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OCR for page 428
24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Complele Cancellation of Ship Waves
in a Narrow Shallow Channel
Xue-Nong Cheni, Som Deo Sharma2, Norbert Stuntz3
(i Forschungszentrum Karlsruhe, 2 Gerhard Mercator University Duisburg,
3 VBD-European Development Centre for Inland and Coastal Navigation Duisburg,
Germany)
ABSTRACT
In this paper we report our recent experimental find-
ing of the complete wave cancellation for a particular
configuration of a ship in a narrow shallow channel.
Chen & Sharma (1996,1997) predicted theoretically
that a twin-oblique-soliton wave pattern can be fitted
in the space between a ship and each of the two chan-
nel sidewalls such that no waves exist either upstream
or downstream and, therefore, the ship does not suf-
fer any wave resistance. Recently, we carried out a
fundamental model experiment in the VBD towing
tank to validate this prediction. For designing the ex-
periment an updated stationary shallow-water wave
model of Boussinesq type was applied in the far-
field, which allows only an approximate twin-soliton
solution, while in the near-field an improved slender
body theory was applied to obtain the required ship
cross-sectional area and waterline inversely from the
far-field solution. The experiment showed that in the
exact design condition the ship's (reflected) bow and
stern waves cancelled each other so completely that
almost no waves were observed behind the ship and
the measured total resistance was reduced so much
that the estimated wave resistance became practically
zero. In fact, an apparently slightly negative wave re-
sistance provoked a scrutiny of the common viscous-
resistance formulation. Additionally, numerical cal-
culations of wave profile and wave resistance using
an Euler solver were performed to supplement the
theoretical and experimental results.
INTRODUCTION
By purely theoretical analysis, albeit inspired partly
by previous experimental results, Chen & Sharma
discovered and reported in (1996, 1997) that when
a slender ship moves in a narrow shallow channel at
a supercritical speed, its bow and stern waves can be
made to cancel each other so completely by a proper
choice of hull-channel geometry that there are no free
waves behind the ship and so it experiences theoret-
ically no wave resistance. This result was subject to
the restriction that the theory accounts only for weak
nonlinearity and ignores viscosity. The present paper
reports results of our recent model experiments in the
VBD Shallow Water Towing Tank that confirm the
existence of a practically waveless hull-channel con-
figuration in reality.
The present model experiment was designed
by means of an analytical theory, taking advantage
of the improvements achieved since the original dis-
covery in 1994-95, cf. Chen (19991. The predicted
waveless state occurred exactly at the design speed
to an extent never observed before.
At supercritical speeds, i.e., V > /, where
V is the ship speed, 9 the acceleration due to gravity
and h the water depth, the ship wave pattern looks
like the shock waves of a 2-D airfoil in supersonic
flight. Both bow and stern waves extend aft obliquely
along their characteristic lines. But normally the bow
wave is a free surface elevation; the stern wave, a
depression. By the nature of nonlinear shallow wa-
ter waves, the elevation can form a permanent pure
oblique soliton, provided the bow has a certain shape,
but the depression can never form a permanent "neg-
ative soliton". If the ship now moves symmetrically
OCR for page 429
along the centerline of a narrow shallow channel of
rectangular cross-section, its waves must be reflected
by the vertical sidewalls. If the hull-channel geom-
etry is adapted to the chosen depth Froude number
as required by the theory, the bow wave after reflec-
tion from the channel sidewall would hit the after-
body and cancel the stern wave so that at the design
speed the resultant wave in the ship wake would dis-
appear totally, see Fig. 1. We have previously pro-
posed the name "shallow channel superconductivity"
for this phenomenon.
BY
,~,,,,,,,,,~""";
,~
,...........................................
\~N
~~ V
.~ > 1
Nigh
z~,~.
/
/
,,, ~~ ~~ z~
Figure 1: Schematic of ship wave pattern at super-
critical speed in a narrow shallow channel.
