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OCR for page 23
24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Wave Patterns and Minimum Wave Resistance for
High-Speed Vessels
E. O. Tuck, D. C. Scullen, and L. Lazauskas
(The University of Adelaide, AustraTia)
ABSTRACT
Flow fields and wave patterns are computed in both
the near and far field of a moving ship-like disturbance
of small amplitude, using efficient routines for the 3D
Havelock source. Classical thin-ship wave patterns can
be computed for conventional ships, catamarans or sub-
marines by this method, accurately and with extremely
fine detail, in minutes on inexpensive desktop comput-
ers. Similar methods can be used for surface pressure
distributions, modelling either hovercraft or planing ves-
sels of small draft. In contrast to the thin-ship case, the
ability to compute accurate near-field flows is essential
to the solution process for a prescribed planing surface,
which requires solution of an integral equation to deter-
mine the equivalent pressure distribution. Such pressure
distributions can also be optimised to minimise wave re-
sistance, from which follows a formally simpler design
process for planing surfaces, where the shape of the sur-
face is output rather than input.
INTRODUCTION
Objects of length L travelling steadily at speed U at or
near a free surface under gravity 9 tend to make large
waves if and only if the Froude number F = U/~
takes values of the order of 0.55. This is because their
length is then about half of the transverse wavelength
2~rU2/g, and (roughly) equal and opposite waves from
bow and stern add together. At such speeds, wavemaking
dominates over other physical processes, and its accurate
prediction is critical.
Most conventional marine vessels travel much more
slowly than this, since generation of large waves costs
energy, and in that low-speed range (say F < 0.35) the
tendency has been to assume that theoretical estimates
of wavemaking are unreliable. Although this is to a cer-
tain extent true, it is also somewhat unreasonable to ask
theory to predict a small effect in the presence of much
greater (e.g. viscous) effects.
Some special marine vessels, e.g. hovercraft, also nor-
mally operate at speeds that involve little wavemaking,
because the Froude numbers are high, e.g. F > 1.3.
Again, hydrodynamic theory seems not to have a large
role to play; such vessels at their design speeds are es-
sentially airplanes in ground effect. However, they must
accelerate through the large wavemaking range ("hump
speed") in order to get to their higher design speed.
The present paper is concerned with speed ranges
corresponding specifically to large wavemaking, mostly
(say) 0.4 < F < 1.0. In that range, theoretical predic-
tion of waves and wave resistance is potentially valuable,
much more so than is often appreciated. For example,
Figure 1, taken from Tuck (1987), shows a comparison
between computations based on Michell's (1898) thin-
ship wave resistance theory and experiments of Chap-
man (1972), for a parabolic strut with beam/length ra-
tio 0.15. This indicates quite remarkable agreement for
F > 0.4, even though this is not a particularly thin
body. Although there were no experimental results for
F < 0.4, one might suspect that the relative accuracy in
predicting the small waves made in that range would not
be as good as it is for higher speeds.
Another example where one might expect good ac-
curacy is for travailing pressure distributions of "small"
magnitude, modelling hovercraft and potentially also
planing surfaces or flat ships. Our interest in thin or flat
ships is not only in their wave resistance, which is one
measure of the wave energy in the far-field waves, but
also in the actual wave elevation pattern itself, and then
not only in the far field, but also in the near field close to
the vessel.
We first discuss a mature computational tool for thin
ships, extending Michell's (1898) theory in a number of
directions, but with emphasis on efficient computation of
a detailed and accurate wave pattern over a large region
including the immediate neighbourhood of the vessel. A
similar code is then discussed for the wave patterns of
travelling pressure patches. The latter is potentially use-
ful not only in a direct sense for vessels such as hover-
craft, but also in an inverse sense for planing surfaces or
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flat ships, whose hull shape is equated to the free-surface
shape immediately beneath a patch of pressure.
One of our aims is to reduce or minimise wave re-
sistance. We summarise here new results for pres-
sure patches of minimum resistance at fixed total load,
and give examples illustrating the corresponding near-
field wave patterns, which are candidate hull shapes for
planing surfaces. Interestingly, although optimal pres-
sure distributions are necessarily fore-aft symmetric, this
does not imply such symmetry of the near-field pattern,
nor of the flat-ship hull, whereas the corresponding opti-
mal thin-ship theory demands fore-aft hull symmetry.
We also provide discussion and preliminary results on
the inverse problem, namely that of finding the pressure
distribution corresponding to a given flat-ship hull. This
is a very difficult computational task, and we are con-
tinuing to work on it. However, design of a pressure
distribution for low wave resistance, followed by direct
computation of the shape of the corresponding flat-ship
hull, is a computationally simpler task which is already
complete.
HAVELOCK SOURCES
The topic of this paper is detailed and accurate computa-
tion of steady flow fields, wave patterns and wave resis-
tance, for bodies moving at constant speed U at or near
a free surface under gravity 9, in calm water of infinite
depth. The bodies must be small in some sense, so that
the free-surface condition can be linearised, and there
are many examples of such bodies, including thin ships,
catamarans, submarines, hovercraft and other types of
surface-effect ships, planing surfaces or flat ships, etc.
Subject to the usual assumption of an inviscid in-
compressible fluid moving irrotationally, all such flows
can be generated by distributions of Havelock sources,
which are point sources in the presence of the free
surface. The velocity potential of a unit Havelock
source (Havelock 1917, 1928, Wehausen and Laitone
1962, p. 484) located at (x, y, z) = (0, 0, () is
~7r/2 /too
Y' ; () 47r2 j-7r/2 JO
e—ik(xcos0+ysin0) Le—klZ—(I k + ko sec2 ~ k(z+~)
L k- kOsec2oq
(1)
with ho = g/U2. The path of k-integration passes above
the pole at k = ko sec2 0, so guaranteeing that waves oc-
cur only for x > 0 . The first term inside the square
bracket of (1) contributes the potential of an ordinary
1.2
1.0
0.S
0.6
0.4
0.2
R-
2pU2B2
,~
Present (&Chapman) Theory
x Experiment (Chapman)
-
-
. . . . . .
0.5 O., 0.9 1.
Froude number F
,
0.1 0.3
Figure 1: Comparison between theory and experiment
for wave resistance of a parabolic strut.
infinite-fluid Rankine source, since
A_2 1
r~/2 r~
dd / dke-ik(= cOs B+ysin B)-klz-<
41r J-7r/2 J0
4~/x2 + y2 + (`z _ (~2
(2)
The second term inside the square bracket of (1) is the
correction for the free surface, and it is easy to verify
that the Kelvin linearised free-surface condition
GXX + koGZ = 0
(3)
holds on z = 0.
Although the ability to represent free-surface flows by
Havelock sources has been available for about a century,
an apparent inhibition for routine use has been the sheer
computational task of evaluating the double integral (11.
When Havelock sources are distributed over a spatial
region, at least two further numerical integrations have
to be performed, and if detailed flow fields are then re-
quired at many (x, y, z) values, some billions of values
of G may be required! There is therefore a premium on
efficient evaluation of this double integral.
