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OCR for page 421
Numerical Grid Generation and Upstream Waves for Ships Moving
in Restricted Waters
R. C. Ertekin and Z. -M. Qian
University of Hawaii
Honolulu, USA
Abstract
The shallow-water wave equations of
Boussinesq type are employed to
numerically solve the problem of a
vertical strut moving in a channel.
Since the most important parameter in
soliton generation by moving
disturbances is the blockage
coefficient, a strut can be made
equivalent to a finite draft ship. A
boundary-fitted curvilinear coordinate
system based on elliptic equations is
generated to deal with the difficulties
due to the body-boundary conditions in
a channel containing an
arbitrarily-shaped ship boundary which
extends to sea floor. The strut problem
is solved numerically in a transformed
computational plane which contains
uniform grid size. A finite-difference
method is applied to the equations to
march in time. The surface elevation
and wave resistance are computed and
compared with the available experimental
data. The agreement between the
calculations and experimental data is,
in general, very good.
1. Introduction
The phenomenon of ship-generated
solitons was rediscovered experimentally
(see Huang et al. [12]) during the
experiments done by Sibul et al. t28] on
an unrelated subject. Huang et al. tl2]
observed that when a ship model is set
into motion, starting from rest and
quickly reaching a constant velocity,
two-dimensional waves that precede the
model are generated one after the other
in addition to the usual
three-dimensional waves behind the
model. The waves that move ahead of the
model were completely above the still
water line and their speeds were critical
or supercritical. These waves have been
termed solitons or solitary waves which,
unconventionally, refer to individual
waves in a train of waves. The
subcritical, critical and supercritical
421
wave speed refer to the depth Froude
number, Fh =U/ ~ , being less than,
equal to and greater than 1.0,
respectively, where U is the ship-model
speed, g is the gravitational
acceleration and h is the undisturbed
water depth which is constant. No
published mention of the phenomenon of
ship-generated solitons could be found
until the reports by Thews and Landweber
[30], Sturtzel and Graff t29], and Graff
[9] that describe the continuous solitary
wave generation experimentally were
brought to attention in 1984.
Wu and Wu t35] reported first on some
numerical calculations of a
two-dimensional pressure distribution
moving steadily over the water surface
in which the same phenomenon of soliton
generation was predicted. These
calculations were based on generalized
Boussinesq equations derived earlier by
Wu t34]. Some of the calculations were
also reported in Huang et al. [13].
The most striking feature of these
nonlinear waves is that they are almost
perfectly two-dimensional, spanning the
tank walls, even though the generating
source is a three-dimensional ship model.
Only a few qualitative features of these
solitons could be observed during the
experiments of Huang et al. [12] since
the experiments were not systematic.
Ertekin [3] carried out a series of
experiments in which certain parameters
such as water depth, model draft and tank
width were changed systematically.
During the experiments, a ship model
(Series 60, Block 80) was towed along
the centerline of the tank with a constant
velocity. The total resistance
experienced by the model and the
run-away-soliton amplitudes were
measured simultaneously in these
experiments.
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The experimental results of Ertekin
[3] and Ertekin et al. t4] showed some
very important qualities of the
ship-generated solitons. Among several
are: the dependence of soliton
amplitudes on the blockage coefficient
(the ratio of the cross-sectional area
of the model at midships to the cross-
sectional area of the water mass, see
Eq. (33~; the phenomenon is not
associated with equipment
malfunctioning; at critical and
supercritical speeds a steady-state flow
cannot, in general, be obtained; below
critical speed soliton amplitudes die
out leaving a shelf in front of the ship
model; and no solitons can be generated
as blockage coefficient goes to zero.
The last feature of the phenomenon
implies that as the tank width or the
water depth goes to infinity, no upstream
waves can be generated. However, we note
that a recent work by Pedersen [24]
challenges this conjecture by en argument
related to the existence of Mach stems
as discussed in Ertekin t3].
Ertekin t3] also investigated the
theoretical cases of a two-dimensional
pressure distribution, and a
two-dimensional bump on the sea floor
moving with a constant velocity. Both
the Green and Naghdi [10] equations for
a thin fluid sheet (see also, Ertekin
t6]), and the shallow-water equations
derived by Wu [34] were used. Some of
the results were included in Ertekin et
al. t4]. Akylas [1] and Cole t2] have
considered a two-dimensional bottom bump
by using the Korteweg de Vries equation
(KdV). Lee t16] and Lee et al. [17]
considered a two-dimensional bottom bump
both experimentally and numerically.
Some of their preliminary results were
included in the Discussion Section of
Ertekin et al. [4].