In this paper we present briefly the mathemat-
ical model, its refined solution and a new ship de-
sign based on it. The experimental results are shown
in detail as measured resistance and wave profiles,
illustrated by photographs. A comparison with the-
oretical wave profiles is carried out. Some discus-
sion of the friction line used to estimate the experi-
mental wave resistance is included. Additionally, a
fully nonlinear free-surface Euler numerical solution
is presented. A video clip of the model experiment,
documenting the waveless state, will be shown at the
Symposium.
MATHEMATICAL MODEL AND SOLUTION
Here we describe the mathematical model and its so-
lution only briefly and refer the reader for details
to a parallel journal paper Chen, Sharma & Stuntz
(20021. Unless otherwise stated, all variables are
nondimensionalized by reference to water depth h,
acceleration due to gravity 9, and water density p.
The flow is governed by a 2-D steady shallow-water
wave equation,
(1—U2)ixx + My + U~x~yy + 3Uix~xx
where
+2U~yi~y + 3 (a==== + ~==yy) = 0, (1)
where So is a depth-averaged velocity potential and
(nondimensional) ship speed U = V/ >/~ is identi-
cal to the depth Froude number Fnh. The wave ele-
vation ~ can be approximately expressed as
(= Unix
(2)
In an improved slender body theory the bound-
ary condition at the ship position becomes
S9Y(X, +0) = ~ 2~1 + 5) { die t(U—(P=)Sf (x)],,
J
(3)
Sf (x) = So(x) + (s + ~X)b(x) + ((X)b(x), (4)
s is sinkage and ~ is trim angle (positive bow down-
ward).
The no-flux boundary condition on the parallel
vertical channel sidewalls simply reads,
(x, iw/2) = 0. (5)
It is difficult to find an exact twin-soliton solu-
tion for Eq. (1) but one can obtain a good approxima-
tion after Miles' weak-interaction solution (1977~. It
is of the form
~ = F (A + feed (a)) + G (9 + fc~2 (I)) (6)
with
= x + cot °~1Y—x1,
71 = x + cot ct2Y + X2,
where F and G are single-soliton or solitary wave-
train solutions, Al and ~2 are phase-shift functions
due to the twin-soliton interaction, x1 and x2 are ini-
tial phase constants, and °~1 and a2 are the angles be-
tween the phase lines and the positive x-axis. It is un-
derstood that the two component solitons are propa-
gating in directions with equal positive x-component
and possibly unequal but opposite y-components,
implying that cot out and cot C>2 have different signs.
The phase shift functions are found to be
i = 2(U2~+ t2 ) 1) Gin),
U¢3—Cot2 ~~ ) Fig)
2(U2 + cot2 ~1 - 1)
2
OCR for page 430
The single-soliton solutions for F and G are of the
same form, that is,
F(~) = A1/kl tank,
G(r') = A2/k2 tanh~k271),
cottony = I + A ~~ -1, ki = 2U '
where Ai is the amplitude and ki the wave number of
the soliton. To ensure exact symmetries about x = 0
and y = w/2, we set AT = A2 = A, kit = k2 =
k, x~ = x2 = so, cot Ott = cot or and cot 0~2 =
—cot a, where
cot a = >/(U2 - 1—AU)/(1 + AU).
The boundary condition on the channel sidewalls Eq.
(5) is satisfied if the channel width is
W = Pro/ cot or.
Finally, fC in Eq. (6) is an improvement factor that is
here assigned a value of 2.3 in the design condition.
A theoretical solution corresponding to the following
design is shown in Fig. 2.
Figure 2: Theoretic solution of the ship wave pattern
in the narrow channel. Here by virtue of symmetry
only half of the free surface for w/2 ~ y > 0 is
shown (height magnified ten times).
DESIGN
The design task consists of two steps. First, the cross-
sectional area curve of the ship and the appropriate
channel width are determined by theory for a chosen
depth Froude number and a trial twin-soliton, requir-
ing some iteration to achieve a prescribed displace-
ment. Second, the detailed section shape, not pre-
scribed by the theory, is selected to meet the practical
demand of a fair and simple hull geometry. Also for
simplicity, a "fixed" towing mode is assumed, i.e.,
no running sinkage and trim are permitted. This is
not an inherent limitation imposed by the theory. But
it does facilitate the design of the experiment by ex-
cluding possible differences between the theoretical
and real values of sinkage and trim in the "free" tow-
ing mode. The trim, by the way, is always zero in
theory for a fore-and-aft symmetric hull form in the
waveless state.
