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OCR for page 173
Nonlinear Free Surface Waves Due to a Ship Moving Near
the Critical Speed In a ShaBow Water
H.S. Choi, K.J. Bai, I.W. Kim, I.H. Cho
(Seoul National University, Korea)
ABSTRACT
This paper describes two methods of solution
to the nonlinear freesurface waves generated by a
ship moving steadily with a transcritical speed in a
shallow water. As a mathematical model, a non
linear initial/boundaryvalue problem is formulated
within the scope of potential theory. One method is
based on matched asymptotic expansion techniques
and the Kadomtsev  Petviashvili equation is ob
tained as the leadingorder solution for a slender
ship. The other one is based on classical Hamil
ton's principle and the finite element method is im
plemented for numerical calculations. In order to
examine the effect of the tank width on the wave
field and resulting hydrodynamic forces, computa
tions are made systematically for the Series 60 ship
model with Cb = 0.8 by these two different methods.
For wider tanks, the pressure distribution on the free
surface, equivalent to the ship model, is treated. The
results obtained by two different methods are com
pared each other and with experimental measure
ments available. Also discussed are the appearance
of stem waves at the tank wall and the evolution of
the crestline of diverging waves in a wide tank.
NOMENCLATURE
A
cb
D
Fh
9
J
L
N
NO
Ni
: typical wave amplitude
: block coefficient of ship
: fluid domain
: waterdepthbased Froude number
: gravitational acceleration
: water depth
: functional
: Lagrangian or ship's half length
: total number of nodes
: total number of free surface nodes
: trial function basis
non, nil, no) : outward unit normal vector
p
q
S (x)
SO
SF
SF
sm
SO
t
t*
T.
g
U
W
X,y~z
xc
CY
p
INTRODUCTION
. pressure
: source strength
: longitudinal distribution of cross
sectional area of ship
: blockage coefficient
: free surface
: projection of SF
: maximum cross sectional area
of ship
: ship surface
: time
: final time
: generation period between first
two solitons
: ship's speed
: tank's half width
: rectangular coordinates
: xlocation of the crestline at y = 0
: speed parameter
or unwinding parameter
: blockage index
: slenderness parameter
or variational operator
: nonlinear parameter
: surface elevation
: tank width parameter
: dispersion parameter
: water density
A freesurface flow of an ideal fluid caused by a
ship translating with a constant speed near the shal
low water celerity is described by an initial/ bound
ary value problem governed by the Laplace equation
with the free surface as a part of solution.
In the past, problems of this type were nor
mally treated after the boundary conditions on the
unknown freesurface had been linearized. Recently,
however, there are growing interests in solving the
Hang S. Choi, Kwang J. Bai, Jang W. Kim, Il H. Cho
Department of Naval Architecture, Seoul National University, KwanakKu, Seoul 151742, Korea
173
OCR for page 173
nonlinear Resurface problems more exactly. In
some cases, it is of vital importance since linearized
solutions fail to predict experimentallyidentified phe
nomena. One example is the generation of upstream
advancing solitons by moving disturbances in shal
low water. A comprehensive explanation on the
physics involved is given by Wu A.
There is a line of investigations on this non
linear freesurface problem based on shallow water
approximations which result in a variety of theories
such as the Korteweg de Vries (KdV), Kadomtsev
 Petviashvili (KP), Boussinesq equations and the
Green  Nagdhi formulation (GN); Many references
in this area can be found in Ertekin and Qian [2~.
To name few, Mel and Choi Al, Katsis and Akylas
A, Wu and Wu [5] and Ertekin, Webster & We
hausen t6] considered thre~dimensional problems.
There is another line of approach based on a nu
merical method as finite difference or finite element
methods. Bai, Kim & Kim [7] were the first who ap
plied the finite element method to a 3dimensional
nonlinear shallow water wave problem.
In the present paper, we concern with theo
retical and numerical methods for solving a nonlin
ear threedimensional Resurface flow problem in
a shallow water. Specifically, a ship moving near
the critical speed is treated to numerically simulate
the experimental condition in the towing tank. It
is formulated as an initial/boundary value problem
within the scope of potential theory. As the solu
tion procedure for the nonlinear problem, two dif
ferent methods are described herein. In the first
method, the given problem is reduced to a homo
geneous KP equation with flux conditions on the
boundaries. The ship is simplified to an equivalent
slender body. Then the KP equation is numerically
solved in the twodimensional horizontal Resurface
plane by an explicit finite difference scheme. In the
second method, the original problem is replaced by
an equivalent variational problem based on Hamil
ton's principle applied to water waves derived by
Miles [8~. Then the variational functional defined as
an integral in the unknown threedimensional fluid
domain is solved numerically by the finite element
method. The variational functional used here is
basically the same as the wellknown Luke's vari
ational principle A. However, the present func
tional is more advantageous in numerical computa
tions compared to Luke's principle.
Recently, these two methods have been suc
cessfully applied to the generation and emission of
solitons in the upstream and complicated waves in
the downstream due to a moving ship in shallow wa
ter [7,10~. In these papers, however, no systematic
investigations on the effect of the side walls have
been undertaken. In the present study, it is our
intention to clarify the effect of the width of the
side walls on the wave response and hydrodynamic
forces. Thus the numerical results of the free sur
face elevations, hydrodynamic forces (i.e. wave resis
tance, lift and trimming moment) acting on a ship
obtained from the both methods are presented and
compared partially with the experimental findings
of Ertekin [11~. The formation and development of
stem waves is illustrated, when the generated waves
are reflected at the tank wall. Also discussed are
the evolutions of the crestline of diverging waves in
a quite wide tank.
I N I T I A L / B O U N D A R Y  V A L U E
FORMULATION
We consider a ship advancing steadily with a
transcritical speed U along the centerline of a shal
low tank. A rectangular Cartesian coordinate sys
tem moving with the ship's speed U is used, in which
the xaxis coincides with the longitudinal axis of the
ship and the z=0 plane is the undisturbed free sur
face. The ship directs toward the negative xaxis
and the positive zaxis points upward (see Fig.1~.
