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OCR for page 502
Twenty-Fourth Symposium on Naval Hydrodynamics
Fukuoka, Japan, July 8-13, 2002
Nonlinear Free-Surface Effects
on the Resistance and Squat
of High-SpeecI Vessels with a Hansom Stern
Lawrence ]. Doctors (The University of New South Wales, Australia)
Alexander H. Day (The Universities of Glasgow and Strat;hclyde, Scotland)
Abstract
The inviscid linearized near-field solution for
the flow past a vessel with a transom stern is de-
veloped within the framework of classical thin-ship
theory. The hollow in the water behind the stern is
represented here by a virtual extension to the usual
hull-centerplane source distribution. The shape
and length of this hollow are permitted to change
in a realistic manner with increasing forward speed
of the vessel, as well as with any consequent sinkage
and trim that the vessel might suffer.
To this end, the near-field solution to the
flow using the thin-ship approximation must be
computed, in contrast with the traditional far-field
method developed by Michell (1898~.
The computer program includes the facility
to adjust the sinkage and trim of the vessel un-
til it is in equilibrium. The latter feature of the
computation utilizes an integration of the resulting
pressure distribution over the wetted surface of the
vessel in an entirely consistent manner.
Developments reported in this paper are the
inclusion of nonlinear free-surface effects, by intro-
ducing a vertical straining or distortion of the hull,
in order to account for the changing submerged
wetted volume, resulting from the profile of the dis-
turbed free surface. In addition, enhancements to
the analysis, due to the influence of viscosity, by in-
corporating the displacement thickness on the hull
surface, are considered here.
Finally, a semi-empirical theory for the water
flow within the transom-stern hollow at low speeds
is introduced here. This theory can approximately
predict the nature of the partially ventilated flow
and its resultant "back-pressure" on the transom.
Introduction
Previous Work
Previous work on the subject of prediction
of resistance of marine vehicles, such as monohulls
and catamarans, has shown that the trends in the
curve of total resistance with respect to speed can
be predicted with excellent accuracy, using the tra-
ditional linearized Michell (1898) wave-resistance
theory, together with a suitable formulation for the
component of frictional resistance.
A recent justification for this research, in
which linearized free-surface conditions are em-
ployed, is the very encouraging comparisons that
were made by Doctors and Renilson (1993) for
monohulls and catamarans with closed or poirated
sterns and by Sahoo, Doctors, and Renilson (1999)
for monohulls with opera or transom sterns.
Of course, there has also been much devel-
opment of extensive computer codes that attempt
to model, in an approximate manner, the nonlinear
and viscous-wave effects. There has been excellent
progress with such computer programs and they
may eventually be developed to the stage where
they can be used for hull-form development.
Unfortunately, the execution time is too long
for one to contemplate any realistic optimization of
hull forms using such complex computer programs.
Consequently, any realistic type of optimization is
not feasible. This is because of the necessity to
evaluate the object function of the vessel resistance
many times during the practical design process.
A further point is that more sophisticated
computer codes, such as those briefly alluded to
OCR for page 503
z
:` ~
l
An
S'>
I ZTP
G_
RP+~d ~ 1 _ .
Hollow
y
~ anon
z
. BY
A r
it, .
Figure 1: Definition of the Problem
(a) Geometry and Forces
above, do not always lead to more accurate or re-
liable predictions for numerically sensitive quanti-
ties, such as resistance. This is because resistance
can be affected markedly by minor inaccuracies in
the computed pressure distribution over the sur-
face of the hull. A revealing study of this troubling
possibility was published by Sahoo, Doctors, and
Renilson (1999~. It was demonstrated there that
more reliable predictions for the resistance were
obtained from the consistent linearized approach,
than from a modern nonlinear code, for a set of
fourteen modern high-speed vessels with transom
sterns. Indeed, the linearized approach gave pre-
dictions which were within 5% for most of the test
cases, while the errors from the competing nonlin-
ear method were typically an order of magnitude
greater.
In recent years, further improvements to this
work, which dates back to the research published
by Doctors and Day (1997), have been considered.
That initial paper showed that a simple, but ef-
fective, heuristic model of the shape of the hollow
behind the transom stern would suffice to represent
the flow. This model properly estimated the form
of the hollow as a function of such parameters as
the geometry of the transom stern and the forward
speed of the vessel. This approach was a devel-
opment of the original concept for the transom-
stern hollow introduced by Molland, Wellicome,
and Couser (1994~.
Doctors and Day (2000a) included near-field
effects, so that the sinkage and trim (squat) of the
vessel could also be estimated, demonstrating very
encouraging agreement with experiments. Later,
z
Inside hollower Transom
I ' /
==
———~]
.. I.\ ..... _
Paneling Y
_ ~-
~ _
Hull surface
Profile
~~ t
~ I Boundary layer
Figure 1: Definition of the Problem
(b) Centerplane Paneling
Doctors and Day (2000b) added a considerable el-
ement of sophistication, by including an algorithm
to iterate the shape of the transom-stern hollow
so that the pressure on the surface of the hollow
would be atmospheric. This approach involves ap-
olYin~ the Kutta condition to the edge of the tran-
som stern and also adjusting the length of the hull
to minimize the deviations of the pressure from at-
mospheric.
Current Work
The primary aim of the current work is to
study the possibly of incorporating nonlinear wave
effects in the analysis of the vessel. The logic is
that for most practical cases of interest for a high-
speed vessel, the elevation of the generated waves is
small compared to their length; this should permit
us to continue to employ the linearized free-surface
boundary conditions, thus representing a consider-
able advantage.
