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OCR for page 581
Computations of 3D Transom Stern Flows
Bill H. Cheng
David Taylor Research Center
Bethesda, USA
Abstract
A practical computational method for 3D
transom stern flows is presented. The theory and
numerics of computing transom stern flows are
described in detail. The special treatment of the
linearized free surface boundary condition is
included. In particular, the boundary condition for
a dry transom is derived within the framework of a
free surface potential flow. The transom is treated
as an inflow boundary and the transom boundary
condition is then used to specify the starting values
of a linearized free surface calculation. This
computational method has been incorporated into a
Rankine source panel method, the XYZ Free
Surface (XYZF,S) program version 2.0. The
XYZES program has been used to predict the wave
resistance for a large number of transom stern
ships. The compute d wave resistance has com pare d
favorably with measured wave pattern resistance,
correctly predicting the relative merit of competing
hulls. The agreement with experimental
measurements is remarkably good for Froude
numbers between 0.35 and 0.50, corresponding to
the normal speed range of high-speed transom
stern ships.
1. Introduction
A practical computational method for
ship
by
the
the
transom stern flows is of special interest to
hydrodynamicists. Such interest is intensified
the peculiar property of the flow pattern. If t
ship speed is high enough, the transom clears
surrounding water and the entire transom area is
exposed to the air. The transom flow detaches
smoothly fro no the underside of the transom, and a
depression is created on the free surface behind the
transom. Saunders (14 described this flow pattern
in his book on hydrodynamics and his description
of a dry transom is reproduced in Fig. 1. This flow
pattern has been credited with the reduced wave
resistance for high-speed transom stern ships, as
compared to their cruiser stern equivalents.
581
Early numerical studies encountered
problems in modeling transom stern flows.
Rankine source and Havelock source methods were
hampered by the difficulty of treating the
intersection curve between the transom and the
free surface. Many investigators replaced the fluid
domain behind the transc)'n faith solid surfaces.
For example, Gadd t2] and Ghang [3] simulated
the surface depression by adding a tapering
extension to close the body behind the stern.
Realistic flow patterns cannot be obtained in this
way and more accurate computations of transom
stern flows are needed.
This paper describes an in pro ed method
for com puting transom ste r n flows. The theory and
numerical method for computing 3D transom stern
flows are presented step by step. The special
treatment of the linearized free surface boundary
condition is given in detail. The boundary
condition for a dry transom is derived within the
framework of a free surface potential how. The
physical constraints imposed by this transom
boundary condition require that the static.pressure
be atmospheric and that the flow leave tangentially
at the transom. This computational technique has
been successfully incorporated into a Ranltine
source panel method originally developed by
D awson t4,5~ and further developed as the XYZ
Fre e Su rface ~ XYZFS ~ program Ve rsio n 2.0.
The XYZFS program has been used to
analyze the wave resistance and local flowfield for a
large number of transom stern ships. Comparisons
of wave resistance prediction and experimental
measurements have been made and have shown
general agreement. Cheng et al. t6: described the
hydrodynamic analyses of DTRC Model 5403 and
Model 5404, which represent typical transom stern
ships with bow domes. A parametric study on
stern wedges, as an example of transom stern
variations, was described by Cheng et al. t74. An
FFG-7 hull with a 15-degree stern wedge was
OCR for page 582
TRANSOM
MEAN
WATER LEVEL
~ ~ ---I
in_
DEPRESSION IN WATER
BEHIND HANSOM STERN
UNDISTURBED
WA1 - SURFACE
Fig. 1. Flow pattern behind a transom stern
(sketch from Saunders t 1 ] ).
analyzed by Hoyle et al. [8~. Cheng et al. t94
presented a comparison of the compute d prope lie r
inflow with the corresponding data from wake
survey experiments for a variant of the research
vessel, R/V Athena hull.
Transom stern studies prior to 1983 were
summarized by Cheng et al. A. Subsequently,
Wilson and Thomason [10] published a parametric
study on transom sterns. The effects of transom
depth and transom width variations were identified
using a combined numerical and experimental
approach. More recent studies on transom sterns
included residual resistance computations for Series
64 by Tulin and Hsu [11~. These computations used
a strip theory which is applicable in the limit of an
infinite F'-oude Amber.