In principle, three nondimensional parameters,
Go, A and U. can be freely chosen. The theory would
then yield a nondimensional hull-channel configura-
tion, i.e., a family of geosims with the property of
being waveless at the design Froude number U. In
practice, the physical model experiment has to be
conducted on an object of given absolute size. The
VBD towing tank available to us is 200 m long, 9.8 m
wide and 1.3 m deep. The water depth is varied eas-
ily and routinely by pumping water into another tank,
while the width can be varied on demand by erect-
ing a temporary intermediate wall with considerable
yet justifiable effort. (It suffices to erect the interme-
diate wall over the middle 80 m of the tank length
where measurements are taken, leaving the run-in
and roll-out stretches of the tank undivided.) Since
it was obvious from previous designs that promising
depth Froude numbers lay around 1.5 and the ratio
of channel width to ship length around 0.5, we ar-
bitrarily specified U = ok, along with h = 0.2 m
to ensure a feasible towing speed, and w = 3.8 m
to ensure a reasonable model length. With absolute
size and two nondimensional parameters, namely U
and with, now fixed, we were still free to manipu-
late the pair A and JO to obtain any desired hull dis-
placement within a certain range. We finally settled
for A = 0.15 and x0 = 7.65636 as a compromise
between extreme hull slenderness and unacceptable
wave nonlinearity.
The dynamic cross-sectional area curve Sf (x)
can be obtained by integrating Eq. (3~:
Aft) =—U ~ ~y(1 + ()dx, y = +0.
(7)
However, in order to get the static cross-
sectional area curve SO (x) from Eq. (4), we need to
prescribe somehow the beam box) at the waterline.
Partly anticipating the later determination of section
shapes on practical grounds, we assume a uniform
draft d and a uniform sectional area coefficient CM
3
OCR for page 431
over the entire hull length. It follows immediately
that
b(x) = So(x)/(CMd) (8)
Substituting this into Eq. (4) we obtain easily
So(x) = Sf (x) [1 +
((x, +O) ~ (9)
Returning now to the prescription of section
shapes, we ensure fair waterlines by declaring all sec-
tions to be affine transforms and, hence, defined by a
single nondimensional function
y/(b(x)/2) = f (z/d), z =—d, ...0.
Further, we ensure mathematical simplicity by arbi-
trarily choosing an exponential function
f (z') = t1—expel—7.5(z' + 1)~/~1 - exp( - 7.5)],
which can be integrated in closed form to yield a uni-
forrn sectional-area coefficient
To
CM= J f(z')dz'.
-1
Now, we can either freely choose uniform draft d and
determine maximum beam b(0) to comply with Eq.
(8) or vice versa. Our specific choice was a round
draft d = 15 cm.
One final detail remains to be explained. The
theoretical sectional-area curve S(x) extends from
minus infinity to plus infinity, implying an unrealis-
tic ship of infinite length. Luckily, the curve decays
exponentially so that an approximate practical hull
form of finite length can be acquired by simple trun-
cation of the bow and stern cusps. In absolute terms,
we decided upon a round ship length of 6 m leaving
the stem and stern as sharp edges of 2.7 mm thick-
ness.
,
Item Symbol Value/Unit
Length at waterline 6 m
Beam at midship bm 0.3892 m
Draft d 0.15 m
Area of Sm 0.05063 m2
midship section
Displacement V 0.1283 m3
Wetted surface area So, 2.437 m2
Block coefficient CB = V/l bm d 0.3663
Midship coefficient Crf = Sm/bmd 0.8672
Wetted Cw s = So / ~ 2.7776
surface coefficient
Length/Depth I /h 30
Draft/Depth d / h 0.75
.
