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NONLINEAR MODELLING OF LIQUID SLOSHING IN A MOVING RECTANGULAR TANK

M. Serdar CELEBI* and Hakan AKYILDIZ!!!!

Abstract

A nonlinear liquid sloshing inside a partially filled rectangular tank has been investigated. The fluid is assumed to be homogeneous, isotropic, viscous, Newtonian and exhibit only limited compressibility. The tank is forced to move harmonically along a vertical curve with the rolling motion to simulate the actual tank excitation. The volume of fluid technique is used to track the free surface. The model solves the complete Navier-Stokes equations in primitive variables by use of the finite difference approximations. At each time step, a donar- acceptor method is used to transport the volume of fluid function and hence the locations of the free surface. In order to assess the accuracy of the method used, computations are verified through convergence tests and compared with the theoretical solutions and experimental results.

Key Words: Sloshing, Free Surface Flow, Navier-Stokes Equations, Volume of Fluid Technique, Moving Rectangular Tank, Vertical Baffle, Finite Difference Method

1. INTRODUCTION

Liquid sloshing in a moving container constitutes a broad class of problems of great

practical importance with regard to the safety of transportation systems, such as tank trucks

on highways, liquid tank cars on railroads, and sloshing of liquid cargo in oceangoing vessels.

It is known that partially filled tanks are prone to violent sloshing under certain motions. The

large liquid movement creates highly localized impact pressure on tank walls which may in

turn cause structural damage and may even create sufficient moment to effect the stability of

the vehicle which carries the container. When a tank is partially filled with fluid, a free

surface is present. Then, rigid body acceleration of the tank produces a subsequent sloshing of

the fluid. During this movement, it supplies energy to sustain the sloshing. There are two

major problems arising in a computational approach to sloshing; these are the moving

boundary conditions at the fluid tank interface, and the nonlinear motion of the free surface.

* Assoc.Prof.Dr., Istanbul Technical University, Faculty of Naval Architecture and Ocean Engineering, 80626, Maslak, Istanbul-TURKEY ! Dr., Istanbul Technical University, Faculty of Naval Architecture and Ocean Engineering, 80626, Maslak, Istanbul-TURKEY

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Therefore, in order to include the nonlinearity and avoid the complex boundary conditions of

moving walls, a moving coordinate system is used. The amplitude of the slosh, in general,

depends on the nature, amplitude and frequency of the tank motion, liquid-fill depth, liquid

properties and tank geometry. When the frequency of the tank motion is close to one of the

natural frequencies of the tank fluid, large sloshing amplitudes can be expected.

Sloshing is not a gentle phenomenon even at very small amplitude excitations. The

fluid motion can become very non-linear, surface slopes can approach infinity and the fluid

may encounter the tank top in an enclosed tanks. Hirt and Nichols (1981) developed a

method known as the volume of fluid (VOF). This method allows steep and highly contorted

free surfaces. The flexibility of this method suggests that it could be applied to the numerical

simulation of sloshing and is therefore used as a base in this study. On the other hand, analytic

study of the liquid motion in an accelerating container is not new. Abramson (1966) provides

a rather comprehensive review and discussion of the analytic and experimental studies of

liquid sloshing, which took place prior to 1966. The advent of high speed computers, the

subsequent maturation of computational techniques for fluid dynamic problems and other

limitations mentioned above have allowed a new, and powerful approach to sloshing; the

numerical approach. Von Kerczek (1975), in a survey paper, discusses some very early

numerical models of a type of sloshing problem, the Rayleigh-Taylor instability. Feng (1973)

used a three-dimensional version of the marker and cell method (MAC) to study sloshing in a

rectangular tank. This method consumes large amount of computer memory and CPU time

and the results reported indicate the presence of instability. Faltinsen (1974) suggests a

nonlinear analytic method for simulating sloshing, which satisfies the nonlinear boundary

condition at the free surface.

Nakayama and Washizu (1980) used a method basically allows large amplitude

excitation in a moving reference frame. The nonlinear free surface boundary conditions are

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addressed using an “incremental procedure”. This study employs a moving reference frame

for the numerical simulation of sloshing.

Sloshing is characterized by strong nonlinear fluid motion. If the interior of tank is

smooth, the fluid viscosity plays a minor role. This makes possible the potential flow solution

for the sloshing in a rigid tank. One approach is to solve the problem in the time domain with

complete nonlinear free surface conditions (see Faltinsen 1978). Dillingham (1981)

addressed the problem of trapped fluid on the deck of fishing vessels, which sloshes back, and

fort and could result in destabilization of the fishing vessel. Lui and Lou (1990) studied the

dynamic coupling of a liquid-tank system under transient excitation analytically for a two-

dimensional rectangular rigid tank with no baffles. They showed that the discrepancy of

responses in the two systems can obviously be observed when the ratio of the natural

frequency of the fluid and the natural frequency of the tank are close to unity. Solaas and

Faltinsen (1997) applied the Moiseev’s procedure to derive a combined numerical and

analytical method for sloshing in a general two-dimensional tank with vertical sides at the

mean waterline. A low-order panel method based on Green’s second identity is used as part of

the solution. On the other hand, Celebi et al. (1998) applied a desingularized boundary

integral equation method (DBIEM) to model the wave formation in a three-dimensional

numerical wave tank using the mixed Eulerian-Lagrangian (MEL) technique. Kim and Celebi

(1998) developed a technique in a tank to simulate the fully nonlinear interactions of waves

with a body in the presence of internal secondary flow. A recent paper by Lee and Choi

(1999) studied the sloshing in cargo tanks including hydro elastic effects. They described the

fluid motion by higher-order boundary element method and the structural response by a thin

plate theory.

If the fluid assumed to be homogeneous and remain laminar, approximating the

governing partial differential equations by difference equations would solve the sloshing

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problem. The governing equations are the Navier-Stokes equations and they represent a

mixed hyperbolic-elliptic set of nonlinear partial differential equations for an incompressible

fluid. The location and transport of the free surface in the tank was addressed using a

numerical technique known as the volume of fluid technique. The volume of fluid method is

a powerful method based on a function whose value is unity at any point occupied by fluid

and zero elsewhere. In the technique, the flow field was discretized into many small control

volumes. The equations of motion were then satisfied in each control volume. At each time

step, a donar-acceptor method is used to transport the fluid through the mesh. It is extremely

simple method, requiring only one pass through the mesh and some simple tests to determine

the orientation of fluid.

2. MATHEMATICAL MODELLING OF SLOSHING

The fluid is assumed to be homogenous, isotropic, viscous and Newtonian and

exhibits only limited compressibility. Tank and fluid motions are assumed to be two

dimensional, which implies that there is no variation of fluid properties or flow parameters in

one of the coordinate directions. The domain considered here is a rigid rectangular container

with and without baffle configuration partially filled with liquid.

2.1 Tank Motion

Two different motion models that result in a two-dimensional tank excitation will be

considered in the following subsections.

2.1.1 Moving and Rolling Motion

In this study, we will consider the tank excitation due to the motion of a tank moving

with speed V(t) along a vertical curve Y = ξ(X) as shown in Figure 1. The following relations

can be written:

5

R)t(V)t(

R)t(V)t(U

)t(V)t(U2

y

x

−=Ω

−=

=

!

