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OCR for page 207
Analysis of Transom Stern Flows
A. Reed, I. Telste (David Taylor Research Center, USA)
C. Scragg (Science Applications International Corporation, USA)
ABSTRACT
The boundary value problem for a transom stern ship
at moderate and high Froude numbers is formulated and
solved. The solution is obtained using lifting potential flow
techniques, and involves satisfying a Kutta condition at
the after edge of the hull. The full problem is linearized
about the free stream velocity, and this linearized problem
is solved using two different approaches. One method uses
Havelock singularities, and the other uses Rankine singular-
ities. Both of these approaches are applied to the solution
for the flow about a high-speed transom stern ship, with
encouraging results.
C
cd
CR
Cw
C'
w
Cwp
Fn
NOMENCLATURE
Wave spectral function due to sources
Wave spectral function due normal dipoles
Residuary resistance coefficient,
CR = RR/_PSU2
Wave resistance coefficient, Cw = Rw/-pSU2,
computed from wave spectral energy
Wave resistance coefficient, computed by inte
grating predicted pressure over the surface of
the hull
Wave pattern resistance coefficient derived from
measured wave pattern
Froude number, Fn = U/~/~
g Gravitational acceleration
G Green function
i, j, k Unit vectors in the x, y, and z-directions, re
spectively
k Wavenumber
ho Fundamental wavenumber, ho = g/U2
k=, k' Longitudinal wavenumber
ky Lateral wavenumber
Span Length of panel in the x-direction
L Length of ship
n Normal vector, taken into the fluid
no, ny, nz Components of the normal vector, n, in the x-,
y-, and z-directions
Neumann-Kelvin
Distance from singular point to field point, r =
,/(X _ ()2 + (A - ?1)2 + (A _ ()2
Distance from image of singular point to field
-
point, r' = ~/(x _ ()2 + (Y _ rl)2 + (Z + ()2
Surface of a panel on the hull, free surface, or
wake
N-K
r
r'
si
207
U
V
An
At
x
Sa Ship hull surface (zero sinkage and trim)
SF Free surface (mean free-surface level)
Sw Vortex wake surface
a, v, a) Perturbation velocity components in the x-, y-,
and z-directions
Ship speed; magnitude of free stream velocity
in ship fixed coordinate system
Total velocity vector
Component of the perturbation velocity in nor
mal direction
Component of the perturbation velocity in tan
gential direction
Vector coordinate of a field point, (x, y, z)
x, y, z Coordinates of field point x in a right-handed
ship fixed coordinate system, x-axis forward, y
axis to port, z-axis upward
x`, ye, z' Coordinates of the perimeter of the transom
Z Wave elevation
Dipole strength
Vector coordinate of a singular point, ((, 71, ()
I, 7', ~Coordinates of a singular point, I, in the x-, y-,
and z-directions
Density of water
Source strength
Perturbation velocity potential
p
INTRODUCTION
Naval architects have been familiar with the use of
transom sterns on high-speed displacement ships for well
over 50 years. The concept of a transom stern applied to
displacement ships appears to be an outgrowth of its use
on planing craft where it has been applied since around
the turn of the century. An understanding of when the
use of transom sterns is most appropriate was developed in
the years around the Second World War, when extensive
systematic-series experiments were performed. In the more
recent past, this knowledge concerning the proper applica-
tion and design of transom sterns seems to have been lost.
Modern naval ship designs have used larger and larger tran-
soms while the maximum speed of ships has decreased and
the size of ships has increased. Both of these factors should
weigh against the use of transom sterns on modern naval
vessels.
That the above statements are true is illustrated by
the fact that model tests have shown that the resistance
of modern naval ships such as the DD 963 is significantly
OCR for page 208
higher than it needs to be, due largely to the excessive tran-
som area. In addition to the significantly elevated resistance
of modern transom stern ships, these ships have significant
wave breaking at the stern. This is manifest in two ways:
first, by a large turbulent "rooster tail" aft of the transom,
and second, by a large breaking transverse wave extend-
ing approximately one ship beam off to each side of the
transom. Both of these features have a significant negative
impact on the wake signatures of modern high-speed naval
displacement ships.
While there has been substantial research in the field
of wave resistance over the last twenty-five years, and while
there has been substantial progress in the prediction of
Kelvin wave flows (see for instance Lindenmuth et al. 1990),
the treatment of transom stern flows has been largely ne-
glected. As is obvious from the poor flow predictions in the
region of the transom for Model 5415, given in Lindenmuth
et al., the Neuman-Kelvin and Dawson method solvers need
major improvements if the flow about a transom stern is to
be predicted adequately. However, there have been a few
encouraging developments in this area; one of these, which
may show a proper approach to this problem, will be dis-
cussed later.
At moderate and high speeds, the flow about transom
stern ships is characterized by smooth separation of the
stream lines at the transom. The manner in which the flow
separates from the hull at the transom immediately suggests
an analogy to the flow at the trailing edge of a lifting surface
(see for example Newman 1977~. In the extreme, for high
Froude numbers, the problem becomes a planing problem.
This limit provides some insight into the physics of the prob-
lem, and indicates some physical phenomena which must
be modeled in order that a correct solution to the prob-
lem be obtained. At more moderate Froude numbers, the
flow about a transom stern appears to have many analogies
with the flow about a ventilated hydrofoil (again, see New-
man 1977~. This observation also provides insight into the
proper solution to the transom stern flow problem.