Design speed V 1.9803 ms~
Design depth U = V/ ~ 1.414
Froude number
Design water depth 0.2 m
Design w 3.8 m
. Width/Depth W / h 19
Table 1: Principal dimensions of the model hull and
channel
The principal dimensions of the final design are
compiled in Table 1. The body plan is reproduced in
Fig. 3; the cross-sectional area curve, in Fig. 4.
0.15
n ~
_
0.05
N
o
-
-0.05
-O.1
rrr
: ~ i~
~ ~ it.
1 _
-
-0.15 0.2 -0.1 0.1 0.2
y [m]
Figure 3: Body plan of the model hull with 21 uni-
formly spaced sections
4
OCR for page 432
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OCR for page 440
Representative terms from entire chapter:
frictional resistance
1E,
0.8
0.6
-
o
us o.4
0.2 .
n
. . . . . . . . . . . . . . . . . .
0 10 20
n
30 40
Figure 4: Cross-sectional area curve from stern (sta-
tion 0) to stem (station 40)
MODEL EXPERIMENT
The physical model experiment was carried out ac-
cording to the foregoing theoretical design. The ship
model was towed at a number of steady speeds en-
compassing a wide range around the design value.
Beside the design configuration (h = 0.2 m,
w = 3.8 m) two off-design configurations were also
tested, namely, full tank width (h = 0.2 m, w =
9.8 m) to approximate laterally unrestricted water as
a reference, and a larger water depth (h = 0.3 m,
w = 3.8 m) to study the effect of depth variation.
All runs were executed in the so-called "fixed" mode
to preclude any complications due to running trim or
linkage.
Results at Design Depth
Fig. 5 shows our main result, namely, the curves of
specific total resistance RT/(P9V) measured in the
narrow channel of design width (solid line connect-
ing dots) and in the undivided tank of full width
(dashed line connecting circles), both at the same
design depth. For the purpose of estimating wave
resistance, which unfortunately cannot be measured
directly, two curves of specific frictional resistance
RF/(
~o:o~,- L: 6-~_ ~~
-0.2 ~ ·~-~` --' A I- rid
-0.4
~ -9 -1 0 1
_ ~) ~
. . . .
0.4
0.2 -f" -id
O ~7
-0.2
-0.4
0.4
0.2
-0.2
-0.4
0.4
0.2
-0.2
-0.4
0.4
0.2
-0.2
-0.4
.'t ,%.
7 ~
1
IN a\
" ,,_,_~% ~ `~- J \
At," -~ %
'_ ail'
-__
.
, , , . . , . . . . . , . . . ,
-3 -2 -1 0 1
. . . . . , , . . . , . . , .~. . . , ~
~ 1 - I I - Jl' ~
1
. - . . , , . . , . , . . . . . , . 1 , . . . ~
-3 -2 -1 0 1
~ ' ' ' ' ~
. . . .
0.41
0.2t i-
O ~:~
Figure 6: Measured wave profiles at the design depth Froude number Fob = 1.414 (h = 0.2 m and V =
1.98 m/s) in the 9.8 m wide tank (dashed lines) and the 3.8 m narrow channel (solid lines); graphs from top to
bottom are cuts at y = 0.3, 0.6, 0.9, 1.2, 1.5 and 1.8 m. Note: Probe at y = 1.5 m was missing in the wide tank.
6
0.5
z~0.4
0.3
0.2
0.1
o
~ \~'
-20 -10 0
,... .... ....
.
/ _
it"
_
-20 -10
. . . 1 . . . . ~
0 10 20
0 5
0.3
02 E:—— 1 ~ ~
o
~-
10 20
0.5
z~0.4
0 2 1 ~3
0 1 ~
O ~ ~
.,, .. ....
~ _
1 . . . . . . . ~ . . . .
-20 -10 0 10 20
Am'%
~~-
0.5
z~0.4
0.3
0.2
0.1
o
0.5
z~0.4
0.3
0.2
0.1
o
0.5
z~0.4
0.3
0.2
0.1
. . . . I . , . , . .