!!

(1)

where the radius of curvature is,

2/32X

XX

)1(R

ξ+ξ

= , (2)

and the geometric constraint is,

X)1()t(V 2/12x

!ξ+= . (3)

For a given tank speed V(t) and vertical motion profile ξ(X), the horizontal movement of the

tank X(t) can be evaluated by solving Equation (3). Then, the basic modes of excitation xU! ,

yU! , and Ω where jUiUU yx +="

can be obtained. To be more specifically, we shall assume

a periodical motion profile for a study of harmonic excitation

kXcosk

)X( oθ=ξ (4)

where koθ is the elevation amplitude of the profile, k (=2π/λ) is the wave number of the

profile and λ is the wavelength of the profile. The velocity is described by

)tkVcos1(V)t(X o2

oo δθ+=! (5)

where Vo is a characteristic tank speed and δ is a parameter of which characterized the

response to the grade change. From Equations (1) – (5), it can be shown that:

)t2sintsin(V)t(U 212

oox ω−ωδωθ−=! (6)

tcosV)t(U ooy ωωθ=! (7)

tcos)t( o ωωθ=Ω (8)

6

where ω = (k Vo) is the characteristic frequency of excitation.

2.1.2 Rolling Motion

For a rolling motion about an axis on (X, Y) = (0, -d) in Figure 2, Ωand,U,U yx!! are

specified as:

θ=Ω

Ω−=

Ω=

!

!

!!

2y

x

dU

dU

(9)

where θ is the angular displacement. It is assumed that tcoso ωθ=θ , where θο and ω

represent the rolling amplitude and rolling frequency respectively.

2.2 General Formulation

Before attempting to describe the governing equations, it is necessary to impose the

appropriate physical conditions on the boundaries of the fluid domain. On the solid boundary,

the fluid velocity equals the velocity of the body.

0Vtu,0V

nu

tn ==∂∂==

∂∂ ""

(10)

where nV"

and tV"

are normal and tangential components and u and v are the horizontal (x)

and vertical (y) components of the fluid velocity respectively.

The location of the free surface is not known priori and presents a problem when the

boundary conditions are to be applied. If the free surface boundary conditions are not applied

at the proper location, the momentum may not be conserved and this would yield incorrect

results. Tangential stresses are negligible because of the larger fluid density comparing with

the air. The only stress at such a surface is the normal pressure. Therefore, the summation of

the forces normal to the free surface must be balanced by the atmospheric pressure. This

yields the dynamic boundary condition at a free surface

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0x

)v(mnx

)v(y

)u(mnx

)u(mn2PP yyyxxxATM =

∂ρ∂+

∂ρ∂+

∂ρ∂+

∂ρ∂ν+= (11)

where PATM, ν and ρ are atmospheric pressure, kinematic viscosity and density of the fluid

and nx, mx are the horizontal components of the unit vector, normal and tangent to the surface

respectively and similarly, ny, my are the vertical components of the unit vector, normal and

tangent to the surface respectively. In addition, it is necessary to impose the kinematic

boundary condition that the normal velocity of the fluid and the free surface are equal.

In this study, unsteady motion takes place and the characteristic time during the flow

changes may be very small. Therefore, compressibility of fluid may not be ignored. In some

cases, it is desirable to assume that the pressure is a function of density.

2cddp =ρ

(12)

where c is the adiabatic speed of sound. Expanding the mass equation about the constant

mean density ρο and retaining only the lowest order terms, yields

.0Vtc

102 =⋅∇ρ+

∂ρ∂ "

(13)

The forces acting on the fluid in order to conserve momentum must balance the rate of

change of momentum of fluid per unit volume. This principle is expressed as

( ) ( ) ( ),VFpVVVt

2"""""

ρ∇ν++−∇=ρ∇⋅+ρ∂∂ (14)

where p is the pressure and F"

is the body force(s) acting on the fluid.

2.3 The Coordinate System and Body Forces

In order to include the non-linearity and avoid the complex boundary conditions of

moving walls, the moving coordinate system is used. The origin of the coordinate system is

in the position of the center plane of the tank and in the undisturbed free surface. The moving

coordinate is translating and rotating relative to an inertial system (see Figure 2). The

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equilibrium position of the tank relative to the axis of rotation is defined by γ. For instance,

the tank is rotating about a fixed point on the y-axis at γ = 90o. Thus the moving coordinate

system can be used to represent general roll (displayed by θ) or pitch of the tank.

We suppose that the moving frame of reference is instantaneously rotating with an

angular velocity ( )θθθθ!"Ω about a point O which itself is moving relative to the Newtonian frame

with the acceleration U"! . The absolute acceleration of an element is then,

*aUA ""!

"+= (15)

where *a" is the acceleration of an element relative to the point O. Here *a" is represented by

( )rrtt

r2tra 2

2* """"

"""""

×Ω×Ω+×∂Ω∂+

∂∂×Ω+

∂∂= (16)

where atr2

2 ""

=∂∂ is the acceleration of an element relative to the translating and rotating frame

of reference and *utr ""

=∂∂ is the velocity of the element in this frame. The absolute

acceleration of an element is thus,

( )rru2aUA *"!!"!!"!""

!"

×θ×θ+×θ+×θ++= (17)

This expression may be equated to the local force acting per unit mass of fluid to give

the equation of motion in the moving frame. Here, U"! is simply the apparent body force such

as drift force; u2 "! ×θ is the deflecting or coriolis force; r"!! ×θ is referred to as the Euler force

and ( )r"!! ×θ×θ is the centrifugal force. Thus, the body force term in the Equation (14) is

expressed in component form as

v2U)cossin(dxysingF x22

x θ−−γθ−γθ−θ+θ−θ−= !!!!!!!! (18a)

u2U)sincos(dyxcosgF y22

y θ+−γθ+γθ+θ+θ+θ= !!!!!!!! (18b)

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where g is the gravitational acceleration, d is the distance between the origin of the moving

coordinate and the axes of rotation, xU! and yU! are the accelerations of the tank in the x and

y directions given in Equations (1) and (9).

The governing Equation (13) of the fluid motion with the limited compressibility

option (Equation 12) yields to following equation by normalizing the fluid mean density to

one

0yv

xu

tp

c1

2 =∂∂+

∂∂+

∂∂ (19)

where all variables are now defined in the tank-fixed coordinate system. The modified

momentum equations yield the required expressions for two-dimensional flow in a rotating

tank

( ) ( ) ( ),ucosdxUsindyv2singxp

yuv

xuu

tu 22

x ∇ν+γ+θ+−γ+θ−θ−θ−=∂∂+

∂∂+

∂∂+

∂∂ !!!!! (20a)

( ) ( ) ( )vsindyUcosdxu2cosgyp

yvv

xvu

tv 22

y ∇ν+γ+θ+−γ+θ+θ+θ=∂∂+

∂∂+

∂∂+

∂∂ !!!!! . (20b)

3. NUMERICAL STABILITY AND ACCURACY

In this section the strengths and weaknesses of the numerical technique that effect the

stability and accuracy as well as the limitation on the extent of computation will be discussed.

