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OCR for page 439
Wave Resistance and Squat of a Slender Ship Moving Near
the Critical Speed in Restricted Water
H. S. Choi
Seoul National University
Seoul, Korea
C. C. Mel
Massachusetts Institute of Technology
Cambridge, USA
Abstract
The wave resistance and implied squat of a
slender ship advancing near the critical speed in
restricted water are studied. Employing matched
asymptotic expansion techniques, it is shown that
the response can be described by the
homogeneous Kadomtsev - Petviashvili (KP)
equation with flux conditions on boundaries,
when the channel is wide compared to the ship
length. Numerical results show the generation and
radiation of straight-crested solitons in a periodic
manner ahead of the ship, when it moves at
transcritical speeds with moderate blockage.
The solitons are initially three dimensional, which
are followed by a depressed region and a train of
complicated ship-bound waves in the wake.
Hydrodynamic forces are computed by using
slender body approximation, and the implied
sinkage and trim are estimated based on
hydrostatic relations. These quantities vary with
time and strongly depend on the ship's speed and
blockage. Near the critical speed, the wave
resistance and the trim oscillate around mean
values in phase with the emission of solitons,
while the sinkage takes place out of phase. Ibe
calculated results are in crude agreement with the
measurements.
1. Introduction
A couple of investigators have reported the
fascinating phenomenon observed during shallow
tank tests that a ship model towed near the
critical speed ~ (g = acceleration due to gravity,
h = water depth) radiates a succession of
upstream-propagating waves in an almost
periodic manner (Thews & Landweber 1935;
Izubuchi & Nagasawa 1937; Graff 1962; Huang
et al. 1982~. As a result, the ship experiences
439
considerable changes in resistance, trim and
linkage, or better known as squat. One of most
exciting aspects in this now identified
phenomenon is that a three-dimensional
disturbance, such as a ship, generates 2
dimensional waves propagating upstream in a tank
of finite width. In addition to it, the
propagation speed of the upstream waves, named
solitons, is faster than the constant towing speed
so that a steady state cannot be attained.
Linear theories fail to predict the flow.
Katsis and Akylas (1987) clarified it in the light
of the linearized dispersion relation. Among all
waves radiated from a disturbance advancing at a
speed u, those which may remain stationary with
the disturbance in the direction ~ must have the
wave number k such that F kh cost = (kh ta~2~2'
where F is the depth Froude number (=u/~.
It means that at a transcritical velocity,
F = ~ + O(kh)2, long waves must be in nearly the
same direction as the moving disturbance, i.e.
cost= ~ + of. Furthermore the group velocity
tends to vanish in the moving frame.
Consequently the long waves become almost
nondispersive and the associated wave energy
cannot be radiated. It implies, in order to deal
with the problem, we have to include a balanced
interplay by the nonlinear and dispersive effects
to the leading-order wave equation, and to keep
it in mind that transient waves evolve slowly.
Wu and Wu(1982) were the first who
calculated the generation and propagation of
solitons for a moving disturbance spanning
uniformly across the channel by using the
generalized Boussinesq equation. Akylas
(1984) also considered a two-dimensional
pressure band travelling on the free surface but
focused attention to the immediate neighborhood
of the critical speed. He showed that the physics
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can simply be described by an inhomogeneous
Kortewegde Vries (KdV) equation. In a joint
theoretical and experimental study, Lee et al.
(1989) found a broad agreement between the
experiment and two theoretical models, the
generalized Boussinesq and the forced KdV
equations, for a moving two-dimensional bottom
topography in a shallow water tank. All these
works are concerning with 2 dimensional
disturbances and thus one-dimensional wave
fields.
For a rectangular patch of surface pressure
whose width is comparable with the tank width,
Ertekin et al. (1986) solved Green - Nagdhi's
directed - sheet model numerically. Their
calculations have yielded two-dimensional flow
upstream and three-dimensional flow downstream,
in qualitative agreement with the measurements
they carried out systematically (Ertekin, 1984 and
Ertekin et al.,1984~. Wu and Wu (1987)
obtained similar results using the generalized two-
dimensional Boussinesq equation. Katsis and
Akylas(1987) calculated some 3 dimensional long
waves bounded by side walls using the
Kadomtsev-Petviashvili (KP) equation. Quite
recently, Lee and Grimshaw (1989) studied the
three-dimensional slowly-varying evolution of
wave fields ahead of a bottom topography in a
honzontaDy unbounded fluid domain by using a
forced KP equation. Meanwhile, Bai et al.
(1989) applied finite element method to a
vertical strut sliding in shallow water.
