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Sloshing motions in excited tanks
Jannette B. Frandsen
*
Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USA
Received 17 December 2002; received in revised form 28 October 2003; accepted 28 October 2003
Abstract
A fully non-linear finite difference model has been developed based on inviscid flow equations. Numerical experi-
ments of sloshing wave motion are undertaken in a 2-D tank which is moved both horizontally and vertically. Results of
liquid sloshing induced by harmonic base excitations are presented for small to steep non-breaking waves. The sim-
ulations are limited to a single water depth above the critical depth corresponding to a tank aspect ratio of hs=b¼0:5.
The numerical model is valid for any water depth except for small depth when viscous effects would become important.
Solutions are limited to steep non-overturning waves. Good agreement for small horizontal forcing amplitude is
achieved between the numerical model and second order small perturbation theory. For large horizontal forcing, non-
linear effects are captured by the third-order single modal solution and the fully non-linear numerical model. The
agreement is in general good, both amplitude and phase. As expected, the third-order compared to the second-order
solution is more accurate. This is especially true for resonance, high forcing frequency and mode interaction cases.
However, it was found that multimodal approximate forms should be used for the cases in which detuning effects occur
due to mode interaction. We present some test cases where detuning effects are evident both for single dominant modes
and mode interaction cases. Furthermore, for very steep waves, just before the waves overturn, and for large forcing
frequency, a discrepancy in amplitude and phase occurs between the approximate forms and the numerical model. The
effects of the simultaneous vertical and horizontal excitations in comparison with the pure horizontal motion and pure
vertical motion is examined. It is shown that vertical excitation causes the instability associated with parametric res-
onance of the combined motion for a certain set of frequencies and amplitudes of the vertical motion while the hor-
izontal motion is related to classical resonance. It is also found that, in addition to the resonant frequency of the pure
horizontal excitation, an infinite number of additional resonance frequencies exist due to the combined motion of the
tank. The dependence of the non-linear behaviour of the solution on the wave steepness is discussed. It is found that for
the present problem, non-linear effects become important when the steepness reaches about 0.1, in agreement with the
physical experiments of Abramson [Rep. SP 106, NASA, 1966].
Ó2003 Elsevier Inc. All rights reserved.
Keywords: Sloshing motion; Finite differences; Moving liquid tanks
www.elsevier.com/locate/jcp
Journal of Computational Physics 196 (2004) 53–87
*
Tel.: +1-225-578-0277; fax: +1-225-578-0245.
E-mail address: frandsen@lsu.edu (J.B. Frandsen).
0021-9991/$ - see front matter Ó2003 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcp.2003.10.031
1. Introduction
The present paper investigates numerically steep free surface sloshing in fixed and base-excited rectan-
gular tanks with a focus on moving liquid tanks. Numerical modelling is necessary because neither linear
nor second-order potential theory is applicable to steep waves where high-order effects are significant.
Recently, Cariou and Casella [6] strengthened this concern through an extensive comparison study of
numerical sloshing predictions in ship tanks. The study comprised 11 viscous codes. They concluded and
urged the need for further research on accurate free surface predictions.
Prediction of free surface motions of liquids in tanks is of practical importance. Sloshing effects of free-
surface motion in tanks driven by external forces may have serious consequences for a range of engineering
applications. For example, sloshing effects in the ballast tanks of a ship may cause it to experience large
rolling moments, and eventually capsize. Also, if the forcing frequency is near the natural sloshing fre-
quency, the high dynamic pressures due to resonance may damage the tank walls. Further applications in
the aerospace industry has been reviewed and discussed comprehensively by Abramson [2], both analyti-
cally and experimentally, and recently numerical sloshing motion experiments were carried out by Gerrits
[17]. Another example is the use of tuned liquid dampers designed to suppress wind-induced structural
vibrations experienced in tall buildings. This type of damping device has been recently installed in a few tall
buildings, e.g. the 105 m high Hobart Tower in Tasmania and the 158 m Gold Tower in Japan, as described
by Kareem et al. [26]. Designers are faced with the task of understanding complex fluid–structure inter-
actions when attempting to estimate the energy dissipation performance of tuned liquid dampers. To this
end, a numerical wave tank can provide useful information on the free surface motions, resonant fre-
quencies, etc.
Sloshing effects in fixed tanks have been the subject of a great deal of past research. For example, Telste
[35] modelled inviscid sloshing motion in a 2-D fixed tank by means of a finite difference model. Ferrant and
Le Touze [15] applied an inviscid pseudo-spectral model to predict 3-D free sloshing. Ushijima [38] used an
arbitrary Lagrangian–Eulerian method on boundary-fitted grids to analyse viscous sloshing and swirling
effects in a 3-D cylindrical fixed tank.
There are also several examples of previous studies devoted to investigation of the sloshing waves in
moving tanks; both inviscid and viscous formulations. Recently, Bredmose et al. [5] report on ‘‘flat-topped’’
experimentally observed free-surface profiles caused by vertical harmonic forced accelerations. Chen et al.
[8] use a finite difference model to examine large sloshing motions in 2-D tanks excited by the horizontal
component of four seismic events. For non-overturning waves their model demonstrated that non-linear
effects during some earthquakes are responsible for damage of liquid tanks. Chern et al. [9] and Turnbull et
al. [37] simulated 2-D forced sloshing in horizontally excited tanks of inviscid liquid (near resonance) using
simple r-transformed mappings in pseudospectral and finite element schemes. Celebi and Akyildiz [7]
developed a viscous solver to capture non-linear free surface flows using the volume of fluid technique,
originally developed by Hirt and Nichols [22]. They simulated 2-D sloshing motion in tanks which was
forced to roll or to move vertically. Wu et al. [43] use an inviscid finite element model to study the behaviour
of non-breaking waves in 3-D tanks. They focus on near resonance cases primarily based on tanks excited
by both sway and surge motions and report on the effects of 3-D motions in comparison with 2-D standing
waves. Wu et al. show a few tests with pure prescribed heave excitation and one test case included combined
sway/surge/heave motions of the tank.
The motivation of the present numerical work is to explore the behaviour of liquid motions in a forced
excited tank prescribed to move simultaneously in horizontal and vertical directions. To the authorÕs
knowledge, no investigation of the combined motion has been done systematically. Moreover investigators
have previously focused on either horizontal or vertical driven excitation. In particular, the vertical tank
excitation, as originally explored in FaradayÕs experiment [14], has had much attention (e.g. the review
papers by Miles and Henderson [30]; Perlin and Schultz [32] and the recent work by Jiang et al. [24,25]). The
54 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
literature also reveals extensive research on pure horizontal excitation, in general, especially in relation to
the application of idealised seismic events [8] and ship stability [10].
Previous numerical models generally treat the moving free surface boundary in one of two ways: either
by using Lagrangian tracking of free surface nodes with regridding, or by mappings. The former has the
disadvantage that the surface velocities are difficult to predict correctly, and so free surface smoothing is
often required. Although mappings inherently overcome this problem, they are less flexible to apply to
irregular geometries or to cases where submerged bodies are present in the flow domain. Moreover mapping
types of schemes cannot predict run-up/overturning due to the single value formulation [3].
In the present paper numerical experiments of liquid motion in 2-D tanks excited by periodic loadings
are undertaken. The fully non-linear numerical model is based on inviscid flow equations and solutions are
obtained using finite differences. This paper discusses sloshing motion behaviour in a numerical wave tank
based on potential theory mapped according to a modified r-transformation that stretches the grid from the
bed to the free surface. The r-transformation has been widely applied, recently to shallow water flows [27]
and to simulate waves in relatively deep water [37]. The present mapping ensures that cell increments have
unit dimensions in the discretised mapped domain, and hence simplifies the discretised formulation. The
flow equations are solved on a rectangular grid. The sigma-transformation has two major advantages.
Remeshing due to the moving free surface is avoided and the mapping avoids the need to calculate the free
surface velocity components explicitly. Extrapolations are unnecessary, and so free surface smoothing by
means of a spatial filter is often not required. However, it should be noted that the mapping has to be single-
valued in the vertical direction, and so the formulation does not permit the free surface to become vertical
or overturn. Equivalent solutions on 2-D grid with sigma-transformation are known to be extremely stable,
unlike other schemes which have to use free-surface smoothing [37]. Herein complicated free surface be-
haviour are investigated based on a finite difference scheme which is simple, accurate and computationally
efficient. The numerical model is valid for any water depth except for small depth when viscous effects
would become important. Moreover, the present model can readily be extended to 3-D waves.
The results presented herein have been limited to a single liquid depth (for reasons of brevity). However,
it is very important to note that the liquid depth has a profound influence on non-linear free surface effects.
It has been established that there is a critical liquid depth that delineates two non-linear regimes of the
liquid free surface referred to as soft and hard spring characteristics. Gu and Sethna [19], Gu et al. [20],
Virnig et al. [40] have examined the role of the liquid critical depth in rectangular tanks subjected to vertical
sinusoidal excitation. Another important feature is that there is an excitation frequency range over which
the free surface exhibits chaotic motion [23]. It should be noted that Ibrahim et al. gives a comprehensive
review on sloshing motion predictions with more than 1000 references. The role of critical depth in tanks
subjected to horizontal motions have also been studied [10–13,21]. Moreover, Waterhouse [41] investigated
liquid behaviour near critical depth and gives a complete fifth-order analysis of soft/hardening spring
characteristics.