In this paper, the computed flow pattern
behind the transom is presented and compared with
experimental measurements by Jenkins t12] for the
R/V Athena hull. In addition, a comparison of
wave resistance prediction and experimental
measurements is presented for D TRC Model 5416.
2. Physical problem
Consider a steady flow directed from bow to
stern of a ship. Attention is focused on the region
Just forward and aft of the stern as shown in Fig. 1.
Figure 2 is a schematic diagram representing a
sideview of a transom stern flow in a centerplane.
The keel line typically slopes upward relative to the
mean water level, which is designated by a
horizontal dashed line in Fig. 2. The wavy solid
line represents a streamline which passes under the
transom, detaches from the stern, and forms part
of the free surface. Immediately downstream of
the transom, the water rises rapidly toward the
mean water level, overshoots, and then reaches a
maximum elevation before descending under the
-
KEEL LINE
Fig. 2. Schematic of transom stern flow for the
centerplane shown in a sideview.
influence of gravity. Such ascending and
descending motion is repeated further downstream.
The wave motion described here can be seen in the
wave pattern computed by Coleman t 134 using a
finite difference method.
3. Numerical method
3.1 Panel method for cruiser stern flows
Before the computational method for 3D
transom stern flows is introduced, a panel method
for cruiser stern flows is described. The usual
treatment of cruiser sterns is presented since the
special treatment for transom sterns is similar and
is an extension of the cruiser stern method.
In the Rankine source panel method, a hull
surface is m athem atically subdivided into hundreds
of small source panels. Each panel is characterized
by its centroid, normal unit vector, and area. The
velocity at the control point (i.e., centroid) of a
given panel i induced by another panel j is a
function of the geometry of panel j and of the
distance between panel i and panel j . For
example, the x-component of the velocity induced
by panel j on panel i is equal to the induced
velocity (per unit source strength), denoted as the
influence coefficient CXi;, multiplied by the source
strength Sj, which is a constant for each panel.
The sum of the free stream velocity Uoo and the
velocity induced by all other panels j gives the
velocity at panel i:
N
Ui = Uoo + ~ CXi,jSj · (1)
j =1
In Eq. 1, the summation is carried out for all the
panels on the hull surface, and the quantity N
denotes the total number of panels. The unknowns
on each panel are the source strength to be
determined from a solution of the boundary value
problem subject to the Neumann boundary
condition on the hull surface. This panel method
scheme was developed by Hess and Smith [14~ for
the flow of an infinite fluid past a ship-like body.
582
OCR for page 583
Another way of modeling a ship is to
combine the submerged ship hull with its mirror
image above a plane of symmetry at the mean
water level. This combination is referred to as the
double model in the literature. The flow past a
double model is obtained by a numerical solution
of the boundary value problem subject to the
Neumann boundary condition on the double model
hull. This double model solution is the
approximate solution to the free surface problem in
the limiting ease of zero Froude number when the
free surface is approximated by a rigid wall.
After the double model solution is obtained,
the double model streamlines are traced on the
mean water level. These streamlines will not
penetrate the hull surface and are used to set up a
free surface grid. Constant source panels are placed
on this free surface grid. Then, the free surface
boundary condition is linearized using the double
model solution as the basic solution in the sense
that the deviation from the double model flow is
considered small:
`f = ~ + ˘', (2)
where ~ denotes the velocity potential for the free
surface flow, ~ the velocity potential for the
double model, and ˘' the perturbation velocity
potential. D awson t4] gave the resulting free
surface boundary condition with the double model
linearization as
~ ~ 2˘ ~ + go = 2 ~ l 2~ ~ I ~ 3 ~ SPECIAL SECTION At', ~~ =~\
BEHIND THE—@ 3 ~ , ~ ~
TRANSOM fit _ : ~ ==
~ ~ ~ .~ ,~77>
at the mean water level z O . The free surface
boundary condition involves the gradient of the
velocity potential along a streamwise direction
designated by I, and differentiation is carried out
along the corresponding double model streamlines.
For example, the streamwise velocity on the free
surface is computed by
~l= x ox + Y by . (4)
j~X2 + ~y2- j~X2 + ty2
Note that this differentiation scheme approximates
the free surface flow direction by the double model
flow direction. This appre~im ation is a crucial ste p
to be used later in the derivation of the transom
boundary condition.
