0 0.5 1.0 1.5 2.0

Mode 2

Mode 1

M

FIGURE 10. Normalized oscillation frequencies for the first two modes of an annulus ofradius ratio η = 0.3644 computed from (3.50) (solid lines) and corresponding experimentaldata (solid symbols). The reference frequency ωR = 3.505 rad s−1 computed from (3.22b) isthat for η = 0 for which m0 = 1129 g; details for this annulus are given in table 2.

reminiscent of that produced by the head-on collision of solitary waves reported byMaxworthy (1976).

Motivated by the crossing of the η = 0.8 frequency curve with the lower η curves inour sample calculation for the annulus given in figure 2, we fabricated a new annulusat η = 0.777. The outer radius for the two annuli and the large cylinder was verynearly 13.00 cm; see table 2. With this in mind, we normalize all results, those for thecylinder and the annuli at η = 0.364 and η = 0.777, with the shallow-water referencefrequency ωR = 3.505 for the cylinder. The theoretical curves are shown in figure 11along with the data for the cylinder (open circles) and the annuli (solid diamondsand squares). All experimental data agree very well with the theoretical predictionsand indeed there is strong experimental evidence that the curves for η = 0.364 andη = 0.777 will indeed cross in the neighbourhood of M = 2. In this presentation thecurve for η = 0.364 crosses the cylinder curve (η = 0), in contrast to the results givenin figure 2, where none of the higher η curves crosses the cylinder curve. This isexplained by the fact that the values of m0 = 2000 g are identical for each radiusratio displayed in figure 2 but have different values m0 = 1129, 1207, 1644 g forcomputation of the theoretical results for η = 0, 0.364, 0.777 in figure 11.

4.6. Waveforms and amplification ratiosThe free surface wave profiles ζ/X0 for the large box (equation (3.4)), large cylinder(equation (3.17)), wedge and cone (equation (3.25b)) are compared at the commonvalue M = 1 in figure 12. These are calculated at t = 0 for all containers and alongθ = 0 for the cylinder and the cone. The waveforms are plotted against the normalizedcoordinate ξ = x/D for the box, ξ = r/R for the cylinder, ξ = x/H for the wedge andξ = r/H for the cone. Note that both the wedge and cone surfaces are flat, but that,owing to the larger magnification ratio for the wedge, its rise height is larger thanthat for the cone. We have taken photographs, and in some instances videos, of thefundamental sloshing waveforms viewed from the side and find qualitative agreementbetween the surface profiles with those presented in figure 12. In particular, free


0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 0.5 1.0 1.5 2.0 2.5

M

0.364

0.777

FIGURE 11. Normalized oscillation frequencies for the first modes of annuli with radiusratios η = 0, 0.3644 and 0.777 computed from (3.50) (solid lines) and correspondingexperimental data (solid symbols). The reference frequency ωR = 3.505 rad s−1 computedfrom (3.22b) is that for η = 0 for which m0 = 1129 g; details for these annuli are given intable 2.

–3

–2

–1

0

1

2

3

–1.0 –0.5 0 0.5 1.0

Wedge

Cone

Box

Cylinder

FIGURE 12. A comparison of free surface waveforms for the large box, large cylinder, wedgeand cone computed at M = 1. The normalized horizontal coordinate is ξ = x/D for the box,ξ = x/H for the wedge, ξ = r/R for the cylinder and ξ = r/H for the cone. Details for thesegeometries are given in table 2.

surfaces appeared completely flat for sloshing waves in the cone and wedge, except forsmall capillary effects around the wetted perimeter of the containers.

As a consistency check, we made one measurement of the amplification ratio for thelarge box. For a liquid filling m= 815 g corresponding to M = 0.919, we took a videoof the oscillating wave near the end of the box on which was mounted a millimetrescale to estimate the vertical displacement of the liquid at the endwall x = D. The


formula for the amplification ratio for a box at fixed ω is given by∣∣∣∣ ζ0

X0

∣∣∣∣= Dω2

g

[1+ 2ω2

∞∑n=1

1

(knD)2(gkn tanh knH − ω2)

]. (4.1)

Note that the amplification ratio is not defined relative to the maximum free surfacedeflection, which occurs near ξ = 0.75 for the M = 1 large-box profile shown infigure 12. Evaluation of (4.1) at H = m0/ρWL for the large box gives ζ0/X0 = 1.4192.Our measured amplification ratio ζ0/X0 = 1.46 ± 0.05 is thus in excellent agreementwith the theoretical prediction.