Analyses of small to steep non-breaking waves are carried out for free and forced sloshing motion in a
rectangular tank with constant still water depth providing benchmark tests. The main measure of impor-
tance of non-linearity for problems with a free surface is the wave steepness, which for regular waves can be
defined as S¼peak-trough=wave length. The higher the steepness is, the more important non-linear
phenomena become. This may result in interaction between different frequencies or non-linear dispersion
effects, as the velocity of wave motion becomes dependent on the amplitude. Usually non-linearity reflects
itself through relatively higher peaks and relatively smaller troughs of the surface elevation. The present
paper shows results from these free and forced sloshing tests and indicate that the model is capable of
simulating highly non-linear free surface motions which is known to occur in steep waves.
First, in Sections 2–4 we present the governing equations, approximate forms and the numerical model.
The first test studies are shown in Section 5. Simulations based on fundamental analyses of standing waves
in a fixed rigid tank are carried out. The numerical model is validated for different wavelengths. Increasing
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 55
wave steepnesses are simulated in order to demonstrate cases where the fully non-linear model provides
solutions not obtainable with the approximate forms. In Section 6 the tank is excited vertically and the
stability of the sloshing motion is discussed. Benjamin and Ursell [4] investigated the problem theoretically.
Their analyses were based on an inviscid flow model with surface tension, and they found that small
amplitude wave motion is governed by the Mathieu equation. Benjamin and Ursell concluded that the
linearised solutions are always unstable for an external forcing frequency equal to twice the sloshing fre-
quency. The present model investigates the consequences of eliminating the non-linear terms. In Section 7
we focus on the case of pure horizontal tank motion. The solutions for resonant and off-resonant frequency
of horizontal excitation for various amplitudes are presented. The influence of non-linearity for high
amplitude solutions are illustrated on examples of the surface elevation behaviour, power spectra and
phase-plane trajectories. In Section 8 the results are extended for the case of combined horizontal and
vertical tank excitation, and emphasis is made on the new flow features generated. It is shown that the
vertical tank motion is responsible for the instability of the solution for specific values of motion param-
eters. In addition to the resonant frequency of the horizontal tank excitation, it is shown that an infinite
number of additional resonance frequencies exist due to the combined motion of the tank. The importance
of the non-linear effects is discussed for resonant, non-resonant and unstable solutions.
The results presented herein have been computed on a SUN Ultra 60 workstation with 450 MHz CPU
(SPECfp95: 32.7). The CPU time required did not exceed 2 h for the fixed tank studies whereas the heave/
surge tank tests were the most intensive with an average CPU time of 24 h. Only 12 Mb RAM was required
for any test case.
2. Governing equations of ideal free-surface waves in moving tanks
Investigations of 2-D non-linear motion of liquid in moving tanks are undertaken. Rectangular tanks
which move with respect to an inertial Cartesian coordinate system (X;Z) with horizontal X-axis and
vertical Z-axis, and tank position at time tof X¼XTðtÞ,Z¼ZTðtÞare considered. The Cartesian coordi-
nates (x;z) are connected to the tank, with the origin at the mean free-surface at the left-hand side of the
tank. The fluid is assumed to be incompressible, irrotational and inviscid. The fluid motion is therefore
governed by LaplaceÕs equation,
o2/
ox2þo2/
oz2¼0;ð1Þ
where /is the velocity potential. In the coordinate system fixed to the tank the fluid velocity components
normal to the fixed boundaries are equal to zero. Hence, on the bottom and the walls of the tank we have
o/
oxx¼0;b¼0;o/
ozz¼hs¼0;ð2Þ
where bis the length of the tank and hsdenotes still water depth. On the free surface the dynamic and
kinematic boundary conditions hold, which are
o/
otz¼f¼1
2
o/
ox
2
"þo/
oz
2#ðgþZ00
TðtÞÞfxX 00
TðtÞð3Þ
and
of
otz¼f¼o/
ozo/
ox
of
ox;ð4Þ
56 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
respectively, where fis the free surface elevation measured vertically above still water level, and Z00
Tand X00
T
are the vertical and horizontal acceleration of the tank, and gdenotes acceleration due to gravity.
Initially the fluid is assumed to be at rest with some initial perturbation of the free-surface. Thus, the
initial conditions are:
/¼0;f¼f0ðxÞat t¼0:ð5Þ
It should be noted that (5) is difficult to reproduce in a physical experimental treatment because it is not
possible to achieve simultaneously r/¼0 and a non-infinitesimal f0. So it is somewhat hard to imagine
that any physical experimental way can generate a single mode in a tank. Nevertheless, understanding the
time evolution of the single mode is very important because it can help to predict certain features of the
multi-mode forced motion. An example is the vertical excited tank (Section 6) where this ‘‘non-physical’’
initial condition is used. Examination of the spectra of the uni-modal motion helps to predict the existence
of the side resonances related to for example the combined horizontal/vertical tank motion experiments
(Section 8). It is less complicated to deal with non-physical initial conditions in a numerical experimental
set-up. The real advantage of numerical methods is the possibility to model situations which are hard to
reproduce in physical experiments, but are important from methodological or theoretical view point. But
the method of course should be accurate and reliable.
Finally, we should also note that the mean water level in the tank remains constant, that is:
Zb
0
fdx¼0:ð6Þ
3. Approximate solution for sloshing motion in moving tanks
Analytical approaches and related asymptotic solutions to predict sloshing motion in fixed and moving
tanks have been explored by several investigators, e.g., Faltinsen et al. [11], Ockendon and Ockendon [31],
Hill [21]. Ockendon and Ockendon [31] presents analytical schemes for resonant sloshing due to either
vertical or horizontal excitation. They have for example explained mathematically why the schemes require
Moiseyev-like detuning and consequently yield a third order secular equation to find the dominant wave
amplitude response. Proceeding this way, numerous third-order asymptotic solutions have been derived.
For example, Faltinsen [10] derived a third order asymptotic solution for horizontal tank excitation. We
should also mention that although Faltinsen et al. [11] present test cases for horizontal tank motions, they
have developed a multimodal algorithm for arbitrary tank motions, including roll motion. Hill [21] also
investigated transient resonant waves but based on a different third-order analytical algorithm.
As mentioned, the literature reveals that sloshing motion has been investigated with either vertical or
horizontal excitation. This section describes an asymptotic solution where combined heave/surge excita-
tions are considered. In the approximations herein, we assume small amplitudes of horizontal motion and
initial surface perturbation. The third-order solution presented is limited to single mode resonance cases.
We should also mention that our approach is similar to Faltinsen et al. [11] although they use the Ham-
iltonÕs principle to obtain the evolution equations.
Let us introduce the non-dimensional variables in the following way
/¼acg
xc
/;f¼acf;XT¼acX
T;ZT¼g
x2
c
Z
T;
ðx;z;b;hsÞ¼ g
x2
cðx;z;b;hsÞ;t¼xct;x¼x=xc;
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 57
where the asterisk is used to denote the non-dimensional values, and acand xcare the characteristic wave
amplitude and frequency, respectively. Later in this section we shall omit asterisks assuming all the values
to be non-dimensional. The choice of the characteristic amplitude and frequency depends on the particular
problem in hand and can be related to either the frequency and amplitude of the tank motion or the
frequency and amplitude of one of the sloshing modes. The Laplace equation and the surface boundary
conditions keep the original forms in the non-dimensional coordinates, while the surface boundary con-
ditions can be rewritten as follows
o/
otþð1þZ00
TðtÞÞfþxX 00
TðtÞ¼1
2ðr/Þ2
z¼f
;of
oto/
oz¼o/
ox
of
oxz¼f
;ð7Þ
where ¼acx2
c=gis the characteristic wave steepness. In this section we shall consider the limit of small
steepness !0. Expanding the surface boundary conditions into the Tailor series near the mean water level
we can rewrite (7) as
o/
otþð1þZ00
TðtÞÞfþxX 00
TðtÞ¼1
2r/ðÞ
2
þfo
ot
o/
oz2fr/ro/
oz
þ1
2f2o
ot
o2/
oz2þOð3Þz¼0
;
of
oto/
oz¼of
ox
o/
oxþfo2/
oz2þ21
2f2o3/
ox3
fof
ox
o2/
oxozþOð3Þz¼0
:
ð8Þ
The Laplace equation can now be solved in the rectangular domain with the boundary conditions (8) on the
mean water level f¼0.
The general solution of the Laplace equation in the rectangular domain satisfying the boundary con-
dition on the rigid surfaces can be represented in the form of expansion with the linear sloshing modes
/¼X
1
n¼0
coshðnkðzþhsÞÞ
coshðnkhsÞcosðnkxÞFnðtÞ;f¼X
1
n¼0
cosðnkxÞZnðtÞ;ð9Þ
where k¼p=bis the wavenumber corresponding to the first sloshing mode.
The functions FnðtÞand ZnðtÞdescribing the time evolution of individual components can be found after
substituting the general solution (9) into the free-surface boundary conditions (8) and collecting the terms
corresponding to different wave numbers nk.
First, let us consider the classical perturbation approach. We shall represent the functions Fnand Znfrom
(9) in the form of the asymptotic expansion with respect to the powers of the small parameter
Fn¼Fð1Þ
nþFð2Þ
nþ;Zn¼Zð1Þ
nþZð2Þ
nþ:
Substituting into (8) and collecting the terms with the same powers of we obtain the equations for each
order of approximation. In the main approximation (Oð0Þ) we obtain the following equations describing
the linear sloshing of each of the modes
Fð1Þ0
nþð1þZ00
TÞZð1Þ
n¼bnX00
T;Zð1Þ0
nx2
nFð1Þ
n¼0;ð10Þ
where xn¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kntanhðknhsÞ
pare the linear sloshing frequencies, and bnare the nth coefficients of the Fourier
expansion of xwith respect to cosðnkxÞ:b0¼b=2, bn¼0 for even nand bn¼4b=ðpnÞ2for odd n.