Under the usual assumptions in potential theory,
fluid motions are expressed in terms of a velocity
potential, ¢(x,y,z,t), which is the solution of the
Laplace equation
V2¢ = 0 h < z ~ ~ (1)
in the fluid domain D, where h and ~ are the wa
ter depth and the freesurface elevation, respectively.
The kinematic condition is imposed on the ship sur
face So
inUn2 ~
(2)
where n = (no, no, no) denotes the outward unit nor
mal vector on So. No net flux condition also holds
at the tank bottom and side walls
(z=0 z=h,
+~=0 y=+W.
(3)
(4)
The kinematic and dynamic boundary condi
tions on the free surface SF must be satisfied
id = (t+(U+¢z)(z+~(v , (5)
9~+¢t+U¢z+ 2 1 Vat 12 +P =0 , (6)
where 9 is referred to the gravity constant, p to
the fluid density, and p = p(x,y, t) to the pressure,
which is taken zero when the pressure distribution
on SF is absent.
By assuming that the fluid is initially at rest,
the initial condition may be given as
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¢=¢=0 at t=0, (7)
and the radiation condition yields to
¢) ~ O as X2 + y2 ~ oo (8)
It is to note that a modified radiation condition is
utilized in computations for the downstream bound
ary.
METHODS OF SOLUTION
Since the concept of the two methods employed
here is quite different by their nature, a brief descrip
tion on each method is necessary. We begin with the
theoretical part,then the numerical part follows.
Matched Asymptotic Expansion Technique
In order to analyze the above nonlinear prob
lem, further assumptions and limitations are required
The first step is to introduce appropriate smallness
parameters, with respect to which the expressions
given in the previous section are to be perturbed.
Hereby we define two small parameters
~ = A/h' ~h/L (~9~)
and assume ~ = p2, where A and 2L mean the typi
cal wave amplitude and the ship length, respectively.
It corresponds to the Ursell number of order of unity,
which implies that the nonlinearity and the disper
sion are both important to the leadingorder solu
tion, i.e. we are dealing with a weakly nonlinear
dispersive wave system. However, it may not be
a serious restriction because the above assumption
seems to be valid in a wide range of Ursell numbers
as shown by Lee, Yates & Wu t12~. Since we are in
terested in the ship's speed in the neighborhood of
the critical Froude number, it is expanded as follows:
Fh = 1  2ap2 with ~ = 0~1) . (10)
In order to include the lateral dispersion as well
as the longitudinal dispersion, we have to choose a
wide tank in comparison with the ship length
W/L = 1/~n with ~ = 0~1) . (11)
It is in general recognized that the governing
parameter of the problem is the blockage coefficient,
which is simply the area ratio of the midship to the
tank crosssection. As pointed out by Mel t13], the
order of magnitude of the blockage coefficient must
be O(p ~
SB = Sm/2Wh = 0~4), (12)
where Sm is the maximum crosssectional area of a
ship. If we assume the ship to be slender, of which
the characteristic transverse dimension is denoted
by RO, then the slenderness parameter becomes
~ = Ro/L = o(~2) . (13)
It indicates that the nonlinearity arises directly from
the disturbance caused by a slender ship.
As a result, we have four characteristic lengths
in this problem; water depth (h), tank width (2W),
ship's length (2L) and transverse length (RO), which
have vastly different scales each other.
h/L = 0~), W/L = 0~~~), RO/L = 0~2~. (14)
To accommodate these in our analysis in a consis
tent manner, it is adequate to divide the fluid do
main into three regions; near the ship, far from the
ship and an intermediate region therebetween. The
procedure of the derivation has been reported in de
tail in t10~. Hence we cite here only the results.
In the far field, the geometry of the tank affects
the propagation of waves, but the generation mech
anism of the waves is not known. The wave field is
described by a homogeneous twodimensional KdV
or KP equation [14]
2 6(ZZ2 + 20 / ~VdX . (15)
It shows a balanced interplay between the nonlinear
and the twodimensional dispersion. It is a three
dimensional counterpart of the KdV equation, be
cause it contains the lateral dispersion as well as the
longitudinal dispersion. In the above, the variables
are made dimensionless by
X = LX~, YWY~, ~ = A(', t = 2~ ~ (16)
For the sake of brevity, the primes are dropped here
after.
In the near field, i.e. in the flow region closely
around the ship, the kinematic boundary condition
on the ship surface should be invoked. For a slender
body, Eq.~2) can be replaced by
l~n ~(U + ~Z)R2~1 + (R`/R)2~~12 ~ (17)
where the normal derivative on the ship surface is
approximated by that on its transverse plane. Here
RO stands for the circumferential derivative of the
cross section. The presence of a ship and its mo
tion can be represented by source distributions. By
applying the law of mass conservation to a fluid do
main surrounded by the ship surface, the free surface
and a control surface located far away from the ship,
but still within the near field, the source strength is
readily determined
q= 2pSzfx) With '= _SB4, (18)
where S(x) is the longitudinal distribution of the
cross sectional area of ship.
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In the intermediate region, solutions of the far
and the near fields are matched. As a result, the
boundary condition for v at y = 0 turns out to be
IVY=, 0, r) =  12,BSzztx) . (19)
And we have the leadingorder hydrodynamic pres
sure
P = /`,2psh; + °(113) . (20)
The wave response can be computed by the
KP equation given in Eq.(15) with the boundary
conditions given in Eq.(4) and Eq.(19). The hydro
dynamic forces and moment can be estimated based
on the slender body approximation A. To do it,
a simple explicit finite difference scheme is imple
mented for the KP equation, in which forward dif
ferences are chosen for time derivatives and central
differences for spatial derivatives. But at the wall
and the centerline of the tank, onesided differences
are used in order to incorporate with boundary con
ditions. Unidirectional Sommerfeldtype radiation
conditions are imposed on both open boundaries.