2. Formulation
We consider a ship advancing with a constant
speed in a shallow channel. For simplicity, the
ship is assumed to possess the lateral symmetry
and to move amid the channel so that it is
enough to take only the half of fluid domain into
account. A rectangular coordinate system is
introduced, which is fixed on the waterplane of
the ship. The x-axis coincides with the
longitudinal axis of the ship and the centerline of
the channel (Fig.1~. Under the usual assumptions
of potential theory, fluid motions are described in
terms of the velocity potential Mix*, y*, z*, ,*y,
which is the solution of the following initial -
boundary value problem. The Laplace equation
holds in the fluid region
92+ =0 ~ -it 5 Z S ~ ). (1)
The kinematic and dynamic boundary conditions
on the free surface at z = `* are
~ *=t,*+(u+l *it *++ *< *, (2)
9~*+~ +U~ +~/2 (vl*~2=o. (3'
, X
It is, however, as yet unclear how these
methods with pressure distributions can be
applied to three dimensional bodies such as a ship.
In this respect, Mel and Choi (1987, hereinafter
referred to as I) extended the theory of Mel
(1986) to treat the transient forces on and
responses of a slender ship. They found that the on the channelwall
waves in the far field can be described by one-
dimensional inhomogeneous KdV equation for a
special class of channel width (=2w) and ship's
slenderness parameter (= 8) as follows;
w / ~ = 0(,u-m) with 0 5 m 5 1,2 and 8=o(,~25),
where 2~ stands for the ship length and
~ = ~ / ~ = oLl) for the dispersion. It correctly
predicts the upstream solitons, and the estimated
sinkage and trim for a destroyer favourably
compare with the time - averaged experimental
values of Graff et al. (1964~. But the theory
fails to render three-dimensional waves in the
wake. In viewing the result of Katsis and Akylas
(1987), it strongly suggests us to modify our
theory in their direction. In fact, it was already
pointed out in I that the KdV equation is to be
replaced by the KP equation, when the canal is
much wider, i.e. w, ~ = of. We have done
it in this paper by following the same scheme
described in I, but for a wider channel.
440
No net flux condition holds on the channel
bottom
~ *= 0 (z =h ),
’**=0 6*=W),
and also on the ship's hull r*=R*(x*,0)
(4)
(5)
4>~* (U++x*)R *[1+(Rg ~R*~2]-1,2 (6)
where r* and ~ are polar coordinates on the
cy*, z*) - plane. Under the assumption of
slender body,the normal derivative on the ship
surface has been approximated by that on the
transverse plane at constant x* along the ship.
Before the initial instant ,=o, there was no
disturbance
+*=o, t*=o `~*=oy. (7)
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Representative terms from entire chapter:
critical speed
In order to solve the problem approximately,
we employ perturbation techniques by defining
two smallness parameters
e=A/h , ~=~2/L,
and assume without loss of generality E= ~2 which
implies that the nonlinearity and the dispersion
are both important to the leading order solution
of the problem. In eq.~), A means a typical
wave amplitude. Vanables are normalized as
below:
~ =AL,, ~ =(gAL/~’, 2' =~- (9)
Herein we rather focus our attention to the flow
near the critical speed. Thus the Froude number is
expanded
F2=1 - 20~2 Wit/2 ~=0~1~.
To cope with the slow variation of flow, we have
to rescale time
T= I/,
The channel is assumed wide in comparison with
the ship length
W/L=~/ ~0 Wit/2 ~o=O(~. (12)
As pointed out by Mel (1986), the blockage
coefficient SB must have a magnitude of 0~4)
5~,RO2 / TWIT = 0 (~4), (13)
where RO denotes the characteristic transverse
radius of ship and the blockage coefficient is
simply the area ratio of the midship section to the
channel cross-section. Consequently the
slenderness parameter must be
b=Ro/L=O(~2~.
(14)
It implies that the nonlinearity arises from the
slenderness of disturbance.
Because of these vastly different length
scales involved, it is relevant to divide the channel
cross-section into three regions as:
(i) the far field ; X* ~ < =, y* = 0 (W) = 0 (L/~),
a:
=o(~=°(~L), (15)
(ii) the intermediate field; x*=O(L),
~ ~ z )=O(h)=O(~L), (16)
(a) (iii) the near field; X*=O(L),
~ ~ z '=0(Ro'=0(,~2L). (17)
We shall first analyze each field separately and
then conduct matching by following the line
reported in I. Accordingly most parts must be
very similar to those in I and it is already obvious
that modifications which will appear are solely
attributed to the different definition on the length
scale of win. Neverthless steps are explicitly
given here in order to keep this paper self-
contained.
3. The Far Field
In accordance with the scheme defined by eq.
(15), the far field variables are made
(11) dimensionless
x =Lx,y =Wy,z =hz. (18)
The normalization of other variables holds
effective as defined in eqs.~9) - (11~. In this
field, the banks and the bottom of the channel
directly affect the propagation of waves.
