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OCR for page 191
Steady-State Hydrodynamics of High-Speed Vessels
with a Hansom Stern
Lawrence J. Doctors (The University of New South Wales, Australia)
ATexancler H. Day (The University of Glasgow, Scol:lan(l)
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. However, the innovation in the current ap-
proach is that the hollow in the water behind the
stern is represented here by a virtual extension to
the usual hull-centerplane source distribution.
In the present work, the near-field solution
to the flow using the thin-ship approximation must
be computed. This idea clearly demands a consid-
erable addition to the complexity of the numerical
solution which contrasts with the traditional far-
field method employed by Michell (1898~.
The computer program functions by iterat-
ing the geometry of the hollow until the criterion of
atmospheric pressure on the surface of the hollow is
zero. In addition, the iteration procedure includes
adjusting the sinkage and trim of the vessel until
it is in equilibrium. The latter component of the
computation utilizes an integration of the resulting
pressure distribution over the wetted surface of the
vessel in an entirely consistent manner.
Comparison of the theoretical results with a
systematic series of twelve models shows excellent
correlation with the towing-tank data. Indeed, the
behavior of this approach appears to be much more
robust than, for example, that of the Neumann-
Kelvin problem and some fully nonlinear analyses.
Introduction
Literature Review
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 Michell (1898) wave-resistance theory, to-
gether with a suitable formulation for the compo-
nent of frictional resistance. There have been fur-
ther enhancements to this wave-resistance theory.
These enhancements include the influences of finite
depth and finite width of the canal by Lunde (1951)
and Sretensky (1936~.
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 pointed
sterns and by Sahoo, Doctors, and Renilson ( 1999)
for monohulls with open or transom sterns.
One difficulty has been that nonlinear and
viscous-wave effects are not included and, conse-
quently, the correlation between theory and exper-
iment has not been sufficiently good for the purpose
of practical ship design. For this reason, a consid-
erable effort has been invested in recent years in
the development of fully nonlinear computer codes.
The complete nonlinear kinematic body-boundary
condition and the nonlinear free-surface kinematic
and dynamic boundary conditions are satisfied in
these programs. There has been excellent progress
with such computer programs and they may even-
tually be developed to the stage where they can be
used for hull-form development.
Currently, the execution time is too long for
one to contemplate any realistic optimization of
hull forms. Consequently, any type of optimiza-
tion, such as the genetic-algorithm method of Doc-
tors and Day (1995) and Day and Doctors (1997b)
is not possible. This is because of the requirement
to evaluate the resistance many thousands of times
during the practical design process.
OCR for page 192
::
R I
Pi r
:` .
Mp
Sp
w
i
,, REP op
~ Lxpxro~ i Y
Hollow 1
.~-~1= 4_~
~L;oU 1 ~ ~
Figure 1: Definition of the Problem
(a) Geometry and Forces
A further point is that more sophisticated
computer codes, such as those briefly alluded to
above, do not always lead to more accurate or reli-
able predictions for sensitive quantities, such as re-
sistance. This is because resistance can be affected
markedly by minor inaccuracies in the computed
pressure distribution over the surface 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 pre-
dictions for the resistance were obtained from the
consistent linearized approach, than from a mod-
ern nonlinear code, for a set of fourteen modern
high-speed vessels with transom sterns. Indeed, the
linearized approach gave predictions which were
within 5% for most of the test cases, while the
errors from the competing nonlinear method were
typically an order of magnitude greater.
Further reductions in the errors inherent in
the linearized theory can be obtained in a most
practical manner by means of very easily esti-
mated correction factors for the wave resistance
and for the frictional resistance. Examples of this
research were published by Doctors (1998a, 1998b,
and l99Sc).
Current Work
These principles were advanced in the re-
search presented by Doctors and Day (1997~.
Firstly, transom-stern effects were included in the
theory by accounting for the hollow in the water
behind the vessel. This work was essentially a de-
velopment of the ingenious and practical approach
2:
Legend for panel types
x Hull
+ Hollow
Hull and hollow
Null
Figure 1: Definition of the Problem
(b) Centerplane Paneling
first presented by Molland, Wellicome, and Couser
(1994) and Couser, Wellicome, and Molland (1998~.
There, a simple virtual extension to the hull
behind the transom was constructed in the com-
puter program. The wave resistance for the vessel
was deduced on the basis of an application of the
Michell integral to the entire model and its exten-
sion, together with an estimate of the hydrostatic
resistance due to the existence of the dry transom.