On the other hand, the wave elevation (to-
gether with the local sinkage of the vessel) can be
relatively large in terms of the local draft of the
vessel. Thus, it is proposed to include corrections,
or distortions, to the hull geometry, so that the lon-
gitudinal distribution of the immersed volume will
be correct. Some initial results from this part of
the work have already been presented by Doctors
and Day (2002~.
A secondary aim here is to implement an im-
provement to the viscous model of the flow past the
hull. That is, the influence of the boundary-layer
2
OCR for page 504
will be included by means of adding the displace-
ment thickness to the hull. It would be anticipated
that the outcome of this change is to increase the
pressure drag on the vessel, because of the greater
effective local beam.
Finally, a tertiary aspect of the study is to
develop a semi-empirical model for the water flow
withers the transom-sterrz hollow at low speeds. The
need for this improvement was made apparent from
previous work on this subject, where it was recog-
nized that assuming that the water was fully sep-
arated, even at low speeds, resulted in predictions
for the resistance which were impractically high.
Theory
Definition of the Problem
Figure lta) shows a typical arrangement for
a vessel traveling at a constant speed U in calm
water. The x,y,z coordinate system is also de-
picted. The water is unbounded laterally (in the
y direction) as well as having infinite depth. The
components of the forces acting on the vessel are
indicated. The vessel is defined by means of the
local beam bin, z).
The vessel can either be self propelled or be
towed. In the former case, the thrust from the pro-
peller or the water jet acts along a defined line of
action relative to the coordinate system attached to
the vessel. Thus, the direction and position of the
thrust line vary with the speed of the vessel. In the
latter case, the vessel is towed at the specified speed
from a particular point in the hull. Hence, the line
of action of the thrust is longitudinal, but the line
moves vertically in sympathy with the sinkage and
trim.
Discretization of the Hull
Figure lobe shows how the centerolane pan-
eling is used to represent the hull and the hollow in
the water behind the transom stern. The panels or
elements possess a flat facet and a rectangular base.
They are employed, in particular, for the purpose
of the numerical calculation of the pressure, or pro-
file, resistance. These elements are chosen in order
to approximate the centerplane area of both the
hull and the hollow as closely as possible.
This type of panel is algebraically simpler
than the "pyramids" or "tents" which were pre-
viously employed by Day and Doctors (1997) and
Doctors and Day (1997), for example. The use of
flat facets implies a higher level of discontinuity on
the hull surface. However, numerical tests indicate
that the required number of panels is not at all
excessive.
Potential-Flow Solution
We start in the classical manner by utiliz-
ing the potential function whose gradient gives the
perturbation velocity. The potential satisfies the
Laplace equation throughout the fluid domain. In
addition, the standard linearized free-surface kine-
matic and free-surface dynamic conditions are to
be satisfied.
The solution for the flow past the vessel and
its transom-stern hollow is obtained by using the
equivalent centerplane-source distribution. This
distribution is assembled from panels, as noted ear-
lier, while the panels themselves are constructed
from the elementary Kelvin point source. The
potential due to a Kelvin point source, obtained
by Wehausen and Laitone (1960, p. 484, Equa-
tion (13.36~), is the starting point of the analysis.
Since the hull panel is assumed to be flat,
it is equivalent to a constant-source distribution.
Hence it is a straightforward matter to analytically
integrate this influence over the area of the source
panel. Similarly, this influence is also integrated
analytically over the unit-constant-longitudinal-
slope field panel in the so-called Galerkin manner.
The result for the induced longitudinal gra-
dient of the potential at the field panel was pub-
lished in detail by Doctors and Day (2000a and
2000b) and will not be repeated here. It was nec-
essary for this purpose to define special wave func-
tions, which are closely related to the exponential
integral of a complex argument. In this way, it is a
possible to express the final result, in which one is
only required to numerically integrate with respect
to the wave angle 8.
Furthermore, because of the abovemen-
tioned Galerkin approach, the integrand is ex-
tremely well behaved. Hence, an appropriate num-
ber of points in the wave-angle integration, over the
range—7r/2 < ~ < 7r/2, is just 64.
3
OCR for page 505
Symbol Theory
Field Michell field integral
Linear Near-field with no squat
NL-1 Near-field with squat
NL-2 Near-field with squat and hull
distortion
NL-3 Near-field with squat, hull dis-
tortion and hull-pressure correc-
t~on
Table 1: Five Wave Theories
Nonlinear Wave Effects
With respect to the importance or otherwise
of nonlinear wave effects, the five different theories
employed for this current work are:
The "Field" approach, which is based on the
Michell integral together with a transom-stern
hydrostatic drag correction. This method, of
course, is too elementary to predict sinkage
and trim. It was referred to as the simplistic
approach by Doctors and Day (2000b).
2. The "Linear" near-field approach in which the
actual pressure is computed, with the vessel
fixed. Nevertheless, the sinkage and trim can
still be computed, using the forces together
with the hydrostatic stiffnesses of the vessel
in sinkage and trim.
3. A partly nonlinear approach, denoted by
"NL-1", in which the vessel attitude is prop-
erly iterated.
4. A more nonlinear approach, introduced here
and denoted by "NL-2", in which the hull is
also strained, or distorted, according to the
formulas:
x' — x
_ ,
A' = Y.
z' = z—(.
(1)
(2)
(3)
The primed coordinates indicate their new val-
ues and ~ is the local elevation of the free sur-
face.