However, the boundary condition Qn the ship hull
is of no meaning and the forcing agency is not
known. The governing equation and the
boundary conditions are rewritten in terms of the
dimensionless farfield variables:
(6Pxx+~ ~lo2di,yy)+4pzz=o (-l 5 Z 5 ~2~) (19)
2~1 _ 2
Substituting the expansions
+~~(0)+ ~2~(V+ ~4~(4)+, . . (24)
~ t(0) + ~2~(2) + ~4~(4) + . . . (25)
into eq.~19), we obtain
+(o)=o,
(I )= - ·~(x),
+(4)=_~(2)+~2~(0)) (26c)
With help of no flux condition on the channel
bottom, the general solution straightforwardly
yields to
- ilr'2~;(0)=o (Z=o). (3 ~
It is the homogeneous KP equation, which is the
three-dimensional counterpart of the KdV
equation, since it contains the transverse
dispersion as well as the longitudinal dispersion. If
t(°)=o, the KdV equation is recovered.
An observer In the far field is too far away
(26a) from the ship so that he just observes waves
without knowing the details of wave generation,
which can only be found through matching with
(26b) the intermediate field. For this purpose, we
need the inner expansion of the far field potential
for small y
’=~(O)(X O. T)+y+(°)(X,O,7~+~ ~ l(V(X,O,T)
_ ]rZ+~2~(0)~+or~3~ (31)
.$~°~=~°3(x y ~) (27a) 4. The Near Field
+(2)=~(2)(xyT)_~(z+~2~(0) (27b)
+(4)= +(4)(X y T) _ ~ (Z + i)2~’ (2)+ ~ 2’ (0)) + y =Roy , z =Roz,
In a fluid domain close to the ship, the
characteristic length is Ro and thus let us
nondimensionalize the near field variables
+ ~ (z+~4~0) (27c) r =ROr, R =RoR, (32)
where +(n) are unknown functions to be determined
later by matching with the intermediate field
potential. The free surface elevation is derived
from the dynamic condition, eq.~21)
but retain the rest of the normalizations. The
Laplace equation
x < z s (~ /~L
is now transformed for
=_g,~o' (z=o), (28a) 82’xx+~,+~--°- (33)
tt2' _~2~_~0~_ ~ ’~0~_ ~ `~0~2 ~Z=O) `2gb) ~ the free surface at Z-=~3/~t, we have
Utilizing the above relations, the kinematic
condition of 0~4) on z=o turns out to be
~ (4) ~ (2)+ ~ (0)2a ~ (0)2~( )t ( ~ (Z = 0) . (29)
Combining eqs.(27) - (29) and differentiating with
respect to x, we finally get the leading order wave
equation
Liz= ~ 8~1 - 2~ Ace+ ~1 - 22 +
+ ~38’xtx+ (~3/~, (34)
(1 - 2~2~+ SIX'+ ~2~1 _ 2~ Aft+
+ 1 ~2[<,2+(,,2+<,,2~82]=o. (35)
On the ship hull r=R(x,0), the condition is
tax)~`Xx) - _~(0)~2xx~ ~ (8 )2~1 - 2~2+~21 jR [1+(Re/R) ] . (36)
442
As already discussed,the slenderness parameter is determined
a sarnll quantity of o(~2) and thus we suppose
the following expansions:
,4, A, (0)+ ,<~(1)+ ~2~(2)+ . . . (37)
~ `(0)+ ~;(1)+ 2~;(2)+ (38)
Then the Laplace equation and the boundary
conditions are readily expressed as below:
~,(n~)+~’~,Ln)=0 ~x < z 5 O. n=0~l,2), (3 ~
chin)= o (Z = 0, n = 0,1 ,2), (40)
{;(o)=_,-,(°) (z-=oy, (41)
`1,(")=0 (r=R, n =0,1), (42a)
~>(2)=R ~~ + OR / R~2 ~-1,2 ~r=R~, (42b)
From eqs.(39) and (40) it is clear that ’(~) may be
absorbed by ’(°3. The leading-order problem is of
homogeneous Neumann type, of which solution
formally takes the fonn
’(°)=f(°)(X T)
(43)
The next-order solution contains a particular part
owing to the inhomogeneous term in eq. (42b)
`~(2)=f(2)`X At+ ~>~2~`` y A,. (44)
The particular solution represents the disturbed
flow due to the motion of the ship,of which outer
approximation for large r will prove to be more
meaningful in the viewpoint of matched asymptotic
expansions. For large r, the ship shrinks to a line
source and thus +` is expressed
4, (2)= _q (x ,T) In r + c (x ,7), (45)
where q stands for the source strength and c may
be regarded as a part off(2)(x,T).
Applying the law of mass conservation to the
fluid domain surrounded by the ship, the free
surface and a control surface located far away
from the ship, the source strength is readily
443
2 F2
q = 2 SBSXKX),
To ~
(46)
where SB jS referred to the blockage coefficient
and six) to the longitudinal distribution of the
cross-section area of the ship. It is reminded that
the blockage coefficient has a magnitude of o(,u~4)
near the critical speed, i.e. F = 1 + o(,u2~. For the
sake of matching with the intermediate field
solution, it is necessary to expand eq.(44) for large
r
~ ~ ~f(°)~x Ty + p.,2tf(V(x ,T) +Lenr)] + · (47)
1
5. The Intermediate Field
Here the proper reference length for y* and r*
is h, hence we introduce the dimensionless
intermediate coordinates
y = Ily^, r hr.