The extension was allowed to grow in length with
increasing forward speed of the vessel in a physi-
cally plausible manner as detailed by Doctors and
Day (1997~.
A more sophisticated enhancement of this
work was presented recently by Doctors and Day
(2000~. The local flow field was computed on the
basis of the linearized theory. This permitted the
squat (linkage and trim) to be determined; ex-
cellent correlation with experimental data was ob-
tained in this way.
However, the shape of the hollow was still
determined in a heuristic manner. It is the purpose
of the current research effort to iterate the shape of
this transom-stern hollow in order to improve the
accuracy of the approach even further.
Theory
Definition of the Problem
Figure lta) shows a typical arrangement for
a vessel traveling at a constant speed U in calm
OCR for page 193
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 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
Further details of the problem are provided
in Figure lobe. Here, the centerplane paneling is
seen to overlay 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 profile, re-
sistance. These elements are chosen in order to
approximate the centerplane area of the hull and
the hollow as closely as possible. The longitudi-
nal and vertical slope of the facets of the elements
match the corresponding values of the hull surface
in a root-mean-square sense.
This type of panel is algebraically simpler
than the "pyramids" or "tents" which were previ-
ously employed by Day and Doctors (1997b) and
Doctors and Day (1997a), for example. The use
of flat facets implies a higher level of discontinuity
on the hull surface. On the other hand, numerical
convergence tests for wave resistance, based on the
two types of panels, showed that a similar number
of panels was required in either case; namely, 40
panels in the longitudinal direction and 8 panels in
the vertical direction.
Equations for the Potential
We start in the traditional manner by uti-
lizing the potential ¢, whose gradient gives the
perturbation velocity. The potential satisfies the
Laplace equation throughout the fluid domain:
Axe + ~~y + fizz = 0 . (1)
The linearized kinematic free-surface condition,
namely
~z+U~ = 0 on z=0, (2)
states that any water particles on the free surface
remain there. Here, ~ is the elevation of the free
surface measured upward from its undisturbed po-
sition z = 0. In addition, the Bernoulli equation
provides the linearized dynamic free-surface condi-
tion
Ups—9;—,ll; = 0 on z = 0, (3)
in which 9 is the acceleration due to gravity. The
Rayleigh (1883) artificial viscosity ,u, which is as-
sumed to be vanishingly small and positive, has
been introduced in this technique in order to im-
pose the radiation condition, which states that
waves must be propagated downstream. De Prima
and Wu (1957) gave a clear description of this con-
cept. Elimination of ~ from Equations (2) and (3)
yields the linearized combined free-surface condi-
tion:
U2¢x2 + 9¢'z—,u~x = 0 on z = 0 . (4)
Finally, the "bed" kinematic boundary condition
states that
As = 0 as z ~—no . (5)
Potential-Flow Solution
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 are constructed from the el-
ementary Kelvin point source. The potential due
to a Kelvin point source of strength Q. obtained
by Wehausen and Laitone (1960, p. 484, Equa-
tion (13.36) ), is
47rT + her, + (P ~ (6)
where we have defined the radial distances
r = >/(x—x/~2 + (y _ y/~2 + (z _ z/~2 `7y
r' = x,/(x—x/~2 + (y _ y/~2 + (z + z/~2 (8)
OCR for page 194
Marl ~1 n
Model,
Figure 2: Sample Ship Models
(a) Lego Ship Models 7 and 8
and the wave term is
7r oo
~ 47r2 | do | do (exp{k[z + zl +
—7r 0
Model 12
it_
Figure 2: Sample Ship Models
(b) Lego Ship Models 11 and 12
and the required arguments in the special function
G4 are defined by
x = Xi—xj + lax,
+ id—x') cost + iffy—y') sinew/ Z = pi—Zj + mi\z .
/(ko—k cos2 ~—i,llcos0) .