For each component n, Eq. (10) can be reduced to a single equation for the surface elevation
Zð1Þ00
nþx2
nð1þZ00
TÞZð1Þ
n¼bnx2
nX00
T:
58 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
In the case of harmonic excitation XTðtÞ¼ahcosðxhtÞ,ZTðtÞ¼avcosðxvtÞthis equation can be reduced
to the non-homogeneous MathieuÕs equation
Zð1Þ00
nðsÞþðp2qcosð2sÞÞZð1Þ
nðsÞ¼4bnx2
nx2
hahcosð2xhsÞ;ð11Þ
where s¼t=2, q¼2avx2
vx2
n,p¼4x2
n. We note that Eq. (11) represent sloshing motions in a pure vertically
excited tank when the right hand side is zero [4].
Solution of the homogeneous Mathieu equation can be represented as a linear combination of two
linearly independent Floquet solutions FrðzÞ,FrðzÞhaving the form
FrðzÞ¼eirzPðzÞ;
where rðp;qÞis the Mathieu characteristic exponent and PðzÞis periodic with period p[1]. Solution of
MathieuÕs equation is stable when the value of ris real, and it becomes unstable when the value of ris
complex. For small values of the parameter qin the stability regions, the characteristic exponent has the
following asymptotic expansion
rðp;qÞ¼ ffiffiffi
p
pq21
4p1ðÞ
ffiffiffi
p
pq415p235pþ8
64 p4ðÞp1ðÞ
3p3=2þOðq6Þ;ð12Þ
which can be used to estimate rfor small and moderate values of q. The contour plots of the real part of the
characteristic exponent rðp;qÞare represented in Fig. 1, where the thick lines indicate the boundaries of the
instability regions.
Furthermore, the function FrðzÞcan also be written as a superposition of harmonic oscillations with
frequencies rþ2k,k¼0;1;2..., as follows:
FrðzÞ¼ X
1
k¼1
c2keiðrþ2kÞ:
If the non-homogeneous Eq. (11) contains solutions in the stable regions, resonance will occur when the
frequency of the right hand side equals one of these frequencies. It should be noted, that the strength of
Fig. 1. Stability map for the first-order linear solution including contour plots of Mathieu characteristic exponent rð4X2
n;2jvX2
nÞ.
r-values are listed internally on the vertical axis of the figure. The thick lines show the boundaries of instability regions.
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 59
these resonances decays faster for large absolute values of k. Thus, the first-order solution will exhibit
resonant behaviour when the frequency of horizontal excitation xhsatisfies the condition
xh¼xres
k¼r
2þk;k¼0;1;2;... ð13Þ
So in contrast to pure horizontally forced excited tank motion, which exhibits one distinct case of reso-
nance, the case of combined forced tank motion (horizontal and vertical) contains an infinite number of
resonances. As it can be seen from (12), for small vertical excitation, as q!0, the value of r=2 goes to xn,
which in the limit gives the linear resonant frequency for pure horizontal motion.
The right hand sides (Oð1Þ) of the second order equations of Eq. (8) include products of the derivatives
of the first-order solutions, which are the infinite sums of sines or cosines. Therefore, for derivations of the
second-order solution it is necessary to estimate the nth Fourier components of these products. This can be
carried out by using the following expression:
X
1
n¼0
AncosðknxÞX
1
n¼0
BncosðknxÞ¼X
1
n¼0
CnðA;BÞcosðknxÞ;
CnðA;BÞ¼cnð1
signðnÞÞA0B0þA0BnþAnB0þX
1
j¼1
AjjjBjnjj!;
c0¼1
4;cn¼1
2n¼1;2;...
and
X
1
n¼0
AnsinðknxÞX
1
n¼0
BnsinðknxÞ¼X
1
n¼0
SnðA;BÞcosðknxÞ;SnðA;BÞ¼1
2signðjðjnÞÞ X
1
j¼1
AjjjBjnjj:
The equations for the time evolution of the second-order Fourier components of the surface elevation and
velocity potential can now be written as follows:
Zð2Þ0
nx2
nFð2Þ
n¼Cnðk2Fð1Þ;Zð1Þ
nÞSnðkF ð1Þ;kZð1ÞÞ;
Uð2Þ0
nþð1þZ00
TÞZð2Þ
n¼1
2SnðkF ð1Þ;kF ð1ÞÞ1
2Cnðx2Fð1Þ;x2Fð1ÞÞCnðx2Fð1Þ0;Zð1ÞÞ:ð14Þ
The importance of the cubical non-linearity for the behaviour of dynamical systems is well known [28].
Third order terms can influence the solution in the main approximation leading to an important effect as
change of the natural frequency of the system with amplitude. This leads to dramatical changes in be-
haviour of the non-linear system near resonance compare to a linear one. In the classical perturbation
theory such influence can be taken into account, for example, by applying the solvability condition for the
third order equations [21] which leads to the restrictions for the behaviour of the first order amplitude.
Another way of doing this is to construct a non-linear evolution equation for each of the modes [11]. For
high orders the procedure involves extensive algebra including the multiple infinite sums. Nevertheless, in
the case when there is one dominating mode, the procedure of constructing an asymptotic solution becomes
much simpler. Such a dominating mode will generate only a restricted number of modes in higher orders
[10]. This situation, for example, takes place in the case of resonance or instability of one of the modes.
In the case of resonance small horizontal excitation can produce finite perturbations in fluid. We assume
that the amplitude of horizontal motion has order 2. That is
XTðtÞ¼2^
XTðtÞ;^
XTðtÞ¼Oð1Þas !0:
60 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
The asymptotic representation of the non-linearly interacting modes is then
/¼X
3
n¼1
n1coshðnkmðzþhsÞÞ
coshðnkmhsÞcosðnkmxÞFnmðtÞ;f¼X
3
n¼1
n1cosðnkmxÞZnmðtÞ;ð15Þ
where km¼mp=bis the wave number corresponding to the dominating mode. The term n¼0 is trivial.
Because of the constant mean water level we have Z0¼0, and the corresponding term for the velocity
potential is a function of time only and does not produce the contribution into the velocity field. Therefore
we have omitted these terms from (15). Substituting (15) into (8) and keeping terms up to Oð2Þ, we derive
the following non-linear ODEÕs describing the evolution of the modes in (15)
F0
mþð1þZ00
TÞZm¼bmX00
Tþ21
84ð2k2
mþx2
mx2
2mÞF1F24x2
mZ2mF0
m
8k2
mx2
mZmF2
mð3k2
mZmF0
mþ4x2
2mF2
2mÞZm;
F0
2mþð1þZ00
TÞZ2m¼1
4ðk2
mx4
mÞF2
m1
2x2
mZmF0
m;
F0
3mþð1þZ00
TÞZ3m¼b3m^
X00
Tþk2
m
1
2x2
mx2
2mFmF2m1
8k2
mZ2
mF0
m1
2x2
mZ2mF0
m1
2x2
2mZmF0
2m
ð16Þ
and
Z0
mx2
mFm¼2ZmF2m
þ1
8x2
mZ2
m
1
2Z2mFm;
Z0
2mx2
2mF2m¼k2
mFmZm;
Z0
3mx2
3mF3m¼3k2
mZmF2mþ3
2k2
mZ2mFmþ3
8k2
mx2
mZ2
mFm;
ð17Þ
The modes which are not included in (15) do not take part in the non-linear interaction and can be included
as independent linear modes. They then satisfy the linear equation (10). In the linear limit, when we neglect
all the non-linear terms in (16) and (17), all modes are independent and satisfy Eq. (10).
When considering the particular case studies later in the paper we use specific frequencies and amplitudes
as characteristic ones. They are: non-dimensional scale of surface perturbations for single mode motions
e¼ax2
n=g, where ais the amplitude of the initial surface perturbation; non-dimensional forcing amplitudes
jv¼avx2
v=gand jh¼ahx2
h=g, where av;hand xv;hare the amplitudes and frequencies of vertical and
horizontal excitations respectively; non-dimensional sloshing frequency Xn¼xn=xv; width parameters
B¼bx2
v=g,Bn¼bn=b; and so on.
4. Numerical model
A fully non-linear model for idealised 2-D waves in a numerical wave tank has been developed. A
modified r-transformation is used to map the liquid domain onto a rectangle, such that the moving free
surface in the physical plane becomes a fixed line in the computational mapped domain.
The r-transformation was first used by Phillips [33] in connection with numerical weather forecasting
schemes. Later the sigma coordinate system was used by Mellor and Blumberg [29] for ocean modeling to
improve predictions of both surface Ekman and instabilities in boundary layers. More recently Chern et al.
[9] use a Chebyshev expansion to discretise the r-transformed potential flow equation in their prediction of
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 61
2-D non-linear free-surface motions. The latest model in the literature is described by Turnbull et al. [37]
who simulate inviscid free surface wave motions using a 2-D r-transformed finite element model.
Fig. 2 illustrates the effect of the mapping in the present model, which has been designed so that each
computational cell in the transformed domain is of unit size. This is why we refer to this formulation as the
modified r-transformation. In this model, remeshing due to the moving free surface is avoided. Other
advantages are that the mapping implicitly deals with the free surface motion, and avoids the need to
calculate the free surface velocity components explicitly. Extrapolations are unnecessary, and free surface
smoothing by means of a spatial filter is not required for the results presented here.
With reference to Fig. 2, the mappings from the physical ðx;z;tÞdomain to the transformed ðX;r;tÞ
domain are given by
x$X;X¼m1þðm2m1Þ
bx;z$r;r¼n1þðn2n1ÞðzþhsÞ
h;t$T;T¼t;ð18Þ
where h¼fþhs; the wave amplitude is f, the still water depth is hs, and bis the tank width. We designate
the grid size to span from m1to m2in the horizontal x-direction and n1to n2in vertical z-direction.