Newman (1987) made a significant advance in this
direction by providing economised polynomial approxi-
mations for the "local" portion of the Havelock source,
namely
Gt~x, y, z; () = G(-~x~, y, z; () . (4)
This is an even function of x which is identical to G
when x < 0, i.e. ahead of the source, and so is not
wavelike. Thus G = Gt + GF where the "far-field"
portion GF is identically zero for x 0
is given by—27ri times the residue at the pole, namely
2
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Representative terms from entire chapter:
ship research
Figure 2: Computed wave pattern for a DDG5 1 ship at 30 knots.
G (x, y, z; () = — ° / seC2 yckO(Z+~) sec2 ~
~ —7r/2
sink see b) cosmos sec2 ~ sin §) do . (5)
Although defined by "only" a single integral, the far-
field part GF of the Havelock source in fact presents
greater computational difficulties than the local term G it,
because of the rapidly oscillating character of the inte-
grand, especially when ~~ approaches Art, i.e. for ex-
treme diverging waves, and in practice it is best to avoid
its direct computation.
THIN SHIPS
For a monohull thin ship with offsets y = ~Y(x, z),
the disturbance velocity potential is generated by a dis-
tribution of Havelock sources over the centreplane R in
y = 0, with strength according to Michell (1898) of
magnitude BUYS (x, z) per unit area at the point (x, 0, z).
Thus
(b(X, y, z) = MU// dads Yore, () GO—(, y, z; go.
R
(6)
If we write the Havelock source G = Go + GF as above,
we can then write correspondingly o = ¢~ + oF. The
double integral in (6) can be carried out by direct numer-
ical quadrature for the local part of, but for the far-field
3
part or we proceed indirectly, c.f. Noblesse (2001). In
the following, let the draft of the ship be T and the x
values at the extreme bow and stern be XB and as re-
spectively. Thus the region R is XB < X < AS and
—T < z < 0.
Substituting the integral representation (5) for GF, we
find
2uk2 r7r/2
(F(x y z)= - °,1 ddsec36
AT —7r/2
ekOZ see ~ cosmos sec2 ~ sin §)
[P cosmos see 6) + Q sink see by], (7)
where
1 r° X,
P + iQ = —iko see ~ i/_ T ~XB
d:
y<(: <)eiko~sece+kO
Havelock source contributes only for stations ~ ahead of
the observation point x.
So long as there is no transom, and x > as, integra-
tion of (8) by parts gives a simple formula for P + iQ
involving the actual hull offsets Y. namely
Q i; Y((,()eko~sec 0+2ko~sec~ dads `9'
o.os
°
If there is a transom with Y(xs, () ~ 0, or if ha <
x ~ as, there is an important extra contribution from the
integrated part, which is included in our computations,
but is not quoted here for simplicity.
In any case, given the offsets Y. the ((, () double in-
tegral over the hull centreplane to determine P + iQ can
be done efficiently once and for all by numerical quadra-
tures, the results being stored for subsequent use in the
integral (7) for the far-field potential. This storage must
be for a fixed set of values of it, and so long as x > Us,
results can then be obtained by a single reintegration
with no further cost from the integrals over R. for as
many (x, y, z) values as desired. If XB < X < AS, there
is some further cost, as new values of P + iQ must be
computed for each new x, but this is not usually a large
impost.
The actual free-surface elevation is z = Zip, y) =
—(u/g)¢x (x' y, 0), and our main output is the wave pat-
tern, in the form of contour plots of this quantity Zip, y).
However, the wave resistance is also immediately avail-
able via (8) with HE = US, namely
2 r7r/2
RW =—pU2ko ~
7r —7r/2
[p2 + Q2] sec5 ~ do, (10)
which is Michell's (1898) wave resistance integral. Note
that the task of evaluating the wave resistance Rw is
equivalent to that of evaluation of just one wave eleva-
tion value Zip, y) in the far field; we compute millions
of such values, in the near field as well as the far field.
Generalisations of thin-ship theory
Although developed initially by Michell (1898) for con-
ventional ships, the above theory has somewhat more
general applicability. Recall that the only essential ap-
proximation is that the vessel can be represented by a
centreplane distribution of Havelock sources of strength
proportional to the local hull slope. Thinness of the hull
justifies this approximation, but this thinness need not
be extreme, and for wave pattern purposes, relative er-
rors appear to be comparable to the beam-to-length ratio
(Tuck and Scullen 2002), so accuracies of within i10%
are to be expected for most ships.
Note that the only requirement for thinness is that the
beam is small compared to the length; the draft is irrele-
vant (Tuck 1987~. In particular, thin-ship theory applies
of
i
Expenment ------
SWPE6—
~ \1'': \~'!
10
15
20 25
Figure 3: Comparison between SWPE computations for
a ship model as in Figure 2 and "Wake-Off" experimen-
tal measurements along a cut parallel to the ship's track.
as well to submarine vessels as it does to surface ves-
sels. Indeed it usually performs even better, since the for-
mer usually have a lower beamllength ratio. For conven-
tional submarines, thin-ship results can be computed up
to and beyond the point where the vessel breaks the sur-
face. These results remain in good agreement with "ex-
act" nonlinear computations (Tuck and Scullen 2002) as
the submergence is reduced, right up to the point where
the latter inevitably fail due to incipient breaking.
It is paradoxical (for both submerged and surface ves-
sels) that the seemingly more accurate nonlinear theory,
as implemented in Tuck and Scullen (2002) and in well-
known codes such as RAPID (Raven 1996) and Shipflow
(Larsson 1997), is nevertheless sometimes less useful
because of the potential for such failure (see Abbott 1998
for a case study), than is the linear theory, which never
fails numerically. Of course what the linear theory does
in extreme cases is to predict one or two unreasonably
large wave crests and deep troughs in the near field,
which really should have broken. However, any such
breaking is a highly localised phenomenon, and the wave
pattern everywhere else and hence the wave resistance
remain reasonable (Sculler and Tuck 1995~.
Thin-ship theory also holds for multihull vessels, and
again in most such cases the individual demihulls are
thinner than conventional ships, so the linearization is
more justifiable. However, now there is a further compli-
cation involving interactions between hulls. In general,
each demihull must be represented not only by centre-
plane distributions of Havelock sources, but also by cen-
treplane distributions of lateral Havelock dipoles, the lat-
ter accounting for lateral velocities induced on one demi-
hull by the others. In an aerodynamic analogy, this is a
4
"lifting" as distinct from a "thickness" effect (Newman
1977, p. 168), and also becomes relevant for a mono-
hull at an angle of yaw (Xu 19911. There have been very
few analyses of this type of interaction (see Lin 1974,
Salvesen et al 1985, Suzuki et al 1997~. However, the ef-
fect is expected to be quite small, and in particular is for-
mally small if (as is usually the case) the demihulls are
not only thin but also slender, i.e. have small draft/length
ratio as well as small beam/length ratio (Tuck and New-
man 1974, Tuck 1987~. Its neglect enables considera-
tion (Tuck and Lazauskas 1998) of optimal multihull ar-
rangements to maximise cancellation of far-field waves
and reduce wave resistance.
An apparent "generalization" of thin-ship theory is to
vessels with transom sterns. However, as long as the the-
ory is applied with the source strength given in terms of
hull slopes, e.g. use of (8) rather than (9), or a careful
integration by parts is performed, the original Michell
(1898) formulation is quite capable of handling dry tran-
som sterns. This implies a representation of the subse-
quent trailing wake as a straight impermeable cylinder
extending infinitely far downstream with cross-section
identical to the transom. Somewhat better models of
transom wakes are possible allowing eventual cavity col-
lapse (Couser et al 1998, Doctors and Day 1997), but in
practice the original Michell model works well.