Here, the circular wave number is
k = he sec2 0,
the fundamental wave number is
ho = g/U2,
and the complex horizontal wave number is
The first term in Equation (6) can be inte-
grated for a constant-strength-source panel ok and
a unit-constant-longitudinal-slope field panel in the
so-called Galerkin manner. The result for the in-
duced longitudinal gradient of the potential at the
field panel is
fx,1
= -4 ok ~ w' ~ wm G4(i' j,l,mj,
I=—I m= - 1
(10)
in which the weighting factor is given by the for-
mula
w' = ~ 2
~ -1
—1 for 7
= 0
= 1
The four special functions needed for this
analysis are interrelated by spatial integrations as
follows:
G~(x, z) = 1/~,
G2(x,z) = ~G,(x,z)dz
= sinh-i (z/~x~), (12)
1~ +iLy = 1cexpti8) . G3(x,z) = /G2~,z)(1:~
z sinh-i (x/~z~) + x sinh-i (z/~x~),
G4~(x,z) = ~G3(x,z)dz
2 z2 sinh-i (x/~z~) +
+ xz sinh-i (z/~x~)—
2X~ .
(13)
(14)
The second term in Equation (6) can be in-
tegrated in a very similar manner.
The third term can also be integrated with
respect to the wavenumber it, as well as with re-
OCR for page 195
specs to the spatial coordinates, to yield
~r/2 ~ ~
27~2k2~i / cos36 ~ we ~ Wm.
—7r/2 1= - 1 m= - 1
F4 (Z(i, j, 1, m)) do ~ (15)
in which the argument of the function F4 is given
by
Z(i, j, 1, m) = k(zi + zj + mi\z) +
+ ik~(xi - Xj + If\x) .
The special complex wave function F4 is
closely related to the exponential integral and was
defined by Doctors and Beck (1987). The zeroth
wave function is
co
Fo(z) = f ~P( I)dk-7riexp(z) - 1/z
o
= exp(z) {E1 (z) - 21riH[Im(—z)] } - 1/z
(16)
and the complex exponential integral is
00
E1(z) = | P( )dt,
z
while the other four complex wave functions are
Fl(Z) = /Fo(Z)dZ
Modeling of the Hollow
Fo(z) + 1/z, (17)
F2 (z) = / F1 (z) dz
F1 (z) + ln(—Z), (18)
F3(z) = / F2(z) dz
F2 (z) + z ln(—Z)—Z. (19)
F4(z) = / F3(z) dz
F3(z) + 2z2 ln(_z) _ 4 z2 . (20)
The hollow is considered to be a virtual ex-
tension to the hull of the vessel and is modeled by a
continuation of the centerplane source distribution.
The determination of the shape of the centerplane
of this hollow represents a vital part of the hydro-
dynamic problem.
The length of the hollow is initially esti-
mated by the method detailed by Doctors and Day
(1997~. That is, the eq~bivalent radial position of a
point in the region of the hollow behind the stern
is taken to be
r = ~/~. (21)
We next state that the effective trajectory of a
particle of water on the surface of the hollow is
parabolic in nature and that it has parametric co-
ordinates as follows:
x = x~ran —Ut, (22)
r = r ran—U — t _—gt2, (23
[ d~ ~ ~ 2=:~tran
in which x~ran and r~ran are the coordinates of the
equivalent springing point of the hollow on the
transom girth.
Equations (22) and (23) can be solved by
setting r = 0 to yield the location of the vertex, or
rooster tail, of the hollow ~ho~ Hence, the length
of the hollow is evaluated as:
Lho~ = XGran —Xho~ (24)
In the current enhanced technique, the ini-
tial estimated profile of the hollow is assumed to
be defined by the parabola that springs longitudi-
nally from the bottom of the transom and meets
the vertex on the (un(listurbed) free surface.
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 required source
strength is computed using the standard thin-ship
result, namely,
a. = - U—,
~x
in which b is the local beam. The total gradi-
ent of the potential ~x at the field panel i is next
computed, by summing the contributions from the
source panels presented in Equations (10) and (15~.
From this, one can determine the pressure on the
surface of the hull, as follows:
p = p(U¢~ - gz), (25)
OCR for page 196
where p is the density of the water and z is the
elevation of the centroid of the field panel.
Next, the three components of the general-
ized forces on the vessel (pressure resistance, sink-
age force, and bow-up moment) are then found
from this pressure distribution:
RP = - ||P~ dude, (26)
s
in which the generalized surface slope is given by
the formula
fib
-
Finally, the total resistance can be found by
the simple summation
RT = RP +RF +RA, (28)
in which Rp is the pressure resistance computed
from Equation (26), RF is the frictional resistance
estimated from the 1957 International Towing Tank
Committee (ITTC) formula, described by Lewis
(1988, Section 3.5), and RA is the correlation resis-
tance.