In the numerical study, the flow field is discretized into many small control volumes. The

equations of motion are then satisfied in each small control volume. Obvious requirements

for the accuracy are included the necessity for the control volumes or cells to be small enough

to resolve the features of interest and for time steps to be small enough to prevent instability.

Once a mesh has been chosen, the choice of the time increment necessary for the numerical

stability is governed by two restrictions: One of them is that the fluid particles can not move

through more than one cell in one time step, because the difference equations are assumed the

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fluxes only between adjacent cells. Therefore, the time increment must satisfy the following

inequality,

δδ<δ ++

j,i

2/1i

j,i

2/1i

Vy,

Uxmint (21)

where 2/1ix +δ and 2/1iy +δ are the half sizes of the cell in x and y directions respectively.

Typically tδ is chosen equal to a time between one-fourth and one-third of the minimum cell

transit time. The second restriction is that, for a non-zero value of kinematic viscosity,

momentum must not diffuse more than approximately one cell in one time step. A linear

stability analysis shows that this limitation implies

2j

2i

2j

2i

yxyx

21t

δ+δδδ

<νδ (22)

The other parameter necessary to insure numerical stability is αααα , which is the

upstream differencing parameter. In the absence of physical viscosity, αααα must be included

for the stability. It can also be seen how αααα can be adjusted to minimize diffusion-like

truncation error. The proper choice for αααα is then,

δδ

δδ

≥α≥++ 2/1i

j,i

2/1i

j,i

ytV

,x

tUmax1 (23)

For our computations, αααα is typically set to be 30-50% higher than the Courant number.

Formally, when considering accuracy of a finite difference scheme, the order of accuracy is

defined by the lowest order powers of the increments of time and space appearing in the

truncation error of the modified equation. A higher order scheme, which is second order

accurate, can be used to improve accuracy, but any process, which increases the accuracy of

the results, will also increase the computation time, and in most cases the relationship is non-

linear. Another parameter, which has an effect on the accuracy, is σ . This is the criterion

used to govern the level of mass conservation. For an incompressible fluid,

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σ≤⋅∇ u" (24)

If σ is not zero, then the fluid is numerically compressible. Since it is extremely difficult to

enforce zero divergence, a finite value must be used. Typical values are about 310−=σ . But

it has been found that even larger values of epsilon do not seriously affect the results.

4. THEORETICAL ANALYSIS

The procedure of theoretical solutions used in comparison to the numerical results is

briefly summarized in this section. For a rectangular tank without any internal obstacles under

combined external excitations (e.g. sway plus roll or surge and pitch), analytical solutions can

be derived from the fundamental governing equations of fluid mechanics. These solutions can

be used to predict liquid motions inside the tank, the resultant dynamic pressures on tank

walls, and the effect of phase relationship between the excitations on sloshing loads.

The case is considered as a two-dimensional, rigid, rectangular tank without internal

obstacles that is filled with inviscid, incompressible liquid. It is forced to oscillate

harmonically with a horizontal velocity Ux, vertical velocity Uy, and a rotational velocity Ω.

Since the fluid is incompressible, the velocity potential must satisfy the Laplace equation with

the boundary conditions on the tank walls. The dynamic and kinematic free surface boundary

conditions must, then, be satisfied on the instantaneous free surface.

For the analytical solutions, it is assumed that: (i) the amplitudes of tank motion Ω, Ux

and Uy are of the same order of magnitudes, being proportional to a small parameter

ε << 1, (ii) Ω, Ux and Uy oscillate at the same frequency but in different phases. Once the

velocity potential φ has been determined, the pressure of the fluid can be calculated from the

Bernoulli’s equation.

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5. NUMERICAL IMPLEMENTATIONS

It is assumed that the mesh dimensions would be small enough to resolve the main

feature of liquid sloshing in each case. The step of time advance t∆ , in each cycle is also

assumed to be so small that no significant flow change would occur during t∆ . There is no

case where a steady state solution is reached in the forcing periods used. Either instability set

in or computer time becomes excessive, so the duration of computation is limited for each

case. Therefore, computations are halted when the fluid particles extremely interact and spray

over the top side of the tank during the extreme sloshing. In all cases the tank starts to roll

about the centre of the tank bottom at time t = 0+. Since the major concern is to find the peak

wave elevations on the left wall of the tank on the free surface, and forces and moments on

both walls, the analysis is based on the comparison of the wave elevations above the calm free

surface and corresponding forces and moments exerted.

5.1 Moving Rectangular Tank

For the numerical solutions with the moving rectangular tank along a vertical curve,

the β value (β = )aD

2

2

of 0.0625 is used. For fill depth D = 4 ft, the effective tank length (2a)

corresponding to β value is 32 ft.. The parameter δ (given in Equation 5) is chosen to equal to

–1 showing the variation of tank speed with acceleration downhill and declaration uphill. The

tank speed Vo (given in Equation 5) is taken as 7.5 ft/sec.. In the following numerical

computations, the excitation frequency of the tank ω is varied from 0.1 to 1.3 rad/sec. and the

corresponding wavelength of the periodical motion profile is taken as 1.43 ft.. Tank roll

motion is defined by tsino ωθ=θ where θο is the rolling amplitude.

A typical numerical simulation of sloshing in a rigid rectangular tank with and without

baffle is shown in Figure 3 for the case of resonant frequency of the fluid, ωn = 1.0864

13

rad/sec., and rolling amplitude θο = 8ο. Αs can be seen from the snapshots, the baffled case

decreased the amplitude of sloshing but generated some additional eddies near the baffle.

Figure 4 through 8 show the plots of the wave amplitude, horizontal and vertical

acceleration of the tank, sloshing forces and moments in the longitudinal directions in

connection with the rolling of the tank. For the lower frequency of the tank excitation,

ω = 0.1 rad/sec. and the rolling amplitudes θο = 5.73ο and θο = 8ο, in Figure 4, the normalized

sloshing force and moment are only slightly dependent on the amplitude of excitation. For the

sloshing force without baffle (see in Figure 4c), the numerical and theoretical results agreed

well at the rolling amplitude θο = 5.73ο. The wave profile exhibits linear characteristics due to

the lower excitation frequency. During the simulation, computations show that the numerical

result of maximum force (Fmax) gives %7 overestimate compared with the theoretical solution.

It can be concluded from the baffled results that, for the lower excitation frequency, sloshing

force and moment reduced slightly compared to those of unbaffled case.

As the excitation frequency of ω is increased to 0.5, results start to diverge from linear

characteristics as shown in Figure 5. The normalized sloshing force and moment are deviated

slightly for the rolling amplitudes θο = 5.73ο and θο = 8ο due to the still existing linear effects.

Results show that there is a significant difference between theoretical (linear solution) and

nonlinear numerical force calculations especially near the ωt = 1.5 ~ 2.5 due to the dominant

hydrostatic effect corresponding to the wave amplitude. It is also seen that there is a %36

increase in maximum forces (Fmax) between rolling frequencies ω = 0.1 and 0.5 rad/sec.. In

the case of baffle configuration (in Figures 5g-h), the maximum forces and moments are

reduced %19 and %24 compared to the unbaffled cases respectively. We also observed that

the effect of baffle on sloshing force and moment increased as the rolling frequency changes

from ω = 0.1 to 0.5 rad/sec..