Neverthless, a relatively large computation domain
ahead of the ship is provided to avoid numerically
reflected waves from the open boundary. Then the
computation domain is gradually enlarged in both
directions as the computation proceeds. Based on
numerical experience, the grid size and time incre
ment are chosen as
Ax = 0.1, I\r = 0.00002
and Ay is so taken as the ratio of Ax/~\y remains
unity in the physical plane for better resolution of
dispersion.
The Series 60 with Cb = 0.8 is numerically
modelled in terms of the longitudinal distribution
of its crosssectional area. However, the portion of
both ends has been slightly modified by a parabolic
distribution in order to satisfy the slender body as
sumption.
Finite Element Method
The finite element method has been success
fully applied to nonlinear waterwave problems, for
example, Washizu et al.~15], Ikegawa t16], Nakayama
& Washizu t17], Washizu & Nakayama [18], Betts &
Assaat [19], Bai, Kim & Kim t7] (hereafter referred
to as BKK), Bai, Kim & Lee t20], Bai, Kim & Lee
t21] and Kim & Bai t22~. The finite element method
is based on Luke's variational principle in [16~~19~.
However, in the present paper, the variational func
tional given in Miles [8] is used as the basis of the
finite element computations. This variational form
is simply a direct application of the classical Hamil
ton's principle to the nonlinear waterwave problem.
For the problem at hand we can define the functional
J and the Lagrangian L as follows:
rt*
JO L aft, (~21)
/SF ¢) (by + U(Z~) dSU /s n2 ~ dS
2 /D ~ ¢ dV 2 / ~ dS r`22')
 / P ~ dS,
p so
whereSF is the projection of SF on the Oxy plane
and t* is the final time. ~ denotes the velocity poten
tial on the free surface, i.e. ¢(x, y, t) = flex, y, a, t).
By taking the variations on J with respect to
the unknown functions, ~ and ¢, we obtain
/SF{(¢' 6~)~=~*  (g) [~)~=0) dS (~23)
/0 /sp (~t + USA
+2~V¢~2 + gz + P)Z=! Use dS aft,
/o ~
/SF (O + U<2  n in) bI dS
 /S bin+ Unz)[if dS
+ /D V2¢ {¢' den.
(24)
Here [J = [Jo + [Jo,. Equation (,23`J shows that
the dynamic freesurface boundary condition is re
covered from the stationary condition on J for the
variation of ~ at each time step. The wave eleva
tions at t = 0 and t* are supposed to be specified as
the constraints. Equation (24) shows that the kine
matic condition on SF and the governing equation in
the fluid domain are recovered from the stationary
condition on J for the variation of ¢.
In the numerical procedure for the applica
tion of the finite element method, we discretize the
fluid domain into a number of finite elements. Then
we approximate ~ in Ndimensional function space
whose basis is continuous in D. We denote the basis
of this trial space by {NiJ`=,,...,N. It is convenient to
introduce another set of basis function, denoted by
{Mk~k=l,...,NF, which is defined only on the free sur
face. By the introduction of these basis functions,
one can represent ¢, ~ and ~ as
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+(x, y, z, t) = ~i(t)Ni(=, Y. a; a), (25)
i(x, y, t) = (k(t)Mk(X, a), (26)
((x, y, i) = (k(t)Mk(X, A), (27)
where
Mi(x, y) = Nit (x, y, z; a) A=, (28)
k= 1,...,NF.
Here NF is the total number of nodes on SF and it is
the nodal number of the basis function N`, of which
the node coincides with that of the freesurface node
k. Summation conventions for the repeated indices
are used here. It should be noted that the basis
function {Ni Ji=t,...,N is dependent on the freesurface
shape z = `(x,y,t) but its restricition on SF is the
function of (x, y) and independent of <. This special
property of {Mk~k=~, ,NF is maintained here since
new nodal points in D are shifted only along the
zaxis at each time step.
The tensors Kit, Pi' are the kinetic and po
tential energy tensors and Tk' is the tensor obtained
from the freesurface integral, which can be inter
preted as a tensor related to the transfer rate be
tween these two kinds of energy. It is of interest
to note that in Eq.(29), Pi' = gTki. However, Tk'
will be defined differently from this in the present
computation by introducing the lumping scheme.
The stationary condition on J = r Ldt is equiv
alent to the following EulerLagrange equation
Tii dt ¢` =  UCi`~' (30)
 PiJO Pi! ~Pi ~
Tk'~ =  UCi`0 (31)
Once the trial function is represented by using dt
the above basis function, the Lagrangian L can be + K`,~jIj + A,,
written as
Kiwi =  fi for i ~ it (32)
L = ~kTk! dt ~ + U¢kCk' ~(29)
 2¢`K`i~ifists`
 2 (kPkI ~Pl 0,
where
Tk! = ~ MkM' dS,
Ci`=; MidS,
SF [3X
Kij = / VNi VNj dV,
Pal = 9 JO MkM' dS,
SF
Here Eq.(30) and Eq.(31) are the nonlinear ordi
nary differential equation for Ink, ~k}k=l,,NF and
Eq.(32) is the algebraic equation for {¢i}iii,t which
is the constraint for the above two equations. Here
it should be noted that the second term on the right
hand side of Eq.(31) is computed by the volume in
tegral as originally defined, whereas BKK used the
surface integral reduced from the original volume in
tegral. This change is made in the present work since
the previous computation in BKK is found to be less
accurate in the conservation of energy compared to
the present scheme from our numerical test.