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:
~SI.CF = (Sp + W— Zprop)/(P9AW) ~ (~295)
b) = ~—zpRp + (up—LCF)Sp—ME—
—(xprop—LCF) Zprop +
+ ZpropXprop —
—IF (RF + RA ) —Zstab Rstab +
+ (LCB—LCF)W]/(WGML)
(30)
The additional symbols introduced here are SO the
sinkage pressure force, W the weight of the ves-
sel, Zprop the vertical component of the propulsion
fib
ax
fib
LIZ
z——x— for bow-up moment
ax Liz
for resistance
force (equal to zero in the case of the vessel being
towed), Aw the static waterplane area, zp the ver-
tical lever arm for the pressure resistance (equal to
zero), up the longitudinal lever arm for the pres-
sure 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 longitudi-
nal component of the propulsion force, IF the ver-
tical arm for the frictional force (measured to the
centroid of the wetted surface), Zs~ab the vertical
lever arm to the stabilizers, Rub the resistance of
the stabilizers (zero in the current work), LOB the
longitudinal center of buoyancy, and GM the lon-
gitudinal metacentric height.
for sinkage force. (27) The sinkage at the coordinate origin x = 0
and the trim are
s = sI,cF—~ LCF, (31)
t = -Lid, (32)
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 (29) and (30~.
The use of the ideally consistent hydrodynamic
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.
Iteration of the Hollow
The pressure at any point on the hollow is
given by a formula of the type of Equation (254. If
there are Thou panels that lie on the centerplane
of the hollow, then there will be the same number
of panel collocation points at which we desire the
pressure to be zero (or atmospheric). The Kutta
condition is applied by extending this requirement
to include the last column of panels on the surface
of the hull next to the transom.
In addition, we must impose the closure con-
ditions. Because the panels are of uniform rectan-
gular shape, we just require the source strengths crj
along a longitudinal line of panels (z is constant)
to sum to zero.
A set of overconstrained linear equations is
then set up, in which the pressures are minimized
OCR for page 197
in a least-square sense while the closure conditions
are satisfied exactly. Next, the secant method is
employed in which the length of the hollow is ad-
justed until the average pressure on the surface of
the hollow is minimized.
The error in the average pressure on the sur-
face of the hollow after the iteration process was
typically one percent of the hydrostatic pressure at
the keel of the vessel.
In summary, it can be seen that the method-
ology used in this work is analogous to that under-
lying the solution of the problem of symmetric two-
dimensional cavitating flow behind ~ body possess-
ing a bluff stern, as presented by Newman (1980,
Section 5.13, pp 208-215~. However, the following
extensions have been introduced:
1. Three-dimensions;
2. Free-surface;
3. Equilibrium of the body;
4. Iteration of cavity shape.
Simplistic Resistance
The simpler and traditional approach to
thin-ship resistance has been to utilize the Michell
(1898) result for the resistance applied to the ves-
sel, together with a suitable formulation for the
shape of the hollow. Examples are the work of
Day and Doctors (1997a), Day, Doctors, and Arm-
strong (1997), Doctors and Day (1997), and Doc-
tors (1999~. This approach leads to the following
estimate of the total resistance:
RT = RW + RH + RF + RA, (33)
in which Rw is the wave resistance, computed as
7r/2
RW = PU2 / sec36(P'2 + Q'2)d6, (34)
o
where the complex Michell wave function is
P'+iQ'
= // ~ ' ~ exp~ik~x + hz~dxdz
so
-
= ~ Pi' + iQ', (35)
i=1
Item
Length of bow section
Waterline beam
Draft
Maximum-section coef.
Symbol
Lbow
B
T
_ CM -
Value
0.750 m
0.150 m
0.09375 ret
0.6667
Table 1: Lego Ship Models (Common Data)
with SO being the centerplane area. The complex
wave function for a panel with center (xi, Zi) is
4 1 1
Pi' + iQ' = k l sin(2kx/\x) sinh(2ki\z)
'2
[A ] exptik~xi +hzi) . (36)
Finally, RH is the so-called hydrostatic resis-
tance, resulting from the imbalance of the hydro-
static pressure owing to the transom being "dry":
o
RH =—P9 ~ b(xtran~z)zdz ~ (37)
or
—1 t ran
in which Ttran is the draft at the transom.
This simplistic approach to resistance pro-
vides an identical result to that from the near-field
approach described earlier in this paper, in the case
of a vessel with no transom.