. .
: l
:
:
_
, . . .
of/
Elf/
at_
-20 -10 0 10 20
1 ' ' ' ' 1
1''''~ i... ~ ....
~ ~ 1'`
I D
A= ~
: . . . . 1 . . . . 1 . . . . 1
-20 -10 0 10
1
1
1
O
- 2 0 - 1 0
Jim
20
. ~
it ~
1 1 ~
0 10 20
Figure 7: Comparison of theoretical (dashed lines), experimental (thick solid lines) and numerical (thin solid
lines) wave profiles at design depth Froude number Fnh = 1.414 in the design narrow channel; graphs from
top to bottom are cuts at y/h = 1.5, 3.0, 4.5, 6.0, 7.5 and 9.0.
7
The comparison of primary interest is, of
course, that between the wide tank and the narrow
channel at the design depth Froude number, as car-
ried out in Fig. 6. Evidently, the strong free waves
behind the model in the wide tank are almost totally
absent in the narrow design channel, corroborating
the dramatic drop in wave resistance. Quantitatively,
the highest free-wave amplitude observed in the wide
tank is 100 mm and in the narrow channel only 4 mm.
We think it is fair to call it "complete" wave cancel-
lation.
The most direct test of the theory is to compare
the theoretical and experimental wave patterns in the
design condition. This has been carried out in Fig. 7
for six equidistant longitudinal cuts. There is striking
agreement between theory and experiment except for
a small phase shift, which may be due to bottom fric-
tion or imperfect reflection from the sidewalls. Fig. 8
is a photographic attempt to convey a visual impres-
sion of the wave pattern in the design condition of
zero wave-resistance. Note the high wave crest be-
side the model and the almost absolutely flat free-
surface behind the model.
= . ~
Figure 8: Twin photographs of the wave pattern in
the design condition of zero wave-resistance, sepa-
rately viewing the forebody (top) and the afterbody
(bottom).
Numerical Simulation
The foregoing theoretical and experimental inves-
tigation was supplemented by a numerical simula-
tion using an Euler solver developed at the Merca-
tor University in Duisburg as part of a larger project
to compute fully nonlinear viscous free-surface flows
around ships moving in restricted waters, see Bet et
al. (1999~. Briefly stated, it is a computer code
for solving the nonlinear Euler equations of steady
three-dimensional flows of an incompressible invis-
cid fluid on arbitrary grids employing a nodal finite-
volume method, see Hanel et al. (20011. Major fea-
tures of the algorithm are: (i) coupling of the mass
and momentum equations by the method of artificial
compressibility, (ii) integration in time by an explicit
Runge-Kutta multi-stage time-stepping scheme, and
(iii) free surface tracking by a level-set formulation
on fixed grids.
Fig. 7, already partly discussed, includes be-
sides the theoretical and experimental wave profiles
a third set designated as numerical. These were com-
puted using the Euler solver. As expected, conver-
gence problems arose in the transcritical speed range
where the physical solution is known to be unsteady.
Nevertheless, the numerically simulated wave pro-
files at the design Froude number, almost adjacent
to the transcritical range, are obviously in fair agree-
ment with the theoretical and experimental profiles.
The numerical wave resistance calculated by
integrating the pressure on the hull surface in the
Euler solution is compared in Fig. 9 to the resid-
uary resistance obtained in the experiment by sub-
tracting the frictional resistance after Huhges from
the directly measured total resistance. The agree-
ment is excellent except for a gap in the numerical
results over the transcritical speed range for reasons
just mentioned.
Fig. 10 shows a perspective view of the
starboard-half of the wave pattern in the design con-
dition as numerically simulated by the Euler solver.
The upper right yellow surface is the model running
from left to right. The lower left yellow surface is
the channel bottom with isobars plotted on. The blue
band in the middle is the free water surface with the
channel sidewall at the left edge. More clearly than
in the photograph of the physical experiment (Fig. 8),
we can see here the theoretical semi-rhombic wave
pattern (Fig. 1 and Fig. 2) comprising two oblique
wave crests alongside the model merging into a sin-
8
ale crest of almost double the height at the channel
sidewall. This is a purely local wave bound to the
model. No free waves can be seen behind the model.