First attempts to consider a
three-dimensional disturbance in
computations are due to Mei and Choi
[18], Mei [19], Ertekin et al. tS] and
Mei and Choi [20]. Mei [19] considered
a vertical strut which is slender so that
the rigid-boundary condition on the body
can be applied at the center-line of the
strut within the order of perturbation
expansion. This approximation has
resulted intwo-dimensionalwavesin both
the upstream end downstream regions since
the modified KdV equation derived is
two-dimensional only. A remark may be
necessary with regard to the terminology
used here for the number of dimensions.
In three-dimensional Cartesian
coordinates where x and y are in the
horizontal still-water plane and z is
vertical pointing up, we use the
terminology two-dimensional for flows
confined to the x-y plane and
three-dimensional for flows confined to
422
both the x-z plane and the x-y plane even
though some flow quantities, such as the
velocity potential, do not depend on the
vertical coordinate z because of the
mean-layer approximation. Going back to
the discussion on Mei's results, we note
that the two-dimensionality of
downstream waves as calculated by Mei
[19] was not observed, in general, in
the experiments of Ertekin t3].
Ertekin et al. [5] considered a
three-dimensional pressure distribution
and solved the Green-Naghdi (G-N)
equations in the time domain. The
solution of this nonlinear
initial-boundary-value problem also
showed three-dimensional downstream
waves, qualitatively agreeing with the
experimental results. This confirmed
that the application ofMei's formulation
has a limited range, at least, as far as
the downstream waves are concerned. The
wave resistance is also found to be in
qualitative agreement with the
experimental data. More recently, Katsis
and Akylas t14] and Wu and Wu [36]
obtained results for a three-dimensional
moving surface-pressure distribution by
using forced nonlinear
Kadomtsev-Petviashvili (K-P) and
generalized Boussinesq (gB) equations,
respectively. The stability of the
forced KdV equation as it relates to
run-away solitons is investigated by Wu
[37].
In the present study, we investigate
the nonlinear waves generated by a
vertical strut by using the generalized
Boussinesq equations as derived by Wu
[34]. The no-flux boundary condition is
satisfied by means of a n,~merical
grid-generation technique (see for
instance, Thompson et al. t31]. The
nonlinear and unsteady results are
directly compared with the experimental
data. In Section 2, we formulate the
fluid-dynamics problem to be solved with
all the boundary and initial conditions
to be satisfied. We also transform the
equations from earth-bound coordinates
to moving coordinates in this Section.
In Section 3, the numerical
grid-generation technique used is
discussed and the equations of Section
2 are transformed to a regular
rectangular computational domain. In
Section 4, the numerical-solution method
is given and wave resistance experienced
by the strut is discussed. The
finite-difference method employed is
presented and sample results are shown.
Preliminary results are also given in
Ertekin and Qian t8], and the detailed
derivations of the equations presented
here and some other results can be found
in Qian t25], and Ertekin and Qian t7].
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2. Formulation of the problem
In order to clarify the physical
problem let us consider Fig. 1. This
shows a ship model moving along the
centerline of a shallow-water channel.
The dimensionless speed of the ship-model
is given by the depth Froude nether, Fh,
which is not necessarily critical. The
boundaries consist of the tank walls,
the center-plane symmetry axis if only
half of the physical region is to be
considered due to symmetry (mirror-image
problem), the two inlet and outlet
boundaries (or "open" boundaries) ahead
of and behind the model, and the no-flux
condition on the model. The channel
floor and the free-surface boundary
conditions are not discussed since these
are either exactly (in the case of G-N
equations) or approximately (in the case
of gB, KdV or K-P type equations)
satisfied by the particle velocity
vector. Then the problem can be solved
by using a nonlinear and unsteady
shallow-water wave equation to obtain
the unknown particle velocities created
by the movement of the model. One can
use neither linear nor steady-state
equations because of the nature of the
phenomenon. In fact, it can be shown
(Ertekin [3]), perhaps unsurprisingly,
that the steady form of the gB equations
used in this study predicts no
disturbance in the upstream region. The
same is true for other shallow-water wave
equations.
2.1 Boussines~ equations
The two different sets of
shallow-water equations that have been
applied frequently tosoliton-generation
problems in the past, namely the
Green-Naghdi equations and the
generalized Boussinesq equations, have
both advantages and disadvantages
compared with the other. Even though
the derivation of both of these equations
can start with the assumption that the
fluid is incompressible and inviscid,
only the gB equations require that the
flow be irrotational. This feature of
the gB equations allows one to consider
the layer mean value of the velocity
potential and the surface elevation as
the unknowns to be determined. On the
other hand, such a potential does not
exist in the case of the G-N equations
since the flow is, in general,
rotational. As a consequence, the G-N
equations are expressed in terms of the
unknown velocity components and the
surface elevation. The apparent
advantages of the G-N equations over the
gB equations were discussed in Ertekin
et al. t4, 5]. In a three-dimensional
problem with a large domain, the gB
equations are more efficient to solve
computationally. Therefore, we choose
to solve the following set ofgB equations
for a constant water depth and zero
atmospheric pressure (Wu [34]~:
t°+V ((h+~°)Vl°~=O, (1)
¢,o+21V¢°1 2 + gtO = h v260 (2)
where (x°,y°) are the coordinates of
the fixed coordinate system in which
x° specifies the direction opposite to
the movement of the ship, <° is the
surface elevation of the wave, {° is the
layer mean value of the velocity
potential defined by u =V4° in which
u =(u°,v°) , h is the undisturbed water
depth (constant) of the channel and V is
the two-dimensional gradient vector in
the horizontal plane. Eqs. (1) and (2)
are the statements of conservation of
mass and momentum, respectively.