5. A further modification, denoted by "NL-3",
in which the hull-surface pressure p is cor-
rected so that the new pressure p' is zero on
Symbol Theory
Simple Simple friction line using vessel
speed
V Friction line using RMS velocity
on hull surface
V & 5* Friction line using RMS velocity
on hull surface, which has been
thickened by the boundary-layer
displacement thickness
Table 2: Three Viscous Theories
Local Power Coeff-
Reynolds
Law accent
Number n C2
Rod
< 3 x 106 7 0.3709
> 3 x 106 9 0.2716
Table 3: Displacement Thickness
the (strained) free surface, as required by the
physics of the problem:
I'd, y, z) = pox, y, z) - pox, y, 0) . (4)
These different approaches are summarized
Enhanced Viscous Effects
In previous work by the authors, the fric-
tional resistance RF was estimated using only a
suitable friction line based on a semi-empirical
method, such as the 1957 International Towing
Tank Committee (ITTC) formula, described by
Lewis (1988, Section 3.5~. In the present effort,
we consider the three approaches for incorporating
viscous effects which are summarized in Table 2.
These approaches are:
1. The abovementioned simple friction line.
2. A modified approach, in which the root-mean-
square velocity on the surface of the hull is uti-
lized in place of the speed of the vessel in order
to compute both the effective Reynolds num-
ber and the resulting frictional force. This ap-
proach can only be implemented in a method
that involves the near-field computation.
4
OCR for page 506
- -
Item Symbol Value
Length of bow section Lbow 0.750 m
Waterline beam B 0.150 m
Draft T 0.09375 m
Maximum-section coef. CM 0.6667
Table 4: Lego Ship Models (Common Data)
3. A further modified approach, in which the esti-
mated boundary-layer displacement thickness
[* is added to the local hull beam b, in order
to generate a hull, with an effective local beam
b", for the purpose of computing the potential-
flow solution. This idea has been used in the
past in order to include some viscous ejects
in an otherwise potential-flow analysis. The
concept is shown in Figure lobe.
Thus, the effective local beam is given by the
formula:
b" = b + 25* . (5)
To this end, the approach outlined by Dun-
can, Thom, and Young (1970, pp 311 to 319),
was implemented. This method provides the
following equations:
u/uOO =
516 =
5*15 =
Chin ~
cIR-2/(n+3)
1/(n + 1),
(6)
(7)
(8)
where u is the velocity within the boundary
layer, uOO is the velocity at the edge of the
boundary layer, ~ is the coordinate normal to
the hull surface, ~ is the boundary-layer thick-
ness, ~ is the longitudinal coordinate measured
back from the stem at the relevant waterline,
and Ret is the corresponding local Reynolds
number.
The two constants, C2 and n in Equations (6)
to (8), were obtained from boundary-laver
measurements made by Smith and Walker
(1958). Appropriate values depend on the
Reynolds number and are listed in Table 3.
Low-E'roude-Number Regime
The assumption that the water separates
cleanly at the sharp edge of the transom stern has
Length
~ ~ = rat
1 0.000 0.0000 0.7500 0.6666
2 0.000 0.1875 0.9375 0.7290
3 0.000 0.3750 1.1250 0.7499
4 0.000 0.5625 1.3125 0.7290
5 0.750 0.0000 1.5000 0.8332
6 0.750 0.1875 1.6875 0.8494
7 0.750 0.3750 1.8750 0.8499
8 0.750 0.5625 2.0625 0.8275
9 1.500 0.0000 2.2500 0.8888
10 1.500 0.1875 2.4375 0.8957
11 1.500 0.3750 2.6250 0.8928
12 1.500 0.5625 2.8125 0.8735
Table 5: Lego Ship Models (Variable Data)
been shown to be a reasonable approach for model-
ing the physical situation at most Froude numbers
of practical interest. Indeed, if one were interested
only in the power requirements of such vessels in
their principal operational mode, it would proba-
bly be unnecessary to be concerned about the fact
that this idealized model predicts a resistance equal
to the hydrostatic drag at vanishingly low speeds.
From a practical point of view, of course, it
is often the case that a high-speed vessel must also
be capable of operating at low speeds for extended
periods of times, in order to conserve fuel. With
this in mind, we shall now build upon the basic
ideas introduced by Doctors (1998c).
Figure lobe illustrates the possibility of the
transom hollow being partly filled with water in the
low-Froude-number regime. We shall assume that
the water is essentially stagnant and that the level
of the water is below that of the surrounding sea, as
a consequence of the hydrodynamic suction created
by the flow behind the transom.
The elevation of the stagnant water in the
hollow is derived from the Bernoulli equation as
(troll = CPU /29, (9)
where Cp is the (negative) pressure coefficient that
would be generated behind the transom stern in a
Is
OCR for page 507
Figure 2: Towing-Tank Models
(a) Lego Ship Models 6 and 8
double-body version of the hull and g is the accel-
eration due to gravity. The hydrostatic pressure
force acting on the surface of the transom stern is
estimated on the basis of the relative elevation of
the calculation point. This contribution to the re-
sistance (positive aft) is:
o
RH,1 = Pg Jo b(xtran' Z)(Z—(hO11) dz, (10)
—1 tray
in which p is the water density and Than is the draft
at the transom; this is located at the longitudinal
coordinate Fran
Forces and Moment on the Vessel
One first assumes that the attitude of the
vessel is the same as its static attitude, that is,
with zero sinkage and trim. The source strength is
computed using the standard thin-ship approach.
The total gradient of the potential at the
field panel is next computed, by summing the con-
tributions from the individual source panels. From
this, one can determine the pressure on the surface
of the hull, using the linearized Bernoulli equation.
Next, the three components of the general-
ized forces on the vessel (pressure resistance, sink-
age force, and bow-up moment) are found from
this pressure distribution, in the normal manner.
All these calculations were implemented using the
standard consistent linearized approximations for
areas and normal vectors.