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It is seen, in Figure 6, that the normalized sloshing force and moment are increased

depending on the amplitude of excitation due to the non-linear effects as the parameter range

(ωt) increases in connection with the tank motion along a vertical curve. In baffled case (in

Figures 6g-h), the maximum forces and moments are reduced %114 and %207 (for the rolling

amplitude θο = 5.73ο) compared to the unbaffled cases respectively. It can be concluded from

the results that the maximum effect of baffle occurs near the resonance frequency of the fluid

(ωn = 1.0864 rad/sec.).

On the resonance frequency, in Figure 7, the magnitude of the maximum forces and

moments did not change significantly compared with the rolling frequency ω = 0.9 rad/sec.,

but the effect of baffle becomes less due to the increasing sloshing effects (turbulent eddies,

wave breaking and spraying) near the baffle (see in Figures 7g-h).

On the off-resonant frequency ω = 1.3 rad/sec., in Figure 8, it is observed that the

sloshing effects are significantly reduced (% 86 in force and %97 in moment for the rolling

amplitude θο = 5.73ο) compared with the resonant frequency near the baffle. For the larger

rolling amplitude (θο = 8ο), new values become %42 in force and %37 in moment. One

possible reason may be that the increasing effect of turbulence is reduced the baffle effect due

the increased pressure gradient variations on the baffle surface.

In order to show the impact of the vertical baffle located at the mid-bottom of the

rectangular rigid tank, the percentage of reduction for forces and moments between unbaffled

and baffled cases was computed for different rolling amplitudes as shown in Figures 9 and 10.

The vertical axes which is represented by %∆F and %∆M calculated as unbaffled

baffledunbaffled

FFF

F −=∆ ,

unbaffled

baffledunbaffled

MMM

M −=∆ . It can be noted from the Figures 9 and 10 that the amount of

reduction in force and moment is increased as the rolling frequency ω approaches to the

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resonant frequency of the fluid where ωn is 1.0864 rad/sec.. It can be also observed that the

reduction in force and moment is started to decrease after the rolling frequency passed the

resonant frequency.

5.2 Rolling Rectangular Tank

For the numerical solutions with the rigid tank in roll motion, the β values

(β = )aD

2

2

of 0.0625 and 1.0 are used. The first value of β corresponds to shallow water case,

and the second is of a typical intermediate fill depth. The frequency of roll excitation (ω) and

amplitude (θo) are varied from 0.6 to 1.0 rad/sec. and 4o to 8o respectively. For a tank width

of 60 ft., fill depth gets values of 7.5 ft. and 30 ft. in terms of β value. The parameter d, which

defines the location of the center of roll motion of the rigid tank is chosen to be equal to 1.0

and 2.0 ft. The resonance frequency of the fluid inside the tank is ωn = 1.243 rad/sec..

Starting with a tank of intermediate fill depth (β = 1) and with the rolling axes located at the

bottom of the tank (d = 1), a typical snapshot for the simulation of the sloshing with and

without baffle configuration is given in Figure 11. Figures 12 through 15 show the plots of

wave profiles, angular accelerations, sloshing force and turning moment in a rigid tank due to

the roll motion.

For the lower frequency ω = 0.6 rad/sec., as shown in Figures 12c and 12d, the

normalized sloshing forces and moments for the unbaffled case are only very slightly

dependent on the amplitude of excitations θο = 4ο and 8ο. It is seen that there is a significant

difference between theoretical and numerical results in Figures 12c and d due to the non-

linear and viscous effects in numerical model and perturbation technique used in the

theoretical solutions. In the unbaffled case, in Figures 12g and h, the sloshing effects become

16

slightly dominate in terms of increasing rolling amplitude θο. It can also be concluded that the

correct arrangement of baffle is reduced the sloshing force and moment %2.7 and %11

respectively for θο = 4ο, and %3.2 and %7.3 for θο = 8ο.

As ω is increased from 0.6 to 1.0 rad/sec., which is closer to the resonant frequency of

1.243 rad/sec., theoretical results in Figures 13c and d show non-linear behaviour. The non-

linear behaviour is more pronounced for larger θο. Numerical solutions indicate that the

trough of wave profile gets relatively wider and flat shape with the increasing of rolling

amplitude θο and rolling frequency ω. Figures 13c and d reveal that there are two basic

differences between theoretical and numerical results: first, the phase shifting is more

pronounced, and second, asymmetric behaviour starts to dominate. The severe oscillations of

fluid particles especially around the baffle and right wall of the tank occur with the increasing

rolling amplitude and frequency as shown in Figures 13g and h.

In Figure 14, for the lower fill depth case (β = 0.0625 and D = 7.5 ft.), the non-linear

effects such as narrow zero crossing and larger amplitudes in force and moment and increased

phase shifting are observed in theoretical and numerical results compared to the previous β =

1.0 case. As a result of this, forces and moments in both numerical and theoretical solutions

are obtained larger (for instance; %3.2 and %144 are increased for the case of theoretical

computation of force and moment) and the locations of maximum force and moment are also

shifted for the rolling amplitude θο = 4ο. These effects can be observed in Figures 14c and d

more severely as the rolling amplitude increased to θο = 8ο. It must be noted that the effect of

vertical baffle for the lower fill depth case is greatly reduce the over turning moment (for

instance; %56 decreased between baffled and unbaffled cases for β = 1 and θο = 4ο, %137

decreased between baffled and unbaffled cases for β = 0.0625 and θο = 4ο) and sloshing

effects (see Figures 13-14g and h).

17

The location of rolling center characterized by the parameter d plays an important role

on the magnitude of the sloshing effects as shown in Figure 15. In order to show the effect of

d variation, the typical case of β = 1 and ω = 1.0 rad/sec. is selected. The variation of θο from

4ο to 8o is increased the numerical over turning moment by %5.3. The numerical results

indicate that the overall magnitudes of sloshing forces and moments are hardened with the

increasing of d.

In order to observe the effect of roll frequency on the maximum wave height and

compare the theoretical and experimental results, the case of D/2a = 0.50 and d/2a = 0.50 with

the rolling amplitudes 6o and 8o is selected, as shown in Figure 16. It can be noted that, with

the increasing off-resonance rolling frequency, the theoretical, experimental and numerical

results tend to agree well. On the other hand, around the resonance frequency, the normalized

wave elevation is obtained relatively low than those of experimental and theoretical results.

One possible reason for this may be the effect of viscosity in the numerical model used. It is

known that the theoretical model did not contain the viscous effects and the experimental

model did not match the Reynolds number.

Additional comparisons are presented in Figure 17 for a tank length of 2ft. (2a = 2 ft.)

at a water depth of 0.8 ft.. The analytical, experimental and numerical solutions indicate that

the normalized wave amplitude is linearly proportional to the excitation rolling amplitude.

The agreement between numerical solutions and experimental results is very well comparing

to the analytical solutions.

