Shallow-water sloshing in rotating vessels undergoing

prescribed rigid-body motion in two dimensions

by H. Alemi Ardakani & T. J. Bridges

Department of Mathematics, University of Surrey, Guildford GU2 7XH UK

— October 23, 2010—

Abstract

New shallow-water equations, for sloshing in two dimensions (one horizontal and onevertical) in a vessel which is undergoing rigid-body motion in the plane, are derived.The planar motion of the vessel (pitch-surge-heave or roll-sway-heave) is exactly mod-elled and the only approximations are in the fluid motion. The flow is assumed to beinviscid but vortical, with approximations on the vertical velocity and accelerationat the surface. These equations improve previous shallow water models for sloshing.The model also contains the essence of the Penney-Price-Taylor theory for the higheststanding wave. The surface shallow water equations are simulated using an implicitfinite-difference scheme. Numerical experiments are reported, including simulationsof coupled translation-rotation forcing, and sloshing on a ferris wheel. The extensionto shallow-water flow in three dimensions and the coupled vessel slosh dynamics arealso discussed.

1 Introduction

The theoretical and experimental study of fluid sloshing has a long history and continuesto attract considerable attention because of its importance in applications: the shippingof liquid natural gas, sloshing of trapped water on the deck of a ship, liquid transportalong roads, as well as unusual applications such as sloshing in a swimming pool on-board a passenger ship (Ruponen et al. [63]), sloshing in a fish tank on deck (Leeet al. [46]), sloshing in automobile fuel tanks (Wiesche [69]), and transport of liquids byrobots in industrial applications (Tzamtzi & Kouvakas [67]). Examples where shallowwater sloshing is predominant are sloshing of trapped water on the deck of fishing vessels(Caglayan & Storch [18], Adee & Caglayan [4]), and on the deck of offshore supplyvessels (Laranjinha et al. [45]), and sloshing in wing fuel tanks of aircraft (Disimileet al. [26]). The recent books of Ibrahim [38] and Faltinsen & Timokha [30] whichhave over 3000 references are both testimony to the current interest in the subject, and anencyclopedic review of existing research on sloshing.

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Although there is a rich history of the study of sloshing, there remains a considerablerange of phenomena that are not understood, mainly because sloshing is driven by a freesurface motion which is highly nonlinear. In addition there are the attendant problems offluid-ship interaction, hydroelastic effects, bubble formation, wave breaking and impact,and probabilisitic aspects.

The interest in this paper is in sloshing in shallow water in a vessel that is undergoinga general rigid body motion. In this paper attention is restricted to two dimensions andthe case of three dimensions is considered in Alemi Ardakani & Bridges [12]. Theaim is to use exact equations for the rigid body motion of the vessel but approximate theequations for the fluid motion. The approximations should simplify the equations but withthe minimum number of hypotheses.

The results in the literature on forced (vessel motion prescribed) shallow water sloshingfall into two categories: derive approximate partial differential equations similar to theclassical shallow water equations, or use asymptotics, based on small depth aspect ratio andforcing amplitude, to derive approximate equations which can be analyzed more efficiently.

The first example of the former is the work of Verhagen & van Wijngaarden [68].They write down an approximation of the shallow-water equations of the form

ht + (hu)x = 0 and ut + uux + ghx = δ sin ωt , (1.1)

where δ sin ωt is due to the imposed rotation of the vessel, h(x, t) is the depth, and u(x, t)is the horizontal velocity. The tilde is used to distinguish this choice of velocity from othervelocities used in this paper. A fixed frame is used so time-dependent boundary conditionsare imposed at the endwalls, x = 0, L . They solved the equations approximately usinga form of the method of characteristics. The principle observation is the formation ofa travelling hydraulic jump, and its characteristics were shown to be in agreement withexperiments. In followup work by Chester [21], the effect of dispersion and dissipation isincluded in the shallow water model, showing improved comparison with the experimentsof Chester & Bones [22]. Jones & Hulme [41] start with the full equations relative toa rotating frame and give a new derivation of (1.1); they study the equations numericallyand with the use of a multi-scale perturbation expansion.

The first work using asymptotics to study forced sloshing in shallow water is Ockendon& Ockendon [54]. The forcing is simplified to a harmonic horizontal translation, and thesmall parameters are the depth aspect ratio, the amplitude of the forcing and the frequencydetuning. Rather than start with the shallow water equations, they start with the fullequations for irrotational flow and derive an integrodifferential equation for the surfacemotion of the form

κ2

3(ηtt + η)− λη − 3

2η2 +

2

πcos t = −3

2

∫ +π

−π

η(s)2 ds , (1.2)

where κ and λ are parameters, related to the dispersion and frequency detuning respec-tively, and η(t) is required to be a 2π−periodic function of t . In the limit κ → 0 andλ → 0 this equation simplifies to an algebraic equation for η . From this algebraic equationit can be deduced that there are three regions in the frequency-amplitude plane. For fixedamplitude there is an interval around the natural frequency where (the hydraulic analogyof) compressive shocks can be found, and outside this region the solutions are regular peri-odic standing waves. Experiments of Kobine [43] showed excellent agreement with theseresults (see Figure 7 of [43]). In further work Ockendon et al. [55] it is shown that this

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equation has a wide range of exotic solutions. It has since inspired study from a dynam-ical systems perspective (e.g. [34, 42, 33]). The equation (1.2) has an infinite number ofperiodic and subharmonic solutions. These works show how forced shallow water sloshingcan lead to very complicated fluid motion through a cascade of subharmonic solutions.

The range of validity of the asymptotic approach was extended by Faltinsen & Tim-okha [29] using modal expansion combined with asymptotics. They examine resonantwaves in shallow water (depth to width ratios between 1

10and 1

4) forced by surge and

pitch excitation at frequencies in the vicinity of the lowest natural frequency. The prin-cipal small parameter is forcing amplitude which is of order ε . The depth aspect ratio,velocity potential and wave height are then all required to be of order ε1/4 . This scalingbrings in nonlinearity to fourth order and dispersion. The dispersion gives the theory amodal form of Boussinesq equation, and no additional boundary conditions at the endwallsare required. Secondary resonances are included. This theory allows for large dimensionmodal expansion and they include up to 20 coupled modes. The large modal system ofcoupled ODEs is then integrated numerically. They emphasize the role of dissipation andvalidation by comparison with experiments. They show how modal truncation and choiceof dissipation influence the comparison with experiments.

Forbes [31] combines a modal expansion with dynamical systems techniques. Quasi-periodic motion and chaos are discovered in the neighbourhood of the periodic solutions.The main indicator of this chaos is the discovery of a collision of eigenvalues of oppositesignature in the Floquet multiplier space (see Figure 8(a) in [31]). Such bifurcations arethe starting point for more complex dynamics [17, 39].

The advantage of asymptotic methods is that detailed studies within the range of valid-ity are possible, leading to fundamental observations, and integration of a system of ODEswill in general be much faster than integration of PDEs. However, the disadvantage is thelimited range of validity, requiring the depth aspect ratio, forcing amplitude and frequencyall to remain within a certain asymptotic range.

In this paper the first approach will be followed: derive a new model PDE based on theshallow water equations. With this model there are – in principle – no restrictions on theforcing amplitude or frequency, and the vessel can undertake any rotation and translation,with the full generality of the vessel motion retained. However in the shallow water limitthe rotation amplitude is required to be small (see Appendix A for precise statement).Viscosity is neglected but vorticity is retained in the fluid motion and the only furtherapproximations on the fluid are on the vertical velocity and acceleration – at the freesurface only. The retention of vorticity is more important in 3D but still plays a role in2D. For example Figures 14–15 of Chen [20] show a non-trivial free surface vorticity for aclass of 2D sloshing flows, although viscosity is included in [20].