Eq.(30) through Eq.(32) are less advantageous
in computations with respect to the numerical sta
bility. To remedy this difficulty, we introduce the
unwinding and local lumping schemes, which are
often used in a wide class of computational fluid
dynamics. Following these common steps, we ob
tain the final set of the reduced ordinary differential
equations as follows:
fi = Uis ri~Ni dS, d ¢i = _ UT~lCm`~' (33)
T1 (1~', Kits, + p + )
Pi = p is P(~, y, t)Mi d S. dd ~ = _ UT.l Cm`~` (34)
+ Tkm (6imjij + film) ~
177
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Kijjj =  fi for i := id,
where
(35)
Tk' = ~ (Mi + ~= ~ Mi) M: dS
(36)
ISF ( 2 a=Mi) COME dS, (37)
{omits' ,k=l;
Tki= ~
~ 0 ~ otherwise.
(38)
Here car is the upwinding parameter as defined in
Hughes & Brooks t234. In Equations (33) and (34), a
consistent upwinding scheme (also known as Petrov
Galerkin method) is employed as discussed in t23],
whereas an inconsistent upwinding scheme was used
in BKK. We leave out the detail procedures here
since one can find them in BKK.
In the finite element computations, two numer
ical models are treated, i.e. a Series 60 ship model
with block coefficient Cb=0.8 and a pressure patch
on the free surface equivalent to the above ship model
To simplify the finiteelement grid generation, the
Series 60 ship model is replaced by an vertical wall
sided ship which has the same crosssection area and
the constant draft of the original model along the
ship length. This means that the equivalent numer
ical ship model has rectangular cross section with a
constant draft. The finite element subdivision un
der the ship's bottom is unchanged while the other
finite element subdivision is changed at each time
step to accomodate the new location of the free sur
face. An eightnode isoparametric element is used
and the integration is carried out analytically along
the vertical direction.
In the computations for the pressure patch on
the free surface, a single finite element is taken along
the depth with a higherorder polynomial basis func
tion which satisfies the bottom condition. In this
case the integration along the vertical direction is
also carried out analytically. This is the socalled
Aversion while the former case is the inversion in
the adaptive finite element method.
NUMERICAL RESULTS & DISCUSSIONS
To simulate the tank tests t11], the Series 60
ship with Cb = 0.8 is chosen as the numerical model.
Its length, beam and draft are 1.52m, 0.23m and
0.075m respectively. The water depth is 0.15m. The
tank widths are 1.22m, 2.44m, 4.88m. In addition
to these, a much wider tank is considered to exam
ine the effect of the tank width on the formation
and propagation of upstream waves by the method
based on the KP equation. But in the finite ele
ment computations, an equivalent pressure patch is
treated by increasing the width up to 480 times the
water depth, since this is easier to compute than a
ship model.
Throughout the computations, the motion is
assumed to start with a prescribed constant speed
as a step function. In presenting the two sets of
computed results, we denote those obtained by the
KP equation with a slender body approximation by
KP, and those of the finite element method by FEM.
The physical quantities with dimensions are shown
as functions of the real time for the convenience
to compare with the earlier experimental work t11~.
Accordingly the wave resistance and the lift force are
given in Newton and the moment in Newtonmeter.
The numerical results of the forces and mo
ment for the experimental condition with the tank
width of 2.44 m are shown in Fig.2 through Fig.5.
In these figures, the solid line corresponds to FEM,
while the dotted line to KP, if not indicated other
wise. Fig.2a2c display the time histories of the wave
resistance for the depth Froude numbers Fh = 0~9,
1.0, and 1.1, respectively. For Fh = 0.9, the com
puted values by both methods coincide fairly well
after 5 seconds. However, the discrepancies are con
siderable for Fh=1.0 and 1.1. It is of interest to
note that the magnitude of the oscillatory behav
ior becomes large in the case of FEM for Fh = 1.1.
To investigate this rather large discrepancy, two dif
ferent expressions for the pressure are used for the
computation of forces and moment in FEM; the ex
act Bernoulli's equation and linearized Bernoulli's
equation based on the linear shallow water approxi
mation. These two sets of computations are given in
Fig. 3a3c, where the solid lines indicate the exact
form and the dotted lines the linear approximation.
These figures explain partly the source of the dis
crepancies observed in Fig. 2b and 2c, since the KP
equation is an approximate solution to the problem.
Furthermore, the ship has been assumed to be slen
der and only the leadingorder pressure has been
taken into account. It is to mention that the both
methods give greater wave resistance than the ex
perimental measurement in all the cases studied.
Fig. 4a4c are the computed results of the
lift force for Fh = 0.9, 1.0, and 1.1. Contrary to
the wave resistance discussed above, the lift forces
by FEM are consistently smaller than those by KP.
FEM predicts a longer oscillatory period than KP.
Presumably this is due to the difference in the dis
persion in the two methods.
In the similar fashion, the moment with re
spect to the origin is illustrated in Fig.5aSc. The
moment estimated by KP is almost twice as large
as that by FEM for Fh = 0.9 in Fig 5a. The results
of KP contain considerable oscillatory components,
whereas those obtained by FEM are nearly constant
178
OCR for page 173
after 5 seconds.
The time histories of the forces and moment
at the critical speed for the tank widths of 1.22m,
2.44m and 4.88m, denoted by 1, 2, 3 in this order,
are depicted in Fig.6 through Fig.8. The wave resis
tance is given in Fig.6, and the lift force and moment
are in Fig.7 and Fig.8, respectively. The results ob
tained by KP are indicated by (a) and FEM by (b) in
these figures. As the tank width becomes larger,the
wave resistance and the moment decrease and they
seem to converge to some limit values. But the lift
increases as the tank width increases.
The evolutions of the surface elevation evalu
ated at 90cm ahead of the bow (Gauge No. 3 in
t11~) for the tank width of 2.44m are shown in Fig.9
ac. The asterisk marks correspond to the experi
mental measurements, but the starting time is only
of qualitative meaning due to the different nature of
the initial conditions in the computations and the
experiments. For the subcritical speed, Fh = 0~9,
the measured profile looks closer to FEM, but the
trend is opposed for Ah = 1.0 and 1.1. Over the
three speeds, the generation period of the upstream
solitons is shorter in the results calculated by KP
compared to those by FEM. It is observed that the
mean water level is slightly higher in KP for Fh =
0.9.