In short, the computationally extremely effi-
cient 2D shallow-water theory seems to be at least
as close to physical reality as the more general but
also computationally far more demanding 3D Euler
simulation.
0.04
0.03
U2
,,
Ul
0.02
3
i, 0. 01
au
U2
. ~£ . . . , ~
0 5 0.75 1 1.25 1.5
depth Froude number
1.75 2
Figure 9: Comparison of measured residuary resis-
tance from model experiment (solid line connect-
ing dots) with calculated wave resistance from Euler
simulation (circles) over a wide speed range in the
narrow design channel.
Figure 10: Perspective view of the numerically sim- where
ulated wave pattern in the design condition (height
magnified four times).
and
Discussion of Viscous Resistance
The only reasonably practical and reliable method
available to date for obtaining wave resistance from
a model experiment is to resort to a modified Froude
hypothesis, i.e., to subtract from the directly mea-
sured total resistance the best possible empirical es-
timate of the viscous resistance, thereby ignoring
any wave-viscous interaction. It is current standard
practice in tankery to estimate viscous resistance of
streamlined hull forms by applying a viscous form-
factor to some agreed friction line, usually the 1957
ITTC ship-model correlation line.
Our hull is extremely thin (bm/1 = 0.0649) and
slender (V/13 = 0.000594), entailing a form-factor
only slightly above unity. On the other hand it has
also an extremely small aspect ratio (2d/1 = 0.05)
which would tend to induce edge effects and increase
the form factor. In any case, the purely viscous form-
factor cannot lie below unity. However, we observe
in Fig. 5 that in order to avoid the physical impossi-
bility of a negative wave resistance in the design con-
dition, an implausible viscous form-factor less than
unity would be required.
We think that this paradox can be explained by
taking account of wave-viscous interaction. Our de-
sign condition is characterized by exclusively posi-
tive wave elevation alongside the model (see Fig. 7),
which is most unusual, specially over the afterbody.
This induces two opposite effects: (i) the increased
wetted surface would tend to increase the frictional
resistance and, hence, increase the apparent form fac-
tor, but (ii) the absolute forward motion of the water
under the wave crests would tend to decrease the rel-
ative velocity of water past the hull and, hence, de-
crease the apparent form-factor. The following rough
calculation shows that the latter effect predominates.
By definition, the frictional resistance ignoring
the wave effects is
RFO = 2PV SWCF(RU).
(10)
By analogy, the frictional resistance including the
above wave effects would be
RFW = 2 P(V—U) 2 ~5W + /\ SW ) OF (Rn ), ( 1 1 )
1 {~/2
u = - / ups, 0)dx,
I J-~/2
rt/2
/\Sw= / ((x,0)dx,
-~/2
The ratio of frictional resistance accounting for wave
interaction to the frictional resistance ignoring wave
interaction becomes
RFW ~ _uN2l' ASWN\
RFO = ~1 vJ <1+ sw )
9
This ratio can be interpreted as a Froude-number de-
pendent frictional "form factor", although it is obvi-
ously not a true viscous form-factor. The latter would
have to be determined in the absence of waves, e.g.,
by towing a deeply submerged double model.
The numerical value of this apparent form-
factor in the design condition is easily calculated
from the known theoretical flow and is indeed found
to be less than unity, namely 0.947. This explains
the observed paradox and does not violate any physi-
cally motivated expectation. On the contrary, our de-
sign displays an unexpected fringe benefit, namely, it
not only eliminates wave resistance but also reduces
frictional resistance.
Depth Variation
With a view to possible application of this design in
actual practice, it is of interest to examine how sensi-
tive the state of zero wave-resistance is to unwanted
variation of external parameters. It is already appar-
ent from Fig. 5 that the resistance minimum is rather
sharp and sensitive to speed. The practical conse-
quence is that passively stable steady motion is only
possible at speeds somewhat higher than the point of
minimum resistance.