The key to retaining exact motion of the vessel is to use a body-fitted moving coordinatesystem. The first work in this direction – in the context of shallow water – is the paper ofDillingham [24]. Starting with the full Euler equations in two-dimensions, a set of shallowwater equations for (h, u) , where h(x, t) is the fluid depth and u(x, t) is the depth-averagedhorizontal velocity,

u =1

h

∫ h

0

u(x, y, t) dy , (1.3)

of the following form, were derived

ht + (hu)x = 0 and ut + u ux + a(x, t)Dhx = b(x, t)D , (1.4)

where the terms a(x, t)D and b(x, t)D contain the vertical and horizontal accelerations of

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the moving vessel (precise expressions are given in §6). The boundary conditions are u = 0at the sidewalls.

With the choice of mean velocity u(x, t) the mass equation is exact in (1.4). However,the derivation of the momentum equation requires a number of assumptions. The deriva-tion in Dillingham [24] is somewhat ad hoc, in that terms are neglected without clearimplications. A more careful derivation of the equations (1.4) is given by Armenio & LaRocca [14]. Their derivation leads to an equation of the same form as (1.4) but theircoefficients, denoted a(x, t)ALR and b(x, t; y)ALR , are very different from those in (1.4).

In constructing a shallow-water approximation, the velocity can be chosen to be thehorizontal velocity at any point between the bottom and the surface, or it can be anaverage velocity. In principle, within the shallow-water approximation, all these velocitiesshould give the same equation, and that is the case – without rotation. With rotation,different choices of horizontal velocity lead to different shallow water equations. Our originalintention was to use the depth-averaged velocity and precisely itemize the assumptionsinvolved. The depth-averaged velocity seems most natural because the equation ht +(hu)x = 0 is then exact.

We discovered however that using the horizontal surface velocity,

U(x, t) := u(x, y, t)

∣∣∣∣h , (1.5)

where the surface notation is defined by

f

∣∣∣∣h := f(x, y, t)

∣∣∣∣y=h

= f(x, h(x, t), t) , (1.6)

leads to equations with some surprising and useful properties.In two-dimensions (one vertical and one horizontal space direction), consider the inviscid

– but retaining vorticity – sloshing problem, with fluid occupying the region 0 < y < h(x, t)with 0 ≤ x ≤ L . The governing equations are the Euler equations relative to a movingframe of reference, and conservation of mass, with the usual boundary conditions (thedetails are recorded in §2). Remarkably, the horizontal surface velocity satisfies the exactequation

Ut + UUx +

(a(x, t) +

Dv

Dt

∣∣∣∣h)

hx = b(x, t) + σκxx , (1.7)

where DvDt

∣∣∣∣h is the Lagrangian vertical acceleration at the free surface, a(x, t) is vertical

acceleration due to the rotating frame, and reduces to g , the gravitational constant, whenthe reference frame is stationary, and b(x, t) is a horizontal acceleration due to the rotatingframe (derivation given in §3). The term σκxx is a curvature term and appears only whensurface tension is present, with σ > 0 the coefficient of surface tension.

Equation (1.7) is not closed since it requires the vertical acceleration at the surface, butthe form is illuminating, and neglect of the Lagrangian vertical acceleration gives a closedsystem. On the other hand, equation (1.7) shows how the Lagrangian vertical accelerationdrives the surface dynamics. In the case where tank is fixed, the acceleration term reducesto

aLag := g +Dv

Dt

∣∣∣∣h .

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According to early work of Penney & Price [60] and Taylor [66], aLag is precisely theterm that drives breaking of standing waves. They argue that aLag ≈ 0 is the condition forthe highest standing wave. Experiments of Taylor [66] confirmed the importance of thisquantity. In this paper we will be primarily interested in the case of shallow water wherethe Lagrangian vertical accelerations at the surface are small, but some comments aboutthe case aLag ≈ 0 are in §4.

The kinematic condition at the free surface is

ht + Uhx = V or ht + (hU)x = V + hUx , (1.8)

where V = v(x, h(x, t), t) is the vertical velocity at the surface. Hence, it is clear from(1.7) and (1.8) that a closed system of shallow water equations is obtained by making thefollowing assumptions on the vertical velocity and acceleration at the surface,∣∣∣∣V + hUx

∣∣∣∣ << U0 , and

∣∣∣∣∣Dv

Dt

∣∣∣∣h∣∣∣∣∣ << |a| , (1.9)

where U0 is a reference order one velocity (cf. Appendix A). When the amplitude ofrotation is small the second condition requires the magnitude of the vertical accelerationat the surface to be small compared with g .

It is equations (1.7) and (1.8), with the two assumptions (1.9) which are the basis forthe analysis and numerics in this paper. This shallow-water model for sloshing is simulatedwith an implicit numerical algorithm that is similar to a one-dimensional Abbott-Ionescuscheme. This scheme and its variants are widely used in hydraulics Abbott [2]. A reviewof other schemes that have been or could be used is given in §7. The main reason for usingthis scheme is that it generalizes nicely to two-horizontal space dimensions and is used in[12]. Secondary reasons include the form of the numerical dissipation generated by thetruncation error, the block tridiagonal structure, and its generalization to include dynamiccoupling with the vessel motion. The application of this scheme requires only the solutionof block tridiagonal coefficient matrices at each step, with an iterative process to accountfor the nonlinearity.

An outline of the paper is as follows. The governing equations are recorded in §2, basedon a new derivation of the moving frame equations in Alemi Ardakani & Bridges [13].In §3 a sketch of the derivation of the new shallow water slosh equation (1.7) is given. Theassumptions necessary for closure are given in §5. In §4 the implications of aLag ≈ 0 arediscussed. In §6 a comparison of the new SWEs with previous work is presented, withfurther detail in the technical reports [8, 9]. A sketch of the numerical algorithm is givenin §7, with details on the numerical algorithm in the technical report [10]. The propertiesof rigid body motion of the vessel are discussed in §8. Simulations based on the newalgorithm are then presented in §9. Comparisons with previous work are reported in thetechnical report [11] and new results with coupled forcing are reported in §9. The theoryand computation in this paper point to new directions for 3D sloshing and for dynamiccoupling between the vessel and fluid motion and these directions are discussed in §10.Appendix A shows that the key assumptions are satisfied in the shallow water limit byusing a scaling and asymptotic argument.

2 Governing equations

The configuration of the fluid in a rotating-translating vessel is shown in Figure 1. Thevessel is a rigid body and two frames of reference are used to study the motion. The

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spatial (inertial) frame has coordinates X = (X, Y ) and the body frame has coordinatesx = (x, y) . The distance from the body frame origin to the point of rotation is denoted byd = (d1, d2) and d is a constant. The whole system has a uniform translation q(t) . Theposition of a particle in the body frame is related to a point in the spatial frame by

X = Q(x + d) + q ,

where Q is a proper rotation (QT = Q−1 and det(Q) = 1). The position of the rigid bodyis uniquely described by the pair (q,Q) [57]. Let θ(t) be the angular position between theX and x axes. The angular velocity vector is Ω = (0, 0, Ω) with Ω = θ . The connectionbetween θ and Q is

Q = θJQ , J =

(0 −11 0

).