In the numerical calculation of the
convective term, finite differencing is used to
calculate derivatives between adjacent panels along
a double model streamline. A four-point upstream
finite differencing scheme is used to eliminate
upstream propagating disturbances as recommended
by Dawson ~4~. For the foremost upstream point,
a two-point upstream finite difference operator is
used. The starting values are specified by a uniform
flow and a corresponding wave elevation of zero.
In the panel method for free surface flows,
the source strength distribution must satisfy
simultaneously the linearized free surface boundary
condition and the Neumann boundary condition on
the hull. For a case with N panels representing
the total number of the hull panels and free surface
panels combined, the resulting system of N by N
equations is full, nonsymmetrieal, and not
diagonally dominant. A Gaussian elimination
scheme is used to solve this system of equations.
For the ease of a transom stern, the main
section of the free surface (as shown in Fig. 3) can
be handled in the same way as for cruiser sterns.
The double model linearization and the upstream
finite difference operator are also used. However,
the flow domain behind the transom presents a new
problem and needs special treatment. In particular,
a new section of free surface panels is introduced,
the transom is treated as an inflow boundary, and
the starting values of a free surface calculation
must be specified at the transom. This problem is
addressed in the remainder of this paper.
MAIN SECTION7 FREE SURFACE
Fig. 3. An aerial view of the free surface paneling
for a transom stern hull.
3.2 Theory of the transom boundary condition
As for a cruiser stern ease, a transom stern
solution begins with a double model computation.
The geometric model for a transom stern hull is
shown in Fig. 4. Note that the transom is
intentionally left open and there are no panels on
the transom. The hull surface for the double
model serves as a stream surface separating the
external flow about the hull from a fictitious
internal flow. The double model flow computed in
this way is appropriate for a day transom with the
exit flow detaching at the transom tangential to the
hull surface.
583
OCR for page 584
Fig. 4. Perspective view for the numerical model
of a transom stern hull. Note the open transom.
For the free surface solution, the free
surface boundary conditions must be adapted to the
iansom stern geometry. To simplify the analysis
of transom flows in practical applications, several
assumptions are made in the theory for transom
stern flows. It is assumed that the transom is left
open as in the double model flow computations and
that the potential flow detaches from the transom at
well defined locations. These locations are identified
as the points on a curve which has the same shape
as the transom bottom. The shape of the transom
bottom may be described by specifying the transom
depth ZT as a function of the transve'rse coordinate
y:
ZT = f ( Y ) at x = XT, (5)
where the subscript T denotes the transom and
XT the longitudinal coordinate of the transom.
Along the intersection curve between the transom
and the free surface, the static pressure is equated
to the atmospheric pressure pOO since the transom
clears the surrounding water and is exposed to the
air:
Pr = pOO at x = XT and z = ZT for a given y
In Eq. 6, the effect of an air wake behind the
transom is neglected and the atmospheric pressure
is considered a global constant. With the
dependence on pressure removed, Bernoulli's
equation describes a steady-state balance of kinetic
energy and potential energy:
~ 1 AX + by + ~Z2 ~ + gZT = 1 U2 (7;
where ~ ~x, by, Ha ~ denotes the gradient of the
velocity potential in ~ x, y, z ~ directions. Eq. 7
may be rearranged as
~ 2 + ~ 2 + ~ 2 ~ 2& ZT
(8)
Thus, the kinetic energy for the water at the
transom can be determined by the right hand side
of Eq. 8, where ZT lies below the mean water level
and takes on a negative value. This equation is the
first constraint for transom stern flows.
Since a flow cannot penetrate an
impermeable surface, another constraint to be
satisfied at the transom is that the exit flow must be
tangential to the hull surface. The magnitude of
the velocity at this point is given by the square root
of the right hand side in Eq. 8. The flow direction
is part of the free surface solution and is
determined by the transom geometry, which can be
spe cified by a ( local) tangential unit vector:
X j˘2 + ˘2 + ˘2 ( ~
— he ~ 10)
tax + by + Adz
= go (11)
j˘X2 + ~y2 + ~Z2
A first approximation of the tangential unit
vector is obtained by replacing the potential in Eqs.