The general form of these equations
in which the sea floor topography may
depend on the x°,y° coordinates and time
t° were obtained under the assumption
that the Ursell number is of order unity
(Ursell [32]~. However, they seem to be
valid in a wide range of Ursell numbers
as shown by Lee t16]. The velocity
potential and the surface elevation in
these equations depend on x°,y° and to.
These equations satisfy approximately
the nonlinear free-surface condition and
the sea-floor condition. The
configuration of the physical region is
shown in Fig. 2.
Before elaborating on the boundary
conditions and initial conditions to be
satisfied, we need to justify the use of
a vertical strut to model the conditions
of the experiments done by Ertekin [3].
In those experiments, the tank width,
the model draft and the water depth were
systematically changed to obtain 27
different blockage coefficients, Sb;
S - To
-
(3)
where Ao is the cross-sectional area of
the underwater portion of the full model
at midships at a given draft and W is
the half-width of the tank. Also, Fh is
varied to cover the range of 0.5-1.3.
The most striking finding of these
experiments was the dependence of the
soliton amplitude, speed and the period
of generation (the time that it takes
for the second soliton to generate) on
the blockage coefficient, Sb. Typical
experimental results have been given in
Ertekin et al. t4]. Therefore, it is
clear that one can use a vertical strut
423
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Representative terms from entire chapter:
computed surface
to model the same conditions which
existed during the experiments, i.e.,
hull form is of secondary importance.
Now, we can go back to the discussion
of the boundary conditions and initial
conditions. Because we assume that (see
Fig. 2) AO and BC are part of the symmetry
axis, only half of the physical domain
needs to be considered. The
computational advantage of this scheme
is obvious. We then have the following
boundary conditions. On the symmetry
axis AO and BC, and the channel wall DE,
the no-flux condition is
ye = 0.
On the ship boundary, which is moving in
the negative x° direction, we have
(Oonxo+tyonyo=-uonxo, (5)
where U° is the speed of the moving
boundary and n=(nX0,ny0) is the unit
normal vector of the ship boundary
pointing into the fluid. On the upstream
and downstream open boundaries, we use
Sommerfeld's radiation condition with
constant shallow-water wave celerity,
ides,
(tO~
(t+(U~ 1)
equations which we solve by a Successive H =-(~+1)V26-V; Vt
Over Relaxation (SOR) method. The
solution is accomplished by a point
iteration in which the initial guess is
chosen as a weighted average of the
boundary points. The boundary values of
x and y in the physical plane are
specified. A typical result for a
parabolic strut whose dimensionless
equation is given by
y=2Aw(l- ~ At,
where AW/2 is the half-width of the strut
and L is the length of the strut, is
shown in Fig. S. In this Figure AW/2=
0.3, L= 8.0, Sb= 7.5 %, and Ax=Ay=0.1.
However, only every fifth grid line is
plotted for clarity.
The accuracy of the elliptic
generation system used here has been
checked against a simple ideal flow
problem (doublet + uniform flow) which
has an exact solution, and found to agree
within a tolerance of 1 x 10-8. We note
that x = constant lines and ~ = constant
lines coincide in their respective
planes, and as a result, no coordinate
contraction has been used. The purpose
of this scheme is to preserve the symmetry
boundary AO and BC. If we had allowed
x to depend upon ~ also, then we would
have had to solve the problem in the
entire physical region, not just in half
of it. In that case, the boundary points
would not have been uniform.
3.2 Governing equations in the
transformed plane
and the Jacobian, J. of the
(26) transformation is J=x~y~-x~y'. Note
that even though the general formulas
are given here' x depends on ~ alone
since y = constant and ~ = constant are
parallel everywhere.
= - J2 (<+13(a¢~-20~+yl~-
12 (Yet' Yt, n) (Yntt - Yttn) +
J2 (Xq+t - x`~) (-x~' + x`~),
Similarly, the momentum equation (17)
becomes
(! + At,`` + Bl'6T'+ Char,,, = Q. (28)
where
A=-3 2, B=3 ~2, C=-3y2,
Q = _t, _ ~ (42 + 42 _ U2) + 3 UV Ax = ~< ~
~
CO=2(Y~` yarn)
= x~yr,x~` - x~y~x~r, + YLYT\Y\E - Y' YET] ~
And the dimensionless boundary and
initial conditions (18)-(21) become as
follows.
t~=0 on AO, BC and DE (29)
-~+~=0 on AB. (30)
(t = ZI,2 ~
t! = Ply,
where
ZI.2 = - ~ (Yq<~- Yawn),
PI,2=-(U~ 1~(~(Y~-Ytl~-U),
on ED (subscript 1) and CD (subscript
2),
t=0, t=Ux(~), at t=O-.