The fluid occupies the region

0 ≤ y ≤ h(x, t) with 0 ≤ x ≤ L .

The governing equations relative to the body frame will be used. The body (moving) framefor sloshing has been used by a number of authors (e.g. Lou et al. [51], Faltinsen etal. [28], La Rocca et al. [44], Ibrahim [38], Faltinsen & Timokha [30]). There aresome subtleties, especially in 3D which will be used in [12], and so a new derivation basedon first principles is given in the technical report [13]. The momentum equations, derivedin [13], for the fluid in the vessel relative to the body coordinate system fixed to the vessel,are

DuDt

+ 1ρ

∂p∂x

= −g sin θ + 2Ωv + Ω(y + d2) + Ω2(x + d1)− q1 cos θ − q2 sin θ ,

DvDt

+ 1ρ

∂p∂y

= −g cos θ − 2Ωu− Ω(x + d1) + Ω2(y + d2) + q1 sin θ − q2 cos θ ,(2.1)

where g > 0 is the gravitational constant and

Du

Dt:=

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y.

Conservation of mass relative to the body frame takes the usual form

ux + vy = 0 . (2.2)

The boundary conditions are

u = 0 at x = 0 and x = L , v = 0 at y = 0 , (2.3)

andp = −ρσκx and ht + uhx = v , at y = h(x, t) , (2.4)

where σ ≥ 0 is the coefficient of surface tension and

κ :=hx√

1 + h2x

.

The vorticity is defined by

V =∂v

∂x− ∂u

∂y.

6

Y

X

X’

Y’

q

dθ

y

x

Figure 1: Diagram showing coordinate systems for rotating translating tank. The coordi-nate system (X ′, Y ′) is the translation by q of the fixed coordinate system (X, Y ) .

The equation governing vorticity is obtained by differentiating the x−momentum equationwith respect to y and the y−momentum equation with respect to x , leading to

D

DtV =

∂

∂x

(Dv

Dt

)− ∂

∂y

(Du

Dt

)− (ux + vy)V = −2Ω , (2.5)

with the second equality following from substitution into (2.1) and use of (2.2). Thisequation is important for the derivation of the shallow water equations. The integratedform of the vorticity equation is

∂

∂t

(∫ h

0

V dy

)+

∂

∂x

(∫ h

0

uV dy

)= −2Ω h . (2.6)

Let

V :=

∫ L

0

∫ h

0

(V + 2Ω) dydx . (2.7)

It follows from (2.6) that Vt = 0 and so V is a constant of motion.

3 Reduction of the horizontal momentum equation

The key to the derivation of the shallow water equations is the treatment of the pressurefield. Integrating the vertical momentum equation in (2.1) from any point y to the surfaceh , and applying the pressure free-surface boundary condition gives the following exactequation for the pressure field,

1ρp(x, y, t) =

∫ h

y

Dv

Dtds + 2Ω

∫ h

y

u ds

+(g cos θ + Ω(x + d1)− Ω2d2)(h− y)− 1

2Ω2(h2 − y2)− σκx ,

(3.1)

plus the terms associated with q which don’t affect the derivation and can be brought backinto the end result.

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The typical strategy at this point in the derivation of the SWEs is to drop the verticalacceleration term Dv

Dt. However, when we differentiate the pressure with respect to x , and

use the vorticity equation, this term simplifies, and so it can be retained exactly.An expression for the horizontal pressure gradient can be obtained by differentiating

the exact expression for the pressure (3.1),

1

ρ

∂p

∂x=

(g cos θ + Ω(x + d1)− Ω2(h + d2) + Dv

Dt

∣∣∣∣h)

hx

+DuDt

∣∣∣∣hy

− Ω(h− y)− 2Ωht + 2Ωv − σκxx .

(3.2)

Now substitute (3.2) into the x−momentum equation in (2.1)

Du

Dt

∣∣∣∣h +

(g cos θ + Ω(x + d1)− Ω2(h + d2) +

Dv

Dt

∣∣∣∣h)

hx

= 2Ωht − g sin θ + Ω(h + d2) + Ω2(x + d1) + σκxx .

(3.3)

Conveniently, the material derivative of u at the surface can be expressed purely in termsof the surface horizontal velocity

Ut + UUx = [ut + uyht + uux + uuyhx]

∣∣∣∣h = DuDt

∣∣∣∣h + uy(ht + uhx − v)

∣∣∣∣h = DuDt

∣∣∣∣h.Substituting into (3.3) then gives the x−momentum equation at the free surface

Ut + UUx +

(a(x, t) + Dv

Dt

∣∣∣∣h)

hx = b(x, t) + σκxx , (3.4)

where, after adding back in the terms with q1 and q2 ,

a(x, t) = g cos θ + Ω(x + d1)− Ω2(h + d2)− q1 sin θ + q2 cos θ

b(x, t) = 2Ωht − g sin θ + Ω(h + d2) + Ω2(x + d1)− q1 cos θ − q2 sin θ .(3.5)

The equation (3.4) is exact. Note also that the assumption of finite-depth is never used inthe derivation of (3.4). It is also valid for an infinite-depth fluid. The translation vector(q1, q2) is relative to the spatial frame. The rotation of the acceleration vector (q1, q2) thatappears in (3.5) is due to the fact that these accelerations are viewed from the body framein (3.4). When the vessel is stationary (3.4) reduces to the surface equation for the classicalwater wave problem [16]. The exact surface equations in a rotating frame are extended tothree-dimensions in [12].

There is also an exact equation for the surface vorticity. Let

Γ(x, t) := V

∣∣∣∣h := (vx − uy)

∣∣∣∣h .

Then evaluating (2.5) at y = h ,

Γt + UΓx = −2Ω .

Hence Γ + 2Ω is transported by the horizontal surface velocity. However, in contrast toconservation of V in (2.7), conservation of surface vorticity is nullified in general by theboundary conditions

d

dt

(∫ L

0

Γ dx + 2LΩ

)=

∫ L

0

ΓUx dx .

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4 Vertical accelerations and the highest standing wave

One of the interesting features of the exact surface equation for the horizontal surfacevelocity (3.4) is the way that the vertical acceleration appears. Define

aLag(x, t) = g +Dv

Dt

∣∣∣∣h = −∂p

∂y

∣∣∣∣h . (4.1)

A condition proposed by Penney & Price [60] for the limiting periodic standing wave isaLag = 0. They argued that – in the absence of surface tension – the pressure just insidethe liquid near the surface for a standing wave must be positive or zero and consequentlyat the surface ∂p

∂y≤ 0 which is equivalent to aLag ≥ 0. When this condition is violated then

the standing wave should cease to exist.Experiments of Taylor [66] confirmed the importance of aLag in determining the high-

est wave. Numerical calculations of Saffmann & Yuen [64] also found that aLag ≈ 0for standing waves near breaking. In their discussion of breaking mechanisms for sloshingwaves, the authors of Royon-Lebeaud et al. [62] remark that a destabilization of astanding wave first appears when the wave front acceleration exceeds the acceleration ofgravity. Okamura [56] has computed large amplitude standing waves with crest accelera-tions up to −0.9998 g , with aLag tending to zero at the crest as the limiting standing waveis approached.