9 through 11 bar the double model potential to give
the tangential unit vector in the direction of the
double model flow:
Tx = six ( 12)
j~X2 + ~y2 + ~Z2
Ty = i t 13)
j~X2 + ~y2 + ~Z2
Tz = Z , ( 14)
j~X2 + ~y2 + ~Z2
584
OCR for page 585
FREE SURFACE
BOUNDARY CONDITION
( x, y, z ) by 1/2 L . Equations 15 through 17
can then be rewritten as follows:
i + 1 ~:—A UT = ~ Tx ( 18)
TRANSOM BOUNDARY
CONDITION \'
NEUMANN I BOUNDARY AT = ~ / 1 _ ZTS Ty ( 19 )
CONDITION v n
Flux
Fig. 5. Description of boundary conditions for WT = N/1~ T.:,
transom, free surface, and hull.
where the vector ~ fix, Eye By ~ represents the
velocity of the double model flow for a hull panel
whose centroid lies just forward of the sharp corner
at the transom. This hull panel has the same y-
value as the transom and is denoted by the index
NQ as shown in Fig. 5. The approximation of the
free surface flow direction by the double model
flow direction is consistent with applying the free
surface boundary condition along double model
streamlines on the main section of the free surface,
as spe cifie d by Eq. 4. The validity of this
approximation has been verified by comparing
computed and measured wave profiles behind the
transom.
When the velocity magnitude from Ea. 8 is
(20)
where Fn2=U2 /gL and ZT= ~zT/L . Equations
18, 19, and 20 are applied as the transom boundary
conditions for the free surface calculations in the
case of a dry transom. In so doing, the static
pressure is forced to be atmospheric and the free
surface flow is forced to leave the transom
tangentially in a direction specified by the double
model flow. The transom boundary conditions
show that the velocity at the transom is directly
related to transom depth and the Froude number
and is indirectly related to other transom
characteristics (i.e., the buttock angle, the deadrise
angle, and the run angle) through the direction of
the double model flow.
3.3 Implementation of the free surface boundary
condition
combined with the flow direction from Eqs. 12
through 14. the three velocity components at the In this section, the special treatment for the
transom are approximated by:
UT = N/~TX (15)
VT = A/W,~ Ty (16)
WT = N/~ Tz , (17)
where the vector ~ UT, VT' WT) has been
nondimensionalized by ~ US and denotes the
velocity at the transom in the ~ x, y, z
directions, respectively. The sign of the vector
~ UT, VT, WT ~ IS determined by the tangential unit
vector. The quantity °° represents the square of
gZT
the Froude number based on the transom depth.
In usual numerical calculations, it is more common
to use the Froude number based on the waterline
length L . This convention is achieved by scaling
free surface boundary condition behind a transom
stern is described. The free surface boundary
condition in this region uses the conventional
linearization about a uniform flow:
2Fn2 ux + ·v = 0, (21)
where the length is nondimensionalized by 1/2 L
and the vector ~ u, v, w ~ represents the
perturbation velocity in a dimensionless form. The
free surface boundary condition is applied at the
mean water level, on which flat panels are placed.
Computations are performed for the centroid of a
panel, as indicated by "bullets" in Fig. 5. In the
discretization of the convective term in Eq. 21, an
upstream finite difference operator is used to
eliminate upstream propagating waves. In
particular, a two-point upstream finite difference
operator must be used to start the computations for
transom stern flows:
UX; = CAi Ui + CB; us, (22)
where CBi = x _ = - CAi . ~ 23)
585
OCR for page 586
In Eqs. 22 and 23, the index i denotes the
foremost point behind the transom and the index
t the upstream point at the transom. The quantity
ui is an unknown to be determined and us, the
velocity at the upstream point, must be specified.
The value us is taken from the transom boundary
condition Eq. 18. However, a minor modification is
required here since UT represents the total velocity
and us represents the perturbation velocity
superimposed on the uniform flow. Thus
Ut = ~ + UT . (24)
Substituting Eqs. 22 and 24 into Eq. 21 and
. ~
rearranging gives,
2Fn2 CA; u; + wi = - 2Fn2 CB; [ 1 + UT] . (25)
This equation is the discretized form of the free
surface boundary condition at a particular y-value
for the foremost free surface panel with index i
next to the transom. The unknowns reside on the
left hand side and the knowns on the right hand
side. For the the downstream panel at the same
y-value with index i + 1, the following equation
is used:
2Fn2 ~ CA+ up + CBi+~ ui ~ + win = 0, (26)
where CB = =- CA 27
i+l Xi - Xi+1 i+}
Equations 26 and 27 are applied to succeeding
panels with indices i+ 2, i+ 3, etc. until the
downstream boundary is reached. Equations 22
through 27 are applied for other y-values to cover
the special section of free surface (Fig. 3)
downstream from the transom.