In Table 1, the amplitude and propagation speed
of the first soliton, and the generation period be
tween first two solitons are listed together with the
experimental measurements. Here the tank widths
are indicated by the ratios to the ship length, namely
W/L = 0.8, 1.6 and 3.2 for 2W = 1.22m, 2.44m and
4.88m, respectively. The amplitude measured by
experiments is the smallest among three for W/L
= 0.8, but it is somewhat between the two com
puted results for W/L = 1.6 and 3.2. It is inter
esting to note that KP predicts consistently higher
amplitudes for all cases. For the propagation speed,
the overall behaviour is similar as in the amplitude.
But there is no clearcut trend in the case of the
generation period. The period measured in the ex
periment is longer than the numerical predictions
for Fh = 0.9, whereas the result obtained by FEM
is the longest for Fh = 1.1.
Based on his systematic experiments, Ertekin
t11] concluded that the characteristics of upstream
advancing solitons are determined primarily by the
blockage coefficient and the detailed geometry of
disturbances is of secondary importance. Recently
Pedersen [24] and Ertekin & Qian [2] investigated
the influence of parameters other than the blockage
coefficient on the generation mechanism of solitons.
Along this line, we carried out additional computa
tions by KP for a slender ship with a parabolic dis
tribution of cross sectional area (Cb = 0.667) with
the same blockage coefficient as the Series 60 ship
(Cb = 0.8~. The result for W/L = 1.6 is given in Ta
ble 2. Comparing it to the corresponding result for
the Series 60 ship in Table 1, we can recognize that
the amplitude and the period have been remarkably
changed. The amplitude is decreased by about 0.06
times the water depth and the period is increased by
about 20% for all three speeds considered. We may
conjecture that the hullform, which may be properly
represented in terms of ship's block coefficient, plays
a significant role on the generation of solitons.
Fig.10 and Fig.11 are the snapshots of wave
contour around the ship in a tank of width 4.88m at
the critical speed. Due to the different nondimen
sionalization, the time instances are slightly shifted
in two figures. The solid lines represent a constant
positive surface elevation and the dotted lines denote
a negative surface level. Two adjacent lines differ
the surface elevation by 0.04 times the water depth.
The wave contours obtained by KP looks more com
plicated and upstream waves propagate faster than
those obtained by FEM. It is partly due to the dif
ference in the propagation speed, as shown in Table
1 (b). In these figures, we can observe the formation
of stem waves at the wall, when the waves generated
from the bow are reflected there. The stem waves
are further developed, as time elapses. It suggests
that the formation of straight crestlines is associated
with the stem waves, which supports the conclusion
made by Pedersen t24~.
The perspective views of the wave fields for
the above case are illustrated in Fig.12 for KP and
Fig.13 for FEM, respectively. Cautions should be
paid that the vertical displacement is exaggerated by
5 times compared to the horizontal scales. The up
stream solitons and threedimensional downstream
waves are clearly shown.
Fig.14 shows the wave resistance computed by
FEM by systematically increasing the tank width
for the pressure patch. In this case the pressure
distribution is specified to have the same blockage
coefficient as the ship model in the earlier exper
imental condition. The pressure is assumed by a
trapezoial distribution in both x and y directions
and constant along the length of the parallel middle
body in the xdirection. The length of the pressure
patch is taken to be same as the ship length. Along
the yaxis the pressure distribution is assumed to
be constant along 0.8 times the water depth and
changes linearly to zero at the edges of the patch.
The width of the patch is taken 2.4 times the wa
ter depth. The number indicated to the lines cor
responds to the tank, whose width is consecutively
doubled starting from 2W = 1.22m upto 5. The line
6 is the case for the tank width of 72m.
The oscillatory component in the
179
OCR for page 173
wave resistance is pronounced for small tank width,
whereas it becomes insignificant as the tank width
becomes very large. However, the mean value of the
wave resistance remains nearly constant after 5 sec
onds. It is to note that the values of wave resistance
for the pressure patch are smaller than those for the
ship (see Fig.6b).
For the above pressure patch, the computation
domain is continuously enlarged up to 2W/h = 480.
The maximum tank width treated here may be re
garded as a case of infinite width at that time. Be
cause the tank width is kept sufficiently large by in
creasing it at every time step so that the disturbance
near the side walls is not felt in the computations.
Fig.15 is threedimensional wave profile at Ut/h =
320 obtained from FEM. It is to observe that two
diverging waves have already advanced upstream.
The first crestline is plotted on a logarithmic
scale in Fig.16, where xc is referred to the x loca
tion of the crestline at the centerline of the tank.
Although the slope in this scale varies slightly with
time, it is approximately 0.5. It suggests that the
crestline is almost a parabola, which was also dis
cussed by Redekopp t25] and Lee & Grimshawi26~.
ACKNOWLEDGEMENTS
This work has been supported by the Korean
Science & Engineering Foundation under the Non
linear Ship Hydrodynamics Program, Grant No.
87020703.
REFERENCES
t1 ~ Wu, T.Y., Generation of UpstreamAdvancing
Solitons by Moving Disturbances, Journal of
Fluid Mechanics, Vol.184, 1987, pp.7599.
t2 ~ Ertekin, R.C. and Qian, Z.M., Numerical
Grid Generation and Upstream Waves for Ships
Moving in Restricted Waters, Proceedings of
the 5th International Conference on Numeri
cal Ship Hydrodynamics, 1989, pp.421437.
t3 ~ Mei, C.C. and Choi, H.S., Forces on a Slen
der Ship Advancing Near the Critical Speed
in a Wide Canal, Journal of Fluid Mechanics,
Vol. 179, 1987, pp.5~76.
t4 ~ Katsis, C. and Akylas, T.R., On the Exci
tation of Long Nonlinear Water Waves by a
Moving Pressure Distribution.Part 2: Three
Dimensional Effects, Journal of Fluid Mechan
ics, Vol.177, 1987, pp.4965.