As before, the subscripts ~ and ~ denote
the derivatives with respect to these
variables. We are now ready to solve
equations (27) and (28) subject to the
boundary conditions (29~-~32~.
4. Numerical solution of gB equations
and wave resistance
In this Section, we present the method
employed to solve the nonlinear and
unsteady shallow-water equations as
given in the last Section, and also, give
the formulation for wave resistance.
Some of the results obtained will be
presented and compared directly with the
available experimental data. The
accuracy of the equations and the
numerical method will be revealed when
we discuss the speed, amplitude and the
period of generation of solitons, and
wave resistance.
4.1 Finite-difference method
Because of the success of the Modified
Euler Method in previous applications,
we use this two-step method to march in
time. The spatial derivatives are
approximated by second-order
central-difference formulas. The
truncation error of the resulting
equations is O ~ A X 2 , ~ y 2 , Atop. In this
method, the values of ~ and ~ at the
present time step, indicated by the
superscript (2), may be obtained by
modifying the values obtained in the
middle step, indicated by the superscript
(m). The middle-step values are
evaluated by modifying the
previous-time-step values, indicated by
the superscript (1).
With these preliminaries, the
continuity equation (27) becomes
him) = t(~) + AtHtI)
(33)
t t2' = t t ~ ' + ^ t t H ` ~ ' + H tm, j , ( 34 )
where At is the time interval and H was
31 given by (27). Similarly the momentum
equation (28) becomes
+ A lt~ ~ + B. `n ~ + Ct ~ ~ ~ =
ho'+ Al `~, + Bait'+ Ctt''+ AtQt',, (353
lt2) + A+tt, + B+~2, + cash = t(~) + Attt'+
Bl t2'+ C¢t2, + ^t Its + Qtm,' ~ (36)
where A, B. C and Q were defined by (28).
The upstream and downstream
open-boundary conditions become
<;tm) = itch) + ~tzt~2,
(37)
tt2~=
Haussling [11], and Ertekin et al. t5])
is applied to reduce the effect of the
numerical errors which are caused by
high-frequency cross-channel waves.
and ~ are filtered in the ~ and ~
directions in the transformed plane after
each time step. The SOR parameter varied
between I.5 and 1.9 in our calculations.
4.2 Wave resistance
The wave resistance, R. experienced
by the vertical strut can be obtained by
evaluating the pressure, Ps, on the
strut, i.e.
-R(t)=pgh3= 2 J ( J Ps dZ)YxdX (41)
The pressure on the strut can be obtained
from Euler's integral once the velocity
potential and the surface elevation are
determined. Following Wu t34] and Wang
et al. t33], we obtain
PsdZ=~(1 +~)~1 I+
-h 2
(42+ 24-2)(~+2(62+~2-U2)+~), (42)
in the moving coordinates. Equation (42)
is obtained by expanding the pressure in
a perturbation series where the error
term is O(es) , and ~ is the dispersion
parameter. One can now substitute (42)
into (41) and obtain the wave resistance
as a function of time after transforming
the variables from the (x,y) system to
(5,~) system.
4.3 Discussion on the numerical
Accuracy and results
When the water depth, h, is constant,
the conservation of mass statement given
by (~) is exact. Therefore, this equation
can serve as a check on the numerical
accuracy of the results. We note that
the same argument cannot be made by using
the conservation of linear momentum
statement given by (2) since this
equation is approximate. The
dimensionless water mass, m/ph3, given
by
J ( 1 + t) Jd A5n + J :; Sd ~ - ~ Sd Id t, (43)
where
S(5, A, t ) = ~ ( 1 + t) (yntE - yips),
and lo, tR represent the left and the
right open boundaries, respectively, and
428
All denotes the surface of the
computational region, remains constant
within 1%, showing that numerical
accuracy is very high.
The gB equations are solved by a
computer program which uses the grid
points generated by another computer
program. Two sets of numerical tests
are conducted and the parameters of the
problem are selected to approximately
correspond to the experiments done by
Ertekin [3].