A final generalization is that of allowing viscous
damping of the far-field waves. Since the far field can
be considered as a superposition of plane waves travel-
ling at angles ~ to the x axis, with amplitude P + iQ, it is
only necessary to apply appropriate damping factors to
this plane wave (c.f. Lamb 1932, p. 624~. We have found
(Tuck et al 2002, c.f. Maruo 1976, equation (92~; see
also Cumberbatch 1965, Zilman and Miloh 2001) that
the factor
exp [—U ho sec5 b~xcos~ + ysin§~ (11)
inserted in the integrand of (7) provides good results,
where z, is a measure of the kinematic viscosity. This
quantity z, is not really intended to represent molecular
viscosity as such, but to model the damping effect of the
turbulent shear layer in the wake, and hence must take a
value comparable to the eddy viscosity in the wake.
Choice of an appropriate value of zJ is a difficult mat-
ter; a value of between 100 and 10000 times the molec-
ular viscosity seems to eliminate some of the most ex-
treme short diverging waves (with ~~ close to ~r/2) near
the track of the ship, without unreasonably damping out
genuine features of the wave pattern. Of course damp-
ing of far-field waves is not the only effect of viscosity,
and for example it is also possible (Havelock 1948, Lars-
son 1997) to modify the source strengths to take account
of apparent fattening of the hull due to the displacement
Figure 4: Computed pattern for a catamaran
Figure 5: Photo of a catamaran model
thickness of the viscous boundary layer.
Computational considerations
We have developed (Tuck et al 2002) a computer pro-
gram "SWPE" which takes as input a conventional set of
offsets defining the ship, and yields data suitable for plot-
ting the resulting wave pattern over any specified region.
A typical run of SWPE takes about fifteen minutes on a
current PC to produce a detailed pattern containing about
100,000 points, yielding plots of photographic quality.
One feature is the ability to "zoom" in arbitrarily close
on any interesting flow region, although the mechanism
for doing this is actually to re-sample by running SWPE
again with the 100,000 points distributed over a smaller
region, without loss of accuracy. In contrast, the resolu-
tion of "panel" methods (Raven 1996, Larsson 1997) is
set in advance by the choice of the number and distribu-
tion of panels on the free surface, which is dictated by
over-all accuracy requirements.
The similarity between (7) and (10) suggests that the
triple ((, A, b) numerical integration task of computation
of the far-field potential or at each separate point will
s
0:~
-
~ ~ -
-~ ~ ~ -~
~ -
Figure 6: Wave pattern for a submarine.
be comparable to that of computing one value for the
wave resistance Rw, already a daunting prospect if we
desire information at 100,000 points or more, and made
worse by the fourth k-integration needed for the local
contribution ¢t to the potential.
We have previously (Tuck 1987, Tuck and Lazauskas
1999) developed efficient routines for evaluation of
Michell's wave-resistance integral (10) subject to
(8~. These routines use Filon's (1926) quadrature
(Abramowitz and Stegun 1964, p. 890) in the (-
direction, in order to capture the rapid oscillations of the
integrand of (8) as ~~ ~ /2. Conventional (e.g. Simp-
son) quadratures fail to produce the correct rate of decay
of the diverging part of the wave spectrum. That program
computes the wave resistance of a typical ship to 4-figure
accuracy in less than 50 milliseconds on an inexpensive
2GHz PC.
Essentially the same numerical methods that were
successful for wave-resistance computations have been
adapted in SWPE to compute the far-field flow and wave
elevation. Again, Filon's quadrature plays an important
role, and without it the diverging waves are poorly pre-
dicted. In addition, a special algorithm as in Tuck et
al (1971) also captures the stationary-phase character of
the integral (7) as x, y ~ x, thus allowing uniform ac-
curacy of computation as we move far away from the
ship. The PC time to compute a single far-field point is
then about 63 milliseconds. However, because the P. Q
functions can be stored and used repetitively, the time
to compute a wave field of 100,000 points is only about
the same as for about 4000 separate single-point calcu-
lations, i.e. about 4 minutes.
It remains to compute the local portion ot~x,y),
and in principle this remains a formidable quadruple-
integration task. Fortunately, a significant part of this
task has already been done for us, since Newman (1987)
has provided economised polynomial approximations
for Go,. Hence we need merely substitute this polyno-
mial code into (6) and carry out the ((, () integrations by
Simpson's rule (no Filon treatment is needed as the lo-
cal integrand is not rapidly varying). Nevertheless, one
cannot entirely escape the fact that these computations
require more arithmetic, and the local part of the com-
putation tends to dominate computer times, typically by
a factor of about four. The net effect is that the total PC
time to compute a complete 100,000-point field is about
15 minutes.
Once such a field of wave elevations is computed, var-
ious plotting procedures can be used to display the re-
sults. We have found it convenient to use very finely
graded contour plots. That is, we assign a colour (in the
written paper gray, in the presentation blue) of varying
intensity to each of a large but finite set of stepped levels
of wave elevation Z. In practice we use 256 such levels,
and with as many as 100,000 data points, the effect is
close to what could be seen in a photograph taken verti-
cally above the ship, although a careful examination of
some Figures (especially where the elevation is relatively
small) reveals evidence of actual discrete contours. We
generally here delete information about the actual mag-
nitude of these elevation contours, although this is avail-
able.
Sample results
Figures 2, 4 and 6 show respectively SWPE-computed
wave patterns for a conventional ship, a catamaran, and
a submarine. Other similar results are given in Tuck et al
(2001) and Tuck (2001~.
The ship in Figure 2 is in fact the same DDGS1 de-
stroyer hull as was used for the "Wake-Off" test (Lin-
denmuth et al 1991~. As indicated in Figure 3, SWPE
results for elevations along parallel cuts are in excellent
qualitative and reasonable quantitative agreement with
the experimental data. This agreement is somewhat bet-
ter than was displayed by the various (then) state-of-
the-art computer programs that participated in the 1991
Wake-Off test, and is comparable with that achieved sub-
sequently by the very successful nonlinear code RAPID
(Raven 1996~. Some residual discrepancies are likely
to be attributable to localised breaking in the experi-
ments, which is unlikely to be capturable by any numer-
ical code, linear or nonlinear.
6
Figure 7: Constant-pressure patch.
Figure 4 provides SWPE computations for a catama-
ran hull, and can be compared to the photograph of Fig-
ure 5 (courtesy Australian Maritime College), although
no attempt was made to duplicate the hull shape, and
the speeds and dimensions were only roughly matched.
There are obviously some points of close similarity, and
also some features, mainly to do with the turbulent wake,
that are not captured by the computations.
Figure 6 is for a Los Angeles class submarine hull
moving at 10 knots, at a submergence such that the sail
top is about to break surface. This pattern is especially
interesting in that separate wave structures due to the
sail and the main hull can be distinguished. The Froude
number for the main hull is low enough for that part of
the pattern to be mainly transverse, whereas the Froude
number based on sail length is large, so that part of the
pattern is mainly diverging.
PRESSURE DISTRIBUTIONS
The velocity potential for the flow induced by a unit
delta-function pressure (Wehausen and Laitone 1962, p.