The resulting system of difference equations
is then solved using the Rankine source panel
method. Substituting Eq. 1 into Eqs. 25 and 26
gives
2F`n2 ~ CAj CXj j Sj - firs; = - 2Fn2 CB; ~ 1 + UT]
and
2Fn2~ [ CAj+1 CXi+l,j + CBi+l C~i,j ]Sj - 2~Si+1 = 0
The index N denotes the total number of panels
for the hull surface, the main section of free
surface, Add the special oil behind the Tom.
The influence coefficient CXj j denotes the x-
component of velocity induced on panel i by panel
j per unit source strength. In the computations of
influence coefficient for each hull panel, the
contribution from its mirror image must be added.
The source sty engths s; are the unknowns to be
determined through the solution of a boundary
value problem, and a Gaussian elimination scheme
is used in the numerical solution. The distribution
of source strength must satisfy simultaneously the
free surface boundary condition and the Neumann
boundary condition on the hull, as represented in
Fig. 5.
A two-point upstream finite difference
operator has been used in the special section
behind the transom for wave resistance
computations. These calculations are applied to a
short computational domain covering one half of a
ship length downstream from the transom. The
four-point operator will be introduced for future
calculations, which extend farther downstream, to
improve the numerical accuracy of transom stern
computations.
4. Comparison with experiments
This section presents a comparison of
numerical predictions with the corresponding
experimental measurements for high-speed transom
stern ships. Such a comparison shows the extent to
which the numerical model of transom stern flows
can predict the hydrodynamics of the physical
model. Another objective is to identify
opportunities for future research in transom stern
flows. Two examples are presented here: the local
flow field around a ship and the force on a ship.
The local flow field is represented by the wave
pattern behind the transom for R/V Athena. The
force on a ship is represented by the wave
resistance for D TRC Model 5410.
4.1 Wave profiles
Figure 6 gives a sideview of the computed
wave profile behind the transom for R/\ Athena.
The vertical and horizontal axes are
nondimensionalized by 1/2 L and are plotted to
(28) the same scale. The numerical model of the
transom stern is shown on the right. The dashed
line represents the mean water level and the solid
wavy line represents the free surface computed on
pane is closest to the ce nte rplane. The
computations were performed for a Froude number
of 0.48. The free surface has a small slope, which
t29y is appropriate for this Froude number, and the flow
586
OCR for page 587
0.4875
2 z/L
O _
COMPUTED WAVE PROFILE
_
_ . at . ~ . ~ _
—MEAI`J WATER LEVEL t
TRANSOM
U X ~
~0.4875 . . . . . ~ ~ l ' _
-2.05 - 1 .9S -1.85 -1 .75 -1 .65 - 1.55 -1.45 - 1.35 -1 .25 -1.15 -1 .05 ~0.95 -0.8S -0.75
2 x/L
Fig. 6. Computed wave profile in the centerplane
behind the transom for R/V Athena
at a Froude numbe r of 0.48.
angle can be measured graphically. Notice the
ascending motion followed by the descending
motion as expected. A gap is left between the
transom and the computed free surface to indicate
the centroid location of the foremost panel, where
the free surface calculations are started. An
upstream extrapolation of the free surface shows
that the free surface intersects the transom at the
transom depth ZT.
Figure 7 gives a close-up view of Fig. 6 near
the transom. Again the dashed line and solid wavy
line represent the mean water line and the
computed free surface closest to the centerplane,
respectively. The corresponding wave profile
measured by Jenkins t12] is represented by isolated
points and the rearmost point indicates the extent
of experimental data. Thus, the experiment
indicates that the transom clears the water. and the
assumption of a dry transom is valid for a Froude
number of 0.48. The numerical prediction and
experimental measurements of wave profiles
behind the transom show close agreement,
espe cially just downstre am of the transom. The
double model approximation of the flow direction
seems Justine d in this case.