The derivatives of the potential function /ðx;z;tÞare transformed with respect to x,zand tinto de-
rivatives of UðX;r;TÞ.
The first derivatives of the velocity potential, /, are obtained as
o/
ox¼ðm2m1Þ
b
oU
oX
þa
h
oU
or;
o/
oz¼ðn2n1Þ
h
oU
or;
o/
ot¼oU
oTþc
h
oU
or;
ð19Þ
where a¼ðrn1Þof
oXand c¼ðrn1Þof
oT.
Similarly, LaplaceÕs equation (1) can be rewritten as
o2U
oX2þ1
h
oa
oX
2a
h
oh
oXoU
orþ2a
h
o2U
oroXþa2
h2
"þb2ðn2n1Þ2
h2ðm2m1Þ2#o2U
or2¼0:ð20Þ
The fixed vertical wall boundary condition on X¼m1;m2, and the flat bed boundary condition on r¼n1
(2) become
Fig. 2. The physical domain (a) mapped onto the computational domain (b).
62 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
oU
oX¼a
h
oU
or;ðn2n1Þ
h
oU
or¼0:ð21Þ
The dynamic free-surface boundary condition (3) on r¼n2becomes
oU
oT¼ðn2n1Þ
h
of
oT
oU
or1
2ðm2m1Þ2
b2
oU
oX
"ðn2n1Þ
h
of
oX
oU
or2
þðn2n1Þ2
h2
oU
or
2#
ðgþZ00
TÞfXm1
m2m1
bX 00
T;ð22Þ
where Z00
Tand X00
Tare the vertical and horizontal acceleration of the tank, and gdenotes acceleration due to
gravity. The kinematic free-surface boundary condition (4) on r¼n2becomes
of
oT¼ðn2n1Þ
h
oU
or1
"þðm2m1Þ2
b2
of
oX
2#ðm2m1Þ2
b2
of
oX
oU
oX:ð23Þ
Eqs. (20)–(23) are then discretised using the second order Adams–Bashforth scheme and solved in the
transformed domain iteratively using successive over-relaxation.
The remaining part of the article presents test cases based on the above numerical model. The case
studies will report on sloshing motions in fixed, pure vertical and horizontal forced moving tanks. The final
study include result of sloshing motion in tanks forced to move simultaneously in horizontal and vertical
directions.
5. Standing waves in fixed tanks
Simulation of inviscid free sloshing motion in fixed rectangular tanks is the first benchmark validation
test which will be presented. The numerical model is validated for different wavelengths. Increasing wave
steepness is simulated in order to demonstrate cases where the fully non-linear model provides solutions not
obtainable with the approximate forms. Numerical predictions of the free surface motions are compared
with analytical results from second-, and third-order potential theory. The entire second order free-surface
elevation for the nth sloshing mode along the length of the fixed tank can be derived explicitly:
fðx;tÞ¼acosðxntÞcosðknxÞ
þax2
n
g
1
8
x4
nþg2k2
n
x4
n
þ1
8
3x4
ng2k2
n
x4
n
3
2
x4
ng2k2
n
x2
nð4x2
nx2
2nÞcosð2xntÞ
þ1
2
x2
nx2
2nx4
n3g2k2
n
x2
nð4x2
nx2
2nÞcosðx2ntÞcosð2knxÞ;ð24Þ
where the linear sloshing frequencies are defined as xn¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gkntanhðknhsÞ
pand x2n¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g2kntanhð2knhsÞ
p.
This solution coincides with other investigators [10,42]. The numerical initial conditions which satisfy the
velocity potential and free-surface equations are prescribed as
fðx;nÞt¼0
j¼acosðknxÞand /ðx;zÞt¼0
j¼0;ð25Þ
where ais the amplitude of the initial wave profile, kn¼np=bis the wave number for n¼0;1;2;... and xis
the horizontal distance from the left wall.
Non-linear free-surface motions are investigated by varying the initial wave steepness, defined in the
fixed tank studies as e¼ax2
n=g, where gravity is g¼9:81 m/s2until near breaking conditions are en-
countered. The results presented are for a tank of aspect ratio hs=b¼0:5. The linearly stretched grid in the
physical domain in accordance with the r-transformed Eq. (18) is shown in Fig. 2.
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 63
The results herein are based on the first set of numerical tests done by Frandsen and Borthwick [16] who
checked the sensitivity of the numerical scheme to the time step and grid resolution. Figs. 3 and 4 illustrate
for n¼1andn¼3 the time-dependent free surface motion at the wall of the tank for (a) very small
amplitude sloshing where e¼0:0014, and (b) large amplitude sloshing where e¼0:288. The time histories
of the free sloshing analyses are non-dimensionalised with the sloshing frequency xn, so that the non-di-
mensional time t¼xnt, and the non-dimensional time step Dt¼xnDt. Although a time step of 0.003 s
Fig. 3. Free-surface elevation at the left wall in fixed tank for n¼1, for (a) e¼0:0014 and (d) e¼0:288. - - -, second order solution;
--, third order solution; ––, numerical solution. The corresponding wave phase-plane and spectra of the numerical model (b, c) linear
solution; (e, f) non-linear solution.
64 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
was used in both cases, the grid size was increased in the vertical direction for the larger amplitude test case.
A grid size of 40 40 was sufficient to model accurately small to moderate amplitude waves (approximately
e<0:09), in comparison with the third-order analytical solution. Increasing the grid points in the vertical
direction was found to be more effective in improving accuracy than increasing the grid points in the
horizontal direction. A grid size of 40 80 was used to model steeper waves (e>0:1). For large amplitude
Fig. 4. Free-surface elevation at the left wall in fixed tank for n¼3, for (a) e¼0:0014 and (d) e¼0:288. - - -, second order solution;
--, third order solution; ––, numerical solution. The corresponding wave phase-plane and spectra of the numerical model (b, c) linear
solution; (e, f) non-linear solution.
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 65
sloshing (n¼1, 3) it can be observed (Figs. 3(d) and 4(d)) that the phase-shift grows in time between the
second order analytical solution and the fully non-linear numerical model. The maximum amplitudes are
also higher than those of the approximate solution. This has also been observed by Tadjbakhsh and Keller
[34], Vanden-Broeck and Schwartz [39], Tsai and Jeng [36] and Greaves et al. [18]. The third order ap-
proximate solution compares well with the numerical solution with regard to the phase. The third-order
solution, however, tends to underpredict the peaks and overestimate the through, especially when the
numerical predicted trough are smallest. The peaks following the smallest trough are also not captured well
by the third order solution. This effect is more evident for n¼1 than n¼3 due higher non-linearity.
The influence of non-linearity can be seen more clearly on the phase-plane diagrams in Figs. 3(b), (e) and
4(b), (e). The small amplitude diagrams (b) display linear behaviour of the free surface with repeatable
patterns for the peaks and troughs in bounded orbits, while the large amplitude cases (e) show bounded
solutions with higher peaks caused by non-linearity.
Figs. 3(c), (f) and 4(c), (f) show the associated spectra. The spectra of small amplitude waves (c) display
the fundamental sloshing frequency. It can be observed that small additional frequencies due to non-linear
mode to mode interaction are present for the large initial amplitude cases (f). We note that they are
responsible for the large disturbances in the free surface elevation.
Fig. 5. Free-surface elevation at the left wall in fixed tank for wave steepness of ––, e¼0:0014; - -, e¼0:144; - - -, e¼0:288; for (a)
n¼1 and (b) n¼3.
Fig. 6. Wave profiles for a typical half-period, n¼1, for (a) e¼0:0014 and (b) e¼0:288.
66 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
Fig. 5 shows the effect of increasing wave steepness on the free surface time history at the tank wall. As
expected, the wave motions become progressively non-linear as the steepness increases. This is reflected by
the higher peaks, lower troughs with a growing phase-shift, as time evolves.
Furthermore, the corresponding numerical wave profiles across the tank at different times during a
typical sloshing period for n¼1, 3 are also shown in Figs. 6 and 7. The small amplitude waves (a) display
linear standing waves whereas the steep wave cases (b) exhibit a dispersion effect that is most evident at the
nodes. Other r-methods show identical behaviour as reported by other investigators [9,37].
6. Vertically excited tanks
The second set of validation tests is concerned with forced sloshing of liquid in a rectangular tank
subjected to vertical base-excitation.
The coordinate system of the numerical model is fixed at the left wall of the tank, and moves with the
tank. The only change to the governing Eqs. (20)–(23) of the numerical model is the dynamical free-surface
boundary condition (22) in which X00
T¼0. The tank is assumed periodically excited with the vertical base
acceleration, Z00
T¼x2
vavcosðxvtÞ, where avis the vertical forcing amplitude, tis time and xvis the angular
frequency of forced vertical motion. The initial conditions are equivalent to the sloshing motion simulation
in a fixed tank (25). In the vertically excited tank test cases the parameter jv¼avx2
v=gis a measure of the
importance of the vertical forcing motion and eis a measure of non-linearity. Numerical predictions of the
free surface motions are compared with analytical results from second-, and third-order potential theory.
First we note that the linear solution for the motion of fluid in a vertically excited tank was first obtained by
Benjamin and Ursell [4], who also investigated the stability of this motion. In the case when the initial
surface perturbation includes only one Fourier component fð0;xÞ¼acosðknxÞthe solution does not in-
cludes infinite sums and can be represented in a relatively simple form. The second-order correction for the
surface elevation then consist only of the double wavelength term and the entire second order solution can
be written explicitly as
fðx;tÞ¼acosðknxÞZð1Þ
nðxvtÞþaax2
v
g
cosðk2nxÞZð2Þ
2nðxvtÞ;ð26Þ
where the amplitude of the initial perturbation (a) is used as a characteristic amplitude of the wave. The first
and second-order evolution functions Zð1;2Þ
n;2nsatisfy the homogeneous and non-homogeneous Mathieu
equations:
Fig. 7. Wave profiles for a typical half-period, n¼3, for (a) e¼0:0014 and (b) e¼0:288.