598) exerted on the free surface of a stream U is propor-
tional to G=, where Gin, y, z; 0) is a Havelock source
located at the free surface. That is, a pressure point is
identical to a surface horizontal dipole. The velocity
potential of a distribution of prescribed pressure pax, y)
7
over a region R of the plane z = 0 is then
¢(x, y, z) =—ii dads p((, y) GXFX—(, y—A, z; 0~.
(12)
This is a similar formula to that (6) for a thin ship,
the pressure distribution p replacing the offsets Y as in-
put, but the region R is now part of the plane z = 0
instead of the plane y = 0. Again, separation into local
and far-field portions enables direct computation of the
local portion or with the aid of Newman's (1987) rep-
resentation for Go, and indirect computation of the far-
field portion OF proceeds in a similar manner to that for
the thin ship. In particular, the same far-field potential
(6) and wave-resistance formula (10) apply if we define
P + it = 2pU to fiR pax, y)
eikox see 0+ikoy sec2 ~ sin 8yx`'y (13)
The pressure-distribution version of our program
SWPE computes the same set of wave-pattern and flow-
field outputs as does the thin-ship version, with a simple
replacement of the ship offset data Y(x, z) at stations x
and waterlines z, by pressure distribution data pax, y) at
stations x and buttock lines y.
Figure 8: Single bi-quadratic pressure patch.
Results for simple pressure patches
All pressure-patch results in the present paper are for a
rectangular region R. namely Axe < a, Lye < b. The re-
sults are parametrized with respect to the conventional
length-based Froude number F = U/~. We place
special emphasis on a hovercraft-like beam/length ratio
b/a = 0.5, and a Froude number F = 1/~; ~ 0.7,
which is close to that (the so-called "hump speed", Ever-
est and Hogben 1967) for maximum wave resistance.
Where necessary, as a specific example we have cho-
sen a length 2a = 80m, width 2b = 40m and mean
pressure p0 = 10kPa, so the hydrostatic mean draft
z0 = p0/(pg) is about 1 metre and the displacement
about 3200 tonnes. The hump speed U corresponding
to F = 1/~ is then about 40 knots; a real hovercraft of
this size would normally travel at a greater speed where
wavemaking is negligible, but we are more interested
here in performance of a vessel of this size at a lower
(but still high) speed where large waves are made unless
there is some design optimization.
The simplest type of pressure patch, of relevance to
hovercraft, is one of uniform pressured = po = constant
over a rectangular region. The wave resistance of such
patches was computed by Newman and Poole (1962, see
also Doctors and Sharma 1972~. Figure 7 shows the
computed wave pattern for such a distribution.
Note the diverging wave structure, especially as it
streams away from the step pressure discontinuity at the
sides of the patch, which also induces a lateral step in the
free-surface elevation. The step pressure discontinuity at
bow and stern is of less significance from that point of
view, c.f. Lamb (1932) p. 405, with a smooth continu-
ous free surface. The present computational procedure
is very efficient at displaying features such as these di-
verging waves, which have a very fine rapidly-varying
structure that is hard to capture with "panel" methods,
and this would be even more apparent in a view display-
ing more of the far field of the disturbance.
8
Figure 9: Wave pattern for tandem patches.
In the following it is convenient to define a non-
dimensional wave resistance coefficent
CD= P2b W. (14)
and for this special example, we have CD = 2.265.
Because our numerical integration method has the ac-
curacy of Simpson's rule for the integrals over R. it
is almost exact for such uniform pressures, as well as
for pressures that vary linearly or quadratically along or
across the region.
Figure 8 shows the wave pattern for another exam-
ple in this near-exact category, namely that for a "bi-
quadratic" pressure
p~x,y)= 4p0 [1—(a) ] [1- (b) ~ (15)
which decays parabolically to zero at all sides of the
patch. Here p0 is the mean pressure, i.e. such that the
lift 4abpO is the same as that for a uniform pressure p0.
Although the bi-quadratic example of Figure 8 clearly
yields a much smoother local wave field than the uniform
pressure of Figure 7, it is far from being superior from
the point of view of low (far-field) wavemaking, and has
nearly double the wave resistance, namely CD = 4.270.
However, there are certainly pressure distributions
which perform much better from the wave-resistance
point of view. One such is a tandem pair of patches.
For example, suppose there are two separate patches
of positive pressure located at the bow and the stern, each
occupying 20% of the overall length. Each patch is of
bi-quadratic form, with a pressure similar (on its own
planform) to that in (15), and the remaining 60% of the
length between them is free of pressure. Figure 9 shows
the resulting wave pattern. At fixed net lift, this configu-
ration has CD = 1.330, less than two-thirds of the wave
resistance of the constant-pressure patch of Figure 7, and
less than one-third of that of the full-length bi-quadratic
patch of Figure 8. Figure 10 shows a perspective view of
p
1.5
1
0.5
o
-0.5
-1
-1 .5
-2
-2.5
-3
Figure 10: Elevation beneath the tandem pressures of Figure 9.
the elevation immediately beneath the planform R. The
stern section has an appearance like a "tunnel" hull, an
observation of relevance to use of this type of how for
design of planing surfaces. There is also an interesting
comparison with optimised tandem hydrofoil configura-
tions, e.g. as studied by Payne (1997~.
Other examples are given in Scullen and Tuck (2002~.
Minimum wave resistance patches
The above simple examples of pressure distributions are
not necessarily optimal from the point of view of low
wave resistance. Although for actual hovercraft there
may be severe limitations on achievable pressure vari-
ations, nevertheless it is of significant interest to seek
pressure distributions yielding minimum or at least low
wave resistance, at fixed total lift force balancing the
weight. This is not only of potential use for hovercraft,
but also because such pressures might be achievable on
the hull of a planing craft. That is, if we compute the
wave elevation directly beneath such an optimised pres-
sure distribution, the resulting free surface can be re-
placed by a solid surface, which would then represent
a planing surface designed from the outset for low wave
:.2
resistance.
y/a
Important early works on minimization of wave resis-
tance of travailing pressure distributions include those
of Maruo (1949) and Bessho (1962). More recently
Doctors (1997) (see also Doctors and Day 2000) has
optimised unconstrained families of pressures, involv-
ing up to 4 distinct "subcushion" parameters, and we
have (Tuck and Lazauskas 2001) optimised both un-
constrained and constrained pressure distributions with
many (e.g. 400) parameters, with indications of conver-
gence toward a continuous optimum.
In the more practical case where the pressures are con-
strained to be non-negative, this continuous optimum has
the following general structure, which varies with speed.
The most important feature (at all speeds where wave-
making is significant) is the presence of pressure lines
of zero longitudinal extent concentrated at the extreme
bow and stern ends. These end pressures vary in magni-
tude smoothly across the width, decreasing toward zero
at the sides. In fact, an aerodynamic analogy (with exact
equivalence between wave resistance and induced drag
in the limit as F ~ x) shows that at sufficiently high
speed this lateral variation must be elliptic. However, at
9
Figure 11: Wave pattern for optimum 3-line pressure.
beam/length ratio 0.5, we have found this to be so only
for F > 1.3, with a preference for a more rapid decrease
toward zero at the sides at lower speeds.
So long as F > 0.96 the optimum consists only of
these two bow and stern pressure lines. Such Froude
numbers are high enough that transverse wave cancella-
tion by bow-stern interaction is ineffective, and the best
that can be achieved is to place the disturbing pressures
as far apart longitudinally as possible.
