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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]
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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
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218 A. Herczyński and P. D. Weidman
the container takes the form
m0Ẍ = 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+ xẌ). (2.3)Here ρ is the liquid density, g is
gravity and −ρxẌ 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 + xẌ = 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 usually
require 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 to
dry container mass m0, denoted by
M = mm0. (2.7)
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Periodic oscillation of free containers driven by liquid
sloshing 219
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=1An
cosh kn(z+ H)cosh knH
sin 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+1k2nD
sin knx. (3.2)
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220 A. Herczyński and P. D. Weidman
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=1An 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 ω2 − 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.
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Periodic oscillation of free containers driven by liquid
sloshing 221
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(α2n − 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(α2n − Z2)
=∞∑
n=1
[1
α2n − Z2− 1α2n
]= 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
1α3n(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 2π
3β2 − 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
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222 A. Herczyński and P. D. Weidman
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−1and 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=1AnJ1(knr)
cosh kn(z+ H)cosh knH
cos 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)
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Periodic oscillation of free containers driven by liquid
sloshing 223
and inserting this expression and (3.14) into (2.5) determines
the coefficients as
An =− 2αnRω3X0
(α2n − 1)J21(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−H
p(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=1AnωJ1(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=1Anω 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 RH
∞∑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)
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224 A. Herczyński and P. D. Weidman
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)=−ζ0Hωxz sinωt + C(t)x, (3.25a)
ζ(x, t)= ζ0H
x 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)
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Periodic oscillation of free containers driven by liquid
sloshing 225
into free surface condition (2.5) yields, upon integration,
C(t)=− gζ0ωH
sinωt − Ẋ + 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 Ẋ
using (2.6) and inserting (3.26) into(3.25a) gives the potential
function
φ(x, z, t)= x[
X0ω − gζ0ωH− 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= 1gω2H− 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− gω2H
(3.33)
-
226 A. Herczyński and P. D. Weidman
is obtained. Equating (3.29) and (3.33) gives
ω2 = gH
1+M1+ M
3
. (3.34)We eliminate H using H =√m0M/ρW to arrive at the exact
solution
ω
ωR= 1
M1/4
√√√√ 1+M1+ 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− gω2H
. (3.38)
-
Periodic oscillation of free containers driven by liquid
sloshing 227
Equating (3.29) and (3.38) and eliminating H in favour of M
furnishes the exactsolution
ω
ωR= 1
M1/6
√√√√ 1+M1+ 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 knH cos 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=1AnP(r)
cosh kn(z+ H)cosh knH
cos 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=1BnP(r), (3.43)
-
228 A. Herczyński and P. D. Weidman
which, after a lengthy but straightforward calculation,
gives
Bn = 2knR21{Y′1(knR1)[J2(knR2)− η2J2(knR1)] −
J′1(knR1)[Y2(knR2)− η2Y2(knR1)]}
η2{P2(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=1AnωP(r)
cosh kn(z+ H)cosh knH
+ gz− rω2X0]
cos θ cosωt. (3.47)
Evaluation of (3.46) gives
Fp = ρπR1[
X0R1H(1− η2)η2
ω2 − ω∞∑
n=1An[η−1P(R2)− P(R1)] tanh knHkn
]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)]Bn tanh knH
(gkn tanh knH − ω2)
]. (3.49)
The fluid mass in the annular domain is m = ρπR21H(1 − η2)/η2 so
using (2.7) weobtain the desired eigenvalue equation for the
frequency of tank motion as
1+M(
1+ ω2 η
(1− η2)∞∑
n=1
BnknR1H
[P(R2)− ηP(R1)] tanh knH(gkn tanh knH − ω2)
)= 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
-
Periodic oscillation of free containers driven by liquid
sloshing 229
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
-
230 A. Herczyński and P. D. Weidman
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
-
Periodic oscillation of free containers driven by liquid
sloshing 231
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.
-
232 A. Herczyński and P. D. Weidman
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−2provided 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.
-
Periodic oscillation of free containers driven by liquid
sloshing 233
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.
-
234 A. Herczyński and P. D. Weidman
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
-
Periodic oscillation of free containers driven by liquid
sloshing 235
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
-
236 A. Herczyński and P. D. Weidman
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
-
Periodic oscillation of free containers driven by liquid
sloshing 237
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
-
238 A. Herczyński and P. D. Weidman
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
-
Periodic oscillation of free containers driven by liquid
sloshing 239
formula for the amplification ratio for a box at fixed ω is
given by∣∣∣∣ ζ0X0∣∣∣∣= Dω2g
[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:∣∣∣∣ ζ0X0
∣∣∣∣= 3(1+M)2M (wedge), (4.2a)∣∣∣∣ ζ0X0∣∣∣∣= 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+ WL
)+ 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
-
240 A. Herczyński and P. D. Weidman
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
-
Periodic oscillation of free containers driven by liquid
sloshing 241
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 12 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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242 A. Herczyński and P. D. Weidman
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Experiments on the periodic oscillation of free containers
driven by liquid sloshingIntroductionProblem formulationAnalytical
solutionsRectangular containersFinite-depth solutionShallow- and
deep-water limitsSample solutions
Cylindrical containersFinite-depth solutionShallow- and
deep-water limits
Wedge geometryCone geometryAnnular containersSample
solutions
ExperimentsRectangular containersCylindrical containersWedge
geometryCone geometryAnnular containersWaveforms and amplification
ratiosNote on system damping
Discussion and conclusionAcknowledgementsAppendix. The
sloshing-induced motion of a thermosReferences