Finally, the total resistance can be found by
6
~ ~ ~ Model 12
Model 10
Figure 2: Towing-Tank Models
(b) Lego Ship Models 10 and 12
the simple summation of the pressure resistance
RP, the frictional resistance RF discussed earlier
in this paper, and the correlation resistance RA.
For experiments performed in the towing tank, we
can assume that RA is zero, since the model hull is
hydraulically smooth.
Equilibrium of the Vessel
At any stage of the iteration for the equilib-
rium of the vessel, one can estimate the corrections
to the sinkage and trim angle (positive bow down),
with respect to the longitudinal center of flotation
LCF:
[sLCF = (Sp + W—Zprop)/(p9Aw) ~ (11)
5p = ~—zpRp + (up—LCF)Sp—Mp—
—(xprop—LCF)Zprop +
+ ZpropXprop—
—ZF(RF + RA)—ZstabRstab +
+ (LCB—LCF)W]/(WGML) ~
(12)
The symbols introduced here are SO the sinkage
pressure force, W the weight of the vessel, Zprop the
vertical component of the propulsion force (equal
to zero in the case of the vessel being towed), Aw
the static waterplane area, zp the vertical lever
arm for the pressure resistance (equal to zero), up
the longitudinal lever arm for the pressure sinkage
force (equal to—LCF), Mp the bow-up pressure
moment, xprop the longitudinal lever arm for the
propulsion force, Zprop the vertical lever arm for
the propulsion force, Xprop the longitudinal com-
ponent of the propulsion force, OF the vertical arm
OCR for page 508
14 -
xl~2
12 -
10 -
1
2l
~ Curve .
.
0 0 0
8- .
~ _
~ 6-
O-
2-
O- _
Data
Exp
NL—1
NL—1
NL—1
NL—1
: _
32
32
16
_ 32 .
_
Grid
20X10
dox05
40X10 ~
(OX10 ~
I-'
ox
/ Series = Lego
Mao Model = 8
L = 2~069 m
B/L = 0.07Z73
Visc = Simule
_ 1 1 1 1 1 -
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
F
Figure 3: Convergence Tests
(a) Resistance
Data
Exp
NL—1
NL—1
NL—1
NL—1
. 32
32
16
_ 32 .
Grid
20xlO
dOx~
(OX1O
dOxlO
_ ~ '~~Oc
/o° Model = 8
~ ° L. = 2.063 m
/° B/L = 0.072713
~hc¢~° Visc —Simple
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0
Figure 3: Convergence Tests
(c) Trim
for the frictional force (measured to the centroid of
the wetted surface), Cab the vertical lever arm to
the stabilizers, Rs~ab the resistance of the stabiliz-
ers (zero in the current work), LCB the longitudi-
nal center of buoyancy, and GM the longitudinal
metacentric height.
The sinkage at the coordinate origin x = 0
and the trim are simply
s = sLCF—3 ·LCF, (13)
t = —L,B, (14)
where L is the length of the vessel.
For simplicity, the hydrostatic stiffness coef-
ficients were used for iterating the sinkage and trim
of the vessel, as seen in Equations (11) and (12~.
The use of the ideally consistent hydrodynamic
8—
xlo-s
6—
-
2—
O—
Serisa = Logo
Model = 8
L = 2063 m
B/L = 0.0727~3
Visc = Simple
1 1 1
0 0.1 02 0.3 0.4
F
oooo°
Curve
0 0 0 0
_
Ne
~2
32
16
~2
Grid
20X10
40x05
{OX10
{OX10
0.5 0.6 0.7 0.8 0.9
Figure 3: Convergence Tests
(b) Sinkage
stiffness coefficients would have posed a somewhat
major computing challenge. Relative convergence
of 1 x 1O-4 could be obtained within about eight
iterations; once equilibrium is achieved, there is no
error introduced by the simpler approach.
Lego Ship Mode! Series
This series of hulls was developed with the
intention of studying the hydrodynamics of tran-
som sterns. Doctors (1998a, 1998b, and 1998c) pro-
vided the details of the hull segments from which
the ship models were assembled. There was a total
of seven segments. The bow and stern segments
have parabolic waterplanes. The bow, stern, and
parallel-middle-body segments all possess parabolic
cross sections. Figure 2 shows views of four of the
test models. Table 4 and Table 5 list the details of
all twelve of these so-called Lego Ship Models.
Results
Numerical Convergence Tests
The three parts of Figure 3 show a test of
convergence for Lego Ship Model 8, for three phys-
ical parameters of interest. These parameters are
the total resistance RT, the sinkage s, and the trim
t. These parameters have been rendered dimen-
sionless using the weight of the ship W or its length
L, as appropriate. The abscissa in the plots is the
Froude number F. It can be seen that using 40
7
OCR for page 509
o.ll
0.05
G
0.2 -
0.15
0.o5l
, Rr/W
RH/W
RP/W
RF/W
Rr/W
Data
Exp
NL-1
NL-1
NL-1
NL-1
Serisa = Lego
Model = 6
L = 1.688 m
B/L = 0.08889 0 °
Visc = Simple on
cow
W
--~ in:
- - - - - - - - - - - - 1
1 ~, 1 1
0 0~2 0.4 0.6 0.8 1
F
Figure 4: Resistance Components
(a) Lego Ship Model 6
o 1
. t
Rr/W
RH/W
RP/W
RF/W
Rr/W,
. Data
Exp
NL-1
NL-1
NL—1
. NL-1
Series = Lego
Model = 10
L = Z4~8 m
B/L = 0.06154
Visc = Simple
o
o
__~> ~
~~O—o _ _ ~ ~~ _ _ _
r . _~—
1 1 1 1 1 1 1 ~
0 0.1 0~ 0.3 Ql 0.5 0.6 0.7 0.8 0 0.1 0~2 0.~3
F
Figure 4: Resistance Components
(c) Lego Ship Model 10
panels longitudinally and 10 panels vertically is suf-
ficient for the current purpose. Also, one requires
only 32 points for each quadrant of the wave-angle
~ integration. The model length L and the beam-
to-length ratio B/L are also printed on the plots.