For F < 0.96 it is desirable to include a third patch
of positive pressure, located midships. For 0.65 < F <
0.96 the centre patch should also be of zero longitudinal
extent, i.e. this patch is a pressure line similar to those at
bow and stern. However, it should not in general extend
all the way across the width of the rectangle, its optimal
lateral extent varying in the range of 80~o to 90% of the
width.
AS F decreases from 0.96, the centre patch bears an
increasing fraction of the total load. For F < 0.65,
the centre patch should have nonzero longitudinal extent,
and by F < 0.5 it is essentially bearing the whole load,
the end patches having withered away; however, by then
the (minimized) wave resistance is quite small compared
to that at the hump speed. We have little interest in opti-
mum configurations at such low wave-making speeds.
In the present paper our main attention is paid to
F ~ 0.7 and that is a speed where the optimum is three
pressure lines, the central line being of about 85~o of the
full width, and bearing about 2570 of the total load. This
configuration has CD = 0.884, and its wave pattern is
shown in Figure 11. This is a very significant improve-
ment on the constant-pressure result CD = 2.265, and
involves only physically-acceptable positive pressures.
Finally, if we do allow negative pressures, an even
lower wave resistance is possible in principle, although
the actual optimal pressures and the resulting elevations
beneath the pressure patch are then not physically rea-
sonable. We have (Tuck and Lazauskas 2001) used a 20
by 20 grid of rectangular constant-pressure panels, and
find an optimum with the remarkably low wave resis-
tance CD ~ 0 44.
However, the price paid is quite large pressure vari-
ations, contours of pressure being as in Figure 12,
with enormous maximum positive pressures of 275kPa
(shown white) and minimum negative pressures of
—160kPa (shown black). Figure 13 shows the resulting
wave pattern. In this case, we display contours only for
~ z ~ < 2m, which captures all wave elevations outside the
patch itself. However, inside the patch there are quite un-
realistic troughs (shown black) of 37m and crests (shown
white) of 26m! Clearly these unconstrained optima are
mathematical curiosities only, and the requirement for
non-negative optimal pressures is necessary in order for
the results to be physically useful.
Free-wave spectrum
Some insight into the process of reduction of wave resis-
tance for pressure patches is provided by the free-wave
spectrum, which is proportional to p2 + Q2 where P. Q
are defined by equation (13), and is a function of wave
angle §. Figure 14 presents graphs of the free-wave spec-
trum, here defined as dR/d0, where R is wave resis-
tance. The vertical scale of the graph is irrelevant to the
following discussion, but note that the area under each of
the curves is proportional to the wave resistance R. Thus
the free wave spectrum, in the form given here, immedi-
ately indicates those wave angles where most energy is
shed.
For the single constant-pressure patch, the main peak
in the free-wave spectrum occurs at about ~ = 53°. A
sensible wave-minimisation strategy would be to reduce
this large peak. Clearly the bi-quadratic pressure distri-
bution fails miserably in this regard. It has a large peak
at about ~ = 60°, which is more than double that of
the constant-pressure patch, and it is no surprise that its
wave resistance is also about double. Its edge smooth-
ness results in a much lower envelope of the spectral
peaks of the extreme diverging waves with ~ > 75°, but
these carry little energy.
The other pressure distributions shown in Figure 14
have a local minimum in their free-wave spectrum curves
at wave propagation angles a little higher than that at
which the constant-pressure patch reaches its peak. For
< 35°, the tandem and the three-line pressures shed
about 30% less energy than the constant-pressure case,
and the pressure strips at the bow and stern provide a sig-
nificant degree of cancellation of transverse waves. For
35° < ~0 < 60°, the energy lost to the diverging wave
pattern is considerably less than for constant pressure.
The ultimate reduction in wave-making is evident
from the curve for the unconstrained 20 x 20 optimum,
which makes very low transverse waves, but also sheds
10
almost negligible energy in the range 40° < ~ < 65°,
whereas its spectral peaks for extreme diverging waves
with ~ > 75° are higher than those of the other examples
in Figure 14.
Similar methods of analysis were used by Tuck and
Lazauskas (1998) to examine wave cancellation effects
of multi-hulled displacement vessels.
PLANING SURFACES
Planing surfaces are flat ships, i.e. ships of small draft.
However, unlike thin ships, for which the limiting solu-
tion as the beam goes to zero is explicit (as a quadruple
integral) given the ship offsets, for flat ships we must
still, even in the limit as the draft goes to zero, solve
an integral equation over the wetted planform R. This
integral equation is of a particularly unpleasant charac-
ter in three dimensions (Maruo 1967, Tuck 1975), es-
sentially because of short diverging waves which tend to
induce unwanted oscillations. Early attempts to solve it
numerically include those of Oertel (1975) and Doctors
(1975), and good results have recently been obtained by
Cheng and Wellicome (1994) and by Lai and Troesch
(1995~. The corresponding two-dimensional problem is
much more straightforward, and has a large literature,
partly surveyed in Tuck (1990~. At the other extreme, for
planing surfaces of low aspect ratio there have also been
some explicit high Froude number solutions, e.g. Tulin
(1957), Casting (1978), and Casting and King (19791.
In order to solve the three-dimensional planing-
surface problem for a given flat ship with equation z =
Zig, y), we "merely" need to find a pressure distribu-
tion pax, y) that causes the free surface to take that shape
z = Ztx, y) beneath the region R of non-zero pressure.
The above examples of solutions of the direct problem
when R is a rectangle are already in principle solutions
of the planing surface problem, providing Zip, y) takes
one of the output forms in the Figures. However, it is
potentially a much more difficult task to invert this prob-
lem, thus finding p when Z is given.
Since the relationship between pressure and wave pat-
tern is a linear one, formally this inversion only requires
inversion of the linear operator. More concretely, if we
discretise the pressure data pax, y) into a vector p of
length N. and the free-surface elevation data Z~x,y)
into a vector Z of length M, our computer code provides
a connection (implicitly or explicitly) of the form
A p = Z
for some M x N matrix A.
One possibility is that M = N. so that we compute
exactly as many elevations as there are pressures. It is
convenient to assume then that the elevations Z are com-
puted at the same N nodal points within the planform R
Figure 12: Unconstrained optimum pressure.
Figure 13: Wave pattern for unconstrained pressure.
as those where the pressures p are specified. Then the
matrix A is square and N x N. and it is only necessary
that this matrix A be nonsingular (which it is) in order
that we be able to find p, given Z.
It is also possible to perform an inversion if M > N.
e.g. by using least squares. In that case we no longer de-
mand that the pressure pax, y) exactly reproduce a given
elevation Zip, y) at M points, but rather that the values
of p at N points be such as to minimise the sum of the
squared departures of the computed Z from the given el-
evations at M points. This leads to equations of the form
ATA p = ATZ
(17)
which can be solved by inversion of the N x N square
matrix ATA. Use of M >> N is desirable in that it
averages out effects like the short diverging waves that
have always caused numerical difficulties in this prob-
lem.