The property that the vertical acceleration tends to −g as the highest wave is ap-proached is distinctly different from the case of Stokes progressive waves where the ver-tical acceleration at the crest tends to − 1

2g as the Stokes limiting wave is approached

Longuet-Higgins [49]. Okamura [56] also shows that the crest acceleration provides abetter parameterization than wave height for very steep waves. In his study of vertical jetsemitted by unsteady standing waves, Longuet-Higgins [50] found vertical accelerationsexceeding −10g .

Reformulate the governing equations around the function aLag . The exact governingequations for (h, U, V ) are

ht + U hx = V ,

Ut + U Ux + aLag hx = σκxx ,

Vt + U Vx + g = aLag .

(4.2)

The latter equation follows by noting that

aLag − g :=Dv

Dt

∣∣∣∣h = Vt + UVx .

The equations (4.2) are not closed – unless aLag is given.Taking the limit aLag → 0 and neglecting surface tension, the equation for U reduces

to the inviscid Burger’s equation Ut + UUx = 0 which has solutions that blow up in finitetime. Pomeau et al. [61] argue that this singularity is related to wave breaking (see alsoBridges [16]). Hence the Penney-Price-Taylor criterion is indeed a limiting condition forstanding waves.

By combining the x−momentum equation (neglecting surface tension) with the massequation (assuming |V + hUx| is small) gives

ht + (hU)x = 0 and Ut + UUx + aLaghx = 0 .

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In this case the limit aLag → 0 is similar to the high Froude number limit of classicalshallow water theory, and this limit produces a range of exotic solutions (cf. Edwards etal. [27]).

5 A closed set of SWEs for planar motion

Since the momentum equation is written in terms of the surface velocity U(x, t) , theh−equation should also be in terms of the surface velocity and it is deduced from thekinematic free surface boundary condition

ht + Uhx = V ⇒ ht + (hU)x = V + hUx . (5.1)

The pair of equations (3.4) and (5.1) is not closed since they still require DvDt|h and V to

be specified. The following two assumptions are the minimum required for closure. It isassumed that the vertical velocity at the free surface satisfies∣∣∣∣V + hUx

∣∣∣∣ U0 , (SWE-1)

for some representative order one velocity U0 , and the Lagrangian vertical acceleration atthe free surface is assumed to satisfy∣∣∣∣Dv

Dt

∣∣h∣∣∣∣ ∣∣a (x, t)∣∣ . (SWE-2)

For a non-rotating vessel, the second assumption (SWE-2) is equivalent to assuming thatthe Lagrangian vertical acceleration is small compared with the gravitational acceleration.The first assumption (SWE-1) is natural in the setting of shallow water since,

0 =

∫ h

0

(ux + vy) dy = V +∂

∂x(hu)− hxU ,

where u is defined in (1.3), and so

V + hUx =∂

∂x(h(U − u)) , (5.2)

and u and U are asymptotically equivalent in shallow water. An asymptotic argumentshowing that (SWE-1) and (SWE-2) are satisfied in the limit as ε → 0, where ε is theshallowness parameter

ε =h0

L, (5.3)

is given in Appendix A. An interesting outcome of the asymptotic argument is that theroll-pitch angle θ should be of order ε in the limit ε → 0.

Under these assumptions the shallow water equations for sloshing relative to a rotatingframe are

ht + (hU)x = 0 and Ut + UUx + a(x, t)hx = b(x, t) + σκxx , (5.4)

with a(x, t) and b(x, t) as defined in equation (3.5).

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The coefficient of hx in the second equation of (5.4) can lead to instability and loss ofwell-posedness. Consider the linearized constant coefficient problem relative to a stationaryframe

ht + h0Ux = 0 ,

Ut + ahx = 0 ,

where here a is a constant. Differentiating and combining gives the following wave equation

htt − ah0hxx = 0 .

This equation is well-posed only when a > 0. So an additional necessary condition for(5.4) to be well-posed is

a (x, t) > 0 . (SWE-3)

The term σκxx provides dispersive regularization since

κxx =∂2

∂x2

(hx√

1 + h2x

)=

∂

∂x

(hxx

(1 + h2x)

3/2

)= hxxx + · · · , (5.5)

where the · · · are nonlinear terms. The term σκxx contributes dispersive regularization atthe linear level, much like a Boussinesq term. The equations with σ 6= 0 have third-orderspace derivatives and so additional boundary conditions at x = 0, L would be required (cf.Billingham [15]). The primary interest in this paper is long waves, and so surface tensioneffects will be henceforth neglected

σ = 0 . (SWE-4)

6 Comparison with previous work on rotating SWEs

The first derivation of the SWEs for sloshing in a rotating translating vessel was given byDillingham [24]. The SWEs are of the same form as (5.4) but the expressions for a(x, t)and b(x, t) are different, and Dillingham uses the average velocity u in (1.3). DenoteDillingham’s coefficients by a(x, t)D and b(x, t)D ; then (after adapting to the notationhere, and noting that Dillingham sets d1 = 0)

a(x, t)D = g cos θ + Ωx− Ω2d2 + 2Ωu− q1 sin θ + q2 cos θ ,

b(x, t)D = −g sin θ + Ωd2 + Ω2x− q1 cos θ − q2 sin θ .(6.1)

In Dillingham’s notation, d2 = zd . To see the difference with a and b in (3.5) rewrite (6.1)

a(x, t)D = a(x, t) + Ω2h + 2Ωu ,

b(x, t)D = b(x, t)− Ωh− 2Ωht .(6.2)

In this comparison, the surface velocity is assumed close to the average velocity. In generalthe two velocities may not be close. The precise difference is quantified in (1.3).

The terms Ω2h and Ωh are neglected by Dillingham as being small compared withΩ2d2 and Ωd2 , which is reasonable for large d2 but not for small d2 . See [11] for numericalexperiments confirming this latter property. The terms 2Ωu and 2Ωht are due to Dilling-ham’s neglect of the horizontal Coriolis force (2Ωv ) which is not justified. Indeed, if 2Ωv

11

is retained and v is replaced by the approximate surface velocity ≈ −hux , then the twoformulations are closer.

Since the assumptions in Dillingham [24] are not precise it is not easy to see whaterror is induced by the neglect of the above terms. One implication of these assumptions isthat the conservation form of the SWEs is lost in Dillingham’s formulation, and there isn’ta variational formulation. We also show some numerical experiments in [11] where the twoformulations are compared and clear divergence occurs for large times.

Huang & Hsiung [36] give a derivation of the rotating shallow water equations, buttheir final equations are almost identical to Dillingham’s SWEs. They implicitly use u forthe velocity field, and their functions are related to a and b in (3.5) by (after identifyingU and u)

a(x, t)HH = a(x, t) + 2Ωu ,

b(x, t)HH = b(x, t)− 2Ωht − Ωh .

Comparison with (6.2) shows they include one additional term. However, the criticalCoriolis terms are missing. They also formulate the translations relative to the body framerather than the spatial frame (see discussion in §8). However, within certain parameterregimes the numerical results obtained using the HH SWEs compare very well with thesurface SWES (see results in the technical report [11]). A review of the derivation of theHH SWEs is given in the technical report [8].

Armenio & La Rocca [14] also use the average horizontal velocity but give a moreprecise derivation of the SWEs. The derivation of [14] is reviewed in the technical report[9]. Their derivation leads to the following form for the momentum equation

ut + u ux + a(x, t)ALRhx = b(x, t; y)ALR , (6.3)

witha(x, t)ALR = g cos θ + Ω(x + d1)− Ω2(h + d2) + 2Ωu ,

b(x, t; y)ALR = −g sin θ − 2Ωhux + Ω(y − h + d2) + Ω2(x + d1) .(6.4)

With appropriate change of notation, this is equation (18) in [14]. Note that the vehicleacceleration q is absent in this derivation but it can be easily added. The pair of equationsht + (hu)x = 0 and (6.3) are called the ALR SWEs.