4.2 Wave resistance .
D TRC Model 5416 represents a typical
high-speed transom stern ship without a bulbous
bow. Dr. Michael Wilson of DTRC designed this
hull form as a candidate for a low resistance ship.
The computations for Model 5416 were performed
at least one year prior to model construction and
tank testing at DISC. The model tests involved a
longitudinal wave cut experiment using capacitance
wave probes to measure wave profiles. A wave
spectrum analysis of the measured wave profiles
gave the measured wave pattern resistance. The
0.1875
2 z/L
o
~0.1875 l l l
-1.25 - 1.15 -1.05 - 0.95
2 x/L
—COMPUTED WAVE PROFILE (PRES - T STUDS
° MEASURED WAVE PROFILE (JENKINS 111])
-O 85 ~0.75
Fig. 7. Comparison of the computed and
measured wave profiles for R/V Athena
at a Froude number of 0.48.
measured wave pattern resistance was then
compared to the wave resistance predicted by
integrating the pressure on the hull using the
XYZFS program version 2.0.
- .v
2.5 .
Is
° 2.0 _
z
-
-
o
by
-
Al:
1.5 _
1.0 _
0.5 _
_~ '~
_ I ~ ~ ~ ~
0 0.1 0.2 0.3 0.4 0.5 0.6
CALCULATED
(XYZFS) TV \~
MEASURED
/ Cw(p)
f5 (WILSON)
'8~
FROUDE NUMBER, Fn
Fig. 8. Comparison between calculated wave
resistance and me asure d wave pattern resistance
versus Froude number for Model 5416.
This comparison of computed wave
resistance and measured wave pattern resistance is
presented in Fig. 8. The computations were
performed for Froude numbers from 0.25 to 0.50 at
increments of 0.05. The hull form was held fixed
587
OCR for page 588
at the even keel position for these computations
and for the experiments. The calculated wave
resistance and measured wave pattern resistance
show general agreement. The agreement is
rem arkably good for Froude numbers from 0.35 to
0.50, corresponding to the operating speeds of
high-speed transom stern ships and corresponding
to the case of a dry transom. For Froude numbers
below 0.35, the slope of the wave resistance curves
is in agreement. However, computations predict a
hump around a Froude number of 0.3, which is
absent from the measured wave pattern resistance.
This discrepancy is partially attributed to the
breakdown of the dry transom assumption at lower
Froude numbers. Transom stern flow at lower
Froude numbers is the subject of a future study,
and further numerical and experimental research is
needed.
5. Conclusion
A practical computational method for 3D
transom stern flows has been developed. A dry
transom boundary condition has been derived for
linearized free surface potential flows. The physical
constraints imposed by the transom boundary
condition require that the static pressure be
atmospheric and that the flow leave tangentially at
the transom. The pressure constraint has been
applied to Bernoulli's equation to determine the
magnitude of the velocity at the transom as a
function of transom depth and Froude number.
The velocity tangency constraint has been enforced
by requiring the free surface flow to leave tile
transom tangentially in a direction specified by the
double model flow. The nurrnerical implementation
of the free surface boundary condition involves the
use of an upstream finite difference operator to
eliminate upstream waves. The transom boundary
condition has been used to specify the starting
values for the free surface calculations in a special
section behind the transom. This computational
method has been incorporated into a Rankine
source panel method, the XYZFS program version
2.0.
The computed flow pattern behind the
transom has been presented and compared with
experimental measurements for the R/V Athena
hull. Comparison of wave resistance prediction and
experimental measurements have been presented
for D TRC Model 5416. The results show that the
wave resistance predictions are remarkably good for
Froude numbers between 0.35 and 0.50,
corresponding to the normal speed range of high-
spe e d transo m ste rn ships.
588
Acknowledgments
This study was supported by the Surface Ship
Technology Exploratory Development Program and
managed by the Ship Hydromechanics Department
(SHD ~ of the D avid Taylor Research Center
(i;~'l'fCCj. The author is indebted to Mrs. Joanna
W. Schot of D IRC for her encouragement and
support during the course of this study. The author
wishes to thank Dr. Henry Haussling of DTRC for
his advice and his constructive comments on the
manuscript. D r. Michael Wilson of SHD kindly
furnished the experimental results for Model 5416.