t5 ~ Wu, D.M. and Wu, T.Y., Precursor Soli
tons Generated by Three Dimensional Dis
turbances Moving in a Channel, Proceedings
of IUTAM Symposium on Nonlinear Water
Waves, 1987, pp.69 76.
to ~ Ertekin, R.C., Webster, W.C. and Wehausen,
J.V., Waves Caused by a Moving Disturbance
in a Shallow Channel of Finite Width, Jour
nal of Fluid Mechanics, Vol.169, 1986, pp.275
292.
t7 ~ Bai, K.J., Kim, J.W. and Kim, Y.H., Nu
merical Computations for a Nonlinear Free
Surface Flow Problem, Proceedings of the 5th
International Conference on Numerical Ship
Hydrodynamics,1989 pp.40~420.
t8 ~ Miles, J.W., On Hamilton's Principle for sur
face waves,J. Fluid Mech., 83, pp. 395387.
t9 ~ Luke, J.C., A Vatriational Principle for a
Fluid with a Free Surface, Journal of Fluid
Mechanics, Vol.27, 1967, pp.39~397.
t10 ~ Choi, H.S. and Mei, C.C., Wave Resistance
and Squat of a Slender Ship Moving Near the
Critical Speed in Restricted Water, Proceed
ings of the 5th International Conference on
Numerical Ship Hydrodynamics, 1989, pp.43
454.
t11 ~ Ertekin, R.C., Soliton Generation by Mov
ing Disturbances in Shallow Water: Theory,
Computation and Experiment, Ph.D. Thesis,
University of California, Berkeley, 1984.
t12 ~ Lee, S.J., Yates, G.T. and Wu, T.Y., Experi
ments and Analysis of Upstream Advancing
Solitary Waves Generated by Moving Distur
bances, Journal of Fluid Mechanics, Vol.199,
1989, pp.56~593.
t13 ~ Mei, C.C., Radiation of Solitons by Slender
Bodies Advancing in a Shallow Channel, Jour
nal of Fluid Mechanics, Vol.162, 1986, pp.53
67.
t14 ~ Kadomtsev, B.B. and Petviashvili, V.I., On
the Stability of Solitary Waves in Weakly Dis
persive Media, Soviet Physics DOKLADY,
Vol.15, NO.6, 1970, pp.53~541.
t15 ~ Washizu,K., Nakayama, T. and Ikegawa, M.,
Application of Finite Element Method to Some
Free Furface Fluid Problems, Finite Elements
in Water Resources, Pentech Press, London,
1977, pp.4.2474.246.
t16 ~ Ikegawa,M., 'Finite Element Analysis of Fluid
Motion in a Container, Finite Element Meth
180
OCR for page 173
oafs in Flow Problems, UAH Press, Alabama,
1974, pp.737738.
t17 ~ Nakayama, T. and Washizu, K., Nonlinear
Analysis of Liquid Motin in a Container Sub
jected to Forced Pitching Oscillation, Interna
tional Journal for Numerical Methods in En
gineering, Vol.15, 1980, pp.12071220. t23
t18 ~ Washizu, K., Nal~ayama, T., Ikegawa, M.,
Tanaka, Y. and Adachi, T., Some Finite Ele
ment Techniques for the Analysis of Nonlinear
Sloshing Problems, Finite Elements in Fluids
Vol.5, John Wiley & Sons, 1984, pp.357376.
t19 ~ Betts, P.L. and Assaat,M.I., Larg  Amplitude
Water Waves, Finite Elements in Fluids, Vol.4,
John Wiley & Sons, 1982, pp.10~127.
t20 ~ Bai, K.J., Kim, J.W. and Lee, H.S., A
Numerical Radiation Condition for Two Di
mensional Steady Waves, Proc. Workshop on
Nonlinear Mechanics, Korea Soc. Theoretical
and Applied Mechanics, Seoul, Korea, 1990,
pp.11~132.
t21 ~ Bai, K.J., Kim, J.W. and Lee, H.S., An
Application of the Finite Element Method to
a Nonlinear Free Sruface Flow Problem, The
,/_ y = W ~
\ ~
tax \ a}
(~////////////////////////////////////~////////~.2
~ y=W ~
elf ~
z = h
Fig.1 Definition Sketch

z

10
O ~I ~I
0 5 10 15
Time (See)
Second World Congress on Computational ME
chanics, Stutgart, Germany, 1990.
t22 ~ Kim, J.W. and Bai, K.J., A note on Hamil
ton's Principle for a Free Surface Flow Prob
lem, Journal of Society of Naval Architects of
Korea (in Korean), 1990, (in print).
~ Hughes, T. J. R. and Brooks, A. A Them
retical Framework for PetrovGalerkin Meth
ods with Discontinuous Weighting Functions:
Application to the Streamlin  Upwind Proce
dure, Finite Elements in Fluids, Vol. 4, John
Wiley & Sons, 1982, pp. 4765.
t24 ~ Pedersen, G., Thre  Dimensional Wave Pat
terns Generated by Moving Disturbances at
Uanscritical Speeds, Journal of Fluid Mechan
ics, Vol.196, 1988,pp.3963.
t25 ~ Redekopp,L.G.,Similarity Solutions of Some
Tw  Spac  Dimensional Nonlinear Wave Evm
lution Equations, Studies in Applied Mathe
matics, Vol.63, 1980, pp.185207.
t26 ~ Lee, S.J. and Grimshaw, H.J., Upstream
Advancing Waves Generated by Thre  Dimen
sional Moving Disturbances, Physics of Fluid
A2~2),1990, pp.l94201.
TV ~
1.'