(48)
and keep all other normalized variables
defined in eqs.(9)-(11) and (18). The
nondimensional equations are
,u-2+Xx+ Fly+ fizz= 0 (- 1 s z s ,u-2~), (49)
4~/120t,u~ ~ + ,u~ (1 2a,u~ )`x
+ ~ ()xtx+ ~ (l~y~y ( = ~2~) (50)
(1 - 2alt2)(` + ~X) + ~2~/1 - 2a~+
+ _[~24,x2+~>2+~2]=0 (Z= 1~ A), (51)
4)z=0 (Z=-1),
`~',t = (~/~)(1 - 2(Y. ~2+ ~2’X) R. [1 + (R. / R )2 ] -~2
(52)
(it =R ). (53)
In anticipation of matching with the near field,
we introduce expansions in the form of eqs.(37)
and (38), with ~ and ~ replaced by ~ and
Respectively, it then follows
Aft, (n)+ ~~, (n)= 0 (1 s z ~0, n = 0, i), (54a)
,:~(2)+~(2)=_,`> (0) (54b)
From the Taylor expansions of the conditions on
the free surface, we get
i`> (n)= 0 (Z = 0, n = 0,1), (55a)
`~(2)=~;(0) (z=o). (SSb)
It is immediate to have
+(°)=f(°)(X T)
+(2~=f~2)(X7)_~(z+~)2f(0)+~(2) (56b)
The particular solution ’(2) again corresponds to
the response to a line source at the ship's
centerline, hence takes the following form for
small r
<,(2)= ~ q^(x,T) in r=In (r).
AT ~ ~
The solution has the inner expansion
+~f (X,T)+ ~ ~ f(V(X,7)f (O)+
q O
+ ln( r)]+
(57)
Matching with the outer expansions of the near
field results in the relations
q=q,
f(°)(X T) =f( )(X IT) ~
`~()~ 2 qY (Yowl)' (61)
so that the outer expansion of the intermediate
field is, in terms of the far field van ables
<~~f(°)`x'T)+ q +,2 Aft_ Size ~27(0)]+ . . .~62)
to 2
Matching provides useful information:
`,(°~(X o :~=f(°~(X,~=f(°)(x,~), (63)
(56a) 2l~o 712 ~4
Differentiating eq.~64) with respect to x and
recalling the relation of eq.~28a), we finally get the
boundary condition for `(°3 at y=o
t( )(X,O,T)= 2 ~S=(X). (65)
~0
We realize that the result thus obtained is
basically the same as those Katsis and Akylas
(1987) derived.
Once t;(°3 is calculated, ,,(°3=f~°)=ff°) is known.
Since the leading-order dynamic pressure is
linearly proportional to the surface elevation, the
forces and moment acting on the ship can easily be
evaluated by invoking the slender body
approximation. The implied sinkage and trim
(S8) may be estimated from the hydrostatic relations
(Tuck, 1966), which are given in I.
(59)
(60a)
x,=,_ if(O) (60b)
The source strength measured in the
intermediate field is indeed idendical with that in
the near field. From mass conservation, the
approximation for l(2) must be
444
6. Numerical Results s(x)=l_x2 (1x1 5 1).
To further investigate the wave field and
resulting hydrodynamic forces, we have to rely on
numerical computations for the homogeneous KP
equation with proper boundary conditions. In
contrast to the KdV equation, only a few
literatures are available, in which numerical
methods for the KP equation have been
discussed. Katsis and Akylas (1987)
applied an explicit finite-difference method for the
standard KP equation and investigated the wave
patterns due to normally distributed pressures in
shallow water laterally bounded and unbounded.
We adopted their scheme. To facilitate
calculations, it is convenient to integrate eq.~30)
with respect to x
((a)= aL(°)+ _~(o)~(o)+ _~(o>+ _~2i SIX (66)
where use has been made that t(°3 and its
derivatives must vanish far upstream. If the
integral term on the right hand side of the above
equation is neglected, it becomes the standard
KdV equation.
In the discretization, simple forward
differences are implemented for time derivatives,
and central differences for spatial derivatives.
But at the wall and the centerline of channel
one-sided differences are used instead of the
central difference, and the boundary conditions,
eqs.~23) and (65), have been incorporated.
Integrals are evaluated by trapezoidal rule. From
diverse numerical tests, the scheme has proven to
be stable for
Ax = 0.1 , Ay = 0.1 , Ad= 0.00002
with the ship's length as 2.0 spanning from
x= - Lo for bow to x= Lo for stem. These values
have been used for all computations in this work.
As Katsis and Akylas pointed out, reflected waves
from the numerical boundaries far upstream and
downstream seriously deteriorates the result.
Several devices for radiation condition have been
tried without success. Thus the computation
domain is taken as large as possible and the portion
is discarded, where numerically reflected waves
are apparent. To save computing time, it is
advised to begin with a small domain and to
enlarge it continuously as time passes.