We note that a series expression similar to (4.1) may be written down for theamplification ratio in a freely moving cylinder. For the wedge and the cone geometries,the amplification ratio is given by the following explicit formulae:∣∣∣∣ ζ0

X0

∣∣∣∣= 3(1+M)

2M(wedge), (4.2a)∣∣∣∣ ζ0

X0

∣∣∣∣= 4(1+M)

3M(cone). (4.2b)

Thus, as M→∞, the amplification ratios become independent of M, being 3/2 for thewedge and 4/3 for the cone.

4.7. Note on system dampingThough we do not attempt to derive an expression for the damping of oscillationsin any of our containers driven by asynchronous liquid sloshing, we consider howit compares to the damping of standing waves in a stationary box. Keulegan (1959)derived an expression for the damping due to viscous friction on the walls of arectangular box. He defines α1 as the damping modulus through the equation

a

a0= e−α1t/T, (4.3)

where T is the period of damped oscillations, t is time and a is the decaying amplitudeof the standing wave. Keulegan’s analysis leads to the following expression for thedamping modulus:

α1 =√νT

π

χ

W, (4.4a)

where

χ = π(

1+ W

L

)+ W

L

(1− 2H

L

)π2

sinh(2πH/L). (4.4b)

The energy loss per cycle of oscillation due to viscous dissipation in the liquidproper was computed by Lamb (1932, § 348). Written in terms of Keulegan’s dampingmodulus, we denote this contribution by α2, viz.

α2 = 2π2νT

L2. (4.5)

Using ν = 0.01 cm2 s−1, we find for the large box the values α1 = 0.020 28 andα2 = 0.000 414. From the first seven oscillations of the large-box trace given infigure 3(b) we find the damping coefficient αexp = 0.1089. It is clear that the


internal dissipation given by α2 is negligible compared to both α1 and αexp. Thedamping modulus in the experiment is about five times that for a stationary box. Partof the discrepancy is certainly due to the anomalous decay in lucite basins, welldocumented by Keulegan (1959, figure 9). However, his analysis does not accountfor the fluid–structure interaction present in our moving container, and the remainderof the discrepancy between α1 and αexp is attributed to that effect, with also smallcontributions due to the friction in the cart’s bearings and the rolling friction of thecart wheels.

5. Discussion and conclusionExperiments on the horizontal, rectilinear, sloshing-induced motion of free

containers oscillating over a nearly frictionless surface have been presented. Theapparatus used, made by PASCO, consisted of a four-wheel aluminium cart witha fine suspension system that can move with very low friction on an aluminiumtrack. The measured frequencies for the fundamental and second modes of transverseoscillation for box, cylinder and annulus geometries were obtained over a range ofdimensionless masses M = m/m0, where m0 is the dry mass of the system and m is theliquid mass inside the container. In addition, the frequency of the only sloshing modeavailable for a wedge and a cone with 90 apex angles were obtained over a rangeof M. Additional rectangular and cylindrical containers were designed in an attempt tocapture the predicted maximum frequency that obtains for each geometry, with onlypartial success because of the relatively flat maxima in each case. More successful wasan experiment devised to document the theoretical prediction that a large-radius-ratioannulus frequency curve will cross a lower-radius-ratio curve at some value of M. Inall cases, measurements are considered to be in very good, if not excellent, agreementwith the theoretical predictions.

We attempted to excite higher modes in all of our containers. In three of them(large box, large cylinder and η = 0.364 annulus) we were able to observe the secondmode, though sometimes over only a limited range of filling ratios M. In none of ourcontainers could we observe the third (or any higher) harmonic, presumably becausethese oscillations would be at frequencies too high to excite manually. They wouldalso have very small amplitudes, making them hard to discern.