Resistance Components
Figure 4 depicts the theoretical computa-
tions of the various resistance components referred
to earlier, for four of the ship models. These show
the hydrostatic resistance RH, the pressure resis-
tance RP, the frictional resistance RF, the total re-
sistance RT, and the total experimental resistance.
In general, the correlation between the theory and
the experiments is good at the higher speeds, which
are of practical significance, that is, for a Froude
number F of 0.5 or greater. These results all per-
02 -
Curve
0 0 0 ~
0.15—
0.1 -
0.05 -
O- _
o
Data
, RT/W E=p
RH/W NL—1
RP/W NL—1
RF/W NL—1
RT/W NL—1
Series = Lego
Model = 8
L = 2~063 m
B/L = 0.072713
Visc = Simple
=:~ I' - --__ ~
0.1 02 0.13 0.4 0.5 0.6 0.7 0.8 0.9
F
Figure 4: Resistance Components
(b) Lego Ship Model 8
0~2- Curve ~ ~ ~ Data ~ Series = L`ego
O O O O RT/W RAP Model = 12
RH/W NL—1 L = 2~813 m
0.15 - ~ RP/W NL—1 B/L = 0.05~
~~__~ RF/W NL—1 Visc = Simple
_ RT/W NL—1
0.1—
Q05—
O- - 9~ 1 1 1
0.4 0.5
F
of
0.6 0~7
Figure 4: Resistance Components
(d) Lego Ship Model 12
fain to the theory NL-1, corresponding to the sim-
plest near-field theory incorporating squat.
The disagreement between theory and exper-
iment is greatest at the lower speeds, where the
transom would in reality be partly wetted, thus re-
ducing the drag. This phenomenon has been ig-
nored in the current calculations. An indication of
the error involved at these low speeds is just the
hydrostatic resistance; it can be observed that sub-
tracting this quantity would bring the theoretical
calculations into line with the experimental data.
This process was done in an approximate manner
by Doctors (1998c).
Nonlinear Wave Effects
Figure 5 shows the specific resistance R/W
8
OCR for page 510
0.16 -
0.12—
0.04
Data
Exp
Field
Linear
NL—1
NL—2
NL—~
Series = Lego
Model= 6
1.688 no
B/L = 0.08889
Visc = Simple=,°' in',,''
54~,,'
~ — ~
0.16
0.12
0.08
0.04 -
O—
,_ 7~
_ ~ 1 1 1
1
0 0~2 0.` 0.6 0.8 1 0
Figure 5: Comparison of Resistance
(a) Lego Ship Model 6
oL.-
Data
Exp
Field
Linear
NL—1
NL—2
. NL—3
Series = Lego
Model = 10
L = Z4~8 m
B/L = 0.06154
Visc = Simple
0.16 -
0.12 -
0.08 -
QOl
O—
0.6 0.7 0.8 0
-
~TT
0 0.1 02 0.S 0.4 0.5
F
1
Figure 5: Comparison of Resistance
(c) Lego Ship Model 10
for the four Lego Ship Models. The different meth-
ods used are listed in Table 1. It can be noted that
either the Field method or the simplest nonlinear
method NL-1 is consistently the best. However, it
is noteworthy that nonlinear method NL-2, which
incorporated hull distortion, is superior for the slen-
derest models. These are Model 10 and Model 12.
The dimensionless sinkage s/L is plotted in
Figure 6. It is quite clear that the two superior
methods are the Field method and the simplest
nonlinear method NL-1. Indeed, it is disappointing
to observe that the more sophisticated nonlinear
methods, namely NL-2 and NL-3, are quite poor in
terms of predicting linkage.
Similar comments can be made about the
nensionless trim t/L in Figure 7. As for the case
Data
Exp
Field
Linear
NL—1
NL—2
NL—3
Series = Logo
Model= 8
L = 206~3 m
B/L = 0.0727~3
Visc = Simple
I' K~i
o
O.~
1 1 1 1 1 1
02 0.23 0.4 Q5 0.6 0.7 0.8 0.9
F
Figure 5: Comparison of Resistance
(b) Lego Ship Model 8
Data
Exp
Field
Linear
NL—1
NL—2
_ NL—3
Series = LeB°
Model = 12
L = 2813 m
B/L = 0.053
Visc = Simple
~ ,"
0.1 0~2 0.13 0.4 0.5 0.6 0
F
Figure 5: Comparison of Resistance
(d) Lego Ship Model 12
of linkage, the two more sophisticated nonlinear
theories predict excessively high values of the trim.
It is gratifying, however, to see that quite reliable
results can again be obtained from either of the
linear or the simpler nonlinear theory NL-1.
Enhanced Viscous Effects
We now turn to the question of improving
the correlation between the numerical predictions
and the experiments, by adding some degree of so-
phistication to the modeling of the viscous effects.
In order to carefully study these effects,
which are evidently quite small, we will first ex-
amine in Figure 8 the impact on the pressure resis-
tance Rp. To further clarify the phenomena, the
vessel is not free to sink and trim in Figure 8. For
9
OCR for page 511
28 -
X10 a
24 -
20 -
16 -
12]
8-
o
20 -
xlo-3
1°
16 -
_
12 -
8—
I_
O- _
Curve | Data .
o o o o Exp
Linear
NL—1
NL—2
NL—3
Series = Lego
frv~~ ~Model = 6
J r L = 1.688 m
7/ ~ B/L = 0.08889
f | Visc = Simple
,,. 1,-1
~ ~ o O O _ .