Our primary concern is to examine the
generation and propagation of solitons by a ship.
For this purpose, let us consider a slender ship
whose cross-sectional area varies parabolically
445
In Fig.2, the evolution of the wave field at the
critical speed is illustrated with time interval
T=0.2 Up to ~=~.o. The parameters are chosen
as below:
~=h / L =0.25, W / L= 1.0 (~o=4~0~,
SB=0.105 (~=25.6), F=1.0 (~=0.0~.
The blockage coefficient seems somewhat too
large for our theory to be valid. Neverthless this
case is taken up, because it is possible to
compare with the experimental and numerical
results of Ertekin et al. (1984 and 1986~. They
calculated Green-Nagdhi's model for a rectangular
pressure patch,which should be roughly equivalent
to a ship with blockage coefficient, SB=0 105, in
the viewpoint of the hydrostatically displaced free
surface. The figures are exaggerated vertically by
2.5 times. It is to note that 1 T corresponds to
their nondimensional time UT,=. During this
time span, the ship advances a distance of 8 times
the ship's length.
At T=0.2, three-dimensional waves emerge
ahead of the ship. A depressed region is built
therebehind, which is followed by a train of
complicated ship-bound waves in the wake and
also reflected transverse waves far downstream.
As time elapses, the upstream waves develop
further and gradually become straight-crested as
they are reflected from the wall. The first soliton
almost completes its formation and becomes two-
dimensional at T=0.4. The second soliton starts to
take its shape at T=0.6, while the first soliton
steadily propagates upstream and the depression is
being elongated. At T= i.0, the second soliton is
completed and the third one begins to appear.
Such a trend can also be recognized in Ertekin et
al., but the waves downstream here look more
ship-bound.
In Table 1, a comparison is made for the
amplitude, propagation speed of the first soliton
and the period of first two solitons. It is to note
that the amplitude was taken at T=~.0 (UT/h=~),
since it continuously increases during the
developing phase. Let us first compare present
results with those of Ertekin. The propagation
speed agrees excellently,but significant deviations
exist in amplitude and period. It is not probable
that these are caused simply by numerical
round-off errors. It was concluded in the works
of Ertekin et al. that the amplitude increases
and the period is shorter as the blockage is
larger, and the details of the disturbance is less
important. The blockage coefficient is identical
in both cases, but the presumed ship of Ertekin
is much fuller than our slender ship. It might act
as a stronger forcing agency and result in higher
amplitudes. If it is true, then the present period
should have been longer in order not to contradict
the measurements. But it is not the case and let
us leave it as open question. Now we turn to Bai
et al.~1989) who applied finite element method to
a vertical strut sliding in a shallow channel.
Their result is closer to the present for amplitude,
but to E~tekin for penod. The propagation speed
is slower than both. We may postulate that it is
attributed to the different mathematical models.
But no clear-cut conclusion can be drawn at this
stage.
Table 1 Soliton amplitude,propagation speed
and period for ,s=2s.6 at a=O.O
,i A / h c / Jim |
UTg / it
I, Bai
I Ertekin
| Present
I.
0.553 1.24
0.6248 1.280
n sr2s 1 1.281
30.0
29.6
28.8
Fig.3 shows the wave profiles along the
centerline and the wall of the channel at ~=~.o.
The completed first soliton assimilates each other
closely. But there is a slight difference on the
rear side of the second soliton, because a new
soliton is just about to burst. A train of
modulated wave packets follows a depressed
region, which is directly reponsible for the sinkage
and trim of ships. The downstream configurations
are in general quite dissimilar.
The time evolution of the wave resistance,
sinkage and trim is depicted in Fig.4. The wave
resistance and sinkage are normalized by the
displacement and half length of the ship,
respectively, while the trim is in degrees. Positive
values indicate resistance, downward sinkage and
trim by stern. The computed sinkage and trim
are of qualitative meaning only, because no
dynamic effects are included. Caution should be
paid on three differently-scaled ordinates. All
these quantities rise initially from zero to first
maximum and then oscillate around mean values.
The oscillation period is approximately ~g=0.45 in
coincidence with that of soliton. It reflects the
fact that hydrodynamic forces and their effects on
the ship are dominated by the generation and
radiation ofsolitons. The wave resistance and the
trim fluctuate in phase with the periodic soliton
emission, while the sinkage takes place out of
phase. It seems unlikely that a steady state will be
attained in time.
To assess the effect of ship's speed, we
consider the same slender ship as above but in a
slightly deeper channel
,u=0.333, =3.0 (W /~=~.0),
~ = 5.0 (SB = 0.062)
for five speeds: two subcritical speeds
a=2.5 (F=0.667) and a=~.O (F=0.882), critical speed
a=O.O (F=~.O), two supercritical speeds
a=-~.O (F=~.106) and a=-2.5 (F=~.247~. The
wave resistance,sinkage and trim are plotted in
Fig.5,6 and 7 in this order. At transcritical
speeds, the wave resistance indeed oscillates. The
amplitude of oscillation reaches its maximum not
at the critical but at a slightly faster speed, and the
period becomes longer as speed increases, which
supports the experimental findings. At the low
subcritical speed,the wave resistance increases
upto a certain threshhold with an intermediate
step, and it arrives at a near - steady state. At
the high supercritical speed,the wave resistance
initially reaches a maximum and then diminishes
with time to a small steady value.