In the first set of the numerical
tests, two different blockage
coefficients, 7.5% and 12.5%, are
selected. The grid size on boundaries
is chosen as Ax=Ay= 0.1, and the time
interval is chosen as At=0.0Seven though
Von Neumann's stability method applied
shows that At can be the same as Ax
without any stability problems. Two
different speeds, Fh=U=0.8 and 1.0 , are
used so that precursor solitons can be
generated quickly. The perspective plots
(see Fig. 6 a,b) as well as the contour
plots (see Fig. 7 a,b) are shown here
for Case 3. For other results see Qian
[25]. In perspective plots the strut
below the lowest wave surface could not
be plotted because of the limitation of
the graphics software that we used. The
period between the first and second
soliton or the period of generation, the
amplitude and the speed of the first and
second solitons are calculated (see
Tables 1-3). The period of generation
is obtained from the numerical
moving-gauge results as shown in Fig. 8
(continuous lines). In Fig. 8 the
location of the numerical gauges are at
distances of0.625L (Gauge2) and L (Gauge
3) ahead of the strut on the symmetry
axis. Note that in the first set of
numerical tests (Cases 1,2 and 3), W=4,
L=8 and [R-l' =64. Comparisons with the
experiments (Ertekin t3] and Ertekin et
al. t4] are also made in Tables 1-3.
To understand the effect of strip length
on the computational results, we
conducted another numerical test by
increasing the ship length to 15.2 but
leaving all other parameters of the
problem the same. This test is referred
to as Case 4. Case 4 results are shown
in Figs. 6 c,d and 7 c,d, and in Fig. 8
(dotted lines). As can be seen from
these Figures, shorter strut has
considerably more wave build-up around
the bow and stern than the longer strut,
and also, the downstream waves are
smaller in the longer strut case. From
Fig. 8 we see that it takes longer to
generate the first soliton in the longer
strut case than the shorter strut case.
Also the amplitude of the first soliton
is smaller, and the amplitude of the
second soliton is larger in Case 4
compared with Case 3 (see also, Tables
1-3~.
In the second set of numerical tests,
we used exactly the same Sb=14.2,W=6.1
and L=15.2 used in one set of the
experiments of Ertekin [3]. In the
experiments, h=10 cm, d=7.5 cm and Fh=1.0
were used (Test No. 1936~. The location
of the moving gauges were given inErtekin
etal. [4]. This numerical test is called
Case 5. In Case 5, Ax=Ay= =0.15 and
At=0.08 are used since the computational
region is larger than before. Also! R - I ~
=137.25 in this case. The perspective
plots of Case 5 results are shown in Fig.
9, and the contour plots in Fig. 10. The
numerical wave-gauge results end the wave
resistance results are shown in Fig. 11.
The corresponding experimental results
are shown in Fig. 12. The numerical
results in Fig. 11 are shifted in time
so that the moment the towing carriage
started moving in Fig. 12 (see the sudden
increase in resistance at t=2.5 s)
corresponds to t=0 s in calculations.
This shift in time in experiments is due
to starting data acquisition before the
carriage is put into motion. The
quantitative differences between the
experiments and calculations in Case 5
are given in Tables 1-3. The agreement
can be considered quite good.
The average total resistance measured
during the experiments was 9.3 newtons
for Case 5. In the calculations the
average wave resistance is obtained as
9.6 newtons. We estimate the frictional
resistance as 1.4 newtons from
Schoenherr's flat plate skin friction
formula. Of course, the eddy or form
resistance must be included also. Thus,
one can conclude that the total
resistance predicted is slightly higher
than the experimental datum. Some other
cases which are not presented here also
support this conclusion.
In conjunction with the numerical
schemes used in this study, we mention
that the surface elevation and the
velocity potential are filtered along
both the ~ and ~ directions at each
time step. The average number of
iterations is 15 for the first-step
solution (for a tolerance of 8 x 10-7)
and 6 for the modified or second-step
solution (for a tolerance of 5 x 10-7).
The computations, which were performed
on the X-MP/48 Cray Computer of the San
Diego Supercomputer center, took about
90 minutes of CPU time for 2200 time
steps in Case 3. Single precision is
used on this 64 bit machine. Filtering
was absolutely necessary in our
calculations. Removal of the filtering
scheme caused very high waves at the
429
stern, and eventually, the computations
could not be continued. The unphysical
waves in the absence of filtering
occurred for various grid sizes and time
intervals.
From the results presented here, we
see that general appearance of the waves
agree with the results of earlier
research (e.g., Huang et al. [13],
Ertekin [3], and Ertekin et al. t5]~-
For each soliton, including the first
and the second one in all cases, the
amplitude increases rapidly while the
speed increases. The relation between
the amplitude and speed of the solitons
(computed) satisfies approximately the
dispersion relation for a general two
dimensional soliton propagation, such as
the second-order formula proposed by
Laitone [15], and a formula derived by
Schember t26]. However, solitons
generated by ships do not have an exact
permanent form. The amplitude of each
soliton gradually increases after
generation. This, of course, may be
partially due to the governing equations
which satisfy the free-surface
conditions only approximately. Also,
the rather small differences between the
computations and experiments may be also
due to the neglect of viscosity (although
this effect must be very small), and, of
course, computational as well as
experimental errors, especially in the
case of period of soliton-generation.