(16) Edge conditions
There is an extra complication for actual planing sur-
faces, in that we must demand smooth detachment at
the trailing end of the surface. That is, we only accept
pressure distributions pax, y) which vanish (return to at-
mospheric pressure) at the trailing end of the planing
11
2.5
2 1.5
2
1
0.5
Single Constant
Single Bi-quadratic
Tandem Double Parabolic
Three-line Parabolic
~ Unconstrained 20x20
/
l
\
/, ~
0 10 20 30 40 50
~ (degrees)
60 70 80 90
Figure 14: Free wave spectrum for various arrangements of pressure patches.
surface. This condition is referred to as a Kutta con-
dition, by analogy with that (Newman 1977, p. 164)
for lifting surfaces. The linear two-dimensional version
of this planing-surface problem is similarly analogous
(Tuck 1990, Bessho 1994) to thin-airfoil theory.
There is a price to pay for such an extra demand, since
for any given Zip, y) we cannot expect the Kutta condi-
tion to hold upon direct inversion. There are two ways
to pay that price. The most computationally straightfor-
ward way is, while retaining a fixed given planform R.
to allow a degree of freedom in specification of Zip, y),
e.g. a vertical shift Z = ZO(x, y) + C(y) where ZO is
the given hull and C(y) must be determined by the pro-
gram. Thus the actual given hull shape is not preserved,
but is "bent" about a longitudinal axis, which may not be
acceptable for applications to real hulls.
Alternatively, and potentially more practically, we can
allow a degree of freedom in specification of the wetted
domain or planform R. while retaining on that new do-
main the exact given hull Z = ZO. In practice, with
a fixed trailing edge, this is normally an adjustment in
the location of the leading edge, and such significant
leading-edge adjustments are natural and inevitable for
12
real planing surfaces. However, this is a computationally
difficult task, and is not attempted in the present paper.
In general, having demanded zero pressure at the trail-
ing edge, the leading-edge pressure will not only not
vanish, but will formally become unbounded within the
present linearised theory. This leading-edge singular-
ity is again analogous to the corresponding leading-
edge singularity (Newman 1977, p. 168) in aerodynamic
lifting-surface theory. In the latter case, this is an arte-
fact associated with rapid velocities at which fluid passes
from the lower to the upper surface of a lifting wing, but
in the planing-surface case it models a splash (Wagner
1932, Tuck 1994, 19954.
Since splashes contribute to the total drag, there is a
premium on reducing or even eliminating this leading-
edge singularity, and thus seeking pressures that vanish
not only at the trailing edge but also at the leading edge.
We then would need another degree of freedom, e.g. we
may wish to modify the original hull ZO to
Z(x, y) = ZO(x, y) + C1 (y) + xC2(y) . (18)
This hull modification involves not only bending but also
"twisting" the given hull about a longitudinal axis, with
an upshift Car (y) and an added angle of attack—C2 (y) at
each lateral position y, both determined by the program.
This procedure generalises that of Cumberbatch (1958)
to three dimensions and finite Froude number.
This does not end the list of possible constraints on
the inversion task for planing surfaces. For example, we
may consider it desirable to require the pressure to van-
ish along all of the boundary of the region R. not only at
the leading and trailing edges, but also at the sides. This
could be because (as seen in Figure 7 for the constant-
pressure patch) sudden lateral terminations of the pres-
sure tend to produce undesirable local diverging waves
which may break, so producing further drag. It is also
relevant that our work on minimization of wave resis-
tance suggests that optimal pressures tend to vanish at
the sides.
For a rectangular planform R with side boundaries
parallel to the stream, it is also not entirely obvious
whether the side boundaries should be considered as
"leading edges" or "trailing edges"; if the latter, we are
required to demand a Kutta condition of zero pressure
on them in any case. It is not hard to conceive of fur-
ther degrees of freedom in modification of the original
hull, allowing such a constrained inversion, but we leave
that for future studies. Interestingly, Cheng and Welli-
come (1994) seem to have been able to find solutions
with zero side pressure without modifying the original
hull for that purpose. One important feature noted by
Tuck and Scullen (2002) is that when the side pressure
steps from a finite value to zero, the free surface also
steps by exactly the corresponding hydrostatic amount.
Hence we cannot expect zero side pressure for bodies
like flat plates where there is non-zero submergence at
the sides.
This discussion about edge pressures highlights what
is a serious numerical difficulty with the inversion prob-
lem for any planing surface, analogous with that for
aerodynamic lifting surfaces, e.g. Tuck (1993~. In gen-
eral, such inversions (if performed exactly without dis-
cretisation) do not produce finite non-zero leading or
trailing edge pressures. The edge pressure is either zero
(e.g. when the Kutta condition is enforced) or infinite
(e.g. at most leading edges). Hence when working with
a discretised model, we must always in some sense be
"approximating infinity". This difficulty can be over-
come (see Tuck and Standingford 1997), but we must
expect to see large numbers in the output pressures near
edges, which can be accompanied by unacceptable grid-
scale oscillations unless care is exercised in the numeri-
cal methods.
Pressure (Pa)
35000
30000
25000
20000
5000
0000
5000
o
4
Flat plate example
-au
x (m)
1 ~
~~ ~5
-20
Figure 15: Pressure on flat plate
my 20
:15
'10
y (m)
In the present written paper we present just one example
computation, for the input flat plate Z0(x, y) =—x/a,
with our standard rectangular planform and Froude num-
ber. This plate initially has its midship station x = 0 at
the undisturbed free-surface level, with the bow raised to
height 1 m and the stern submerged to lm. However, the
program then generates an almost constant downshift,
with C(y) ~ - 0.7m, in order to satisfy the Kutta condi-
tion.
Figure 15 shows the resulting pressure distribution.
The computations were performed by least squares, with
M = 2000 input data points and N = 176 unknown
pressures. The results are preliminary in that although
the over-all trend and size of the pressure output is ac-
ceptable, small grid-scale oscillations are present, simi-
lar to those in other solutions such as those of Doctors
(1975) and Cheng and Wellicome (1994~. It is how-
ever notable that there is no indication of the pressure
approaching zero at the sides of the plate, as assumed
by Cheng and Wellicome (19941. Work is continuing on
this challenging numerical task.
CONCLUSIONS
In this paper we have discussed three applications of lin-
earised water-wave theory to efficient and accurate com-
putation of flows and wave patterns for high-speed ma-
rine vessels. The first is a 21 st-century computational re-
alisation of the l9th-century thin-ship theory of Michell
(1898~. The resulting code is fast and accurate, and ca-
pable of revealing fine detail of the pattern that is diffi-
cult to capture by some other methods. In particular, its
linearity conveys a number of advantages, not least of
which is that it never fails.
The second application is to moving prescribed pres-
13
sure distributions, where again fine detail can be cap-
tured, in both near and far field. This fine detail includes,
especially for distributions which do not vanish at the
sides, very short diverging wave crests streaming away
from corners and sides. Pressure distributions minimis-
ing wave resistance are also discussed and the impor-
tance of a non-negativity constraint is emphasized.
Finally, preliminary work on the inverse problem of
finding a pressure distribution to match a given flat-ship
hull is discussed. This is a notoriously difficult task, es-
sentially because of the above-mentioned short diverg-
ing waves, and presents significant computational chal-
lenges. The present type of code shows considerable
promise of yielding robust solutions, but we are not quite
there yet.
ACKNOWLEDGEMENT
This work was supported by the Australian Research
Council, and also by the Surveillance Systems Division
of DSTO Australia. We thank Gregor McFarlane of the
Australian Maritime College for Figure 5.