With q neglected, and assuming U ≈ u , the ALR coefficients relative to a and b in(3.5) are

a(x, t)ALR = a(x, t) + 2Ωu ,

b(x, t; y)ALR = b(x, t) + Ω(y − 2h)− 2Ω(ht + hux) .

The comparison can be simplified by using the mass equation to eliminate the second termin a(x, t)ALR and the third term in b(x, t; y)ALR . After this change the two systems arevery close (assuming U ≈ u). The coefficient b(x, t; y)ALR still depends on y . There area number of choices for y : y = 0, y = 1

2h (obtained by averaging), y = h and y = 2h .

The most natural choice is y = 12h which is obtained by averaging. Another interesting

choice is y = h . In this case the equations have an interesting variational principle (AlemiArdakani & Bridges [6]). However, the conservation form is lost unless y = 2h which isnot physically reasonable. Henceforth in discussing the ALR SWEs we will use the choicey = h .

It is remarkable that the ALR SWEs are very close to the surface equations, when Uand u are identified, especially since the surface equations have only two assumptions, andthe ALR SWEs have four assumptions. Although the two sets of SWEs are similar, there

12

are still two principal advantages to using the surface SWEs: first it is very clear what theassumptions are in the derivation, and secondly, the derivation extends in a straightforwardway to the case of three-dimensional rotating shallow-water flow, whereas deriving theSWEs in 3D with the average velocity is very difficult and not always unambiguous (AlemiArdakani & Bridges [12]).

7 A robust numerical algorithm for rotating SWEs

Sloshing in shallow water relative to a rotating frame has been simulated numerically usinga number of different methods. A number of authors have used Glimm’s method to simulateshallow water sloshing (e.g. [24, 47, 59, 58]). Glimm’s method is very effective in treatinga large number of travelling hydraulic jumps, but the solutions are discontinuous. Armeniopoints out in the discussion of the paper by Huang & Hsiung [35] that Glimm’s methodalso suffers from a strong mass variation over time in a simulation. Jones & Hulme [41]used Lax-Wendroff, but did not find it to be effective. Lax-Wendroff is explicit with atime step restriction and unacceptably large truncation errors (Abbott [2]). Armenio& La Rocca [14] have used space-time conservation elements, developed by Chang [19],which have excellent conservation properties. They find that the method works very wellfor the simulation of both standing waves and hydraulic jumps, and the speed of travel-ling hydraulic jumps is well predicted even for large amplitudes of excitation. Huang &Hsiung [35, 36, 37] have used flux-vector splitting. This method, which is originally dueto Steger & Warming [65], involves computing eigenvalues of the Jacobian matrices,and is effective for tracking multi-directional characteristics. Their numerical results arequalitatively in agreement with Dillingham [24].

Our strategy for computing shallow-water sloshing is threefold. We want a methodwhich extends easily to the case of two-horizontal space dimensions. Secondly, we wanta method that is both implicit and has some numerical dissipation. Thirdly, we wanta method which generalizes easily to the case of coupled vessel-fluid motion. For onehorizontal space dimension there are a number of methods that could be used. We firstimplemented the Preissmann scheme (Abbott [2]) and found it to have excellent prop-erties. The Preissmann scheme is also very effective for transcritical flows (Freitag &Morton [32]). However, there are problems with extending the Preissmann scheme totwo horizontal space dimensions (Abbott & Basco [3]).

Instead we use a fully-implicit spatially-centred finite difference scheme which leads toa block tridiagonal coefficient matrix. It is similar to a one-dimensional version of theAbbott-Ionescu scheme and the Leendertse scheme [48] which are both widely used in openchannel hydraulics. The representation of waves by this scheme is studied in §4.10 of [2] inone and two horizontal space dimensions.

The scheme has numerical dissipation, but the form of the dissipation is similar to theaction of viscosity. The truncation error is of the form of the heat equation and so is stronglywavenumber dependent. Moreover the numerical dissipation follows closely the hydraulicstructure of the equations. See the technical report Alemi Ardakani & Bridges [10] foran analysis of the form of the numerical dissipation. The numerical dissipation is helpfulfor eliminating transients and spurious high-wavenumber oscillation in the formation oftravelling hydraulic jumps.

13

Rewrite the governing equations (5.4) in a form suitable for the scheme,

ht + Uhx + hUx = 0 ,

Ut + (α(x, t)− Ω2h + 2ΩU)hx + (U + 2Ωh)Ux = β(x, t) + Ωh ,(7.1)

where α and β are the terms that are independent of h and U ,

α(x, t) = g cos θ + Ω(x + d1)− Ω2d2 − q1 sin θ + q2 cos θ ,

β(x, t) = −g sin θ + Ωd2 + Ω2(x + d1)− q1 cos θ − q2 sin θ .(7.2)

In this scheme all space derivatives are approximated by 3−point centred differenceand time derivatives by forward difference. In order to treat the nonlinearity, an iterationscheme is used. In the first sweep, values of h, U from the previous time step are used.These are then updated and a second sweep is implemented. This procedure is repeateduntil the previous and current values of h and U are within a prescribed tolerance at allpoints. Denote the current intermediate values of h, U by h? and U? .

The x− interval 0 ≤ x ≤ L is split into JJ − 1 intervals of length ∆x = LJJ−1

and so

xj := (j − 1)∆x , j = 1, . . . , JJ ,

andhn

j := h(xj, tn) and Unj := U(xj, tn) ,

where tn = n∆t with ∆t the fixed time step.The discretization of the equations (7.1) is then

hn+1j − hn

j

∆t+ U?

j

hn+1j+1 − hn+1

j−1

2∆x+ h?

j

Un+1j+1 − Un+1

j−1

2∆x= 0 ,

Un+1j − Un

j

∆t+(αn+1

j − (Ωn+1)2h?j + 2Ωn+1U?

j

) hn+1j+1 − hn+1

j−1

2∆x

+(U?

j + 2Ωn+1h?j

) Un+1j+1 − Un+1

j−1

2∆x= βn+1

j + Ωn+1hn+1j ,

(7.3)where

αnj := α(xj, tn) and βn

j := β(xj, tn) .

Setting

znj =

(hn

j

Unj

),

equation (7.3) becomes

−An+1j zn+1

j−1 + Bn+1zn+1j + An+1

j zn+1j+1 =

(0 11 0

)zn

j + ∆t βn+1j

(10

), (7.4)

with

An+1j =

∆t

2∆x

[αn+1

j (U?j + 2Ωn+1h?

j)U?

j h?j

], Bn+1 =

[−Ωn+1∆t 1

1 0

], (7.5)

andαn+1

j = αn+1j − (Ωn+1)2h?

j + 2Ωn+1U?j .

14

The equations at j = 1 and j = JJ are obtained from the boundary conditions. The onlyboundary condition at x = 0 is U = 0. The discrete version of this is

Un1 = 0 and

Un0 + Un

2

2= 0 , for all n ∈ N . (7.6)

To obtain a boundary condition for h , use the mass equation at x = 0

ht + h?Ux = 0 ,

with discretizationhn+1

1 − hn1

∆t+ h?