6. CONCLUSIONS

The volume of fluid technique has been used to simulate two-dimensional viscous

liquid sloshing in moving rectangular baffled and unbaffled tanks. The VOF method was also

used to track the actual positions of the fluid particles on the complicated free surface. The

liquid was assumed to be homogeneous and to remain laminar. The excitation was assumed

18

harmonic, after the motion was started from the rest. A moving coordinate system fixed in

the tank was used to simplify the boundary condition on the fluid tank interface during the

large tank motions.

The general features of the effects of baffles on liquid sloshing inside the various

partially filled tanks were studied. Analytical solutions for liquid sloshing under combined

excitations were compared with both numerical and some experimental results. The following

conclusions can be drawn from our numerical computations:

i) The liquid is responded violently causing the numerical solution to become unstable

when the amplitude of excitation increased. The instability may be related to the fluid

motion such as the occurrence of turbulence, the transition from homogeneous flow to

a two-phase flow and the introduction of secondary flow along the third dimension.

Thus, the applicability of the method used in the present study is limited to the period

prior to the inception of these flow perturbations.

ii) The liquid sloshing inside a tank revealed that flow over a vertical baffle produced a

shear layer and energy was dissipated by viscous action.

iii) The effect of vertical baffles was most pronounced in shallow water. For this reason,

especially the over turning moment was greatly reduced.

iv) The increased fill depth, the rolling amplitude and frequency of the tank with/without

baffle configuration directly effected the degree of non-linearity of the sloshing

phenomena. As a result of this, the phase shifting in forces and moments occurred.

v) The larger forces and moments were obtained with the reducing fill depth due to the

increasing free surface effect.

Finally, the effects of turbulence and two phase flow (sprays, drops and bubbles in the

post impact period) as well as three dimensional effects need to be incorporated to assure a

stable and reliable modeling for such cases. For future work, second-order representation of

19

derivatives may be employed to better approximate to the rapid change of divergence in the

fluid. The effect of speed of sound, on the case of extreme sloshing, has to be checked to see

the compressibility effect in some degree. Model studies for sloshing under multi-component

random excitations with phase difference should be carried out to investigate sloshing loads

under more realistic tank motion inputs. Additionally, an integrated design synthesis

technique must be developed to accurately predict sloshing loads for design applications.

NOMENCLATURE

:nV"

The normal component of the fluid velocity

:tV"

The tangential component of the fluid velocity

:P Fluid pressure

:ATMP Atmospheric pressure

:ν Kinematic viscosity

:ρρρρ Fluid density

:, xx mn The horizontal components of the unit vector, normal and tangent to the surface

:, yy mn The vertical components of the unit vector, normal and tangent to the surface

:F"

Body forces

:θθθθ Roll angle

:oθ Roll amplitude of the tank

:γ The equilibrium angle of the tank relative to the axis of rotation

:d The distance between the origin of the moving coordinate and the axis of rotation

:D Fill depth

:2a Tank length

:Ω"

Angular velocity

:U"! Acceleration of the moving frame

:*a" Acceleration of an element relative to the point O

:tδδδδ Time increment

:αααα The upstream differencing parameter

:εεεε The perturbation parameter

20

:σ The compressibility parameter

:ω Roll frequency of the tank

:nω Natural frequency of the fluid inside the tank

:φ The velocity potential of the fluid

δδδδ: The response parameter of the grade change

Vo: The characteristic tank speed

k: The wave number of the motion profile

X(t): The horizontal movement of the tank

V(t): The speed of the moving tank along the motion profile

xU! : The tank acceleration in x direction

yU! : The tank acceleration in y direction

R: The radius of curvature of the motion profile

)X(ξ : The vertical motion profile

η: The wave amplitude

ACKNOWLEDGEMENT

The authors would like to thank to Research Fund of Istanbul Technical University for the

financial support of this study.

REFERENCES

Abramson, H.N., 1966. Dynamic Behavior of Liquids in Moving Containers with

Application to Space Vehicle Technology. NASA-SP-106.

Celebi, M.S., Kim, M.H., Beck, R.F., 1998. Fully Non-linear 3-D Numerical Wave Tank

Simulation. J. of Ship Research, Vol.42, No.1, pp 33-45.

21

Kim, M.H., Celebi, M.S., Kim, D.J., 1998. Fully Non-linear Interactions of Waves With a

Three-Dimensional Body in Uniform Currents. Applied Ocean Research, Vol.20, pp 309-321.

Dillingham, J., 1981. Motion Studies of a Vessel with Water on Deck. Marine Technology,

SNAME, Vol.18, No.1, pp 38-50.

Faltinsen, O.M., 1974. A Non-linear Theory of Sloshing in Rectangular Tanks. J. of Ship

Research, Vol.18, No.4, pp 224-241.

Faltinsen, O.M., 1978. A Numerical Non-linear Method of Sloshing in Tanks With Two-

Dimensional Flow. J. of Ship Research, Vol.22, No.3, pp 193-202.

Feng, G.C., 1973. Dynamic Loads Due to Moving Liquid. AIAA Paper No: 73-409.

Hirt, C.W., Nichols, B.D., 1981. Volume of Fluid Method for the Dynamics of Free

Boundaries. Journal of Computational Physics, Vol.39, pp. 201-225.

Lee, D.Y., Choi, H.S., 1999. Study on Sloshing in Cargo Tanks Including Hydro elastic

Effects. J. of Mar. Sci. Technology, Vol.4, No.1.

Lou, Y.K., Su, T.C., Flipse, J.E., 1980. A Non-linear Analysis of Liquid Sloshing in Rigid

Containers. US Department of Commerce, Final Report, MA-79-SAC-B0018.

Lui, A.P., Lou, J.Y.K., 1990. Dynamic Coupling of a Liquid Tank System Under Transient

Excitations. Ocean Engineering, Vol.17, No.3, pp.263-277.

Nakayama, T., Washizu K., 1980. Nonlinear Analysis of Liquid Motion in a Container

Subjected to Forced Pitching Oscillation. Int. J. for Num. Meth. in Eng., Vol.15, pp 1207-

1220.

Solaas, F., Faltinsen, O.M., 1997. Combined Numerical and Analytical Solution for

Sloshing in Two-Dimensional Tanks of General Shape. Vol.41, No.2, pp.118-129.

Von Kerczek, C.H., 1975. Numerical Solution of Naval Free-Surface Hydrodynamics

Problems. 1st International Conference on Numerical Ship Hydrodynamics, Gaithersburg,

USA.

22

List of Figures

Figure 1. The motion profile of the tank

Figure 2. The moving coordinate system

Figure 3. A snapshot for numerical simulation of the shallow water sloshing (ω =1.0864

rad/sec., θo = 8o)

Figure 4. The comparison of the unbaffled and baffled cases with the various parameters (ω =

0.1 rad/sec., β = 0.0625, d = 1 ft.)


0.5 rad/sec., β = 0.0625, d = 1 ft.)


0.9 rad/sec., β = 0.0625, d = 1 ft.)


1.0864 rad/sec., β = 0.0625, d = 1 ft.)


1.3 rad/sec., β = 0.0625, d = 1 ft.)