Lego Ship Model Series
This series of hulls was developed with the
intention of studying the hydrodynamics of trarl-
som sterns. Doctors (1998a) provided the details
of the hull segments from which the ship models
were assembled. There was a total of seven seg-
ments. 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 1 and Table 2 list the details of all twelve of
these so-called Lego Ship Models.
Results
Numerical Convergence Tests
The wave r esistance of a vessel without
a transom stern, computed according to Equa-
tions (34) to (36), is identical to that computed
OCR for page 198
\l ~ R ~ · \ 1~ ~ ~ ' ~
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 2: Lego Ship Models (Variable Data)
using the considerably more elaborate near-field so-
lution. This assumes the same discretization of the
hull centerplane and the use of the same mesh in
the wave-angle integration. Thus, one has an inter-
esting and powerful check of the computer coding.
The four parts of Figure 3 show a test of
convergence for Lego Ship Model 7 for four phys-
ical parameters of interest. These parameters are
the total resistance RT, the sinkage s, the trim t,
and the hollow length Lam. These parameters have
been rendered dimensionless using the weight of the
ship W or its length L, as appropriate. It can be
seen that using 40 panels longitudinally and 8 pan-
els vertically is sufficient for the current purpose.
In the same vein, one requires 32 points for half
the range of the ~ integration in Equation (15) to
obtain reasonable convergence of the results.
.
The annotation Holl—P,K indicates that
in this case the pressure condition on the surface of
the hollow and the Kutta condition on the transom
were satisfied. Also, Free = Yes states that the
model was free to sink and trim. The slenderness
coefficient L/Vi/3 is also printed on the plots.
Resistance Components
Figure 4 depicts the theoretical computa-
tions of the various resistance components referred
to above, for four of the ship models. These show
the hydrostatic resistance RH, the pressure resis-
tance RP, the frictional resistance RF, the total
resistance RT, and the total experimental resis-
tance RT,E- 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 0.6 or greater.
The disagreement is greatest at the lower
speeds, where the transom would in reality be
partly wetted, thus reducing the drag. This phe-
nomenon has been ignored in the current calcula-
tions. An indication of the error involved at these
low speeds is just the hydrostatic resistance; it can
be observed that subtracting this quantity would
bring the theoretical calculations into line with the
experimental data. This process was, in fact, done
by Doctors (1998d) in an approximate manner.
Comparison with Experiments
Figure 5 shows the specific resistance R/W
for the four Lego Ship Models. The different meth-
ods used are respectively: the simplified method of
Doctors and Day (1997) (Holl = Simp), the simpli-
fied method of Doctors and Day (2000) with the
vessel also free to sink and trim (Free = Yes), the
current method with the pressure and the Kutta
conditions applied (Holl = P. K), and the current
method with the pressure and Kutta conditions ap-
plied and also the hollow length Lho~ iterated to
minimize this pressure (Holl = P. K, L).
It is noteworthy that the main factor of i~r~-
portance is the need to permit the vessel to sink
and trim in order to find its equilibrium position.
The other various enhanced versions of the theory
appear to offer little, if any, improved correlation
with the experiments.
We next examine Figure 6, which shows a
comparison of the theoretical predictions and ex-
periments for the dimensionless sinkage s/L for
the four Lego Ship Models. The second, third,
and fourth methods are considered here. The first
method, of course, does not include sinkage and
trim at all. Again, we observe that the simpli-
fied method which permits determining the equi-
libriurn of the vessel, namely that of Doctors and
Day (2000), is as effective as the current more so-
phisticated approaches.
1— —
OCR for page 199
x 1ol2 - Curve No Nx No
10- ~ 32 20 8
32 40 4
8- 32 40 8
~ a'
6-
2-
O-
D
1
0 0.2 0.4
D,
4-
Series = Lego
Model= 7
L = 1.875 m
Holl = P,K
Free = Yes
1
0.6 0.8 0
F
x101° - Curve Ng Nx
_
_ _ 16 40
8 - ~ 32 20
32 40
32 40
6-
2-
O- _
No
_
8
8
4
8
Liz
_~'
///~
// _
//
//
,//
Series = Lego
Model = 7
L = 1.875 m
Holl = P,K
Free = YeF'
Figure 3: Convergence Tests Figure 3: Convergence Tests
(a) Resistance (b) Sinkage
x 103-5 - Curve No _
30—_ _ _ 16 40
~ 32 20
25- ~ 32 40
32 40
20 -
15 -
5-
1C
O-
. _
. No
8
8
4
8
'/!