Mrs. Janet Dean of DTRC has contributed helpful
suggestions and discussions.
References
. Saunders, H.E., "lIiydrodynamics an Ship
Design, " SNAME, Vol. 1 i i pp.326-327 ( 1957) .
2. Gadd, G.E., "A Method of Computing the
Flow and Surface Wave Pattern Around Full
Forms, " Trans. RICE. Vol. 118, p. 207
~ 1976~.
3. Chang, M.S., "Wave Resistance Predictions
by Using a Singularity Method," Proc. of He
Workshop on Ship Wave-it esistance
Computations, D TNSRD C, Bethesda' MD
(Nov 1979~.
4. D arson, C.VV., "A Practical Computer
Method for Solving Ship Wave Problems," in
Proc. Second International Conference on
Numerical Ship Hydrodynamics, University of
California, Berkeley ~ Sep 1977~.
D awson, C.W., " Calculations with the XYZ
Free Surface Program for Five Ship Models,"
Proc. of the Workshop on Ship Wave-
Resistance Computations, D TNSRD C,
Bethesda, Maryland, (Nov 1979~.
6. Cheng, B.H., J.S. Dean and J.L. Jayne, "The
XYZ Free Surface Program and Its
Application to Transom Stern Ships with Bow
Domes," in Proc. Second Workshop on Ship
Wave-Resistance Computations, D TNSRD C,
Bethesda, MD (Nov 1983~.
7. Cheng, B.H., G.G. Borda, J.S. Dean and S.C.
Fisher, "A Numerical/ Experimental
Approach to Wave Resistance Predictions," in
Computer Aided Design, Manufacture and
Operation in the Marine and Offshore Industries,
Washington, D .C. ~ Sep 1986~.
8. Hoyle, J.W., B.H. Cheng, B. Hays, B.
Johnson, and B. Nehrling, "A Bow Bulb
Design Methodology for High Speed Ships,"
Transactions SHAME) Vol. 94 (Nov 1986).
OCR for page 589
9. Cheng, B.H., J.S. Dean, R.W. Miller, and W.
L. Cave III, "Hydrodynamic Evaluation of
Hull Forms with Podded Propulsors," Naval
Engineers Journal, Vol. 101, (May 1989~.
10. Wilson, M.B. and T.P. Thomason, "Study of
Transom Stern Ship Hull Form and
Resistance," D TNSRD C report 85/072,
D TNSRD C, Bethesda, MD ~ Apr 108Gi).
11. Tulin, M.P. and C.C. Hsu, "Theory of High-
Speed D isplacement Ships with Transom
Sterns," Journal of Ship Research, Vol. 30,
No. 3 ~ Sep 1986~.
12. Jenkins, D.S., "Resistance Characteristics of
the High Speed Transom Stern Ship R/V
ATHENA in the Bare Hull Condition,
Represented by O TNSRD C Model 5365,"
D TNSRD C report 84/024, D TNSRD C,
Bethesda, MD ~ June 1984~.
13. Coleman, R.M., "Nonlinear Flow about a 3-
D Transom Stern, " Fourth International
Conference on Numerical Ship Hydrodynamics,
Washington, D.C. (Sep 1985~.
14. Hess J.L. and A.M.O. Smith, "Calculation of
Potential Flow About Arbitrary Bodies, "
Pergamon Press Series, Progress in
Aeronautical Science, Vol. 8 ( 1966~.
589
OCR for page 590
DISCUSSION
by K. Nakatake
I appreciate your paper treating the
transom stern flow. Fig.8 shows a good
agreement of calculated and measured Cw. But,
in this high Fn range, the effects of trim and
sinkage become important. Did your calculation
include such effect? If possible, please show
the wave height contour around the model.
Author's Reply
Dr.Nakatake asked about the effects of
sinkage and trim in the computations. The
results presented in Fig.8 of my paper
correspond to the fixed case for both the
computations and experiments. To include
sinkage and trim effect, we can reposition the
hull according to sinkage and trim predictions
from fixed case calculations. We did not
analyze and plot wave height contours for a
sunk and trimmed case and is presented in
Fig.A1. The corresponding wave pattern
resistance curve, as measured by Dr. Michael
Wilson of DTRC, is presented for comparison.