Similar trends are to be observed for the
sinkage and trim in Fig.6 and 7. It is to note
that the sinkage oscillates around zero at the
critical speed and it becomes negative (lift up) at
supercritical speeds. Generally speaking, the
overall behaviour of the wave resistance, sinkage
and trim is quite similar to two-dimensional cases
described in I, as long as the channel width is not
too wide and thus the flow around a ship is chiefly
affected by upstream solitons.
The variation of soliton amplitude, propagation
speed and period according to ship's speed is
listed in Table 2. The amplitude given here is
referred to the computed value at ~=~.o for the
first soliton. The numbers in parentheses
designate the experimental counterparts. The
difference in emission period is again remarkable.
However, it can be said that theory provides at
least crude predictions.
446
Table 2 Variation of soliton amplitude, propagation
speed and period for three ship's speeds
A/h ! call
.. .. .
I a= - 1.0
~x= 0.0
i a= 1.0
I _
0.64
(0.60)
0.45
(0.49)
1.31
(1.26)
1.21
(1.20)
0.29 1.13
(0.26) (1.12)
. . _ . _ _ .-
UTg / h
__
50.8
(53.~)
40.5
(49.5)
31.0
(41.6)
Next we examine the wall effect on the
response at the critical speed. Three channel
widths are chosen as w/~=o.s, Lo and 3.0, while
the channel depth and the ship are kept unchanged
(~ 0.333~. It implies that the corresponding
blockage coefficients are 0.123, 0.062 and 0.021.
Fig.8 a - c show the wave fields at T=~.O, when
the ship moved a distance of 4.5 times the ship's
length. The vertical scale is stretched by 3.2
times in comparison with those on the horizontal
plane. Since the lateral coordinate is normalized
by the half width of the channel in all cases,the
banks are designated by -1.0 and 1.0. For a wider
channel, it is necessary to reduce by for a better
resolution. For w/~=o.s, there is an indication
that the second upstream wave develops on the
back of the front waves, whose crest line is spear-
headed. The depressed region is relatively long
and the downstream waves are pronounced. For
w,~=3.0, there is no sign for upstream-
propagating waves and diverging waves with large
run angle prevail. The downstream waves are
hardly two-dimensional. Katsis and Akylas
(1987) suggested that the maximum canal width
for which the downstream waves remain to a
reasonable approximation two-dimensional
depends on crudely the source characteristics.
They found that the maximum channel width is
about 20 h for an elongated pressure distribution.
Since we are dealing with a slender ship, it may be
possible that the downstream waves remain
practically two dimensional upto a certain range
of channel width. But we have not attempted to
confirm it.
The wave resistance, sinkage and trim are
summarized in Table 3. Again the resistance and
sinkage are made dimensionless with the
displacement and the half length of the ship. Trim
is in degrees. First two extremes are given with
the nondimensional time in parenthesis at which
they occur. For resistance and trim, the first
447
extreme corresponds to the first maximum and
the second extreme to the first local minimum.
The extremes and their deviations take greater
values as the channel becomes narrower. For
linkage, the first extreme represents lift up for
w / r=o.s and 1.0, but downward sinkage for
w/~=3.0. The ship sinks more In average in a
wider channel. The time intervals between two
extremes for all three quantities are consistently
0.4 and 1.7 for w/~=o.s and 1.0, respectively.
But there is no such a correlation for w ,~=3.0.
Hang S.Choi would like to thank the Korean
Science & Engineering Foundation for financial
support. He also wishes to thank I.H.Cho, a
graduate student at Seoul National University, for
his drawing pictures.
References
Akylas,T.B. 1984 On the excitation of long
nonlinear water waves by a moving pressure
distribution. J.Fluid Mech. 141,455-466.
Bai,K.J., Kim,J.W. and Kim,Y.H. 1989
Numerical computations for a nonlinear free
surface flow problem. to be presented at the 5th
Inter''. Conf. on N~anerical Ship Hydrodyn.
September, Hiroshima.
Ertekin,R.C. 1984 Soliton generation by moving
disturbances in shallow water : theory,
computation and experiment. Ph.D. Thesis, Univ.
Calif. Berkeley.
Ertekin,R.C., Webster,W.C. & Wehausen,J.V.
1984 Ship - generated solitons. Proc. 15th Sym:p.
Naval Hydrodyn. Hamburg, 347-364.
Ertekin,R.C., Webster,W.C. & Wehausen,J.V.
1986 Waves caused by a moving disturbance in a
shallow channel of finite width. J.Fluid Mech.
169,275-292.