11
l
v
o
(a) Fh= 0.9
· I n
20 25 v
181
5
1 1 1
10 15
Time (See)
(b) Fh= 1.0
1 1
25
/
I ~ 1 ~i ~I
10 15 20 25
Time (See)
(C) Fh= 1.1
Fig.2 Wave Resistance (2W = 2.44m)
: FEM,: KP
OCR for page 173
1
10
10
80
10 15 20 2'5 0
Time (See) O
(a) Fh= 0.9
0 5
5 10 15 20 25
Time (See)
(a) Fh= 0.9
One
1
10 15
Time (See)
(b) Fh= 1.0
, 1 ~1 O
20 25
60
~/ ~Z20
~ 0d
2
10 15 20 25
Time (See)
(c) Fh= 1.1
Fig.3 Exact and Linear Approximate Wave Resis
tances by FEM (2W = 2.44m).
: Exact,: Linear ap
proximation
'1 , 1 1
0 5
1 1 1
10 15
Time (See)
(b) Fh= 1.0
, I , I
20 25
, ~
/ \ ~ it: :
1 ' 1 1 1
5 10 15
Time (SecJ
(C) Ah = 1.1
Fig.4 Lift Force (2W = 2.44m)
: FEM,: KP
182
1 1 1
20 25
OCR for page 173

z;

¢1~
o
o
~ 1~
o
u
~ l
r~
 
 ~
~1
o
~ 
5 10 15
Time (See)
(b) Fh= 1.0
1 1  1
20 25
~' 1 1 1 ' 1 ' I ' 1
0 5 10 15 20 25
Time (See)
(c) Fh= 1.1
Fig.5 Trirruning Moment (2W = 2.44m)
: FEM,: KP
~_
,
81~ ,'' '''.
V\,
' ~I ~, , I , I , . I
( ) 5 10 15 20 25
Time (See)
(a) Fh= 0~9 ~}
, ~
~.
~3
10 15 20 25
Time (See
(a) KP
¢1~
l
~h = 1.0
' 1
0 5 10 15 20 25
Time (See)
(b) FEM
Fig.6 Wave Resistances for Three Different Tank
Widths ~ 1: 1.22m, 2: 2.44m, 3:
4.88m ) at the Critical Speed
R(h
_
60

40
2
l
/ ' /~{ ' / ' ~ 't ~' ~ ~'
Fh = 1.0
l
0 5 10 15
Time ( Sec
(a) KP
' 1 1
20 25
60
~' \ ' ~
1 1
5
o

1 ' 1 ' 1 ' 1
10 15 20 25
Time (See)
(b) FEM
Fig.7 Lift Forces for Three Different Tank Widths
~ 1: 1.22m, 2: 2.44m, 3: 4.88m
at the Critical Speed
183
OCR for page 173
~ 1~
o
1~ ~
r _
o
0~
_ _
~1~
o
~
~\~/~ 1 c,$
~_ ~ 2 =8
~4
1 1 1 1 1 ' 1
5 10 15 20 25
Time (See)
(a) KP
g~>V~~N
5 10 15
Tirne (See)
(b) FEM
· ~
20 25
Fig.8 1Yimming Moment for Three Different Tank
Widths ~ 1: 1.22m, 2: 2.44m, 3:
4.88m ) at the Critical Speed
~_
v

~8
o
.. 
ct
_ 4 , ~ ~
_`
v
~c
o
. 
o4
p
~C
~ )*' ~ 10 1~s ~D O.
1 5 10 15 20 25
Time (See)
(a) Fh= 0.9
'''''
0 5 10 15 20 25
Time (See)
(b) Fh= 1.0
~0
. ,\ ~ *
~ *'7 0~ ~'
*********5 10 15  20
Time (See)
(c) Fh= 1.1
Fig.9 EreeSurface Elevation at the Guage (9Ocm
Ahead of the Bow) : FEM,
: KP, *******: EXP
6
~'
20 10 0 10
o
(a) Ut/h = 19.65
2n 1n
1C
(b) Ut/h = 39.3
Fig.10 Wave Contour at the Critical Speed (KP, 2W
= 4.88m)
184
OCR for page 173
(a) Ut/h = 20.0
20 10 0 10
I ~ ~/~ ! i ,if ~
(b) Ut/h = 40.0
Fig.ll Wave Contour at the Critical Speed (FEM,
2W = 4.88m)
Ut/h = 78.6
R
185
Ut/h = 157.2
Ut/h = 196.5
Fig.12 Wave Evolution at the Critical Speed (KP, 2W
4.88m)
OCR for page 173
Ut/h = 40.n
Ut/h = Born
Ut/h = 120.0
~ 1~
¢1~
Z 1: ~ 7;.\_/ , an. _ 3
Fh = 1.0
1 1 1 1 ' 1
10 15
Time (See)
(Y ' ' ' ' ' ! ' ~' ' ' 1 ' 1
0 5 10 15 20 25 30 35 40 45 50
Time (See)
2
20 25
Fig.14 Wave Resistance for Various Tank Widths with
Ut/h = 160.0 Pressure Patch (Fh = 1.0)
Ut/h = 200.0
Fig.13 Wave Evolution at the Critical Speed (FEM,
2W = 4.88m)
186
OCR for page 173
 ~ ~ ~
~ ~ 
Fig.15 Wave Field for ~ Wide Tank (Ut/h320)
5
4
~3
a
2
1
.J~ I/
~ An'
*: ~
*
*/ *****t = 5.0
***** t  19.8
*~* t  39.6
Sec
Sec
Sec
_ ~ ~ ~ r ~ I ~ l ' I " " " ~ I ~ T I I I I l l l l r I I I I I ~ I ~ I I I I I ~ I ~ i I
1 O 1 2 3 4
Log (xxc)
Fig.16 Plot of the Crestline of the First Upstream
Diverging Wave
187
OCR for page 173
Table 1 The Amplitude and Speed of the First Soli
ton, and the Generation Period Between First
Two Solitons for Series 60 (Cb = 0.8)
(a) Amplitude
A/h W/L=0.8 W/L= I.6 W/L=3.2
KP .480 .315 .202
Fh = 0.9 FEM .384 .234 .134
EXP .367 .273 .143
KP .623 .445 .322
Fh = 1.0 FEM .566 .397 .285
EXP .551 .438 .303
KP .785 .625 .490
Fh = 1.1 FEM .686 .566 .475
EXP .608 .585 .480
(b) Speed
C/ ~W/L=0.8 W/L=1.6 W/L=3.2
KP 1.218 ~1.155 1.078
Fh = 0.9 FEM 1.175 1.100 1.050
EXP 1.170 1.100 1.060
KP 1.293 1.216 1.153
Fh = 1.0 FEM 1.250 1.175 1.125
EXP 1.240 1.190 1.130
KP 1.367 1.304 1.227
Fh = 1.1 FEM 1.300 1.250 1.200
EXP 1.280 1.260 1.210
(c) Generation Period
UT,/b W/L=0.8 W/L= 1.6 W/L=3.2
KP 20.04 31.83 41.26
Fh = 0.9 FEM 30.20 37.40 53.60
EXP 32.70 48.10 65.10
KP 24.89 39.29 60.26
Fh = 1.0 FEM 35.40 47.30 88.40
EXP 37.80 49.80 85.20
KP 33.14 54.75 90.78
Fh = 1.1 FEM 44.20 57.40 128.70
EXP 39.00 50.11 103.60
Table 2 The Amplitude and Speed of the First Soli
ton, and the Generation Period Between First
Two Solitons for a Slender Ship (Cb = 0.667)
A/h
C/~fik
UT9/h
Fh = 09
.253
1.123
38.91
Fh= 1.0
.384
1.216
48.47
Fh=1.1
.565
1.275
66.28
188
OCR for page 173
DISCUSSION
William C. Webster
University of California at Berkeley, USA
In reference [6], we presented numerical results based on Green
Naghdi theory for the same problem. Did you compare your results
with these computations?
AUTHORS' REPLY
We are well aware of your excellent paper coauthored with Profs.
Ertekin and Wehausen, where valuable numerical results are