The computed results could be further improved
by using the experimentally measured sinkage
and trim to reposition the hull.
at,
0.6
dl
Fig.A1 Comparison between calculated wave
resistance and measured wave pattern
resistance versus Froude number for
Model 5416 (sunk and trimmed case)
DISCUSSION
by J. Ando
I'd like to congratulate for your good
results for 3D transom stern flow, and I'd
like to discuss on the radiation condition.
When we try to satisfy the radiation
condition, if we use finite difference
operator, we have some troubles. In my
experience, point-to-point oscillation of the
source strength occurred near the downstream
boundary. Sometimes, the source strength
oscillates so large that the wave pattern is
affected. And the results change due to the
kind of finite difference operator which uses
number of points. Did you have such
experience?
So I'd like to present a method which does
not use any finite difference operator in
order to satisfy the radiation condition. In
this method, the radiation condition is
satisfied automatically by shifting the source
panel in the downstream direction by one panel
length. We call it Kyushu University method
which is abbreviated as KU method.
~-
CALCULATED //
(mFS) MA ~
I REV
I r~
/~` MEASURED
/ cw(p)
/ (WILSON)
Next, I show some results for Wigley hull.
Fig.A2 shows panel arrangement on the still
water surface. By KU method, we don't use any
finite difference operator along the stream
line of the double model flow. Fig.A3 shows
comparison of wave patterns. This pattern by
KU method looks like the experimental result.
Dawson's method gives wider propagation of
f rue waves;
0~ . ~
-1.0 0.0 1.0 2x/L 2.0
KU Method
2z/L
-Q8- .
llllll!llllL-~, , , -
,P -1.0 0.0
J I -- 01 02 0.3 04 0.6 06 ;7 Dawson's Method
FROUDE NUMBER, Fn
-Q4
O
Fig.A2 Comparison of panel arrangement
590
I, ., ~ I, . . .
Cj ~ I · I
I 11 I j ~ I al jll
1 T rTrrTr~ red
I I a, T.
j ~
jI I I I jl ~ ~ jl jt I jI
ret r r I I I r I r
, T [,, T r] ~ r~
_ ~ CLI
i I I ~ I ~ i jl jr ~ jI
Trot I r r rT I I
I I ~ I 1 ~ r1 1 r I
I, ......
~ ~ j ~ ~ j j
i ~ j ~ ~ j j
I ~ r ~ ~ T ~ I I
T ~ r ~ . T ~ ~ I I
. 1 . .1 . .~ ~ ~
11,,,, 111 ~
2x/L 2.0
OCR for page 591
1
Fn=0.289
-
~.t'~-"'t"~ ~ 'I I. ~ I ~~<
Measurement (by SRI)
KU Method
Dawson's Method
Fig.A3 Comparison of wave pattern
Author's Reply
I would like to thank Mr. Ando for his
interest in my paper. It is true that the XYZ
Free Surface problem uses upstream finite
difference operators to eliminate upstream
propagating waves, thus satisfying the
radiation condition numerically. However, we
have not experienced the point-to-point
oscillations of the source strength near the
down stream boundary as referred to by Mr.
Ando. The reason is that we handle the
downstream boundary in the following manner.
As the downstream boundary is approached, a
four-point operator is switched to a three-
point operator and then to a two-point
operator for the rearmost panel. The two-point
operator gives considerable numerical damping
when the rearmost panel is relatively large.
Mr.Ando's "Dawson method" calculation seems to
include a reflection from the side boundary.
Such a reflection can be avoided by extending
the side boundary of the computational domain
to one ship length, measured from the ship's
centerline.
The Kyushu University (KU) method for
satisfying the radiation condition sounds
interesting and seems to be similar to the
method by Jensen[A1]. It would be helpful for
the research community to know more about the
KU method than has been presented in Mr.
Ando's discussion. The results of the Wigley
hull look promising and I encourage Mr. Ando
to continue his studies on the radiation
condition.
[Al] Jensen, P.S.: On the Numerical Radiation
Condition in the Steady State Ship Wave
Problem, J. of Ship Research, 1988.
,` _ ~ ~ _ %. . ~ _ .
591
OCR for page 592
Representative terms from entire chapter:
transom stern