Figure 9. The effect of the vertical baffle on forces and moments (θo = 5.73o)

Figure 10. The effect of the vertical baffle on forces and moments (θo = 8o) Figure 11. A snapshot for numerical simulation of the intermediate fill depth sloshing (ω =1.0

rad/sec., θo = 8o)

Figure 12. The comparison of the unbaffled and baffled cases with the various parameters

(ω = 0.6 rad/sec., β = 1, d = 1 ft.)


(ω = 1.0 rad/sec., β = 1, d = 1 ft.)

23


(ω = 1.0 rad/sec., β = 0.0625, d = 1 ft.)


(ω = 1.0 rad/sec., β = 1, d = 2 ft.)

Figure 16. The effect of roll frequency on maximum wave height

Figure 17. Comparisons of analytical solutions, experimental results and numerical solutions.

Maximum wave amplitude vs roll angle.

24

X

Y

D2a

x, Ux

y, Uy

Motion Profile

V(t)

Figure 1. The Motion Profile of the Tank.

2aD

O

θγ

d

x

y

Figure 2. The Moving Coordinate System

Y

X

x0

y0

x0 - y0 : Equilibrium Positionx - y : Instantaneous Position

25

Figure 3. A Snapshot for Numerical Simulation of the Shallow Water Sloshing.( ω=1.0864 rad/sec, θ0=80 )

Time = 0.509 sec.

(a)

Time = 0.509 sec.

(e)

Time = 2.0 sec.

(b)

Time = 2.0 sec.

(f)

Time = 4.01 sec.

(c)

Time = 4.01 sec.

(g)

Time = 5.5 sec.

(d)

Time = 5.5 sec.

(h)

26

Figure 4. The Comparison of the Unbaffled and Baffled Cases with the Various Parameters.( ω = 0.1 rad/sec, β = 0.0625, d = 1 ft )

0 1.5 3 4.5 6

ωt

-0.144

-0.108

-0.072

-0.036

0

0.036

0.072

0.108

0.144

0.18

ζ/2

a

Wave Profile ( θ0=5.730 )Wave Profile ( θ0=80 )

(a)

0 1.5 3 4.5 6

ωt-0.12

-0.096

-0.072

-0.048

-0.024

0

0.024

0.048

0.072

0.096

0.12

ζ/2

a

Wave Profile-Baffled ( θ0=5.730 )Wave Profile-Baffled ( θ0=80 )

(e)

0 1.5 3 4.5 6

ωt-0.22

-0.176

-0.132

-0.088

-0.044

0

0.044

0.088

0.132

0.176

0.22

Acce

lera

tion

Horizontal ( θ0=5.730 )Vertical ( θ0=5.730 )Horizontal ( θ0=80 )Vertical ( θ0=80 )

(b)

0 1.5 3 4.5 6

ωt-0.22

-0.176

-0.132

-0.088

-0.044

0

0.044

0.088

0.132

0.176

0.22

Acce

lera

tion

Horizontal-Baffled ( θ0=5.730 )Vertical-Baffled ( θ0=5.730 )Horizontal-Baffled ( θ0=80 )Vertical-Baffled ( θ0=80 )

(f)

0 1.5 3 4.5 6

ωt-32

-25.6

-19.2

-12.8

-6.4

0

6.4

12.8

19.2

25.6

32

F/(

0.5

ρg

D2

θ 0)

Force ( θ0=5.730 )Force ( θ0=80 )Force-Theoretical ( θ0=5.730 )

(c)

Fmax1=16.82Fmax2=16.81Fmax3=15.68

0 1.5 3 4.5 6

ωt-32

-25.6

-19.2

-12.8

-6.4

0

6.4

12.8

19.2

25.6

32

F/(

0.5

ρg

D2

θ 0)

Force-Baffled ( θ0=5.730 )Force-Baffled ( θ0=80 )

Fmax1=15.84Fmax2=15.77

(g)

0 1.5 3 4.5 6

ωt

-14.4

-10.8

-7.2

-3.6

0

3.6

7.2

10.8

14.4

18

M/(

0.5

ρg

D3

θ 0)

Moment ( θ0=5.730 )Moment ( θ0=80 )

Mmax1=8.25Mmax2=8.74

(d)

0 1.5 3 4.5 6

ωt-15

-10

-5

0

5

10

15

M/(

0.5

ρg

D3

θ 0)

Moment-Baffled ( θ0=5.730 )Moment-Baffled ( θ0=80 )


(h)

27


0 1.5 3 4.5 6

ωt

-0.144

-0.108

-0.072

-0.036

0

0.036

0.072

0.108

0.144

0.18

ζ/2

a


(a)

0 1.5 3 4.5 6

ωt

-0.144

-0.108

-0.072

-0.036

0

0.036

0.072

0.108

0.144

0.18

ζ/2

a


(e)

0 1.5 3 4.5 6

ωt-1.2

-0.96

-0.72

-0.48

-0.24

0

0.24

0.48

0.72

0.96

1.2

Acce

lera

tion


(b)

0 1.5 3 4.5 6

ωt-1.2

-0.96

-0.72

-0.48

-0.24

0

0.24

0.48

0.72

0.96

1.2

Acce

lera

tion


(f)

0 1.5 3 4.5 6

ωt-45

-36

-27

-18

-9

0

9

18

27

36

45

F/(

0.5

ρg

D2

θ 0)

Force ( θ0=5.730 )Force ( θ0=80 )Force-Theoretical ( θ0=5.730 )

(c)

Fmax1=23.02Fmax3=22.51Fmax2=13.67

0 1.5 3 4.5 6

ωt-45

-36

-27

-18

-9

0

9

18

27

36

45

F/(

0.5

ρg

D2

θ 0)


Fmax1=19.28Fmax2=24.23

(g)

0 1.5 3 4.5 6

ωt-20

-16

-12

-8

-4

0

4

8

12

16

20

M/(

0.5

ρg

D3

θ 0)


Mmax1=12.06Mmax2=12.78

(d)

0 1.5 3 4.5 6

ωt-20

-16

-12

-8

-4

0

4

8

12

16

20

M/(

0.5

ρg

D3

θ 0)



(h)

28


0 1.5 3 4.5 6

ωt

-0.144

-0.108

-0.072

-0.036

0

0.036

0.072

0.108

0.144

0.18

ζ/2

a


(a)

0 1.5 3 4.5 6

ωt

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

ζ/2

a


(e)

0 1.5 3 4.5 6

ωt-2

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

2

Acce

lera

tion


(b)

0 1.5 3 4.5 6

ωt-2

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

2Ac

cele

ratio

nHorizontal-Baffled ( θ0=5.730 )Vertical-Baffled ( θ0=5.730 )Horizontal-Baffled ( θ0=80 )Vertical-Baffled ( θ0=80 )

(f)

0 1.5 3 4.5 6

ωt-50

-40

-30

-20

-10

0

10

20

30

40

50

F/(

0.5

ρg

D2

θ 0)

Force ( θ0=5.730 )Force ( θ0=80 )

(c)

Fmax1=37.13Fmax2=46.50

0 1.5 3 4.5 6

ωt-50

-40

-30

-20

-10

0

10

20

30

40

50

F/(

0.5

ρg

D2

θ 0)