1 1
0 0.2 0.4
lay
F
Figure 3: Convergence Tests
(c) Trim
,~ 1~
q
c
Series = Lego
Model = 7
L = 1.875 m
Holl = P,K
Free = Yes
1
0.6 0.8 0
Similar sentiments can be expressed with re-
gard to Figure 7, which presents the dimensionless
trim t/L for the four Lego Ship Models. Again,
the simplified method which includes the determi-
nation of the equibrium of the vessel, is as accurate
as the current methods.
Finally, we present Figure 8, which shows
the dimensionless hollow length Lho~/L for the four
Lego Ship Models. It is curious to note that there
is little significant difference between the length of
the hollow computed in the current hydrodynamic
manner and the length of the hollow resulting from
the simplified geometric approach of Doctors and
Day (2000), for example. At the very minimum,
the general trend of the variation of the length of
the hollow with the Fioude number is properly pre-
dicted, but would seem to be rather too high.
/
o-
Nx
40
20
40
40
/
/
1
0.2 0.4
1
F
Figure 3: Convergence Tests
(d) Hollow Length
Series= Lego
Model = 7
L = 1.875 m
Holl = P,K
Free = Yes
1
0.6 0.8
It is also vital to point out the obvious fact,
with regard to Figure 8, that the experimental data
is at least as subject to debate as the theoretical
calculations. The length of the hollow was deter-
mined by means of a visual observation using a wire
mesh as a reference frame. The accurate location
of the rooster tail of the hollow Thou was not possi-
ble because of various factors; it would even seem
that the definition of this length itself poses a major
difficulty.
Concluding Remarks
The present research has been instructive in
showing that the precise length of the hollow be-
hind the transom stern is not critical to predicting
the resistance, sinkage and trim, of the vessel. In
many ways, this is quite remarkable, because the
OCR for page 200
R—
Xlol2 - Curve Comp L/V1/3 = 7.615 1 x10-2 Curve Comp L/V1/3 = 8.187
10 ~ ~ Free — Y.s 10 Free — Yes
R
6—Series = Lego to ,, 6- Series = Lego GIG
Model = 7 /O ° ~ ~ Model = 8 / °
~ ~ -8-° ° -I '< oO
2 ~ _ _— _ _ _ D_ - - - - =
o
0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
F i
Figure 4: Resistance Components Figure 4: Resistance Compc
(a) Lego Ship Model 7 (b) Lego Ship Model 8
x 101° - Curve Comp
o o o o T,E ]
~ F
6— T
Series = Lego
4—Model = 11
L = 2.625 m
2-
O-
L/V1/3 = 9.375
Hall = P,K °
Free = Yes
0 /
0 0.'
/c
2.625 m ~°
/ 0
1
0.2 0.3 0.4
F
Figure 4: Resistance Components
(c) Lego Ship Model 11
10 -
xlo-2
A—
_
6-
' V<4- .
it' ________ 2-
1 1
0.5 0.6
flow of the water in the region of the hollow must
surely affect the pressure distribution on the hull
itself. There should be a resulting influence on the
behavior of the vessel. It now appears that the
principal features of the hollow can indeed be mod-
eled in the simplified manner, allowing one to find
(within engineering accuracy) the essential hydro-
dynamic performance factors of interest.
Other models for the profile of the hollow
have been considered. For example, it is thought
that an S-shaped profile might also be worthy of
study. Here, one end of the S corresponds to the
springing point from the bottom of the transom and
the other end corresponds to the rooster tail. Both
ends of the S would be horizontal in this alternative
model.
O-
Curve Comp
o 0 0 0 T,E
~ _ H
_______ P
_ F
T
Series = Lego
Model = 12
L =2.813 m
L/V1/3—9889
Holl = P,K
Free = Yes
o
~ 0-~
0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
F
Figure 4: Resistance Components
(d) Lego Ship Model 12
It is necessary to emphasize that, at least
with regard to resistance, the current approach still
uses a traditional estimator for the frictional com-
ponent. That is, ideally, one might consider utiliz-
ing a form factor to correct the frictional resistance.
Alternatively, and preferably, a more sophis-
ticated approach could be considered. That is,
the influence of the actual geometry of the hull on
the frictional resistance would be computed. Such
an approach would take into account the non-zero
pressure gradient on the surface of the hull and
would be consistent with the current thin-ship the-
ory.