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 67
Zð1Þ00
nðxvtÞþX2
nð1þjvV00ðxvtÞÞZð1Þ
nðxvtÞ¼0;
Zð2Þ00
2nðxvtÞþX2
2nð1þjvV00ðxvtÞÞZð2Þ
2nðxvtÞ¼ 1
2X2
n
1
2
X2
2n
X2
n
þ2!K2
n
B2X2
nX2
2n
2!Zð1Þ0
nðxvtÞ2
ð1þjvV00ðxvtÞÞ K2
n
B2
X2
nX2
2n
2Zð1Þ
nðxvtÞ2;
where B¼bx2
v=g,X¼xn=xv,K¼pn, and with the following non-dimensional initial conditions
Zð1Þ
nð0Þ¼1;Zð2Þ
2nð0Þ¼0;Zð1Þ0
nð0Þ¼Zð2Þ0
2nð0Þ¼0:
The free surface motions are examined in cases of increasing wave steepnesses, inside and outside of the
regions of parametric resonance (instability regions). The six test cases considered herein are marked on the
stability map in Fig. 8. The results presented are for a tank of aspect ratio hs=b¼0:5.
The first set of tests are carried out in a stable zone, with frequency ratio X1¼1:253, and a non-di-
mensional forcing amplitude, jv¼0:5. The time histories for the free surface elevation for small and high
wave steepnesses are shown in Fig. 9. Good agreement between the approximate solution and the numerical
model is achieved for small amplitude waves. The behaviour of the free surface motion in the vertically
excited tank is similar to the standing waves observed in the fixed tank (Fig. 3) but for this magnitude of jv,
irregular peaks and troughs are generated in time. For higher wave steepness, as the solution evolves in
time, a discrepancy in phase-shift between the numerical model and the approximate solution is evident; the
fully non-linear model predicts waves of slightly longer period than the approximate solution. Differences in
amplitudes, of both peaks and troughs, can also be observed. It should be noted that for small non-lin-
earities (e¼0:0014), a grid size of 40 40 resulted in sufficient accuracy in comparison with the second-
order approximation (Fig. 9 (a)). However, the steeper wave case (e¼0:288) required a finer grid resolution
of 40 80 (Fig. 9(d)). It was again found to be more effective in ensuring accuracy by increasing the mesh
density in the vertical direction than by using higher resolution horizontally. The non-dimensionalised time
is defined as t¼x1t, and the non-dimensional time step is Dt¼x1Dtwhere x1is the fundamental first
sloshing frequency in a fixed tank. A non-dimensional time step of 0.011 was used for the test case in the
stable region for both the small and steep wave cases. Fig. 9(b) and (e) show the corresponding phase-plane
Fig. 8. Linear stability map of sloshing motion in vertically excited tank. Test cases (): (1) X1¼1:253; jv¼0:5, (2) X1¼0:5;
jv¼0:3, (3) X1¼1:0; jv¼0:5, (4) X3¼0:5; jv¼0:2, (5) X1¼0:6; jv¼0:5, (6) X1¼0:55; jv¼0:5.
68 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
plots for the small and steep wave cases. The small amplitude wave phase-plane plot (b) displays linear
behaviour of the free-surface through the closed orbit whereas the non-repeatable non-closed orbits of the
large amplitude sloshing in (e) show that the free-surface exhibits more complicated behaviour typical of
non-linear systems.
Fig. 9(c) and (f) shows the spectra corresponding to small and large amplitude wave cases (Fig. 9(a) and
(d)). The linear solution (Fig. 9(c)) displays a dominating frequency of r1=2=X1near to the first fundamental
Fig. 9. Free-surface elevation at the left wall in vertically excited tank for n¼1 in stable region, X1¼1:253, jv¼0:5 (Fig. 8: test case
1) for (a) e¼0:0014 and (d) e¼0:288. - - -, second order solution; ––, numerical solution. The corresponding wave phase-plane and
spectra of the numerical model: (b, c) linear solution; (e, f) non-linear solution.
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 69
sloshing frequency (x1) and secondary frequencies at r1=21=X1. The non-linear solution (Fig. 9(f))
contains additional frequencies related to low energy content, including the second fundamental sloshing
frequency (r2=2=X1), which contributes to the non-linear generated waves.
Fig. 10 shows the free surface elevation time histories in unstable regions. The wave steepness parameter
was kept constant at a low value of e¼0:0014 and a grid size of 40 40 was used. A non-dimensional time
Fig. 10. Free-surface elevation at the left wall of vertically excited tank in unstable regions for small initial amplitude (e¼0:0014): (a)
X1¼0:5, jv¼0:3 (Fig. 8: test case 2); (b) X1¼1:0, jv¼0:5 (Fig. 8: test case 3); (c) X3¼0:5, jv¼0:2 (Fig. 8: test case 4). - - -, second
order solution; - -, third order solution; ––, numerical solution. The corresponding phase plots for the numerical model (d, e, f).
70 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
step of 0.011 was prescribed. Fig. 10(a) shows the free surface elevation time history for X1¼0:5, jv¼0:3.
The test, no. 2 in Fig. 8, corresponds to a first sloshing mode in the first instability region. From t¼0to
approximately 80, the wave amplitudes and phase predicted by the fully non-linear model are found to be in
close agreement with the second order solution (26). Then the amplitudes begin to grow rapidly, dis-
crepancies in amplitudes and phase between the numerical model and the second order solution increases,
due to the enhanced non-linearity of the free surface motions. The third-order approximation (Section 3)
compares almost exact with the numerical solution. However, the third-order solution predicts slightly
deeper troughs and lower peaks as time evolves. The associated numerical predicted phase-plane diagram is
shown in Fig. 10(d). Fig. 10(b) shows the free surface elevation time history X1¼1:0, jv¼0:5 (test case 3
in Fig. 8). Following parameters were assumed: e¼0:0014, Dt¼0:011 and a grid size of 40 40 was used.
This is also an example related to the first sloshing mode but this particular free surface test case lies in the
second instability region. As expected, the amplitudes do not grow rapidly in this region compared to the
first instability region (Fig. 10(a)). We observe that the second order solution deviates from the numerical
solution as times evolves whereas there is almost an exact agreement with the third order solution. Further
we observe that the amplitude of the first mode start to grow in a resonance mode. As the amplitude in-
creases the natural frequency changes. The change of the natural frequency with amplitude creates low
frequency amplitude oscillations. This non-linear detuning effect is also in agreement with the numerical
solution. Similar free-surface behaviour has also been observed by others, e.g., Hill [21]. Fig. 10(c) corre-
spond to the third test case (X3¼0:5, jv¼0:2, e¼0:0014, Dt¼0:020 and grid: 40 40) which represents
the second fundamental sloshing mode and lies in the first instability region (test case 4 in Fig. 8). Because it
is a second mode, again the amplitudes are not found to grow rapidly in comparison with the first mode
case (Fig. 10(a)). Again the second order solution deviates in peaks, troughs and phase compared to the
numerical solution. The third order and the numerical solutions agree well, capturing the detuning effect.
There is almost exact in-phase behaviour at all times. The peaks of the third order solution compare well
with the numerical solution from t¼0 to approximately 150 at which time the amplitudes decays into a
lower oscillation mode. Then the third order solution underestimate the peaks and troughs. Moreover, it
can be observed in the associated numerical predicted phase-plane diagrams (Fig. 10(e) and (f)) that the free
surface exhibits standard linear behaviour for an unstable system.
The final test of pure vertical tank excitation is carried out near the stability boundary (inside and out-
side). Test 5 in Fig. 8 represents a free surface problem which is just outside the instability region. Test 6
contains a set of parameters close to the ones of test case 5, but is located just inside the instability region. In
Fig. 11 the free-surface elevation is simulated for small and steep amplitudes for these two cases. Although
Fig. 11. Free-surface elevation at the left wall of vertically excited tank near stability boundary. (a) X1¼0:6; jv¼0:5 – stable region
(test case 5); (b) X1¼0:55; jv¼0:5 – unstable region (test case 6). - - -, small initial amplitude (e¼0:0014); ––, large initial amplitude
(e¼0:288).
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 71
test case 5 and 6 have closely spaced parameters, the simulations of the numerical model illustrate the ex-
pected stable solutions for X1¼0:6, jv¼0:5 and unstable solutions for X1¼0:55, jv¼0:5. It is observed
that the solution of test case 5 remains stable for both small and large initial wave steepnesses whereas the
small initial wave steepness of test case 6 is sufficient to demonstrate a rapidly unstable solution.
7. Horizontally excited tanks
Investigations of forced sloshing of liquid in a rectangular 2-D tank subjected to horizontal base-exci-
tation is undertaken in this section. The only change to the governing Eqs. (20)–(23) is the dynamical free-
surface boundary condition (22) in which Z00
T¼0.