Two trends in the data are apparent. First, measurements of the sloshing-inducedfrequency of the axisymmetric containers (cylinder, cone, annulus) were generally inbetter agreement with theory than those for the containers of planar symmetry (box,wedge). We tested to see if this might be some capillary effect by adding severaldrops of PhotoFlo to reduce the surface tension, but no discernible change in thefrequency was noted for these long, damped standing waves. Second, while dampingmight be expected to reduce the oscillation frequency, our experimental resultsare sometimes slightly above, but almost never below, the theoretical predictions.Also, in all geometries except the wedge and the cone, evaporation would lowerthe observed frequencies, whereas our measurements are nearly always above thepredicted values. We contend that this systematic trend is probably due to a slightrestoring force provided by the cart’s suspension system, especially at high fillingswhen the depressed springs are more susceptible to coupling with the oscillations ofthe liquid in the container.

Of particular interest is the fact that the transverse oscillation of sloshing-inducedmotion in an annulus can be determined using only the fundamental azimuthal mode,cos θ , with one nodal diameter. The shape of the oscillating free surface, however,cannot be determined unless higher modes are included. Observations of the free


surface motion revealed waves propagating around opposite sides of the annulus thatmet in head-on collisions at θ = 0,π. Collisions with splashing was observed duringthe first couple of oscillations during which the wave amplitudes were relatively large,reminiscent of those produced by the head-on collision of solitary waves.

For the cylinder we have observed the sloshing-induced motion in a natural settingdescribed in the Appendix.

Acknowledgements

We have benefited greatly from discussions with Dr M. Cooker and ProfessorJ. Yu during all phases of this work. We thank the two referees whose commentsled to a much improved manuscript. We appreciate the precision work of J. Butler(Colorado Plastic Products, Inc.) in fabricating the boxes and the wedge, and ofJames Tucker (Tucker Precision Machining) for turning the cone on a lathe from solidstock. M. Sprague provided special guidance in programming of Mathematica. Wethank Y. Peng who assisted in taking videos of our experiments and in some of themeasurements, and also J. Golden for providing us the precise internal diameter of theNissan thermos.

Appendix. The sloshing-induced motion of a thermos

While on a trip to climb Mt. Vinson in Antarctica during December 2010, P.D.W.observed the transverse oscillations of a thermos induced by the sloshing motionwithin. The results given in this appendix are all due to P.D.W. and will be describedfrom his point of view.

Each evening, the clients of the expedition were given a nearly full thermos of hotwater to take to their tents in order to stay hydrated. One evening a guide filled mybottle about seven-eighths full of hot water and set it down on the horizontal hard-packed snow bench that formed part of the cooking shelf. It spontaneously began tooscillate to and fro at relatively high frequency and then stopped suddenly. I estimatedthe frequency to be two to three oscillations per second. Evidently, the hard-packedsnow provided a sufficiently smooth surface to enable sloshing-induced motion of thethermos.

Back in Colorado I made an estimate calculation of the sloshing-induced frequency.The measured dry weight of the vacuum insulated Nissan thermos (model FBB 1000P6) is m = 501 g and its internal diameter provided to us by Nissan is D = 2 1

2 inch(R = 3.175 cm). Assuming the 1.0 litre thermos was filled with 875 g of water, theestimated value M = 1.75 is obtained. Using the nominal values ρ = 1.0 g cm−3 andg = 980 cm s−2, computation for the first mode of oscillation gives ω = 24.3 rad s−1.Since the oscillation frequency was not measured in situ, I will use the average valueω = 2.5 Hz of the perceived frequency. Thus the theoretical value is to be comparedwith the average value ω = 15.7 rad s−1 estimated for the Antarctica observation, some35 % lower than theory. This is to be compared with our preliminary results for acylinder obtained using an air-bearing table, which were 10–15 % lower than theory.I conclude that, while the hard-packed snow surface did provide a means to viewthe sloshing-induced oscillation of the thermos, it is not an ideal frictionless surface.Nevertheless, it was instructive to see the sloshing-induced motion occur in a naturalsetting.


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