Finally, it would be feasible to incorporate
the influence of the deformation of the free-surface
on the actual submerged volume of the vessel. This
OCR for page 201
x 1 ol 2 - Curve Holl Free Comp ~ x 1 o-2 Curve Holl Free Comp
o o o o Yes T,E o o o o Yes T,E
10 - _ Simp No T ° ~ 10 - _ _ _ Simp No T .
Simp Yes T ~ , - ~ Simp Yes T At,/ ,, ~
~ 8 1 _ | P,K,L | Ye9 | T | ~ ' | ~ 1~ | P,~ ~ Yes | T | i, -a l
ty6- igloo''' 6- / '',o,,-
~_ ~~ 9~- 4- /~o'
_,~Oo ° Series = Lego ,~,O'o Series = Legs
o° Model = 7 2— :'' o ° Model = l
,~ L = 1.875 m ~ L = 2.064
~, O ¢b L/V1/3 = 7.615 L/V1/3 = 8.187
O- 1 1 ' 1 O- 1 1 1
0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
F ~
Figure 5: Comparison of Resistance Figure 5: Comparison of Resist
(a) Lego Ship Model 7 (b) Lego Ship Model 8
10 -
xlo-2
8-
Curve | Holl | Free | Comp
o o o o Yes T. E
__ __ _ __ Simp No T
__ _____ Simp Yes T
_ _ _ _ _ P,K Yes T
_ P,K,L Yes T
I:
me'
—''o ~ Series = Lego
to ° Model = 11
O ° L = 2.625 m
a O ° L/Vi/3 = 9376
4-
O-
o
0 i'
,'
0 0.1
0.3
F
0.4
0.5
0.6
Figure 5: Comparison of Resistance
(c) Lego Ship Model 11
could be effected approximately by means of a local
vertical distortion of the sectional geometry. The
resulting influence may well be of similar signifi-
cance to that of permitting the vessel to find its
equilibrium.
Acknowlecigments
The authors would like to thank the Direc-
torate of Naval Platform Systems Engineering, De-
partment of Defence, for its support through Con-
tract 9627MZ. They also gratefully acknowledge
the assistance of the Australian Research Coun-
cil (ARC) Large Grant Scheme (via Grant Num-
ber A89917293~. The support of this work by The
University of New South Wales is also greatly ap-
preciated.
0.7
10
xlO_2 - Curve
0 0 0 0
6-
4-
2-
Holl
Simp
Simp
P,K
_ P,K,L
Free | Comp |
Yes T. E
No T
Yes T
Yes T
Yes T
o
of
art'
Series = Lego
to Model = 12
~Oo°° L =2.813 m
O ° ° L/V1/3 = 9.889
T I I T I I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
F
Figure 5: Comparison of Resistance
(d) Lego Ship Model 12
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OCR for page 202
10—
X10-3
03
X10-9
6-
Curve Holl
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8 - Simp
P,K
P,K,L
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4-
2-
Data
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Figure 6: Comparison of Sinkage
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O-
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-
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75 Series = Lego
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Figure 6: Comparison of Sinkage
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10
x 10-3 Curve Holl
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P,K
P,K,L
6-
4-
Series = Lego
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L = 1.875 m
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Free = Yes
1 O- 1
0.6 0.8 0 0.2
—
—
o o x10-3 Curve Holl
a,, 5- 0 0 0 0
o 0 ~ Simp
P,K
4— P,K,L
3-
-
xlO-3
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n-
Data
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,Y
to
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L/V1/3 = 8.187
Free = Yes
1
0.4 0.6 0.8
F
Figure 6: Comparison of Sinkage
(b) Lego Ship Model 8
Data |
Exp
Theory
Theory
_ Theory
too
O o °
OO0 ~
o,'~
o,? 0
d5~ Series = Lego
I) Model = 12
L =2.813 m
L/V1/3 = 9 889
Free = Yes
1 1 1 1 1 1 1
0.7 0 0.1 0.2 0.3 0.4
F
0.5 0.6 0.7
Figure 6: Comparison of Sinkage
(d) Lego Ship Model 12
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OCR for page 203
8
4
1
—~ 1
30 -
X10-8
25 -
20 -
1'
10 -
o
Hell
Simp
P,K
P,K,L
Data
Exp
Theory
Theory
Theory
By,
Boo
// o
my/ °
,,~c~=~==<'
~ ~ ~ tare °~ _
_—-~~ ~ X1
,~o °
~ oo
0
o
Series = Lego
Model = 7
L = 1.875 m
L/V1/3 = 7.615
Free = Yes
0.2 0.4 0.6 ~
Figure 7: Comparison of Itin.