Fmax1=17.28Fmax2=21.20

(g)

0 1.5 3 4.5 6

ωt-50

-40

-30

-20

-10

0

10

20

30

40

50

M/(

0.5

ρg

D3

θ 0)


Mmax1=25.86Mmax2=43.60

(d)

0 1.5 3 4.5 6

ωt-50

-40

-30

-20

-10

0

10

20

30

40

50

M/(

0.5

ρg

D3

θ 0)


Mmax1=8.41Mmax2=16.50

(h)

29


0 1.5 3 4.5 6

ωt

-0.144

-0.108

-0.072

-0.036

0

0.036

0.072

0.108

0.144

0.18

ζ/2

a


(a)

0 1.5 3 4.5 6

ωt-0.16

-0.128

-0.096

-0.064

-0.032

0

0.032

0.064

0.096

0.128

0.16

ζ/2

a

W. Profile-Baffled ( θ0=5.730 )W. Profile-Baffled ( θ0=80 )

(e)

0 1.5 3 4.5 6

ωt-2.4

-1.92

-1.44

-0.96

-0.48

0

0.48

0.96

1.44

1.92

2.4

Acce

lera

tion


(b)

0 1.5 3 4.5 6

ωt-2.4

-1.92

-1.44

-0.96

-0.48

0

0.48

0.96

1.44

1.92

2.4

Acce

lera

tion


(f)

0 1.5 3 4.5 6

ωt-45

-36

-27

-18

-9

0

9

18

27

36

45

F/(

0.5

ρg

D2

θ 0)

Force ( θ0=5.730 )Force ( θ0=80 )

(c)

Fmax1=37.60Fmax2=40.50

0 1.5 3 4.5 6

ωt-35

-28

-21

-14

-7

0

7

14

21

28

35

F/(

0.5

ρg

D2

θ 0)


Fmax1=29.32Fmax2=30.93

(g)

0 1.5 3 4.5 6

ωt-40

-32

-24

-16

-8

0

8

16

24

32

40

M/(

0.5

ρg

D3

θ 0)


Mmax1=27.37Mmax2=33.68

(d)

0 1.5 3 4.5 6

ωt-25

-20

-15

-10

-5

0

5

10

15

20

25

M/(

0.5

ρg

D3

θ 0)


Mmax1=16.24Mmax2=17.96

(h)

30


0 1.5 3 4.5 6

ωt

-0.144

-0.108

-0.072

-0.036

0

0.036

0.072

0.108

0.144

0.18

ζ/2

a


(a)

0 1.5 3 4.5 6

ωt-0.16

-0.128

-0.096

-0.064

-0.032

0

0.032

0.064

0.096

0.128

0.16

ζ/2

a


(e)

0 1.5 3 4.5 6

ωt

-2.24

-1.68

-1.12

-0.56

0

0.56

1.12

1.68

2.24

2.8

Acce

lera

tion


(b)

0 1.5 3 4.5 6

ωt

-2.24

-1.68

-1.12

-0.56

0

0.56

1.12

1.68

2.24

2.8

Acce

lera

tion


(f)

0 1.5 3 4.5 6

ωt-45

-36

-27

-18

-9

0

9

18

27

36

45

F/(

0.5

ρg

D2

θ 0)

Force ( θ0=5.730 )Force ( θ0=80 )

(c)

Fmax1=30.76Fmax2=32.17

0 1.5 3 4.5 6

ωt-35

-28

-21

-14

-7

0

7

14

21

28

35

F/(

0.5

ρg

D2

θ 0)


Fmax1=15.79Fmax2=21.79

(g)

0 1.5 3 4.5 6

ωt-40

-32

-24

-16

-8

0

8

16

24

32

40

M/(

0.5

ρg

D3

θ 0)


Mmax1=21.52Mmax2=23.96

(d)

0 1.5 3 4.5 6

ωt-25

-20

-15

-10

-5

0

5

10

15

20

25

M/(

0.5

ρg

D3

θ 0)


Mmax1=8.21Mmax2=13.08

(h)

31

Figure 10. The Effect of the Vertical Baffle on Forces and Moments.( θ0 = 80 )

Figure 9. The Effect of the Vertical Baffle on Forces and Moments.( θ0 = 5.730 )

0 0.3 0.6 0.9 1.2 1.5

Rolling Frequency, ω

-50

-25

0

25

50

75

100

125

150

%∆

F,%

∆M

Reduction of Sloshing ForceReduction of Sloshing Moment

Tank Length, 2a = 32 ftD / 2a = 0.125d / 2a = 0.03125

ResonantFrequency( ωn = 1.0864 )

0 0.3 0.6 0.9 1.2 1.5

Rolling Frequency, ω

-50

-25

0

25

50

75

100

125

150

%∆

F,%

∆M

Reduction of Sloshing ForceReduction of Sloshing Moment

Tank Length, 2a=32 ftD / 2a = 0.125d / 2a = 0.03125

ResonantFrequency( ωn = 1.0864 )

32

Figure 11. A Snapshot for Numerical Simulation of the IntermediateFill Depth Sloshing ( ω=1.0 rad/sec, θ0=80 )

Time = 0.758 sec.

(a)

Time = 0.758 sec.

(e)

Time = 2.51 sec.

(b)

Time = 2.51 sec.

(f)

Time = 4.5 sec.

(g)

Time = 6.01 sec.

(d)

Time = 6.01 sec.

(h)

Time = 4.5 sec.

(c)

33

Figure 12. The Comparison of the Unbaffled and Baffled Cases with the Various Parameters.( ω = 0.6 rad/sec, β = 1, d = 1 ft )

0 1.5 3 4.5 6

ωt

-0.16

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

0.16

0.2

ζ/2

a

Wave Profile ( θ0=40 )Wave Profile ( θ0=80 )

(a)

0 1.5 3 4.5 6

ωt-0.32

-0.256

-0.192

-0.128

-0.064

0

0.064

0.128

0.192

0.256

0.32

ζ/2

a

Wave Profile-Baffled ( θ0=40 )W. Profile-Baffled ( θ0=80 )

(e)

0 1.5 3 4.5 6

ωt

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Acce

lera

tion

Horizontal ( θ0=40 )Vertical ( θ0=40 )Horizontal ( θ0=80 )Vertical ( θ0=80 )

(b)

0 1.5 3 4.5 6

ωt

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Acce

lera

tion

Horizontal-Baffled ( θ0=40 )Vertical-Baffled ( θ0=40 )Horizontal-Baffled ( θ0=80 )Vertical-Baffled ( θ0=80 )

(f)

0 1.5 3 4.5 6

ωt-13

-10.4

-7.8

-5.2

-2.6

0

2.6

5.2

7.8

10.4

13

F/(

0.5

ρg

D2

θ 0)

Force ( θ0=40 )Force ( θ0=80 )Force-Theoretical ( θ0=40 )Force-Theoretical ( θ0=80 )

(c)