A linear solution for fluid motions with surface tension in a horizontally base-excited tank was first
obtained by Faltinsen [10]. Herein surface tension is assumed to be negligible. We prescribe harmonic
forced motion of XTðtÞ¼ahcosðxhtÞwhere ahdenotes the horizontal forcing amplitude, tis time and xhis
the angular frequency of forced horizontal motion. The initial conditions are /ð1;2Þ
nð0Þ¼0, fð1;2Þ
nð0Þ¼0,
corresponding to the fluid being at rest. We use the coordinate system with x-axis fixed on the undisturbed
water surface and z-axis fixed on the left-hand wall of the tank. The numerical predictions of the free
surface motions in the horizontally excited tanks will be compared with analytical results from second- and
third-order potential theory. When xv¼0, it can be shown that the entire second order solution for the
free-surface elevation and velocity potential can be written explicitly as
fðx;tÞ¼ahX
1
n¼0
cosðknxÞZð1Þ
nðxhtÞ
þahx2
h
g
X
1
n¼0
cosðknxÞZð2Þ
nðxhtÞ!ð27Þ
and
/ðx;z;tÞ¼ahg
xhX
1
n¼0
coshðknðzþhsÞÞ
coshðknhsÞcosðknxÞUð1Þ
nðxhtÞ
þahx2
h
g
X
1
n¼0
coshðknðzþhsÞÞ
coshðknhsÞcosðknxÞUð2Þ
nðxhtÞ!;ð28Þ
where Zð2Þ
nand Uð2Þ
nare
Zð2Þ0
nðxhtÞX2
nUð2Þ
nðxhtÞ¼ 1
B2CnðK2
nUð1Þ
n;Zð1Þ
nÞ
SnðKnUð1Þ
n;KnZð1Þ
nÞ;
Uð2Þ0
nðxhtÞþZð2Þ
nðxhtÞ¼ 1
2B2SnðKnUð1Þ
n;KnUð1Þ
nÞ1
2CnðX2
nUð1Þ
n;X2
nUð1Þ
nÞCnðX2
nUð1Þ0
n;Zð1Þ
nÞ;ð29Þ
where Kn¼pnare non-dimensional wave numbers. It can also be shown that the first order solution re-
duces to
Zð1Þ0
nðxhtÞX2
nUð1Þ
nðxhtÞ¼0;
Uð1Þ0
nðxhtÞþZð1Þ
nðxhtÞ¼BBnH00ðxhtÞ:
The free surface motions are numerically examined off- and at resonance where resonance is occurring
when the external horizontal forcing frequency (xh) is equal to the natural sloshing frequency (xn)ofthe
liquid. The free-surface behaviour is investigated by varying the external force through the parameter
jh¼ahx2
h=gwhich is a measure of non-linearity. The results presented are for a tank of aspect ratio
72 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
hs=b¼0:5, except for a special comparison study near the critical water depth (shown in the last part of this
section).
Fig. 12(a) and (d) show the free-surface elevation at the left wall in an off-resonance region with
xh=x1¼0:7 for small horizontal forcing amplitude where jh¼0:0036 (a), and for large horizontal forcing
amplitude where jh¼0:036 (d). The time histories of the forced sloshing analyses are non-dimensionalised
Fig. 12. Free-surface elevation at the left wall in horizontally excited tank; xh=x1¼0:7; (a) jh¼0:0036 and (d) jh¼0:036; - - -,
second order solution; ––, numerical solution. The corresponding wave phase-plane and spectra of the numerical model: (b, c) linear
solution; (e, f) non-linear solution.
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 73
with the first natural sloshing frequency. A grid size of 40 40 and 40 80, respectively, were prescribed for
small and large forcing frequency. A time step of 0.003 s was used in both cases. Good agreement with the
second order solution is achieved for both the case of small and large forcing frequency. The associated
wave phase-planes (Fig. 12(b) and (e)) display linear behaviour of the free-surface with irregular patterns
for the peaks and troughs in bounded orbits. Fig. 12(c) and (f) shows the associated spectra of the free
surface elevation. It can be observed for the small forcing amplitude case (c) that there exists energy at two
distinct frequencies, i.e. at the forcing frequency and at the first sloshing frequency. In addition, for the
larger forcing amplitude case (f), a third frequency with low energy content exists (x2) due to non-linear
effects. This second natural sloshing frequency is responsible for the deviation between the numerical model
and second order approximation.
The test case presented in Fig. 13 is also an off-resonance case but with a forcing frequency higher than
the first natural sloshing frequency (xh=x1¼1:3). As shown in the small amplitude spectrum (c) an ad-
ditional (third) natural sloshing frequency is present in the solution of this particular free-surface problem.
The free-surface time elevation for small amplitude waves (jh¼0:0036) is shown Fig. 13(a). Although x3
has a low energy content it contributes to the lower numerical predicted peaks compared to the second
order solution. The associated wave phase diagram (b) displays irregular peaks and troughs in bounded
orbits. Next, the horizontal forcing parameter was increased to jh¼0:072 and the free-surface elevation
simulated (Fig. 13(d)). The increase of jhintroduces non-linearity in the solution resulting in discrepancy in
amplitudes between the fully non-linear model and the second order solution. As time evolves the phase
between numerical model and approximate solution deviates, the numerical model having a longer period.
This is due to the present of two additional secondary frequencies (xhx1), as shown in the spectrum (f),
which are generated by non-linear interaction between modes. For this reason the wave phase-plane (e)
displays more irregular patterns compared to the small forcing frequency case (b).
Fig. 14 compares the small and large amplitude cases of the fully non-linear model for xh=x1¼0:7 and
1.3. For stronger excitation it can be observed that the peaks are higher, the troughs are less deep and the
period is longer than those of the approximate solution, which is typical non-linear effects.
Fig. 15 shows the free-surface elevation at the left wall at resonance, xh¼x1¼3:76 rad/s, for (a) small
jh¼0:0014 and (c) large jh¼0:014 horizontal forcing amplitudes. For the small amplitude case there is
good agreement between the approximate solutions and the numerical model. For the large amplitude case,
at the initial stage of the process (tx1<20) while the amplitude is still small, the numerical solution co-
incides with both the second-order and the third-order solutions. Eventually, as the amplitude increases, the
non-linear effects begin to play a considerable role leading to higher peaks and smaller troughs in the
surface elevation, compared to the third-order solution. As in the previous test cases the third-order so-
lution predict the phase almost in exact agreement with the numerical solution. The second order solution
do capture these non-linear features but discrepancy in amplitude and phase compared to the fully non-
linear model is evident. This process can be observed even more clearly on the numerical predicted wave
phase-planes, when the spiral trajectory of the linear solution (b) deforms gradually from cycle to cycle in
the non-linear case (d), as the centre of the trajectory gradually moves towards higher amplitudes. The
maximum steepness shown for the resonant solution (Fig. 15(c)) can be estimated to be approximately 0.25,
and as shown the numerical solution begins to deviate from the linear one as the steepness reaches about
0.1.
As mentioned in Section 1, other investigators have analysed horizontal tank motion. Herein we com-
pare a specific test case previously done by Hill [21] and Faltinsen et al. [11]. Their tank was 1.73 m wide
with a still water depth of 0.6 m. We define the wave length as k¼2b=nand denotes the critical depth as hc.
At the critical depth the response changes from a ‘‘hard-spring’’ to a ‘‘soft-spring’’. Gu et al. [20], Faltinsen
[11], Waterhouse [41] found hc=k¼0:583 m or in general hc¼0:337 bfor the first mode, respectively.
Since hsis 0.6 m, this particular study is a near critical depth case. Fig. 16 represents an off-resonance case
[11,21] with a forcing frequency higher than the first natural sloshing frequency (xh=x1¼1:283), similar to
74 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
Fig. 13 where jh¼0:069 (large). The fully non-linear solution (based on a grid size 40 80 and a time step
of 0.003 s) is compared with Hill [21] who developed a third order solution assuming one dominating mode.
We also compare with the second and third order solution (Section 3). The troughs of the third order
solution compare well with the numerical results, as shown in Fig. 16. However the third order peaks are
over estimated. The second order solution compare better with the numerical results than the third order, in
the periods where the amplitudes are largest. The opposite is true for the smallest amplitudes. HillÕs [21]
Fig. 13. Free-surface elevation at the left wall in horizontally excited tank; xh=x1¼1:3; (a) jh¼0:0036 and (d) jh¼0:072; - - -,
second order solution; ––, numerical solution. The corresponding wave phase-plane and spectra of the numerical model: (b, c) linear
solution; (e, f) non-linear solution.
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 75
third order analytical-based algorithm generates a free surface elevation with smaller troughs/peaks com-
pared to the fully non-linear solution. Also a phase shift is present compared to the other approximate
solutions and the numerical method. Furthermore, Faltinsen et al. [11] did extensive theoretical and ex-
Fig. 15. Free-surface elevation at the left wall in horizontally excited tank at resonance; xh=x1¼1; (a) jh¼0:0014 and (c) jh¼0:014;
--, linear solution; - - -, second order solution; - -, third order solution; ––, numerical solution. The corresponding phase-plane of the
numerical model: (b) linear solution; (d) non-linear solution.
Fig. 14. Free-surface elevation at the left wall in horizontally excited tank. (a) xh=x1¼0:7; - - -, small amplitude solution
(jh¼0:0036); ––, large amplitude solution (jh¼0:036). (b) xh=x1¼1:3; - - -, small amplitude solution (jh¼0:0036); ––, large am-
plitude solution (jh¼0:072).
76 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
perimental sloshing experiments. Fig. 17 shows the theoretical result of Faltinsen et al. [11] of the above
mentioned test case of Fig. 16. Their solutions generate peaks of 0.113 m/0.136 m and troughs of 0.107 m/
0.1 m corresponding to the experimental and theoretical (impulse) tests. The present equivalent numerical
solution generates 0.134 m/0.108 m at an equivalent time of 9.1 s/8.3 s, closer to the theoretical model.