0.4
Figure 6: Comparison of Sinlcage
(a) Lego Ship Model 6
Curve Data
o o o o Exp
_ Linear
NL—1
NL—2
NL—3
roll
6/ Series = Lego
j~ ~0~o° ~0_--
/ v, - , ~
_~r~ ~ ~~ L = 2.4~8 m
B/L = 0.06154
Visc = Simple
, ^~ it,
f ~ O OOZE
0 0.1 02 0.S 0.4
F
l
0.5 0.6 0.7
Figure 6: Comparison of Sinkage
(c) Lego Ship Model 10
the sake of brevity, the results for just two models,
Lego Ship Model 6 and Model 8, are presented.
The influence of adding the displacement
thickness to the ship hull in order to generate the ef-
fective hull, as detailed in Equation (5), does indeed
increase the pressure (or potential-flow) resistance
of the vessel. Furthermore, this increase operates in
the hoped-for direction, in that the final result for
the predicted total resistance will increase. Thus,
these predictions will be brought more into line
with the experimental data plotted in Figure 5.
Nevertheless, the effects are indeed small and
only amount to around one percent of the pressure
resistance. Hence, the favorable impact of the dis-
placement thickness on the total resistance will be
even less on a relative basis.
Data
Exp
Linear
NL—1
NL—2
NL—3
_-
1
14 -
X10 ~
12—
10 -
8—
6—
I_
2—
O—
0.8 0
Series = Lego
Model = 8
~ _
''
I! ~
~ ~
f /
f ~—~2—=~ of -cam
~ ~ L = 2063 m
,,—~ ~Q~,/ B/L = 0.07273
~ Visc = Simple
1 1 1 1 1 1 1 1
0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9
F
Figure 6: Comparison of Sinkage
(b) Lego Ship Model 8
Data
Exp
Linear
NL—1
NL—2
NL—3
Series = Lego ~ Jo\/
Model = 12 vA> 0OOO g
,~1
At, . f '_,_
~—/~4' out B/L —0.05333
O Jo - °~ Visc = Simple
~ 1 1 1 1 1
0.1 0.2 0.3 0.4 0.5 0.6 0
F
Figure 6: Comparison of Sinkage
(d) Lego Ship Model 12
We now fix our attention onto the more real-
istic case of a vessel which is permitted to sink and
trim. These results are plotted in Figure 9. As in
Figure 8, we present the results for only two Lego
Ship Models. It can now be seen that any minor ef-
fects of the displacement thickness on the pressure
resistance are even less for this case. The reason for
this is presumably that any slight increase in the
general level of the pressure (as noted in Figure 8),
will (in the real case) lift the vessel slightly out of
the water. That is, the sinkage is marginally less.
As a consequence, a smaller part of the hull is
immersed and the final increase in the pressure re-
sistance, due to the displacement thickness, is now
essentially negligible.
Figure 9 also differs from Figure 8 in that
10
OCR for page 512
60 -
X10
50 -
40 -
30 -
20:
10 -
0]
Data
Exp
Linear
NL—1
NL—2
_ NL—:3
~—_/
rJ
I ;,~{~-o =-o-=oo-~
f ~ ~ Leo Series = Lego
~ Model = 6
0 0 0 oo~o
L = 1.688 m
B/L = 0.08889
vim = Timely
0 0.2 0.4 0.6 0.8 1
F
Figure 7: Comparison of Trim
(a) Lego Ship Model 6
40
xlO~3 Curve
0 0 0 0
30
20 -
~ _
10—
O—
—10 -
Data
Exp
Linear
NL—1
NL—2
NL—:3
/ ~
,/
in_,/
~ ~ /x
/~ -I 0
,'~°Series= Lego
,c~ Model— 10
B/L = 0.06154
Visc = Simple
1 1 1 1 1 1
0 0.1 02 0.13 0.4 0.5 0.6 0.7 0
F
Figure 7: Comparison of Trim
(c) Lego Ship Model 1O
three curves are plotted rather than just two. The
additional curve represents the intermediate case
of allowing only for the variation in the velocity
of the water past the hull, without the influence
of the displacement thickness. Ideally, the result-
ing slight difference in frictional drag will alter the
equilibrium condition for the vessel, and hence the
computed pressure resistance. The data shows that
this effect is truly quite negligible.
Low-Ffoude-Number Regime
Finally, we consider the matter of the low-
Froude-number theory for the assumed stagnant
water in the transom hollow, as discussed earlier.
The four parts of Figure 10 show results of
these computations for the four Lego Ship Mod-
60 -
X10 ~
50 -
40 -
30 -
20 -
10 -
O—
—10 -
0 0.1 02 0.3 0.4 Q5 0.6 0.7 0.8 0.9
F
Data
LinEeXaPr /r ~ W/'
NL—1 ~
NL—2 / Series = Lego
_ NL—~ ~/ Model = 8 __
-I
L = 2063 m
W
B/L = 0.07273
Visc = Simple
1 1 1 1 1 1 1
Figure 7: Comparison of Trim
(b) Lego Ship Model 8
30—
X10-3
25—
20—
15—
10—
5—
O—
—5—
Data
Exp
Linear
NL—1
NL—2
NL—:3
of
0 0.1 02 0.S 0.4
F
Figure 7: Comparison of Trim
(d) Lego Ship Model 12
f ~
f \.