Graff,W. 1962 Untersuchungen ueber die
Ausbildung des Wellenwiderstandes im Bereich
der Stauwellengeschwindigkeit im flachem, seitlich
beschraenktem Fahrwasser. Schifftech}~ik,
Bd.9,Heft 47,110-122.
Graff,W., Kracht,A. & Weinblum,G. 1964 Some
extension of D.W.Taylor's standard series.
Tra~zs.Soc.Naval Arch. & Marine E'~gnrs. 72, 374-
401.
Huang,D.-B.,Sibul,O.J. & Wehausen,J.V. 1982
Ships in very shallow water. Festkolloquium zur
Emeritierung von Karl Wieghardt, Institut fuer
Schiffbau, Hamburg Univ. Bericht Nr.427, 29-
49.
Izubuchi,T & Nagasawa,S. 1937 Experimental
investigation on the influence of water depth
upon the resistance of ships. (in Japanese) Japan
Soc. Naval Architects, 61, 165-206.
Katsis,C. & Akylas,T.R. 1987 On the excitation
of long nonlinear water waves by a moving
pressure distubution.Part 2.Three-dimensional
effects. J.Fluid Mech. 177, 49-65.
Lee,S.-J.,Yates,G.T. and Wu,T.Y. 1989
Experiments and analyses of upstream-advancing
solitary waves generated by moving disturbances.
J.Fluid Mech. 199, 569-593.
Lee,S.-J. and Grimshaw,R.H.J. 1989 Upstream-
advancing waves generated by three-dimensional
moving disturbances. to appear in Physics of Fluid.
Mei,C.C. 1986 Radiation of solitons by
slender bodies advancing in a shallow channel.
J.Fluid Mech. 162,53-67.
Mei,C.C. & Choi,H.S. 1987 Forces on a slender
ship advancing near the critical speed in a canal.
J.Fluid Mech. 179,59-76.
Thews,J.G. & Landweber,L. 1935 The influence
of shallow water on the resistance of a cruiser
model. US Exp. Model Basin, Navy Yard
Rep.408.
Tuck,E.O. 1966 Shallow-water flows past
slender bodies. J.FIuidMech.26,81-95.
Wu,D.M. & Wu.T.Y. 1982 Three-dimensional
nonlinear long waves due to moving surface
pressure. Proc. 14th Symp. Naval Hydrodyn. Ann
Arbor,103-129.
Wu,D.M. & Wu,T.Y. 1987 Precursor solitons
generated by three-dimensional disturbance
moving in a channel. Proc. IUTAM Symp. on
No''li''ear Water Waves, Tokyo,69-76.
Table 3 First two extreme values of wave resistance
sinkage & trim for a slender ship in channels
with different width ~ ~ = 0.0, `~0.333)
W/L
l Rw
s
aT
2w
.~_
1 ~1
1 2L -1
Fig.1 Definition sketch of a slender
ship advancing in a channel
448
. 0.5
0.140 (0.7)
0.101 (1.0)
-0.008 (0.5)
0.013 (0.9)
12.082 (0.6)
Q ADA {1 ~N
1.0
0.105 (1.1)
0.074 (1-8)
-0.005 (0.8)
0.014 (1.5)
8.827 (1.0)
A.__ `~.VJ 6.456 (1-7) 1
3.0
0.085 (2.1)
0.082 (2.8)
0.017 (0.5)
0.011 (1.9)
7.900 (1-7)
5.894 (3-9)
(a) T = 0'
Fig.2 Evolution of wave field generated by a slender ship
449
(b) ~ = 0.4
{~1 ~ = Or
(d) ~ = 0.E
(e1T= 1.0
~ o
o -
a,
8-
c~
, 1
o
o
0- l l
-40.00 -24.00
1 1 1
-8.00 8.00 24.00
I ~ I I I ~
40.00 56.00 72.00 X
(a)y = 0
~ 8-
~D
8-
o
o
o
m_
-40.00 -24.00
I I I I I I 1 1 1 1 1 1
-8.00 6.00 24.00 40.00 56.00 72.00 X
(b)y = W
Fig.3 Wave profiles along the centerline & the wall of channel at ~ = 1.0
RW
-
O
O KI ~
.