Later, as time evolves, at 37.1 s/37.8 s the numerical solution predicts peaks and troughs of 0.155 m/0.097
m. Faltinsen et al. predicts peaks/troughs of 0.127 m/0.087 m and 0.143 m/0.079 m corresponding to the
experimental and theoretical (impulse) tests, respectively. Therefore the numerical solution is in reasonable
agreement with the work of Faltinsen et al. for this particular test case.
Fig. 16. Free-surface elevation at the left wall in horizontally excited tank; xh=x1¼1:283; ah¼0:029 m and jh¼0:069; ––, numerical
solution; , second order solution; - - -, third order solution; --, Hill (2003).
Fig. 17. Solution of Faltinsen et al. (2000) showing the free-surface elevation at the left wall in horizontally excited tank;
xh=x1¼1:283; ah¼0:029 m and jh¼0:069. Theoretical predictions include two different initial conditions. The curve ‘‘Zero’’
corresponds to zero initial conditions, ‘‘Impulse’’ means initial impulse conditions.
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 77
8. Horizontally and vertically excited tanks
Let us now investigate the influence of vertical excitation on the solution for pure horizontal motion
considered for hs=b¼0:5 in the previous sections. As mentioned we use the general form of the excitation
laws XTðtÞ¼ahHðxhtÞ;ZTðtÞ¼avVðxvtÞ, where ah;v,xh;vare the characteristic amplitudes and frequencies
of horizontal and vertical motion, respectively. The initial conditions related to the fluid being at rest at
time t¼0 are /ðx;z;0Þ¼0;fðx;0Þ¼0:We prescribe harmonic tank excitation in both horizontal and
vertical directions with excitation law VðsÞ¼HðsÞ¼cosðsÞ. In this case we can describe the individual
sloshing modes in accordance with (11) which becomes unstable for certain values of parameters. It can be
observed that the horizontal component of the motion generates perturbations with wave numbers kn
corresponding to odd values of nand n¼0. If any of the pairs of the parameters ðXn;jvÞ,n¼1;3;... lie in
the instability region, then the corresponding mode grows in time exponentially. However, as we will
demonstrate, if mode interaction occurs detuning effect may be present. The stability map of (11) is shown
in Fig. 18. In the following, results are shown for the selected test cases with kv¼0:2;0:3;0:5 including all
associated individual sloshing modes, as marked in Fig. 18.
All models comprise 40 40 grid points for small horizontal forcing frequency and 40 80 grid points
for large horizontal forcing frequency and a time step of 0.003 s. The tank dimensions and free sloshing
frequencies are the same as in the previous sections (hs=b¼0:5). The fully non-linear numerical results are
compared with second-order and third-order approximate forms for standing waves in a tank moving in
both horizontal and vertical directions. In the horizontally- and vertically-forced tank studies, the pa-
rameter jh¼ahx2
h=gis a measure of the importance of non-linearity. As mentioned, the equation for the
combined motion (11) differs from the equation for pure vertical excitation, by the forcing terms due to
the horizontal motion on its right hand side. These terms can produce resonance, which is recognised by the
linear growth of the amplitude in time. Contrary to the pure horizontal motion we now have infinite
number of resonances instead of one. Figs. 19–21 illustrate the time history of the free surface elevation at
the left wall for three main resonant frequencies of the horizontal motion (xh=x1¼0:18, 0.98, 1.78) in a
Fig. 18. Stability map for the first-order linear solution. Symbols represent points corresponding to the odd sloshing modes for selected
cases. *: X1¼1:253, jv¼0:5; :X1¼0:5, jv¼0:3; :X1¼1:0, jv¼0:5; n:X1¼0:276, X3¼0:5, jv¼0:2.
78 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
Fig. 19. Free-surface elevation at the left wall for the main resonances in horizontally and vertically excited tank X1¼1:253, jv¼0:5,
xh=x1¼0:98. (a) jh¼0:0014 and (b) jh¼0:0069; - - -, second order solution; - - -, third order solution; ––, numerical solution.
Fig. 20. Free-surface elevation at the left wall for the first side resonances in horizontally and vertically excited tank X1¼1:253,
jv¼0:5, xh=x1¼0:18. (a) jh¼4:85 105and (b) jh¼0:0194; - - -, second order solution; - - -, third order solution; ––, numerical
solution.
Fig. 21. Free-surface elevation at the left wall for the second side resonances in horizontally and vertically excited tank X1¼1:253,
jv¼0:5, xh=x1¼1:78. (a) jh¼0:0046 and (b) jh¼0:0228; - - -, second order solution; - - -, third order solution; ––, numerical
solution.
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 79
stable region (first mode () in Fig. 18 or r¼2:453 in Fig. 1). The characteristics of the vertical tank motion
are the same as in test case no. 1 in Fig. 8 (X1¼1:253; jv¼0:5). The strongest of the resonant frequencies
(Fig. 19) relate to the situation in which the horizontal forcing frequency is close to one of the free sloshing
frequencies, here x1. The two other resonant frequencies (Figs. 20 and 21) relate to the coupled frequencies
(xvxh) coinciding with a natural sloshing frequency. We note the resonance at the main frequency
(xh=x1¼0:98) is stronger and thus provides a higher rate of growth of perturbations compared to the
secondary frequencies (xh=x1¼0:18, 1.78), as can be observed by comparing the amplitudes in Figs. 19–
21. For small horizontal forcing amplitude the numerical amplitude coincides with second-order small
perturbation theory (Figs. 19(a) and 20(a)), and there is almost exact in-phase behaviour at all times. For
higher horizontal forcing amplitude, the numerical non-linear solution is close to the approximate solutions
from t¼0 to approximately 60, when the wave steepness is still small (Figs. 19(b) and 20(b)). The influence
of non-linearity grows in time as the steepness increases, which can be observed by higher peaks and smaller
troughs, which is the reason why the approximate solutions begin to deviate from the fully non-linear
numerical model; the third-order being most accurate. The third resonant case, shown in Fig. 21, relates to
the highest forcing frequency case (xh=x1¼1:78) of the three selected tests. This high forcing frequency is
responsible for the non-linearity produced at the free surface even for the small initial amplitude test shown
in Fig. 21(a) (jh¼0:0046). Discrepancy between second/third-order and numerical solution in both am-
plitude and phase occurs as time evolves and becomes more pronounced for the large amplitude case
(jh¼0:0228), as shown in Fig. 21(b). In general, we observe the free surface exhibits more complicated
irregular behaviour than for pure horizontal excitation, due to the influence of vertical tank motion.
Fig. 22 shows the free surface elevation time histories and the associated wave phase diagrams for
unstable solutions. These selected cases for jv¼0:2, 0.3, 0.5 are indicated in Fig. 18. The tests include a
small horizontal forcing amplitude combined with the vertically excited tank cases (test no. 2, 3 and 4 in
Fig. 8). The only difference when comparing the results of Fig. 22 with Fig. 10, is that the tank is moved
horizontally with a small forcing amplitude. The free surface sloshing of the combined forced tank motion,
shown in Fig. 22, illustrates that the instability is still due to the vertically forced motion and that the small
horizontally forced tank motion only causes disturbance and delay in the occurrence of free surface in-
stability. The free surface elevation of Fig. 22(a) represents the first sloshing mode in the first instability
region (Fig. 18: :X1¼0:5;jv¼0:3). From t¼0 to approximately 110, the wave amplitudes predicted by
the fully non-linear model are found to be in close agreement with the second/third order solutions. Then
the amplitudes begin to grow rapidly, discrepancies in amplitudes with the approximate solutions are
found, as expected. The third-order solution is most accurate. The discrepancy is however minor because
the first mode is dominating at all times. The associated wave phase diagram (Fig. 22(d)) displays more
clearly the unstable system, showing the higher peaks and lower troughs as time evolves. Fig. 22(b) is an
example of a first sloshing mode in the second instability region (Fig. 18: :X1v ¼1:0, jv¼0:5). This case
therefore exhibit a less strong resonance than the test of Fig. 22(a). For this particular unstable system, the
amplitudes are small for t¼0 to 200 and peaks and troughs have equal magnitude. The corresponding
wave phase diagram (Fig. 22(e)) also shows the linear free surface behaviour. Therefore good agreement is
found between the numerical model and the approximate solutions. The free surface behaviour shown in
Fig. 22(c) has a stable first sloshing mode (Fig. 18: n:X1v ¼0:276, jv¼0:2). However, the third sloshing
mode (Fig. 18: n:X3v ¼0:5, jv¼0:2) is an unstable solution in the first instability region. Therefore the
entire solution becomes unstable. The amplitudes grow faster compared to the test of Fig. 22(b) because the
solution is within the first instability region but is less in magnitude than the strong resonance case of
Fig. 22(a), as expected. The numerical amplitudes and phase (c) deviate from both second-order and third-
order solutions. The detuning effect of the third order solution is however well predicted in the sense that
the third order non-linearity for the third mode is captured (note that the second-order solution is invalid at
this stage). The shift in the third order solution compared to the fully non-linear solution is due to the
influence of the first mode on the initial oscillations, at a time where the third mode has not started to grow.
80 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
The first–third mode interaction produces perturbations in the third mode which are predicted fairly well by
the third order method (Section 3). However, multimodal algorithms [11] are recommended to be used in
this case. For the corresponding vertical case (Fig. 10(c)), where the first mode is not excited, the agreement
between third order and numerical solutions are better. The fully non-linear numerical predictions are also
illustrated on the corresponding phase-plane plot (f).
Fig. 22. Free-surface elevation at the left wall of horizontally and vertically excited tank for selected unstable cases (parametric
resonance), - - -, second order solution; - - -, third order solution; ––, numerical solution for small horizontal forcing (ah¼0:001 m). (a)
jv;X1¼0:3;0:5 (Fig. 18: ); (b) jv;X1¼0:5;1:0 (Fig. 18: ); (c) jv¼0:2, X1v ¼0:276, X3¼0:5 (Fig. 18: n). The associate wave
phase diagrams of the numerical model (d, e, f).