(a) Lego Ship Model 7
Holl
Simp
P,K
P,K,L
_
Data
Exp
Theory
Theory
Theory
lo
i,
~0°°
Series = Lego
~ /' o Model = 11
o 0 0 oooOo° °OoO L =2.625 m
L/V1/3 = 9.375
Free = Yes
1 1 1 1 1- 1
0.1 0.2 0.3 0.4 0.5 0.6
F
Figure 7: Comparison of Trim
(c) Lego Ship Model 11
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35 -
O ~
30 -
25 -
20 -
15 -
10 -
5—
Curve
0 0 0 c
Holl
Simp
P,K
. P,K,L
Data
Exp
Theory
Theory ,^~
Theory /~~ O ° °
,/ o o
,y 0
,y 0
lo°
~ 0
lo
~Oo 0
Series = Lego
Model = 8
L = 2.063 m
L/Vt/3 = 8.187
=_ I Free = Yes
0.8 0 0.2 0.4 0.6
F
Figure 7: Comparison of Trim
(b) Lego Ship Model 8
Gus
xlo-3
12 -
0.7 0
0.8
8
4—
O—
-4- 1
0.1
. Holl
Simp
P,K
. P,K,L
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Exp
Theory
Theory
Theory
0
~ 0
~ 0
0
/ o
/ 0
/O Series = Lego
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L/V1/3 = 9 889
Free = Yes
1 1 1 1
0.2 0.3 0.4 0.5 0.6 0.7
F
Figure 7: Comparison of Trim
(d) Lego Ship Model 12
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OCR for page 204
-
o
~-
20 -
xlo-2
16 -
12-
8-
~-
.
Curve Holl
0 0 0 0
Simp
P,K
P,K,L
Series = Lego
Model = 7
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.
Data
. Exp
Theory
Theory
. Theory
L/V1/9 - 7.615 /
Free = Yes /
,~ooooooocoo<~oo ° °
. . .
0 0.2 0.4 0.6 0.8
Figure 8: Comparison of Hollow Length
(a) Lego Ship Model 7
16 -
xlo-2
12 -
8—
4-
O—
A
16 -
xlo-2
12 -
_
8-
4-
Curve Holl
0 0 0 0
Simp
~ Pa
P,K,L
Series = Lego
Model = 8
L = 2.063 m
Data
Exp
Theory
Theory
Theory
/
L/V1/3 = 8.187
Free = You
/ 00000000000000 0 0 °
. . .
0 0.2 0.4 0.6 0.8
F
Figure 8: Comparison of Hollow Length
(b) Lego Ship Model 8
_ Curve Holl Data L/V l /3 = 9.875 x 1 ol 2 - Curve Holl Data L /V 1/3 = 9.889
o o o o Exp Free = Yes o o o o Exp Free = Yes
~ Simp Theory / 10 - ~ Simp Theory Off ~
_ ~ P,~ Theory / ~ ~ P,K Theory I
P,K,L Theory /: ~ 8- P,K,L Theory /
Series = Lego / ~ ~ Series = Lego a'
_ Model = l 1 / ~ 6—Model = 12 /
/ 4- /
/OOOooooOOooooooo ° ° ° 1 24 / ooooooooooooo Do 0 0 °
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
F F
Figure 8: Comparison of Hollow Length
(c) Lego Ship Model 11
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Figure 8: Comparison of Hollow Length
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OCR for page 205
DISCUSSION
L. Rahe a
Indian Inshtute of Technology, India
It was a nice presentahon. It is not a
queshon but more of a comment. It
seer s that the primary reason or
monvahon for using such a simp ined
model (i.e. ccuterplaue source
di.vribuuon I is because it is expected or
presumed that the wave drag will be
much smaller than She f ichona pa t. I
propose Chat the model cou d be
improved upon by using a
desingu anzed panel method where She
sources cou d be placed lid e inside the
bounds y which wou d sell preserve She
imp ined ca cu anon aspect and may
work better in She sense of di using She
prearm Fr on.
AUTHOR'S REPLY
I appreciate your comment. Since we
were getting good resu h in this case, so
we continued with the model. However,
we may improve upon it in subsequent
work. Tha k you very much for your
qllo.r on.
Representative terms from entire chapter:
theory theory theory