The soliton generation clearly depends
on the blockage coefficient and speed of
the moving strut, but it also slightly
depends on the length of the strut. An
increase in the Froude number, or the
speed of the ship, can delay the
generation of the first soliton, and thus
increase the period between the first
and second solitons. Soliton amplitudes
and speeds increase correspondingly.
Fast movement of the disturbance
increases the period of soliton
generation and larger blockage
coefficient decreases the period of
soliton generation. The second soliton
almost always comes out with a smaller
amplitude and speed than the first
soliton. As can be seen in Table 2, the
agreement between the experiments and
computations for the amplitude and speed
of the second soliton is very good.
5. Conclusions
The waves generated by a ship in a
shallow-water channel can be modeled by
numerical calculations of Boussinesq
equations, and two-dimensional solitons
may be generated ahead of the ship. The
shallow-water equations of gB type can
successfully simulate this phenomenon.
The generation of a boundary-fitted
curvilinear coordinate system is very
effective for the configuration
containing the ship boundary. The
physical problem is solved in the
transformed plane with a uniform grid
size. The boundary conditions,
especially on the ship boundary, can be
satisfied without the need for a special
treatment of the boundary values. 1.
However, all equations become more
complicated after the transformation,
and thus computations become much more
intensive. Since the grid size is chosen 2.
to be uniform on the physical boundaries,
equal ~ lines become straight and are
perpendicular to the symmetry boundary 3.
and wall boundary. On the symmetry
boundary and wall boundary, the no-flux
boundary condition can be simplified,
and also the symmetry condition can be
used to deal with the third-order
derivatives. If we use coordinate system 4.
control (attraction) for the grid
generation, the grid lines will not be
generally straight, and the no-flux
boundary condition will become more
complicated on all solid boundaries, and
also, no symmetry condition can be used.
The constant wave celerity used on the
open boundaries work out perfectly.
Therefore, it seems that one cannot
easily justify the use of a numerical
scheme in which the phase speed is
calculated (see for instance, Orlanski 6.
[23]~.
More efficient methods of numerical
grid generation can be investigated and
used with three-dimensional ship
surfaces to reduce the errors introduced 7.
by numerical grid generation.
Coordinate-system control is recommended
in order that the coordinate lines can
be attracted by the boundaries,
especially around the ship boundary, so
that the boundary effects can be
considered more accurately. This 8.
obviously requires that we can deal
successfully with the higher-order
derivatives on the boundaries.
Nevertheless, the symmetry boundaries
will be destroyed in this case, causing 9.
the necessity to solve the problem in
the entire physical domain.
Acknowledgements
We are grateful to Prof. John V.
Wehausen for his invaluable comments
throughout this research. Our
appreciation also goes to Prof. Theodore
Y.-T. Wu who made a very timely comment
on the resistance formulation. We would
like to thank the International Business
Machines Corporation for providing the
computer equipment used in this study.
This material is based upon work
supported by the National Science
Foundation under Grant Numbers
430
MSM-8706910 and BCS-8958346. The
computer time was provided under Account
Numbers 941 HAW and 234 HAW awarded by
the San Diego Supercomputer Center.
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Case Sb As us
%
L I ~ l | E | D | c T | ~ %,
1 0.8 7.5 0.18 0.15 20.0 1.05 1.07 -1.9
2 ~ 1.0 ~ 7.5 0.54 0.50 ~ 8.0 ~ 1.20 ~22 -1.6
3 ~ 0.8 12.5 0.29 0.27 7.4 1.12 - 1.11 -
4 0.8 12.5 0.25 0.27 -7.4 1.12 1.11 0.9
5 ~ 1.0 14.2 ~ 0.70 0.62 ~ 12.9 1.27 1.23 - - 3.2
Table 1. Dimensionless amplitude and speed of the
first soliton.
Case Sb As us
No. ~ ~ (%) ~ ~ Ah
I I ~1 T ~ %, T I | ~ %,
1 ~ -0.8 ~ _ 7.5 0.11 T 0.12 1 -8.8 ~ 1.02 1 1-03 1 -1~0
2 ~ 1.0 7.5 0.50 ~ 0.49 ~ 2.0 ~ 1.20 1 1.18 1 1.7
0.8 12.5 0.21 0.23 -8.7 - 1.09 1 1-09 T °~°
4 0.8 12.5 0.24 0.23 4.3 1.09 1.09 0.0
1.0 14.2 0.69 0.56 23.2 1.27 1.23 3.2
,. ~
Table 2. Dimensionless amplitude and speed of the
Notes on the Tables:
C : Present computational results,
E : Experimental data (Ertekin [3]),
D : Difference between the calcula-
tions and experimental data,
Sb: Blockage coefficient,
h : Depth of undisturbed water,
g : Gravitational acceleration,
U : Speed of the ship,
As: Amplitude of the first or second
soliton,
Us: Speed of the first or second soli-
ton in earth-bound coordinates,
Period of generation of solitons.
second soliton.