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 81
Furthermore, we present the off-resonance case in a stable region for the horizontal motion
(xh=x1¼0:7) considered in the previous section (Fig. 12) coupled with a vertical motion at frequency
xv¼3 rad/s (X1¼1:253) and vertical forcing parameter jv¼0:5 (Fig. 9). The horizontal forcing pa-
rameter is varied (jh¼0:0036, 0.036) representing small and large forced motion of ah¼0:01, 0.1 m.
With reference to the pure horizontal forced tank (Fig. 12) and the pure vertical forced tank motion
(Fig. 9), the combined forced tank motion in Fig. 23 illustrates the change in the free-surface behaviour
Fig. 23. Free-surface elevation at the left wall in horizontally and vertically excited tank; xh=x1¼0:7; jv¼0:5; X1¼1:253. (a)
jh¼0:0036 and (d) jh¼0:036; - - -, second order solution; ––, numerical solution. The corresponding wave phase-plane and spectra of
the numerical model: (b, c) linear solution; (e, f) non-linear solution.
82 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
due to the vertical vibrations. The wave phase diagrams show more complicated surface elevation
behaviour for both (b) linear and especially (e) non-linear solutions. The reason for this is the presence
of additional frequencies in the spectrum due to the vertical motion, for example the coupled fre-
quencies xvxh. Figs. 23(c) and (f) show the spectra corresponding to the small and the large am-
Fig. 24. Free-surface elevation at the left wall in horizontally and vertically excited tank; xh=x1¼1:3; jv¼0:5; X1¼1:253. (a)
jh¼0:0036 and (d) jh¼0:072; - - -, second order solution; ––, numerical solution. The corresponding wave phase-plane and spectra of
the numerical model: (b, c) linear solution; (e, f) non-linear solution.
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 83
plitude wave cases (Fig. 23(a) and (d)). The linear solution (Fig. 23(c)) displays two dominating fre-
quencies of which one is equivalent to the horizontal forcing frequency xhand the other r1=2=X1which
is near the first natural sloshing frequency and secondary frequencies at r1=21=X1. The non-linear
solution (Fig. 23(f)) contains additional frequencies related to low energy content, including the second
natural sloshing frequency (r2=2=X1), which besides the large jh, contributes to the non-linear generated
waves.
Next, as shown in Fig. 24, we explore another off-resonance case in the stable region which is similar to
the previous mentioned test case (Fig. 13) but has an increased horizontal forcing frequency of
xh=x1¼1:3. We intend to investigate the effect of combining the forced horizontal tank test case of Fig. 13
and the vertically excited tank jv¼0:5; X1¼1:253 (test case no. 1 in Fig. 8). As experienced in the pure
horizontal case shown in Fig. 13(a)–(c) with small jh, non-linearity causes discrepancy in amplitudes as
time evolves between the fully non-linear model and the second order solution even for small initial hor-
izontally forced tank excitation (Fig. 24(a)). Irregular wave amplitudes are simulated with evidence of
frequency coupling (double closely spaced peaks) between xhand r1=2=X1at approximately every Dtof 20.
Especially at these periods, the second order solution deviates from the fully non-linear model. Again the
combined forced tank motion with a prescribed small forcing frequency contains secondary frequencies at
r1=21=X1. This was also found for the pure horizontally forced excited tank (Fig. 9(c)). The discrepancy
between numerical model and second order solution grows when the horizontal forcing frequency is in-
creased to jh¼0:072, as expected, when the non-linear parameter becomes larger. The non-linear effect is
shown in Fig. 24(d)–(f). An additional frequency exists (xhr1=2) compared to the small jhtest due to
non-linear mode to mode interaction, as it can be observed for the larger amplitude of horizontal motion.
Although the mode has a low energy content, as shown in Fig. 24(f), it is responsible for the deviation
between the approximate form and the numerical model. The second order solution is incapable of cap-
turing this non-linear effect. We note the non-linear effects will become enhanced if xh;x1and xvbecome
more closely spaced.
With reference to the pure horizontal forced tank motion shown in Fig. 14, Fig. 25 compares the
small and the large combined tank motion with the frequency ratio of (a) xh=x1¼0:7 and (b)
xh=x1¼1:3 for jv¼0:5; X1¼1:253. The large amplitude solution deviates from the small amplitude
solution in both peaks, troughs and phase. The effect is more pronounced for the high frequency case
Fig. 14(b).
Fig. 25. Free-surface elevation at the left wall in a horizontally and vertically excited tank; jv¼0:5; X1¼1:253. (a) xh=x1¼0:7; - - -,
small amplitude solution (jh¼0:0036); ––, large amplitude solution (jh¼0:036). (b) xh=x1¼1:3; - - -, small amplitude solution
(jh¼0:0036); ––, large amplitude solution (jh¼0:072).
84 J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87
9. Conclusions
Non-linear effects of standing wave motion of liquid in 2-D fixed and forced excited tanks have been
investigated numerically. A fully non-linear inviscid numerical model has been developed based on po-
tential flow theory with the mapped governing equations solved using finite differences.
Results of liquid sloshing induced by harmonic base excitations are presented for small to steep non-
breaking waves. The simulations are limited to a single water depth above the critical depth corresponding
to a tank aspect ratio of hs=b¼0:5. We note that the numerical model is valid for any water depth except
for small depth when viscous effects would become important. Moreover, solutions are limited to steep non-
overturning waves.
First, simulations of sloshing motion in fixed tanks were carried out. The model was validated for
different wave lengths and steepnesses. Good agreement between second order potential theory and the
numerical model has been obtained for small amplitude wave cases. The numerical model captures high
order non-linearities for the steep sloshing cases which was reflected by higher peaks, lower troughs and
period elongation in comparison with second order potential theory. These are typical non-linear effects. It
was found that the third-order single modal solutions compare in general well with the fully non-linear
numerical predictions with almost exact agreement in phase at all times.
Second, sloshing motion in vertically excited tanks were carried out for stable and unstable solutions.
Sloshing effects in a vertically excited tank in stable regions display similar behaviour to free sloshing
motions in a fixed tank when the forcing parameter, jv, is small. This confirms the periodic behaviour of the
small amplitude solution. When jvgrows, the fluid behaviour is no longer perfectly periodic, and so non-
regular amplitudes result, even for the case of small amplitude waves. Non-linear effects complicate the fluid
behaviour further, making it almost unpredictable. However, in stable regions, the solution remains
bounded at all times. Vertical motions produce drastic effects within the instability regions, where para-
metric resonance takes place. In these regions, even small excitations can cause the growth of small initial
perturbations, if the forcing acts on the tank for a sufficiently long time. We also demonstrate examples
when the frequency changes during growing amplitudes (detuning effects). Good agreement between the
third-order and numerical solution is found for the single mode dominant cases. We also showed that the
second-order solution do not capture detuning effects.
Third, analyses were carried out with pure horizontal forced excitation including excitation frequencies
off-resonance, and at resonant frequencies. The resonance in this case occurs at one of the natural sloshing
frequencies. For the large forcing frequency, the third-order solution compares well with the numerical
prediction. This is especially true with regard to the phase. The peaks were slightly underpredicted and the
troughs slightly overestimated compared to the numerical results. Then, vertical forced vibrations were
added and it was found that they significantly effect the resulting combined motion. In the unstable regions
vertical excitation caused fast exponential growth of the waves generated by the small horizontal tank
motion. Some of the test cases showed evidence of detuning effects. For example, a resonance mode
changing into a low frequency mode of oscillations. It was shown that the discrepancy between third-order
and numerical solution was minor for single dominant modes. Although detuning effect can be captured by
the third-order solution, it was found that it does not work so well when modes interact. A multimodal
algorithm should be used [11]. The combined motion test cases revealed results lying within the first and
second instability regions showing a free-surface elevation with the highest growth rate of perturbations in
the first region. It was also found that in addition to the resonant frequency of the pure horizontal exci-
tation, an infinite number of additional resonance frequencies exist due to the combined motion of the tank.
In summary, it was demonstrated that the fully non-linear model provides solutions not obtainable with
the approximate forms. This is especially true for steep waves, high forcing frequency and mode interaction
cases. Each solution was obtained for both small and large amplitudes of horizontal or vertical or combined
excitation in a tank with still water height of 1 m and tank length of 2 m. In general, the small-amplitude
J.B. Frandsen / Journal of Computational Physics 196 (2004) 53–87 85
free-surface predictions compare well with second-order theory, however, the numerical wave tank captures
steep waves generated by large horizontal forcing amplitudes which differ from the third-order approxi-
mation. The maximum value of steepness, which is the measure of the non-linearity of the solution, ob-
tained during the calculations was 0.3. It was found that for the present problem the relatively high
steepness of 0.1 leads to a solution being significantly non-linear. In all cases, high amplitude solutions
produce higher peaks and smaller troughs than small amplitude linear solutions. Furthermore, non-linear
interaction between individual sloshing modes were demonstrated by extra peaks in the power spectrum of
surface elevation, which leads to complicated irregular behaviour of the free-surface. In general, the liquid
sloshing motion exhibits complicated behaviour due to both the horizontal and vertical forced tank motion.
It is shown that vertical excitation causes the instability associated with parametric resonance of the
combined motion for a certain set of frequencies and amplitudes of the vertical motion while the horizontal
motion is related to classical resonance.
The numerical model is simple, computationally quick and accurate. For the cases presented herein there
was no need for free surface smoothing. The present potential flow model provides a simple way of sim-
ulating steep non-breaking waves, that may be readily extended to the prediction of 3-D wave motion.
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