1

Un+12

∆x= 0 . (7.7)

Combining (7.6) and (7.7) gives the equation for j = 1

zn+11 +

h?1 ∆t

∆x

[0 10 0

]zn+1

2 =

[1 00 0

]zn

1 . (7.8)

Similarly at x = L ,

UnJJ = 0 and

UnJJ−1 + Un

JJ+1

2= 0 , for all n ∈ N . (7.9)

and the discrete mass equation is

hn+1JJ − hn

JJ

∆t− h?

JJ

Un+1JJ−1

∆x= 0 . (7.10)

Combining these two equations gives the discretization at j = JJ ,

−h?JJ ∆t

∆x

[0 10 0

]zn+1

JJ−1 + zn+1JJ =

[1 00 0

]zn

JJ . (7.11)

For fixed h? and U? the following block linear system of equations is to be solved,

zn+11 + h?

1Nzn+12 =

[1 00 0

]zn

1 ,

−An+12 zn+1

1 + Bn+1zn+12 + An+1

2 zn+13 =

[0 11 0

]zn

2 + ∆t βn+12

(10

),

−An+13 zn+1

2 + Bn+1zn+13 + An+1

3 zn+14 =

[0 11 0

]zn

3 + ∆t βn+13

(10

),

......

−An+1JJ−1z

n+1JJ−2 + Bn+1zn+1

JJ−1 + An+1JJ−1z

n+1JJ =

[0 11 0

]zn

JJ−1 + ∆t βn+1JJ−1

(10

),

−h?JJNzn+1

JJ−1 + zn+1JJ =

[1 00 0

]zn

JJ .

where

N :=∆t

∆x

[0 10 0

].

15

The unknown vector is Z := (z1, z2, . . . , zJJ−1, zJJ) ∈ RJJ ×RJJ . Its coefficient matrix is ablock tridiagonal matrix. It is also diagonally dominant, since the norm of Bn+1 is of order1 and the norm of An+1

j is of order ∆t when ∆x is fixed. For fixed h? and U? this is just alinear system and can be solved by standard algorithms. The algorithm is straightforwardto code, and Matlab was used for the programming, with its built in block-tridiagonalsolver. A slight speedup of the algorithm can be obtained by replacing the nonlinear termsUhx + hUx by the linear term −ht and this modified algorithm is discussed in AlemiArdakani & Bridges [10].

The coefficient matrices in the block-tridiagonal system are dependent upon h? and U?

so an iterative process is added to the solution procedure. At the beginning of a time steph?

j is set equal to hnj and U?

j = Unj for each j . Then the linear system is solved, giving new

values for h? and U? which are used to update the coefficient matrices. This process isrepeated until maxj|h?

j −hn+1j |+ |U?

j −Un+1j | is below some tolerance, typically taken to

be 10−8 . We have not studied the convergence properties of this iteration, but away fromcriticality and severe hydraulic jumps the convergence is quick (typically 2-5 iterations).Moreover, according to Abbott [2] and Abbott & Basco [3], this type of iteration iswidely used in the computational hydraulics community.

The initial conditions for the fluid velocity and height at t = 0 are typically taken tobe

U(x, 0) = 0 and h(x, 0) = h0 , (7.12)

where h0 is the still water level.

8 Rigid body motion of the vessel

The fluid vessel is a rigid body free to undergo any motion in the plane. Every rigid bodymotion in the plane is uniquely determined by (q(t),Q(t)) where q(t) is a vector in R

2

and Q(t) is an orthogonal matrix with unit determinant (O’Reilly [57]). In the planethis reduces to two translations q1(t) and q2(t) and an angle θ(t) .

If the translations are zero, then specifying the motion consists of specifying θ(t) ,corresponding to roll or pitch motion of the vessel. Here we will restrict attention toharmonic motion of amplitude δ and frequency ω ,

θ(t) = δ sin(ωt) , Ω = θ , Ω = θ , (8.1)

where, in the shallow water limit, δ is of order ε = h0/L . It is not the value of thefrequency that is important, but its value relative to the natural frequency. In the limit ofshallow water, the natural frequencies of the fluid are

ωm =√

gh0mπ

L, m ∈ N . (8.2)

In the case of pure translations, θ = 0 and q1(t) and/or q2(t) are specified.The case of mixed rotation/translation requires more care. From the viewpoint of a

rigid body, q(t) here is the absolute translation: translation of the origin of the body frameas in Figure 1. However, in ship hydrodynamics spatial translations are relative to thebody motion: translation in the direction of the body axes : that is; surge, sway and heave.Hence one must be careful in interpreting the literature. For example, Dillingham [24]uses absolute translation as here, whereas Huang & Hsiung [36] and Faltinsen &Timokha [29] use translations along the body axes. Either choice is correct, but one

16

must be careful in interpretation. For example, in [29], a coupled pitch-surge excitation isproposed of the form (adjusting to the notation here)

qsurge1 (t) = A1 sin(ωt) , qheave

2 (t) = 0 and θ(t) = A2 cos(ωt) , (8.3)

where A1 and A2 are specified amplitudes. The surge qsurge1 (t) is along the x−axis which

is attached to the body. Hence the absolute acceleration vector is

q1(t) = cos θ(t) qsurge1 (t) ,

q2(t) = sin θ(t) qsurge1 (t) ,

since in general (q1

q2

)=

[cos θ − sin θsin θ cos θ

](qsurge1

qheave2

).

There is an induced vertical displacement relative to the absolute frame. In other words, torepresent a pure pitch-surge excitation of the form (8.3) would require an experiment whichspecified both vertical and horizontal displacements of the vessel relative to a laboratoryframe of reference.

Therefore the natural approach for comparison with experiments (particularly whenboth rotation and translation are present) is to specify the absolute translations along withthe rotations. On the other hand, we are not aware of any experiments which combineboth. The paper of Disimile et al. [26] indicates that their experimental facility has thecapability to produce all 6 degrees of freedom in the forcing, but only planar rotations arereported so far.

When both translations and rotations are specified and are harmonic there is the possi-bility of two frequencies, resulting in quasiperiodic forcing. The simplest analogous modelis that of Duffing’s equation with quasiperiodic forcing. This problem has been studied byWiggins [70] and he shows that chaos results, and experiments of Moon & Holmes [52]on quasiperiodic forcing also indicate chaos. In this paper, the simplest coupled forcing isconsidered, where both frequencies are the same, but the phase differs and results on thisare presented in §9.2.

9 Numerical experiments

To test the algorithm, the first numerical experiments were to compare with previouswork. In the technical report [11] extensive tests are reported, comparing with simulationsof Dillingham [24], Armenio & La Rocca [14] and Huang & Hsiung [36]. Diver-gence is found in parameter regimes where the formulations disagree, and close agreementis found in parameter regimes where the formulations are close. In this section new resultsdemonstrating the flexibility of the formulation are presented. First, the effect of param-eter variation, and then two simulations which include both translations and rotationssimultaneously.