References
t1] Abbott, S. An investigation into shallow water ef-
fects on high speed craft, Thesis, Australian Mar-
itime College, October 1998.
[2] Abramowitz, M. and Stegun, I.A., Handbook of
Mathematical Functions, Dover, 1964.
[3] Bessho, M., "On the problem of the minimum
wave-making resistance of ships," Memoirs of the
Defence Academy of Japan, Vol. 2, 1962, pp. 1-30.
t4] Bessho, M., "On a consistent linearized theory of
the wave-making of ships", Journal of Ship Re-
search, Vol. 38, 1994, pp. 83-96.
[51 Casting, E.M., "Planing of a low-aspect-ratio flat
ship at infinite Froude number", Journal of Engi-
neering Mathematics, Vol. 12, 1978, pp. 43-57.
t6] Casting, E.M. and King G.W., "Calculation of the
wetted area of a planing hull with a chine", Jour-
nal of Engineering Mathematics, Vol. 14,1979, pp.
191-205.
[7] Chapman, R.B., "Hydrodynamic drag of semi-
submerged ships", ASME Journal of Basic Engi-
neering, 1972, pp.874-884.
f8] Cheng, X. and Wellicome, J.F., "Study of planing
hydrodynamics using strips of transversely variable
pressure", Journal of Ship Research, Vol.38, 1994,
pp.30-41.
[9] Couser, P.R., Wellicome, J.F., and Molland, A.F.,
"An improved method for the theoretical prediction
of the wave resistance of transom-stern hulls using
a slender body approach", International Shipbuild-
ing Progress, Vol.45,1998, pp.331-349.
t10] Cumberbatch E., "Two-dimensional planing at
high Froude number", Journal of Fluid Mechanics,
Vol.4, 1958, pp.466-478.
t11] Cumberbatch E., "Effects of viscosity on ship
wakes", Journal of Fluid Mechanics, Vol.23, 1965,
pp.471-479.
[12] Doctors, L. J., "Representation of three-
dimensional planing surfaces by finite elements",
Proceedings of the 1st Conference on Numerical
Ship Hydrodynamics, Office of Naval Research,
1975, pp. 517-537.
t13] Doctors, L. J., "Optimal pressure distributions for
river-based air-cushion vehicles", SchiJfstechnik,
Vol. 44, 1997, pp. 32-36.
[14] Doctors, L.J. and Day, A.H., "Resistance predic-
tion for transom-stern vessels", Fourth Int. Conf: on
Fast Sea Transportation, Sydney, 21-23 July 1997.
Proceedings ed. N. Baird, Baird Publications, Mel-
bourne, pp. 743-750.
t15] Doctors, L.J. and Day, A.H., "Wave-free river-
based air-cushion vehicles". Proceedings of Inter-
national Conference on Hydrodynamics of High-
Speed Craft Wake Wash and Motions Control, Lon-
don, November 2000.
[16] Doctors, L.J. and Sharma, S.D., "The wave resis-
tance of an air-cushion vehicle in steady and ac-
celerated motion", Journal of Ship Research, 1972,
pp.248-260.
[17] Everest, J.T. and Hogben, N., "Research on hover-
craft over calm water", Transactions of the Royal
Institution of Naval Architects, Vol.109, 1967, p.
311.
[18] Filon, L.N.G., "On a quadrature formula for
trigonometric integrals", Proceedings of the Royal
Society of Edinburgh, Vol.49, 1929, pp.38-47.
[19] Havelock, T.H., "Some cases of wave motion due
to a submerged obstacle", Proceedings of the Royal
Society of London, Series A, Vol. 93, 1917, pp.
524-532.
14
[20] Havelock, T.H., "Wave resistance", Proceedings of [34] Noblesse, F., "Analytical representation of ship
the Royal Society of London, Series A, Vol. 118,
1928, pp. 24-33.
waves" (23rd Weinblum Memorial Lecture),
Schi~stechnik, Vol. 48, 2001, pp. 23~8.
[21] Havelock, T.H., "Calculations illustrating the effect [35] Oertel, R.P., "The steady motion of a flat ship, with
of boundary layer on wave resistance", Transac-
tions of the Royal Institution of Naval Architects,
Vol.90, 1948, pp.92-80.
[22] Lai, C. and Troesch, A.W., "Modelling issues re-
lated to the hydrodynamics of three-dimensional
steady planing", Journal of Ship Research, Vol.39,
1995, pp.1-24.
t23] Lamb, H., Hydrodynamics, Cambridge, 1932. t37i
[24] Larsson, L., "CFD in ship design - prospects and
limitations," Schiffstechnik, Vol.44, 1997, pp.133-
154.
[25] Lin, W.C., "The force and moment on a twin hull
ship in steady potential flow", Proceedings of the
10th Symposium on Naval Hydrodynamics, ONR,
Washington DC, 1974. pp.493-516.
[26] Lindenmuth, W.T., Ratcliffe, T.J., and Reed, A.M.,
"Comparative accuracy of numerical Kelvin wake
code predictions - 'Wake-Off' ". David Taylor Re-
search Center, Report DTRC-91/004, Department
of The Navy, Bethesda, MD, USA, 1991.
[27] Maruo, H., (in Japanese), Journal of the Society of
Naval Architects of Japan, Vol. 8 1, (1949~.
[28] Maruo, H., "High and low-aspect ratio approxima- [41]
tion of planing surfaces", SchiJ/stechnik, Vol 72,
1967, pp.57-64.
[29] Maruo, H., "Ship waves and wave resistance in
a viscous fluid", Proceedings of the Ist Interna-
tional Seminar on Wave Resistance, Tokyo, Febru-
ary 1976.
[30] Michell, J.H., "The wave resistance of a ship. 43
Philosophical Magazine, Series 5, Vol.45,1898,
pp. 10~123.
[31 ] Newman, J.N., Marine Hydrodynamics, MIT
Press, 1977.
[32] Newman, J.N., "Evaluation of the wave-resistance
Green function: Part 1 - The double integral", [44]
Journal of Ship Research, Vol. 31, 1987, pp.79-
90.
t33] Newman, J.N. and Poole, F.A.P., "The wave resis- f45]
tance of a moving pressure distribution in a canal",
Schiffstechnik, Vol.9, 1962, pp.1-6.
15
an investigation of the flow near the bow and stern",
Ph.D. Thesis, The University of Adelaide, 1975.
t36] Payne, P., "On the maximum speed of the dy-
nafoil", Fourth Int. Conf: on Fast Sea Transporta-
tion, Sydney, 21-23 July 1997. Proceedings ed.
N. Baird, Baird Publications, Melbourne, pp. 283-
290.
Raven, H.C., "A solution method for the nonlin-
ear ship wave resistance problem", Doctoral The-
sis, Technical University Delft, 1996.
[38] Salvesen, N., von Kerczek, C., Scragg, C.A.,
Cressy, C.P. and Meinhold, M.J., "Hydrodyn-
mamic design of SWATH ships", Transactions of
the Society of Naval Architects and Marine Engi-
neers, Vol.93, 1985, pp.325-348.
[39] Scullen, D.C. and Tuck, E.O., "Nonlinear free-
surface flow computations for submerged cylin-
ders", Journal of Ship Research, Vol.39, 1995, pp.