As already stated, beside the design depth h =
0.2 m we also tested one off-design water-depth h =
0.3 m in the same narrow channel. Fig. 11 shows
the measured specific total resistance at both depths
along with the corresponding specific frictional re-
sistance curves after the ITTC line. We note that the
dramatic drop in total resistance almost down to the
ITTC line still occurs at the 50 To higher off-design
depth but at a reduced depth Froude number corre-
sponding, however, to a 13 % higher absolute veloc-
ity. Measured wave cuts at the speed of minimum
resistance in the off-design depth are reproduced in
Fig. 12. As may be expected, the free waves behind
the model are quite weak but not completely elimi-
nated as in the design condition. We conclude that a
near-optimum state of extremely low wave resistance
can be maintained over a range of water depths if the
ship has sufficient power reserve to adapt its speed.
0.07
0.06
0.05
0.04
0.03
0.02
0.01
. _ _ .3 m 1 ~
_ —7 ~ _
_ ~ 1~;~ =0.2 IT/ '
_ _ _~ ~ f; ~ V
_~ ~~ ~
0.6 0.8 1 1.2 1.4 1.6 1.8 2
depth Froude number
Figure 11: Measured specific total resistance at off-
design water-depth h = 0.3 m and design water-
depth h = 0.2 m along with the corresponding spe-
cific frictional resistance curves after the ITTC line.
CONCLUSIONS
The purely theoretical prediction of a state of no trail-
ing waves and hence zero wave-resistance for a ship
of an ingenious mathematical hull form moving at
a chosen supercritical design speed in a rectangu-
lar channel of appropriate depth and width has been
verified by numerical simulation using a nonlinear
3D Euler solver and validated by physical model ex-
periments conducted in a specially erected narrow
shallow-water channel.
ACKNOWLEDGMENT
We thank the management and staff of VBD-
European Development Centre for Inland and
Coastal Navigation at Duisburg, Germany, for their
valuable support in conducting the model experi-
ments.
10
0.4
urn 0 20
-0.2
-0.4
0.4
z,7l 0 . 20
-0.2
-0.4
0.4
At 0 . 20
-0.2
-0.4
0.4
z/77v 0 · 20
-0.2
-0.4
0.4
0.20
-0.2
-0.4
0.4
z/qrl 0 . 20
-0.2
-0.4
. ~ .. .... .... . .... , - .
~~ A_ ~
:
, .... .... _
-4 -3 -2 -1
. . . .
0 1
_ 1
~
F ~/
.
-1 0 1
... .... .... .... _
-4 -3 -2 -1 0 1
.... .... .... ..·· .
_ J
_
. ., ....
-1 0 1
. . . . . . . . . . . . .
~~
. . . . .
-1 0 1
Figure 12: Measured wave profiles at the off-design depth h = 0.3 m (in the narrow channel of 3.8 m width) at
"optimum" depth Froude number Fnh = 1.30 (V = 2.23 mist; graphs from top to bottom are cuts at y = 0.3,
0.6,0.9, 1.2, l.Sandl.8m.
11
REFERENCES
[1] Bet, F., Stuntz, N., Hanel, D., and
Sharma, S. D., "Numerical Simula-
tion of Ship Flow in Restricted Water,"
7th International Conference on Numerical
Ship Hydrodynamics, Nantes, France, 1999.
[2] Chen, X.-N. and Sharma, S. D., "Non-
linear theory of asymmetric motion of
a slender ship in a shallow channel,"
Proc. 20th Symp. on Naval Hydrodynamics,
Santa Barbara, California, 1994, pp. 386-407.
[3] Chen, X.-N. and Sharma, S. D.,
"On ships at supercritical speeds,"
Proc. 21st Symp. on Naval Hydrodynamics,
Trondheim, Norway, 1996, pp.715-726.
[41 Chen, X.-N. and Sharma, S. D., "Zero wave re-
sistance for ships moving in shallow channels at
supercritical speeds," J. Fluid Mechanics, Vol.