9.1 Effect of centre of rotation

An advantage of the SWEs over the full equations is that parametric studies can be carriedout quickly. Here an example is shown of the effect of change of the centre of rotation.Suppose the vessel is forced to rotate about a point which lies on the vertical line through

17

Figure 2: Computed nondimensional wave height versus the center of rotation for differentfrequencies for the surface SWEs with parameters δ = 1.0 , L = 0.5 m and h0 = 0.075 m ,q1 = q2 = 0 and d1 = −0.25 m . The initial conditions are h(x, 0) = h0 and U(x, 0) = 0and the discretization parameters are ∆x = 0.005 m and ∆t = 0.005 s . The first linearnatural frequency is ω1 ≈ 5.38 rad/sec . The horizontal axis is d2 in m and the verticalaxis is nondimensional wave height in m at x = 0.25m .

the centroid; that is d1 = − 12L and d2 can vary. Taking parameter values as indicated in

the figure caption, Figure 2 shows the nondimensional wave height at x = 0.40 m versusd2 , plotted for different frequencies. The principal trend is that when the centre of rotationis below the tank bottom the response is higher and the farther below the larger theamplitude. Hence tanks on the deck of a ship will have more dramatic fluid response thantanks below deck (other parameters being equal), and tanks situated below the centre ofgravity will in general be more stable.

Pantazopoulos [59] shows numerical results including the effect of centre of rotatation(see Figure 6 in Pantazopoulos [59] and Figure 29 in Pantazopoulos [58]). His resultsshow monotone increase of amplitude with d2 . For most frequency values the results inFigure 2 are in qualitative agreement with [59]. However, a new phenomena shows up herewhen the forcing frequency is at approximately the second natural frequency (the lowestcurve in Figure 2), where initially there is a decrease and then increase. This latter casewas not considered in [59, 58].

9.2 Coupled surge–pitch forcing

Coupled surge-pitch, or surge-pitch-heave forcing is easily modeled with the surface SWEs.In this section results illustrating the coupled surge-pitch motion are presented. For sim-plicity the surge and pitch forcing are chosen to be harmonic with the same frequency butout of phase, and the amplitude of the two is the same order of magnitude. The rotationis

θ(t) = δθ0 sin(ωt + θ1) , 0 ≤ θ1 < π ,

18

Figure 3: Computed nondimensional wave height ( |hmax− hmin|/L) versus the phase shiftfor different frequencies for the surface SWEs with parameters δ = 0.035, θ0 = 0.5 rad ,L = 0.5 m , h0 = 0.075 m , d1 = −0.25 m , d2 = 0.1 m , ∆x = 0.005 m and ∆t = 0.005 s .The horizontal axis is θ1 in rad and the vertical axis is nondimensional wave height atx = 0.25 m .

and horizontal translation isq1(t) = δL sin(ωt) .

The parameters are fixed at

δ = 0.035 , θ0 = 0.5 rad ,

L = 0.5 m , h0 = 0.075 m , d1 = −0.25 m , d2 = 0.1 m ,

and the discretization parameters are ∆x = 0.005 m and ∆t = 0.005 s . Figure 3 showscomputed nondimensional wave height ( |hmax−hmin|/L) versus the phase shift for differentfrequencies for the surface SWEs. The maximum response is when the pitch and surgeforcing are 180o out of phase.

Snapshots of the fluid motion at a sequence of times are shown in Figure 9.2. In thisfigure the parameter values are

δ = 0.244346 , θ0 = 0.5 rad , ω = 5.0 rad/sec , θ1 = π6rad ,

L = 0.5 m , h0 = 0.15 m , d1 = −0.25 m , d2 = 0.0 m ,(9.1)

and the discretization parameters are ∆x = 0.005 m and ∆t = 0.005 s . In these simula-tions transients have settled down (these figures show the flowfield between 460 and 800timesteps). Full simulations including earlier and later times have been combined into avideo. This and other sloshing videos can be seen at the Surrey sloshing website [1].

9.3 Sloshing on the London Eye

The London eye is a large ferris wheel in the center of the city. Our mathematical modelconsists of a vessel, partially filled with fluid, prescribed to travel along a circular path.Even when the speed along the path is constant, sloshing occurs due to change in direction.

19

Figure 4: Snapshots of rigid body motion with interior fluid sloshing for coupled surge-pitchmotion, with parameter values given in (9.1). The snapshots are at t = 2.30 s , t = 2.85 sfor the first row from left to right, t = 3.30 s , t = 3.55 s for the second row from left toright, and t = 3.85 s , t = 4.0 s for the third row from left to right.

20

The base of the vessel remains horizontal along the path. In addition the vessel can alsohave a prescribed rotation about the point of attachment.

The interest in this example is threefold. It is an example with very large displacementsof the vehicle and illustrates the generality of the prescribed vessel motion. Secondly, itis a prototype for the transport of a vessel along a surface. In this case the surface is athe two-sphere and the curve is a great circle. As the vehicle moves along the surface itcan also rotate relative to the point of attachment. Other examples of surfaces of interestare the full two-sphere or a bumpy sphere, which is a model for a satellite containing fluidand orbiting the earth, and a surface modelling terrain. The latter is a model for vehiclestransporting liquid on roads through hilly terrain.

Thirdly, it is an excellent setting to test control strategies for sloshing. For example,suppose the speed along a path in the surface is prescribed. Sloshing will result if the pathis curved due to induced acceleration. The local rotation of the body could act as a control,and roll, pitch or yaw could be induced to counteract any sloshing due to motion along thepath.

The interest in this example is as a prototype for more general trajectories, since theactual London Eye is designed to have extremely low centripetal acceleration. Accordingto the London Eye website, the radius, rc , of the wheel is 65 m , and the travel time of apod is 30 min , giving a frequency of ωc = π/900. The relevant dimensionless parameter isthe ratio of centripetal acceleration to gravity

Fr2 =ω2

crc

g. (9.1)

This parameter is like a Froude number, since the ratio is a velocity squared over grc , butit is a vessel motion parameter and not a a fluid parameter. The Froude number for theLondon Eye is

Fr ≈ 0.009 .

This Froude number is quite low (passengers can disembark from the London Eye withoutit changing speed), and so we will increase it by an order of magnitude in order to inducesloshing fluid motion.

The ferris wheel has radius rc and the circular path is described by the parameterization

q1(t) = −rc (cos θ0 − cos θc) ,

q2(t) = rc (sin θc − sin θ0) ,(9.2)

whereθc = ωct + θ0 ,

and is prescribed to harmonically rotate with frequency ω and amplitude δ about thesuspension point

θ = δ sin ωt , Ω = θ , Ω = θ . (9.3)

The input data is

h0 = 0.15 m , L = 1 m , δ = 0.3047 rad ,

ω = ω1 = 3.8109 rad/sec , θ0 = 0 rad , ωc = 0.5 rad/sec ,

rc = 2 m , d1 = −0.5 m , d2 = −0.60 m , ∆t = 0.01 sec ,

∆x = 0.01 m .

21

Figure 5: Snapshots of a rigid-body with interior fluid sloshing at a sequence of times. Thesnapshots in a counterclockwise order are at t = 1 sec , t = 3.1 sec , t = 5.3 sec , t = 7.2 secand t = 9.7 sec .

The Froude number in this case is Fr = 1/√

2g . Then with the initial conditions (7.12),Figure 5 shows the formation and propagation of a travelling hydraulic jump at a sequenceof times when the tank is rotating about the suspension point harmonically with the forc-ing frequency near the first natural frequency of the fluid. In this example, the angulardisplacement is of order h0/L whereas the translations are of order one, consistent withthe conclusions of the asymptotic analysis in Appendix A.

For the second numerical experiment set the input data as

h0 = 0.15 m , L = 1 m , δ = 0.15056 rad ,

ω = ω2 = 7.6218 rad/sec , θ0 = 0 rad , ωc = 0.5 rad/sec ,

rc = 2 m , d1 = −0.5 m , d2 = −0.60 m , ∆t = 0.01 sec ,

∆x = 0.01 m .