Case
No.
1
2
3
4
5
U (sib) UTV
C ~ E | D
0.8 7.5 31 34 -8.8
1.0 7.5 44- 47 -6.4
0.8 12.5 21 ~ 23 -8.7
0.8 12.5 20 ~ ~23 -13.1
1.0 14.2 27 33 -18.2
Table 3. Dimensionless period between
the first and the second soliton as
observed in the moving coordinates.
432
- ~ ~
~ c
_ ,~ —9
L; ~t
I , ~ ~ ~TAN~WALL ~ (a,
~ ~ ,
i,,,, ,, At, ~ =E
s .... .
l SOLITON CREST l
l ~ " _. _ I
j-N
_ I ~ ~/~/~ _ _ ~ ( b )
, F ~ ~ _ _ _ ~ _ -. , . , , _. ~
Figure 1. (a) Sketches of crest patterns ~t
during the emergence of a soliton in a
moving reference of frame, (b)
superposed sketches of (a) and a train
of solitons (from Ertekin [3]~.
yO
~11 ba~ndazy
~ /-
,,~ / ,,,
/
~ ,~ / up 2~
O \ .
s~~ bout_
~ , 1
~ / c x°
Figure 2. Configuration of the physical
region.
Y' ~
Lit
X'
xo
Figure 3. Fixed (x°,y°) and moving
(x',y') coordinate systems.
j,1
- 1 M1
Figure 4. Physical (x,y) and
computational (hi) planes.
D
_ Dad
4.0-
3.0 —
A:
\2.0 —
>I
1.0 —
0.0 —
4.0
x/h
8.0
Figure 5. Numerically generated grid
for the parabolic strut, Cases 1 and 2.
433
(a) U * T / h = 20
A A ~ A ~ e ~ ~ ~ ~
-24.0 ~1 6.0 - 8.u u.u O.~ 1~.w ^~.v 32.0 40.0
. . . r ~
0.0 I ~ l
-2 t.0 -16.0 -8.0 0.0 8.0 16.0 24.0 t2.0 40.0
~ ~)
\ ~` —1
_ ~
~ __
N~' _
~ \
~ ~ _
(~) U ~ T / h = 20
<~` ~-
_
_ _
_
~' _-
`` \
`
, ~
( b) U ~ T / h = 40
(b) U * T / h = 40
-24.0 ~ ~ ~ _Q ~ n n Rn 16.0 24.0 32.0 40.0
—16.0 - 8.0 0.0 8.0 16.0 2J'.0 32.0 40.0
(c) U ~ T / h = 20
16.8 -8.0 O.B 9.6 18.4 27.2 36.0 44.8
1 ~
( d) U ~ T / h = 40
Figure 6. Perspective plots of computed
surface elevation (a) and (b): Case 3
(shorter strut), (c) and (d): Case 4
(longer strut).
434
(d) U * T / h = 40
~ Pt q ~ 18.4 27.2
0.0-
- ~ 6.S -~.0 0.S 9.6 ~ B.4 27.2 36.0 44.
Figure 7. Contour plots of computed
surface elevation (a) and (b): Case 3
(shorter strut), (c) and (d): Case 4
(longer strut).
0.65
\ 0.4 -
o
a) -
0.2 -
O; 1~- m] ~O ~ ~ ·) ~ ~D d]
0.0
o.o 1 o.o 20.0 30.0 40.0 50.0 60.0 70.0 ~o.o
(gauge2) U *T / h
0.6
5:
\0.4
N
0.2
o.o
/" \\N ,/ \ / ' '
1
o.o 1 o.o 20.0 30.0 40.0 50.0 60.0 70.0 80.0
(gauge3) U * T / h
Figure 8. Computed surface elevation as
observed by numerical
moving-wave-gauges. ~ ): Case 3,
(-----~: Case 4.
U ~T / h = 60
-45.75 -30.s5 -15.35 -0.15 15.05 30.2s 4s.4s 60.65 7s.85
-45.75 -30.5s - 1 s.35 -o. 1 5 ~ 5.~ 30.25 ~s.45 60.65 75.85
l) * T / h = 100
-4s.7s -30.5s -15.35 -0.15 15.05 30.2s ~s.4s 60.65 7s.8s
ooo L v.w
-45.75 -30.5s -15.35 -0.15 15.05 30.2s 45.4s 60.6s 7s.8s
U * T / h = 140
-45.75 -30.5s -15.35 —0. 1 5 1 5 ~ .~n 's
-45.75 -30.55 -15.35 -0.15 15.05 30.25 4s.45 60.65 7s.8s
Figure 1O. Contour plots of computed
surface elevation, Case 5.