contained based on the GreenHaghdi directedsheet model. We have
already compared these results with ours based on the Finite Element
Method and the KP equation in two separate papers, both presented
at the 5th International Conference on Numerical Ship
Hydrodynamics in references [7,10]. We did not include these
comparisons here since we concentrated only on the effect of the tank
width on the wave responses by using the two different methods.
DISCUSSION
Theodore Y. Wu
California Institute of Technology, USA
This paper, delivered lucidly by Prof. Hang Choi, is of basic interest
and bears significance in that two theoretical models of quite different
approach and of different orders in accuracy are here applied to
provide results on this valuable comparative study. I hope the
authors can clarify whether their FEMmethod is indeed equivalent
to the exact Euler flow model on theoretical basis, notwithstanding
numerical errors.
Of particular interest would be a further exploration on the
asymptotic behavior of these two models in two special limits: (i) as
the bodylengthtochannel width ratio tends to zero, (ii) as the
velocity of forcing approaches the upper or the lower bound at which
the upstream emission of solitary waves would evanesce. I would
like to encourage the authors to continue their excellent efforts in
these directions to cast new lights on this very interesting
phenomenon.
AIlTHORS' REPLY
We highly appreciate the discussion raised by Prof. Wu. To the first
question whether our FEMmethod is equivalent to the exact Euler
flow model on theoretical basis, we would like to stress that the basis
of our FEMmethod, i.e., the variational principle in our paper is
equivalent to the exact inviscid irrotational flow with a free surface.
In the procedure of the FEMmethod, the unknown free surface is
also represented as a part of solutions and solved numerically through
iterative scheme. In this sense, the present FEMmethod is
equivalent to the exact potential flow model, except discretization of
continuous functions.
Concerning with the comment on the case of laterally infinite tank,
we tried to numerically follow the similarity solution of Redekopp
[25]. But due to the limited computing capacity, we are able to show
only an intermediate result which indicates the crestline of the first
diverging waves being nearly a parabola.
Based on the present computations, it is hard to predict the upper and
lower bounds of forcing speed at which the quasiperiodic emission
of upstreamadvancing solitons evanesces.
DISCUSSION
John V. Wehausen
University of California at Berkeley, USA
As I understand the authors, the calculations based upon the KP
equations for a vertical strut just touching the bottom and with a
profile determined by the sectioncurve of a Wigley hull or of Series
60, CB = 0.80, whereas those based upon the Laplace equation are
for a ship with the same overall dimensions as the latter hull but with
an altered sectionarea curve appropriate to a wedgeshaped hull; the
blockage coefficients are the same. One is tempted to conjecture that
the differences in results may be due as much to the different
geometries as to the differenct methods of computation. Table 2 may
support this conjecture for the KP equations. Ertekin's (11)
conclusion that blockage coefficient is the most important parameter
in determining properties of the solitons was based upon experiments
with only one hull shape. Later computations for struts, using T. Y.
Wu's generalized Boussinesq equations, have shown also some
dependence upon hull form. Would the authors care to comment?
By chance, calculations have been given in Ertekin, Qian and
Wehausen (Engineering Science, Fluid Dynamics. A Symposium in
Honor of T. Y. Wu. World Scientific Publishing Co., pp. 2943,
Table 1, line 1) for the same configuration as in Table 2, but only for
Fh = 1.0. The generalized Boussinesq equations were used. The
values obtained were A/h = 0.36, cl(gh)'h = 1.14, UTB/h = 61.
The agreement for the first two values is perhaps not unsatisfactory,
but this does not seem to be true for the third. The discrepancies
could be a result of numerical error or of the different equations used.
AUTHORS' REPLY
Thank you very much for your nice comments. To the first
comment, we agree with you. However, in the FEM computations
we used not only the same length and draft but also the same
sectional area with slightly reduced beam.
To the comments in the second paragraph, we also agree with you.
The discrepancies in the numerical results by different methods are
due to the differences in the numerical procedures as well as the
governing equations.
189
OCR for page 173