335, 1997, pp. 305-321.
[51 Chen, X.-N., "Hydrodynamics of Wave-
Making in Shallow Water," Doctoral Disser-
tation, University of Stuttgart, Shaker Verlag,
1999.
[6] Chen, X.-N., Sharma, S. D., and Stuntz, N.,
"Zero wave resistance for ships moving in shal-
low channels at supercritical speeds. Part 2. Im-
proved theory and model experiment," submit-
ted to J. Fluid Mechanics, 2002.
[7] Hanel, D., Dervieux, A., Cloth, O., Fournier,
L., Lanteri, S., and Vilsmeier, R., "De-
velopment of Navier-Stokes ~ '
Hybrid Grids," In: E.H.
al. (Ed.~: Numerical Flow Simulation:
Notes on Numerical Fluid Mechanics, Vol. 75,
Springer Verlag, 2001, pp. 49-66.
[81 Hughes, G., "Frictional resistance of smooth
plane surface in turbulent flow," Trans. INA,
Vol. 94, 1952.
Solvers on
Hirschel et
[91 Hughes, G. "Friction and form resistance in tur-
bulent flow and a proposed formulation for use
in model and ship correlation," Trans. INA, Vol.
96, 1954.
t10] Miles, J. W., "Obliquely interacting solitary
waves," J. Fluid Mechanics, Vol. 79, 1977,
pp.157-169.
12
DISCUSSION
Hang S. Choi
Seoul National University, Korea
When we look at the related waves at the
sidewall, it looks like the so-called Mach
reflection. Have you perhaps examined this
phenomenon during your experiments?
AUTHORS' REPLY
Yes, indeed, there exists a well-known fluid-
dynamic analogy between ship waves at
supercritical speed in shallow water and shock
waves of an aircraft at supersonic speed. Our
experiments can well be used to illustrate Mach
waves and Mach reflection.
DISCUSSION
Turgut Sarpkaya
Naval Postgraduate School, USA
Our experience has shown that the type of the
model described in your paper is very sensitive
to yaw instability and, as a single ship, makes its
use nearly impossible. Have you carried out a
yaw-stability analysis for the single model in
confined as well as unconfined environment?
AUTHORS' REPLY
We thank Prof. Sarpkaya for his corroborative
evidence (oral remarks). It is true for almost any
ship model in a shallow channel of rectangular
cross-section that if ship length and speed are in
the right proportion to channel depth and width,
the bow waves reflected from channel sidewalls
would significantly cancel the stern waves and,
hence, result in a corresponding reduction of
wave resistance. However, to our knowledge a
complete elimination of wave resistance has
never been recorded previously.
The question of stability raised by Prof.
Sarpkaya is indeed very important and goes far
beyond just yaw-stability. It is well known that
various ship types are not passively yaw-rate
stable and that no ship can be passively
directionally stable. Nonetheless, almost every
ship can be actively stabilized on course by
means of a rudder and a controller (misleadingly
called autopilot) enabling it to safely navigate
channels and cross the oceans. This would be all
the easier for a slender ship like ours. What is
really crucial in our design is the stability of
forward motion. The fact that the curve of total
resistance does not monotonically rise with
increasing speed but, instead, dramatically falls
just before reaching the design speed implies a
regime of unstable equilibrium of forward
motion. Assuming a traditional propeller
characteristic, a slight transient increase in
resistance due to an external disturbance would
throw the ship back to a low subcritical speed. In
fact, an enormous power reserve would be
required in the first place to boost the ship from
rest over the huge transcritical resistance hump
on to its favorable supercritical design speed.
This is a problem our design has in common
with various high speed craft, such as planing
hulls, hydrofoil boats, hovercraft and wigs. A
less critical but also practically relevant issue is
the weak transverse static stability of our design
due to a relatively low mean beam-to-draft ratio.
Hence, the present monohull can be only a
stepping stone to progress. A truly practical
design would have to be some kind of a multi-
hull exploiting the principle of mutual
cancellation of non-dispersive bow and stern
waves at supercritical speed.