~ c
U ~ T / h = 1 ~P
. _ ,~,.
U ~ T / h = 140
_
Figure 9. Perspective plots of computed
surface elevation J Case 5 e
435
.o -
~ 8.0 —
_ 6.0—
L~ _
4.0 —
2.0—
0.0 —
/
i I l I l l
.o ~ ~ ~ o.o 20.0 30.0
8.0 — TIME (SEC)
~, _
<~, 60- ~\~M; 1 ,
o.o
0.0 Q ~ 10.0 20.0 Jo.o
8.0 — Tl M E (SEC)
~ 2.0- J:~\
o.o ~
10.0 Q ~ 10.0 20.0 3c.o
:~ 8.0 — Tl M E (S EC)
=60- J~J
o.o — ,
, ~ 1 1 ~ 1
o.o 1 o.o 20.0 ~o.o
r'
l
o.o 1 o.o 20.0 30.0
TIME (SEC)
Figure 11. Computed surface elevation as observed at 4 numerical
moving-wave-gauges and wave resistance, Case 5.
10
u 6
2
<
o
10
6
2
N
10 20 30
{` ~__-
1 - I 1 ~
L'
-
:, 2
0
6
o
6
w 2
o
o 20
:.
w
a
O
10 20 30
~1 340
t_ ~ -
~1 1
1 0
3 0
2 0
10T ~__~
0 10 20 30
Time (see)
Figure 12. The experimental data of surface elevation and total resistance
(Test No. 1936), Fh=l.O, h=10 cm, draft=7.5 cm, W=61 cm, moving gauges.
436
DISCUSSION
by J.W. Kim
The authors should be congratulated on
their successful extension of Boussinesque
type equation to 3-dimensional free-surface
flow problems. We would like to make comments
on the following two points. Author's Reply
The first is the time integration method
adopted in the present paper. As pointed out
in your earlier paper(1984) [Al] numerical
solutions of the gB equation for a disturbance
moving at near the critical speed show the
gradual increase of the amplitude of upstream
solitons whereas the ON equation shows nearly
constant amplitude. We have also reproduced
this numerical calculation based on the 4th
order Runge Kutta method for the time
integration and then we have obtained the
constant amplitude for both equations. The
only difference between ours and your previous
calculation is in the time integration method,
where you used the Modified Euler Method
(MEM). In the present paper you also used MEM
for the time integration. From our experience
in both time integration methods, we believe
that the difficulties you experienced in your
present work before introducing the filtering
process in mainly due to the inadequacy of MEM
for this problem.
The second point I would like to discuss
is on the treatment of the convection term.
Although the gB equation given in your paper
has no convection term explicitly, the
perturbed velocity potential and wave
elevation will have convection effect im-
plicitly since the basic flow is uniform
stream. For the convection operator it is
well-known that the central difference scheme
in spatial discretization has a considerable
phase error on short waves with wave length
comparable to the mesh size. We have also
experienced similar difficulties in our
calculation by the finite element method. As
a remedy for this difficulty we used the par-
tial upwinding scheme.
From the above two points, we believe that the
filtering process in your computation scheme
has presumably played a role of eliminating
the overall combined effect of the two dif-
ficulties due to MEM and convection.
[Al] Ertekin, R.C., Webster, W.C. & Wehausen,
J.V., Ship-generated Solitons, Proc. 15th
Symp. Naval Hydrodynamics, Hamburg, 1984
pp.347-364.
We would like to thank Mr. Kim for a very
useful and timely discussion.
Your comments on the use of MEM and its
effects on the numerical predictions are
agreeable. It is clear to us that a fourth-
order method such as the RK4 method used will
produce more accurate results. However, it is
not very clear that reducing the truncation
errors by using a higher order scheme will
totally eliminate the continuous amplitude
increases observed when one uses the gB
equations. One must keep in mind that the
momentum and, therefore, the energy is not
exactly conserved in gB equations even if the
sea floor is flat. When the disturbance is
small, the amplitude increase may be reduced,
perhaps to a minimum. But we do not think
that it can be eliminated when large
disturbances are used. The same is not true
when the Green-Naghdi equations are used since
these equations satisfy the conservation of
mass and energy exactly, even if the sea floor
is not flat. On the other hand, one may
justifiably argue that if the disturbance is
large, then the assumptions behind the
derivation of the gB equations are violated,
and therefore one should not expect that the
amplitude does not grow. We agree with this
agreement.
It is true that the MEM causes high
frequency waves because of the presence of
central differencing. That is why we used
filtering. Filtering eliminates high
frequency waves which occur around and behind
the hull. But, obviously, it does not effect
the very low-frequency waves ahead of the hull
in the upstream region. Since these waves are
the primary concern to us, we did not worry
about using MEM which proved to be a very
valuable scheme because of its efficiency
compared to a higher-order scheme.
437