185-193.
Scullen, D.C. and Tuck, E.O., "Free-surface eleva-
tion due to moving pressure distributions in three
dimensions", Journal of Ship Research, 2002.
Suzuki, K., Nakata, Y., Ikehata, M., and Kai, H.,
"Numerical prediction on wave making resistance
of high speed trimaran", Fourth Int. Conf: on Fast
Sea Transportation, Sydney, 21-23 July 1997. Pro-
ceedings ed. N. Baird, Baird Publications, Mel-
bourne, pp. 611-621.
t42] Tuck, E.O., "Low-aspect-ratio flat-ship theory",
Journal of Hydronautics, Vol. 9, 1975, pp. 3-12.
Tuck, E.O., Wave resistance of thin ships
and catamarans, Report T8701, Applied
Mathematics Department, The University
of Adelaide, 1987. (downloadable from:
http: //www.maths.adelaide.edu.au /Ap-
plied/staff/etuck/~.
Tuck, E.O., "Planing surfaces", Chapter 7 of Nu-
merical Solution of Integral Equations, ed. M. Gol-
berg,PlenumPress, 1990.
Tuck, E.O. "Some accurate solutions of the lifting
surface integral equation", ANZIAM Journal, Vol.
35, 1993, pp. 127-144.
[46] Tuck, E.O., "The planing splash", Proceedings of
the 9th International Workshop on Water Waves
and Floating Bodies, Kuju, Japan, April 1994, ed.
M. Ohkusu, Kyushu University, pp. 217-220.
[47] Tuck, E.O., "Planing surfaces", Proceedings of the
A. C. Aithen Centenary Conference, Dunedin, N.Z.,
August 1995, ed. L. Kavalieris et al, University of
Otago Press, 1996, pp. 331-340.
t48] Tuck, E.O., "Waves made by surface or underwa-
ter ocean vehicles", Symposium on Ocean Survey
and Resource Development Technology, Proceed-
ings ed. Han-il Park, Korea Maritime University,
Busan, November 2001.
t49] Tuck, E.O., Collins, J.L. and Wells, W.H., "On ship
waves and their spectra", Journal of Ship Research,
1971, pp. 1 1-21.
t50] Tuck, E.O. and Lazauskas, L., "Optimum spacing
of a family of multihulls", Schiffstechnik, Vol. 45,
1998, pp. 18~195.
[51 ] Tuck, E.O. and Lazauskas, L., "Polymich", Depart-
ment of Applied Mathematics, The University of
Adelaide, 1999.
t52] Tuck, E.O. and Lazauskas, L., "Free-surface pres-
sure distributions with minimum wave resistance",
ANZIAM Journal, Vol. 43, 2001, pp. E75-E101.
[53] Tuck, E.O. and Newman, J.N., "Hydrodynamic in-
teractions between ships", Proceedings of the 10th
Symposium on Naval Hydrodynamics, ONR Wash-
ington DC, 1974, pp. 35-70.
[54] Tuck, E.O., Scullen, D.C. and Lazauskas, L., Ship
Wave Pattern Evaluation, Part 6 Report: Viscos-
ity Factors, The University of Adelaide, January
2002. (this and earlier SWPE reports are download-
able from: http: //www.maths.adelaide.edu.au
/Applied/staff/dscullen/~.
t55] Tuck, E.O., Scullen, D.C. and Lazauskas, L.,
"Ship-wave patterns in the spirit of Michell",
Proceedings of the IUTAM Symposium on Free-
Surface Flows, Birmingham, July 2000, ed. A.C.
King and Y.D. Shikhmurzaev, Kluwer Academic
Publishers, Dordrecht, 2001, pp. 31 1-318.
t56] Tuck, E.O. and Scullen, D.C., "A comparison of
linear and nonlinear computations of waves made
by slender submerged bodies", Journal of Engi-
neering Mathematics, 2002.
[57] Tuck, E.0 and Standingford, D.W.F., "Numerical
solution of the lifting surface equation", Bound-
ary Element Technology XII, Knoxville, Tenn, 9-
11 April 1997. Proceedings ed. J.I. Frankel, C.A.
Brebbia and M.A.H. Aliabadi, Computational Me-
chanics Publications, Southampton, pp. 195-206.
[58] Tulin, M. P., "The theory of slender surfaces plan-
ing at high speeds", Schiffstechnik, Vol. 21, 1957,
pp. 125-133.
..
[59] Wagner, H., "Uber sto8- und Gleivorgange an der
Oberflache von Flussigkeiten", Zeitschrifte ange-
wandte Mathematik u Mechanik, Vol. 12, 1932, pp.
1931-215.
[60] Wehausen, J.H. and Laitone, E.H., "Surface
Waves", in Handbuch der Physik, ed. Flugge W.,
Ch. 9. Springer-Verlag, 1962.
t61] Xu, Hongbo, "Potential flow solution for a yawed
surface-piercing plate", Journal of Fluid Mechan-
ics, Vol. 226, 1991, pp. 291-317.
[62] Zilman, G. and Miloh, T., "Kelvin and V-wakes af-
fected by surfactants", Journal of Ship Research,
Vol. 45, 2001, pp. 150-163.
16
DISCUSSION
L. J. Doctors
The University of New South Wales, Australia
Thank you for an interesting paper and
presentation.
Regarding the optimum shape, it is well known
that the linear theory will generate hull forms for
displacement vessels that are symmetric fore and
aft.
Would you kindly confirm that, for the planing
problem, the equivalent result is that the
optimum hull is represented by a pressure
distribution that is symmetric fore and aft-this
leads to a planing shape that is not symmetric
fore and aft.
AUTHORS' REPLY
Your last sentence is correct. However, your
penultimate sentence needs a slight correction,
the word "linear" being replaced by "thin-ship."
The planning theory is also a linear theory, but
unlike thin-ship theory, does not generate fore-
aft symmetric hull forms.
DISCUSSION
Chi Yang
George Mason University, USA
Thank you very much for your interesting
lecture. I have a few questions about the wave
cancellation of a four-hull ship. We studied
optimal arrangement of the center hull and the
outer hulls of a trimaran using four different
methods, and were able to find the optimal
arrangement of the center hull and the outer hulls
with minimum wave drag for a range of speeds.
Have you studied the optimal arrangement of the
four hulls to get minimum wave drag? If so, did
you optimize the arrangement for one design
speed or several speeds?
We have compared four different calculation
methods, ranging from slender ship theory to
fully nonlinear Euler solution, for the trimaran
case and found that the slender ship
approximation can be used for determining the
optimal arrangement of the center hull and the
outer hulls for a trimaran. I would like to have
your comment on the thin ship theory and
slender ship theory and their usefulness in the
hull design optimization.
AUTHORS' REPLY
We considered optimal arrangements of 3 and 4-
hulled vessels (at a single design speed) in a
previous paper "Optimum spacing of a family of
multihulls," in Ship Technology Research vol.
45 (1998) pp. 1 80-195.
Our current view is that slender ship theory (i.e.
beam and draft both small compared to length) is
subsumed in thin ship theory (beam small
compared to length), and therefore to a large
measure unnecessary. In particular, there is no
need for a separate code for slender-ship wave
resistance, as an accurate thin-ship code can
simply be run with an artificial very small draft
and a correspondingly increased beam, such that
the section area is preserved.