Fmax1=7.9Fmax2=7.69Fmax3=5.45Fmax4=5.46

0 1.5 3 4.5 6

ωt-13

-10.4

-7.8

-5.2

-2.6

0

2.6

5.2

7.8

10.4

13

F/(

0.5

ρg

D2

θ 0)

Force-Baffled ( θ0=40 )Force-Baffled ( θ0=80 )

Fmax1=7.65Fmax2=7.45

(g)

0 1.5 3 4.5 6

ωt

-5.6

-4.2

-2.8

-1.4

0

1.4

2.8

4.2

5.6

7

M/(

0.5

ρg

D3

θ 0)

Moment ( θ0=40 )Moment ( θ0=80 )Moment-Theoretical ( θ0=40 )Moment-Theoretical ( θ0=80 )

Mmax1=4.1Mmax2=4.14Mmax3=2.55Mmax4=2.56

(d)

0 1.5 3 4.5 6

ωt

-5.6

-4.2

-2.8

-1.4

0

1.4

2.8

4.2

5.6

7

M/(

0.5

ρg

D3

θ 0)

Moment-Baffled ( θ0=40 )Moment-Baffled ( θ0=50 )


(h)

34


0 1.5 3 4.5 6

ωt-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

ζ/2

a


(a)

0 1.5 3 4.5 6

ωt-0.15

-0.12

-0.09

-0.06

-0.03

0

0.03

0.06

0.09

0.12

0.15

ζ/2

a

Wave Profile-Baffled ( θ0=40 )Wave Profile-Baffled ( θ0=80 )

(e)

0 1.5 3 4.5 6

ωt-0.3

-0.24

-0.18

-0.12

-0.06

0

0.06

0.12

0.18

0.24

0.3

Acce

lera

tion


(b)

0 1.5 3 4.5 6

ωt-0.3

-0.24

-0.18

-0.12

-0.06

0

0.06

0.12

0.18

0.24

0.3

Acce

lera

tion

Horizontal-Baffled ( θ0=40 )Vertical-Baffled ( θ0=40 )Horizontal-Baffled ( θ0=80 )Vertical-Baffled ( θ0=80 )

(f)

0 1.5 3 4.5 6

ωt-40

-32

-24

-16

-8

0

8

16

24

32

40

F/(

0.5

ρg

D2

θ 0)

Force ( θ0=40 )Force ( θ0=80 )Force-Theoretical ( θ0=40 )Force-Theoretical ( θ0=80 )

(c)


0 1.5 3 4.5 6

ωt

-7.2

-5.4

-3.6

-1.8

0

1.8

3.6

5.4

7.2

9

F/(

0.5

ρg

D2

θ 0)



(g)

0 1.5 3 4.5 6

ωt

-11.2

-8.4

-5.6

-2.8

0

2.8

5.6

8.4

11.2

14

M/(

0.5

ρg

D3

θ 0)

Moment ( θ0=40 )Moment ( θ0=80 )Moment-Theoretical ( θ0=40 )Moment-Theoretical ( θ0=80 )


(d)

0 1.5 3 4.5 6

ωt-5

-4

-3

-2

-1

0

1

2

3

4

5

M/(

0.5

ρg

D3

θ 0)



(h)

35


0 1.5 3 4.5 6

ωt-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

ζ/2

a


(a)

0 1.5 3 4.5 6

ωt-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

ζ/2

a


(e)

0 1.5 3 4.5 6

ωt

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Acce

lera

tion


(b)(h)

0 1.5 3 4.5 6

ωt

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1Ac

cele

ratio

nHorizontal-Baffled ( θ0=40 )Vertical-Baffled ( θ0=40 )Horizontal-Baffled ( θ0=80 )Vertical-Baffled ( θ0=80 )

(f)

0 1.5 3 4.5 6

ωt-30

-24

-18

-12

-6

0

6

12

18

24

30

F/(

0.5

ρg

D2

θ 0)

Force-Theoretical ( θ0=40 )Force-Theoretical ( θ0=80 )Force ( θ0=40 )Force ( θ0=80 )

(c)


0 1.5 3 4.5 6

ωt

-22.4

-16.8

-11.2

-5.6

0

5.6

11.2

16.8

22.4

28

F/(

0.5

ρg

D2

θ 0)


Fmax1=11.12Fmax2=12.71

(g)

0 1.5 3 4.5 6

ωt-25

-20

-15

-10

-5

0

5

10

15

20

25

M/(

0.5

ρg

D3

θ 0)

Moment-Theoretical ( θ0=40 )Moment-Theoretical ( θ0=80 )Moment ( θ0=40 )Moment ( θ0=80 )


(d)

0 1.5 3 4.5 6

ωt-20

-15

-10

-5

0

5

10

15

20

M/(

0.5

ρg

D3

θ 0)



(h)

36


0 1.5 3 4.5 6

ωt-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

ζ/2

a


(a)

0 1.5 3 4.5 6

ωt-0.15

-0.12

-0.09

-0.06

-0.03

0

0.03

0.06

0.09

0.12

0.15

ζ/2

a


(e)

0 1.5 3 4.5 6

ωt-0.6

-0.48

-0.36

-0.24

-0.12

0

0.12

0.24

0.36

0.48

0.6

Acce

lera

tion


(b)

0 1.5 3 4.5 6

ωt-0.6

-0.48

-0.36

-0.24

-0.12

0

0.12

0.24

0.36

0.48

0.6Ac

cele

ratio

nHorizontal-Baffled ( θ0=40 )Vertical-Baffled ( θ0=40 )Horizontal-Baffled ( θ0=80 )Vertical-Baffled ( θ0=80 )

(f)

0 1.5 3 4.5 6

ωt-10

-8

-6

-4

-2

0

2

4

6

8

10

F/(

0.5

ρg

D2

θ 0)

Force ( θ0=40 )Force ( θ0=80 )

(c)


0 1.5 3 4.5 6

ωt

-7.2

-5.4

-3.6

-1.8

0

1.8

3.6

5.4

7.2

9

F/(

0.5

ρg

D2

θ 0)



(g)

0 1.5 3 4.5 6

ωt-6

-4.5

-3

-1.5

0

1.5

3

4.5

6

M/(

0.5

ρg

D3

θ 0)

Moment ( θ0=40 )Moment ( θ0=80 )


(d)

0 1.5 3 4.5 6

ωt-5

-4

-3

-2

-1

0

1

2

3

4

5

M/(

0.5

ρg

D3

θ 0)



(h)

37

Figure 17. The Comparison of analytical solutions, experimental resultsand numerical solutions. Maximum wave amplitude vs roll angle

Figure 16. The Effect of Roll Frequency on Maximum Wave Height

0 1 2 3 4 5 6 7 8 9 10

θ0 Degree

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

η/2

a

Analytical SolutionsExperimental ResultsNumerical Solutions

Tank Length, 2a=2.0 ftD / 2a = 0.4d / 2a = 0.4T ( g / 2a )0.5 = 5.0

0 2 4 6 8 10

T ( g / 2a ) 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

η/2

a

Theoretical ( θ0=80 )Experimental ( θ0=80 )Numerical ( θ0=80 )Numerical ( θ0 = 60 )

D / 2a = 0.50d / 2a = 0.50Phase = 00

ResonantFrequency

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