Series = LeB°
Model = 12
L = 2.813 m
B/L = 0.05~ ^~ ,~J
Visc = Simple ~/ ~
, /,~ _~
1
0.5 0.6 0.7
els under study here. It was recognized that these
predictions for the drop in the water level, given
by Equation (9), depend on the value of Cp, and
hence the form of the hull. To this end, one can
consult Hoerner (1965, Figure 21, p. 3-12~. This
figure shows the base-pressure coefficient for bullet-
like bodies in an unbounded flow. Typical values
of Cp lie between—0.2 and—0.1, depending on
the length-to-diameter ratio for this body. There
are indications that the pressure coefficient can be
somewhat more negative for other bluff bodies.
The principal observation is that the theoret-
ical model does predict the correct tendencies very
well. It can be seen that the use of a pressure coeffi-
cient of—0.3 works well for Lego Ship Model 6 and
Model 10, for which the vessel is almost parallel-
sided near the stern. On the other hand, the use of
11
OCR for page 513
40
X 10-3 Series = Lego
Model= 6
L = 1.688 m
B/L = 0.08889
Data = Linear /'
Free = No /'
/ Curve | Visa
/ Simple
Van*
- l l l 1
0 0 ~ 0.4 0.6 0.8
F
36 - _
E
32 -
A:
28 -
24
20 -
Figure 8: Viscous Ejects without Squat
(a) Lego Ship Model 6
52
xio-3
48 -
44 -
40-
36-
:32]
28-
24—
20 - _
_ Series = Lego
Model = 6
L = 1.688 m
B/L = 0.08889 ~/
Data = NL—1
Free = Yes
l
0 02 0.4
F
0.6 0.8 1
Figure 9: Viscous Effects with Squat
(a) Lego Ship Model 6
a pressure coefficient of—0.2 works well for Lego
Ship Model 8 and Model 12, where there is a strong
closing in of the hull ahead of the transom.
Concluding Remarks
The work presented in this paper has shown:
1. The nonlinear theories failed on the most part
to improve the correlation between the predic-
tions and the experimental data. Generally
speaking, either the field approach or the sim-
plest nonlinear theory provided very workable
correlation. That is, the idea of applying dis-
tortion to the hull body in order to generate
a more correct longitudinal distribution of the
immersed hull volume usually gave worse pre-
1
12
32 -
xlo-s
28 -
~ 24-
R 20-
16 -
12 -
8-
Series = IRgo
Model= 8
L = 2063 m
B/L = 0.07273
Data = Linear
Free = No
| Curve | Visa |
— 1~ 1
0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9
F
Figure 8: Viscous Effects without Squat
(b) Lego Ship Model 8
4.5—
X10 ~
40—
35—
~ 30-
20—
15 -
10 -
5—
Series = Len
Model = 8
L = 2~063 m
B/L = 0.07273 of
Data = NL—1
Free = Yes i/
Curve Vise
Simple
~ ..~ ~. Y
Van*
l l l l l l l
0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9
F
Figure 9: Viscous Effects with Squat
(b) Lego Ship Model 8
dictions for resistance, linkage, and trim.
I. The viscous-correction model marginally im-
proved the accuracy for the prediction of resis-
tance, by increasing the values of these predic-
tions. Hence, the correlation with model tests
could be made better. Because the boundary
layer is relatively thinner for a prototype ves-
sel, this effect can be ignored at full scale.
It should be emphasized here that the cor-
rection given by Equation (5) could be made
normal to the hull surface, rather than in the
transverse direction. Similarly, one could de-
fine ~ in Equation (7) as the distance around
the waterline from the stem at the relevant
draft, rather than as the longitudinal distance.
Both these changes would slightly increase the
impact of the displacement thickness on the
pressure resistance.
OCR for page 514
0.1
0.08 -
0.04 -
/~-'
it'
0 0~ 0.4 0.6
Lea = Lego
Model = 6
L = 1.688 m
B/L = 0.08889
vise = v & 6*
1
0.8 1
Figure 10: Model for lYansom-Stern Hollow
(a) Lego Ship Model 6
0.16-
o
_ Curare Data C Series = Lego
LD
Exp Model = 10
NL—1 —0.2 L = 2~4~8 m
B/L = 0.06154,
Visa = Vim*
0.04 -
O- _
0 0.1
o
1 1
02 0.3 0.4 0.5 0.6 0.7 0.8
F
Figure 10: Model for liansom-Stern Hollow
(c) Lego Ship Model 10
3. A frictional-resistance form factor of unity has
been used throughout. It is clear from a study
of Figure 4 that a factor slightly greater than
unity, which increases with the beam-to-length
ratio, would greatly improve the numerical
predictions for the total resistance.
4. The low-Froude-number theory for the tran-
som flow was able to predict the correct trends
in the resistance curve at low speeds. This the-
ory provides predictions for the total resistance
that are vastly superior to those resulting from
the simpler assumption of the fully-ventilated
transom stern.
Regarding the low-Froude-number theory, it
should be worthwhile to investigate this matter fur-
ther. In particular, it would appear that near-
perfect correlation with towing-tank experiments
0.16 - Curve Data _
o o o o Exp
NL—~ 02
0.12- ~ NL-1 —.S 0 /
NL—1 Full /
0.08- ~
o.o4_ ~ , _
o
0 0.1 02 0.S 0.4 0.5
F
Series = Lego
Model = 8
L = Z0613 m
B/L = 0.07273
Visa = V & 6*
1
0.6 0.7 0.8 0.9
Figure 10: Model for Tr~nsom-Stern Hollow
(b) Lego Ship Model 8
Ql6- Curve Data — Series = Lego
o o o o Exp Model = 12
_ NL-1 - ~ L = 2~813 m
0.12 - ~ NL—1 —.S B/L = 0.051333
NL—1 Full Visa = V & 6*
0.08 -
0.04- ~
o- 1~1
0 0.1 0.2 0.S 0.4
F
i< -
0~
I_
1
0.5 Q6 0.7
Figure 10: Model for Trar~som-Stern Hollow
(d) Lego Ship Model 12
will be obtained, if one could just develop a sim-
ple approximate dependence of the transom-stern
pressure coefficient on the form of the ship hull (the
longitudinal rate of change of the sectional area)
just ahead of the transom stern.