The Froude number here is also Fr = 1/√

2g .Figure 6 shows the snapshots of the rigid-body with the interior fluid sloshing when

the frequency of rotation about the suspension point is near the second natural frequencyof the fluid which causes the appearance of additional harmonics in the generated waveshape.

10 Concluding remarks

The starting point of this paper was the surface equations for sloshing in a vessel under-going rigid body motion in the plane. These equations improve on previous shallow waterequations for sloshing in that the vessel motion is exact and there are only two assump-tions on the fluid motion. They show how the breaking criterion of Penney-Price-Taylor

22

Figure 6: Snapshots of a rigid-body with interior fluid sloshing when ω = ω2 .

arises, how surface tension affects the SWEs, and are amenable to numerical simulation.The advantage of restricting to shallow water flow (rather than the full two-dimensionalequations) is that numerical imulations of the SWEs are much faster.

Remarkably, all these properties carry over to the case of shallow water sloshing inthree dimensions. In Alemi Ardakani & Bridges [12] the surface equations for shallowwater sloshing in a vessel undergoing an arbitrary rigid body motion in three dimensionsare derived. The motion of the vessel is exact and the only approximations are on the fluidmotion, again requiring only two natural hypotheses on the surface vertical velocity andacceleration, generalising (1.9).

In this paper the motion of the vessel has been prescribed. The vessel motion canalso be determined by solving the rigid-body equations coupled to the fluid motion. Thedynamically coupled problem is studied in [7, 6].

As an example of the coupled problem, consider for simplicity pure horizontal translationof the vessel (q2 = θ = 0). The exact equation for the dynamically-coupled motion of avessel of dry mass ms and spring constant ν with the fluid is

d

dt

(∫ L

0

∫ h

0

ρu(x, y, t) dydx + (ms + mf )q1

)+ νq1 = 0 ,

where mf = ρh0L is the fluid mass. A derivation of this equation is given in [7]. Now,approximate the horizontal momentum by a shallow water model

d

dt

(∫ L

0

ρhU(x, t) dx + (ms + mf )q1

)+ νq1 = 0 .

Then using the momentum equation with q2 = θ = 0 this equation can be reduced to

msq1 + νq1 = 12ρg(h(L, t)2 − h(0, t))2 , (10.1)

23

which is Cooker’s equation (Cooker [23]). Solution of the coupled problem, where q1(t) isdetermined from (10.1) rather than specified, leads to very different dynamics. The naturalfrequencies change and the dynamic coupling can mollify the fluid motion or enhanceit. Numerical simulation of the coupled problem (5.4) and (10.1) is reported in AlemiArdakani & Bridges [7]. The full dynamically coupled problem including rotation isstudied in Alemi Ardakani [5].

— Appendix —

A Asymptotic validity of (SWE-1) and (SWE-2)

In this section a scaling argument and asymptotics are used to analyze (SWE-1) and(SWE-2) in the shallow-water limit. Let U0 =

√gh0 be the representative horizontal ve-

locity scale. Introduce the standard shallow-water scaling (e.g. p. 482 of Dingemans [25]or p. 26 of Johnson [40]),

x =x

L, y =

y

h0

=y

εL, t =

tU0

L,

u =u

U0

, v =v

εU0

, h =h

h0

, p =p

ρgh0

,

(A-1)

where ε is the shallowness parameter (5.3). The scaled version of the surface velocities are

denoted by U and V .The typical strategy for deriving an asymptotic shallow-water model is to scale the full

Euler equations, and then use an asymptotic argument to reduce the vertical pressure fieldand vertical velocities (e.g. §5.1 of Dingemans [25]). Here however we have an advantageas the full Euler equations have been reduced to the exact surface equations (3.4). Hence thestrategy here is to start by scaling the exact equations (3.4), and then apply an asymptoticargument. This strategy leads to a concise argument for the shallow-asymptotic form for(SWE-1) and (SWE-2).

To check (SWE-1), start by scaling the exact mass equation in (5.1)

het + (hU)ex = V + hUex . (A-2)

At first glance it appears that the left-hand side and the right-hand side are of the sameorder, since ε does not appear. However, the sum of the two terms on the right-hand sideis of higher order. The fact that the right-hand side is of higher order is intuitively clear,since it can be expressed in terms of the velocity differences U − u and in the shallow-water approximation the horizontal surface velocity and average velocity are asymptoticallyequivalent. However, to make this precise go back to the unscaled mass equation and rewritethe right-hand side as

ht + (hU)x =∂

∂x

(∫ h

0

yuy dy

),

using equation (5.2) and the identity

U − u =1

h

∫ h

0

yuy dy .

24

Now substitute for uy using the vorticity field

ht + (hU)x =∂

∂x

(∫ h

0

y (vx − V) dy

). (A-3)

The key to showing the right-hand side is of higher order is the scaling of the vorticity.The appropriate scaling (which leads to a uniform ε2 estimate of uy ) is to assume that thevorticity is of order ε

V =U0

LεV .

Scaling (A-3) then gives

het + (hU)ex = ε2 ∂

∂x

∫ eh0

y

(∂v

∂x− V

)dy . (A-4)

Taking the limit ε → 0 shows that (SWE-1) is satisfied. However, to be precise it isessential that

∂

∂x

∫ eh0

y

(∂v

∂x− V

)dy is bounded in the limit ε → 0 . (A-5)

Assumption (SWE-2) requires that the vertical acceleration, DvDt

∣∣h , in(a(x, t) +

Dv

Dt

∣∣∣∣h)

(A-6)

be small, relative to magnitude of a (x, t) . Scaling the Lagrangian vertical accelerationgives

DvDt

∣∣∣∣h = Vt + UVx

= εU2

0

L

(Vet + U Vex

)= ε

U20

LDevDet∣∣∣∣eh .

HenceDv

Dt

∣∣∣∣h = gε2Dv

Dt

∣∣∣∣eh ,

using U20 = gh0 = gLε . The scaled version of (A-6) is therefore(

a +Dv

Dt

∣∣∣∣h)

= g

(a

g+ ε2Dw

Dt

∣∣∣∣eh)

.

In the shallow-water regime, the assumption (SWE-2) is satisfied if

a

gis of order one and

∣∣∣∣Dv

Dt

∣∣eh∣∣∣∣ is bounded as ε → 0 . (A-7)

25

However, by introducing scaling and taking an asymptotic limit, other anomalies canbe introduced. Introduce the following scaled variables/parameters

q1 =q1

L, q2 =

q2

h0

, d1 =d1

L, d2 =

d2

h0

. (A-8)

Now using (A-1) and (A-8) and scaling the momentum equation in (5.4) considering (SWE-4) and taking an asymptotic limit as ε → 0 gives

Uet + U Uex + cos θhex = −sin θ

ε+ θ2et

(x + d1

)− q1etet cos θ + O(ε) , (A-9)

which shows that the first term on the right-hand side is of order ε−1 and is unboundedas ε → 0. Hence this scaling puts a restriction on the vessel rotation angle. So in theshallow-water regime θ should be of order ε or higher to ensure that the first term on theright-hand side of (A-9) is consistent. In conclusion, this scaling suggest that roll/pitchmotion should be an order of magnitude smaller than q-translations, for ε → 0, in orderto avoid large fluid motions that would violate (SWE-1) and (SWE-2).

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