O l l
0.00 0.20
AT S
~ mi F°°
~ ,
8- on
0
. , ~ ~ ~ I I I ~ I T
0. 40 0.60 0.80 1 .00 1 .20 1 . 40
Fig.4 Evolution of wave resistance, sinkage and trim
for a slender shin
~ a = 0.0, ~ = 1!6-5, TO = 4.0, ,u~ = 0~25)
450
1
~1
o-
oooo
~ 1 1
~ T (degrees )
8
~ 1
o
of
Rw
(U]
o:
_ /<
~ ~ ~o,o
8- ~
g: /, , 1 ,
o
`,.00 0.40 0.60 1.20
Fig.S Evolution of wave resistance on a slender ship
for five different speeds
(,s = 5.0, ~0 = 3.t), `~ = 0.333)
O
o
s
o_
0 ~
1 1
1.60 2.00 2.40 2.80
1
\a = -2.5
1- 1 1 1 1 1 1 1 1 1 1 1 1 '1
0.40 0.~30 1.20 1.60 2.00
Fig.6 Evolution of sinkage for a slender ship
for five different sneeds
( ,s = 5.0, ~0 = 3.t, ,~ = 0.333)
or=-1. O
~ \~, a =1. 0
~ \
2.40 2.~30
`,~ _ _
~ _ ~ _~ a =-2 . 5
8 1 ~ =
0 - I I , 1 1 1 1 1 1 1 , 1 I--- i 1
O.CO 0.40 0.80 1.20 1.60 2.00 2.40 2.BO ~
Fig.7 Evolution of trim for a slender ship
for five different soeeds
( ~ = 5.0, ~0 = 3.~), ,~ = 0.333)
451
-it
(a)W7L=0.5 I
it.
it,
it'
'by\
(C) W/L = 3.0 -I
,~
o ,
,~
Fig.S Wave pattern generated by a slender ship
for three different channel widths at ~ = 1.0 ( ~ = 0.0, ,u = 0.333)
452
,~_
DISCUSSION
.
by R.C. Ertekin
I would like to make a few comments before
asking some questions on the points that are
not clear to me.
Precursor soliton generation is not
restricted to the critical speed. These waves
have been observed and reported for Froude
numbers as low as 0.2. The theory you used
may be restricted to critical speeds but the
phenomenon is lot.
Soliton speeds are not necessarily faster
than the towing speed always. As Fr -1.3
solitons form a bore attached to the bow.
Around critical speed if the soliton amplitude
is very low it is also possible that the
soliton and model speeds are almost the same.
For all subcritical speeds it is not clear
that steady state cannot be reached, since
soliton amplitudes (if 2nd, 3rd, etc.
solitons) decrease as the ship continues to
move forward. In some cases, solitons would
not have been generated if we had a model tank
which is very long.
I am surprised that Sommerfeld's condition
did not work in your case. We have had no
problem so far at the open boundaries.
Perhaps something went wrong in the
implementation process.
I am curious why you have not compared
your resistance results with the experimental
data. Could you please comment on this?
Your theory seems to be valid at Fr=l.
However, most practical ship speeds are well
below that. Can you extend the theory for
subcritical speeds which may be low? If you
can, your method will be much more efficient
than a numerical method which is valid for all
Froude numbers.
Author's Reply
Prof. Ertekin's discussions are highly
appreciated. Since we have no experimental
experience, your comments on the phenomena
observed during tank tests will be much
helpful for our further research.
It is not to mention that the Boussinesq
equations are effective over a wide range of
speed in shallow water, once the corresponding
Ursell number is close to unity. It is also
true that the expressions are rather
complicated and an immense computation is
required. For a simplification, we need an
additional assumption that the depth Froude
number is expandable near the critical value.
We have done it to obtain the two-dimensional
Kdv equation or KP equation. Consequently it
is obvious that our theory is valid for
transcritical speeds.
In reply to the question about the
comparison of wave resistance with
experimental data, we tried to calculate for a
destroyer model. But still more computations
are necessary before we are able to arrive at
a conclusion.
DISCUSSION
by T. Inui
This morning's Session (Session 7)
reminded me my undergraduate diploma thesis
(1943) on "Restricted Water Effect on Ship is
Wake"(1943).
We measured the wake at two lengthwise
positions, i.e. at midship and at the
propeller position, and we also traversed in
beamwise direction at midship. For the
midship wake, which is approximately
"potential" wake, we found an unexpectedly
good agreement between our measurements and
Kreitner's simple 1-D theory. The three
different flow stages, i.e. subsonic, tran-
sonic, and supersonic, were clearly obtained.
Couple years later, I applied linear wave-
making theory to this phenomena (1946), and
found that
i) For purely shallow water dRW/dV is
discontinuous at Fh=1, and
ii) For restricted water Rw is discontinuous
at Fh 1.
However, naturally, I could not succeed to get
theoretically the transonic region. Since
then it was my dream to bridge this gap by
CFD, because it is essential to take into
account the bodily sinkage and squat for the
hull boundary condition.
The authors already obtained the first
approximation for this. Then you may have a
sufficient possibility by applying iteration.
The authors' comments are highly appreciated.
Author's Reply
We thank Professor Inui for the comments
on his own research. In response to it, we
would like to emphasize once again that both
the nonlinearity and the dispersion are
important to the leading-order solution for
the flow caused by a moving disturbance near
453
the critical speed in shallow water, and that
a steady state can hardly be attained.
To the question, direct approaches such as
the works Prof. Bai and Prof. Ertekin
presented at this conference may give a better
answer by taking the instantaneous hull-
boundary condition exactly into their
numerical schemes at the expense of a rapidly
increased amount of computing ti me. In our
slender-body approximation, it can be done
only indirectly, if we include the temporal
variation of the longitudinal distribution of
ship's cross section in terms of source
strength .
454