Further insight into this matter was pro-
vided by Oving (1985), who claimed that the
beam-to-draft ratio of the transom plays an impor-
tant role in answering this question. An alterna-
tive approach, using computational-fluid mechan-
ics (CFD), may also provide understanding of the
transom-stern-hollow flow.
Finally, the matter of the towing-tank exper-
iments should be raised. To this end, additional
tests are planned in order to gauge the accuracy of
some of the experimental data presented here.
OCR for page 515
Acknowledgments
The authors would like to acknowledge the
assistance of the Australian Research Council
(ARC) Discover-Project Grant Scheme (via Grant
Number DP0209656~.
The in-kind support of this work by The Uni-
versity of New South Wales and The Universities
of Glasgow and Strathclyde is also greatly appreci-
ated.
References
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Proc. T~uenty-Second Symposium on Naval Hydro-
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DOCTORS, L.J.: "An Improved Theoretical Model
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cember 1998)
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diction for Transom-Stern Vessels", Proc. Fourth
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tion (FAST '97J, Sydney, Australia, Vol. 2, pp 743-
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DOCTORS, L.J. AND DAY, A.H.: "The Squat
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DOCTORS, L.J. AND DAY, A.H.: "Steady-State
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Workshop on Water Waves and Floating Bodies
(17 IWWWFBJ, Cambridge, England, 4 pp (April
2002)
DOCTORS, L.J. AND RENILSON, M.R.: "The In-
fluence of Demihull Separation and River Banks on
the Resistance of a Catamaran", Proc. Second In-
ternational Conference on Fast Sea Transportation
(FAST '93J, Yokohama, Japan, Vol. 2, pp 1231-
1244 (December 1993)
DUNCAN, W.J., THOM, A.S., AND YOUNG, A.D.:
Mechanics of Fluids, Edward Arnold (Publishers)
Ltd. London, 725+xiv pp (1970)
HOERNER, S.F.: Fluid-Dynamic Drag, Ho-
erner Fluid Dynamics, Brick Town, New Jersey,
438+xiv pp (1965)
Lewis, E.V. (ED. ) Principles of Naval Architec-
ture: Volume II. Resistance, Propulsion and Vibra-
tion, Society of Naval Architects and Marine Engi-
neers, Jersey City, New Jersey, 327+vi pp (1988)
MICHELL, J.H.: "The Wave Resistance of a Ship",
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OVING, A.J.: "Resistance Prediction Method for
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4
OCR for page 516
DISCUSSION
A.F. Molland
University of Southampton, United Kingdom
I agree with authors that are difficult to make
significant changes to basic results of linearised
thin ship (Michell) theory. Transom stem
correction is important as it influences trim.
Current work at Southampton is using a panel
method t1] to find sinkage and trim based on a
hull waterline elevation from thin ship theory.
Results to date are encouraging with the panel
code giving the full surface pressure and velocity
distribution.
Have the authors made wave cuts for these
models to make sure that they are getting the
correct resistance for the correct reason? If wave
form is not correct then this will cause problems
for investigations of wash effects.
Would the authors concur that a viscous form
factor approach for resistance estimate is more
appropriate for their method given implicit
limitations of thin ship approach?
What choices would the authors make for a prior
prediction of the resistance and squat of a
transom stern vessel and what confidence levels
would they ascribe to that prediction?
L1] Turnock, S.R., "Palisupan: user guide and
technical manual," Ship Science Report No. 100,
University of Southampton, 1997.
AUTHORS' REPLY
I would like to thank Dr Molland for his
interesting questions. Dr Molland has done much
original research into the hydrodynamics of
high-speed vessels, in particular catamarans, so it
is a pleasure to receive a considered question
from him.
I understand that much of our work parallels that
of Molland and his colleagues; in particular, the
hydrodynamic model for the transom-stern
hollow that we currently use was inspired by his
early contributions to this field.
At the time of the Fukuoka conference, wave
cuts had not been made for our models, although
these were at the planning stage. Since then, I
have tested a series of catamaran models in the
Ocean Basin at the Australian Maritime College.
In these tests, various demi-hull spacings, water
depths, and model speeds were tested. In each
case, the wave elevation along a set of eight
longitudinal wave cuts was recorded.
While the analysis of these results is still at an
early stage, we have demonstrated that the
associated wave-elevation-prediction computer
program provides wave profiles which are
verified by the experiments within a few percent.
Differences are mainly in the phasing of the
waves rather than in their elevation and, hence,
the energy density in the wave system and the
resulting wave resistance is expected to be
predicted well - as demonstrated by the total
resistance predictions presented in our paper.
We agree that a viscous form factor, applied to a
traditional friction line, can be used to improve
the correlation between theory and experiment.
Generally speaking, we have found that a
viscous form factor between 1.05 and 1.10 and a
wave-resistance form factor between 0.90 and
0.95 provide extremely close correlation, which
can be within 5% for practical hull forms.
In conclusion, experience over a five-year period
indicates that this correlation can be achieved
with the straightforward application of the
Michell (1898) theory, provided the transom
hollow is included in the formulation. The far
more complicated near-field calculation provides
predictions of sinkage and trim, which are
probably sufficient for practical purposes, but the
accuracy in percentage terms is not as high as
that for the Michell far-field resistance
predictions.
Representative terms from entire chapter:
ship model