Tulin and Hsu (1986) developed a model for high-speed
slender ships with transom sterns. By making the assump-
tion that both the beam and draft are small relative to the
length, and examining the asymptotic limit as the Froude
number approaches infinity, they were able to reduce the
three-dimensional problem to a series of two-dimensional
boundary value problems to be solved in the cross-flow
plane. The problem solved at each cross section is similar to
problems addressed in two-dimensional slender wing theory.
On the free surface they employed a trailing vortex sheet
in the wake of the hull to satisfy the free-surface boundary
condition, which in their model was shown to be equivalent
to the Kutta condition applied at the trailing edge of a wing.
Perhaps the most significant result of their model was the
existence of a drag force due to the presence of the trailing
vortex sheet. This "stern-induced resistance" is equivalent
to the induced drag associated with the shed vorticity in
the analysis of three-dimensional lifting surfaces.
Tulin and Hsu calculated the magnitude of the stern-
induced resistance for models from Series 64 and made com-
parisons with the residuary resistance measured at DTRC
for a Froude number, En = U/~fi~;, of 1.49. The excellent
agreement with the measured results suggests that, in the
high speed limit where the wave resistance vanishes, the
primary component of residuary resistance is induced drag
due to vorticity shed from the transom. Furthermore, the
magnitude of the stern-induced resistance is quite large, of
the same order as the wave resistance which occurs at mod-
erate Froude numbers, and consequently any attempt to
model the flow about transom stern ships at finite Froude
numbers must include both the generation of free-surface
waves and the effects of vorticity shed from the hull.
The trailing edge boundary condition which must be
applied in the transom stern problem (actually the bound-
ary condition must be applied on the hull surface just for-
ward of the transom since the transom itself is assumed to
be unwetted), is more restrictive than the Kutta condition
applied in aerodynamics. In wing theory, we require that
the flow at the trailing edge be tangent to the wing sur-
face and that the pressure be continuous across the wake.
For the transom stern problem, we require that the flow
be tangent to the hull and that the pressure be equal to
atmospheric pressure. This is equivalent to placing an ad-
ditional boundary condition on the longitudinal component
of the velocity, which can be satisfied by determining the
appropriate longitudinal gradient of the dipole strength on
the hull and/or by placing sources on the free surface aft
of the transom. These sources aft of the transom would be
equivalent to the sources which are used to model the cavity
aft of a ventilated airfoil.
Cheng (1989) provides a solution to the transom stern
problem in which he satisfies a similar set of boudary condi-
tions at the transom. However, Cheng's solution uses only
sources on the body boundary to satisfy the boundary value
problem. Obviously, with such an approach the trailing vor-
tex wake can not be modeled, nor can the body boundary
condition and the transom boundary condition be satisfied
at all points on the body simultaneously. Furthermore, in
attempting to satisfy both a normal and a tangential flow
boundary condition by varying only the distributed source
strengths, the solution will be very sensitive to the tangen-
tial gradient of the source strengths, and consequently to
panel size and location.
In the analysis of the flow about a surface-piercing strut
operating at a small angle of attack, a direct application of
the linearized Bernoulli equation along corresponding free-
surface streamlines on the high and low pressure sides of the
strut will lead to the conclusion that there can never exist
a discontinuity in the free-surface elevation at the trailing
edge of the strut. This result is, of course, inconsistent
with observation. However, if one applies the full nonlinear
Bernoulli equation to the same problem, it can be shown
that on the two streamlines there will exist a difference in
free-surface elevation at the trailing edge which is propor-
tional to the longitudinal vorticity shed by the strut at the
free surface. Interestingly, near the strut the perturbation
velocities are small and the linearized free-surface equations
remain an adequate approximation for the calculation of the
flow about the body, including the vorticity distribution.
Apparently, this is a situation in which the vorticity gives
rise to nonlinear free-surface effects, but the presence of
these nonlinear effects does not significantly affect the vor-
ticity distribution. Similarly, the nonlinear wavebreaking
which occurs in the wake of a transom stern ship is strongly
dependent upon the vorticity shed from the transom, but
the vorticity distribution at the stern is not greatly affected
by the presence of downstream wavebreaking.
In this paper, we present a brief theoretical foundation
for the transom stern boundary value problem. The full
problem is linearized about the free stream velocity, and this
208
OCR for page 209
linearized problem is solved using two different approaches.
One method uses Havelock singularities, and the other uses
Rankine singularities. Both of these approaches are applied
to the solution for the flow about a high-speed transom stern
ship, at two Froude numbers.
THEORETICAL FORMULATION
Consider a transom stern ship traveling at a compar-
atively high steady forward speed such that the flow sepa-
rates cleanly from the hull at the transom. We assume that
the fluid is inviscid and incompressible, and that the flow
is irrotational everywhere except possibly along a sheet of
trailing vorticity. We can then define a perturbation veloc-
ity potential ¢' which satisfies Laplace's equation
v2¢ = 0
(1)
throughout the fluid domain. Using a ship-fixed coordinate
system with the x-axis forward, the y-axis to port, and the
z-axis upward, the velocity vector V is related to the po-
tential by
V = -pi + vim
= ~-U+uji+vj+wk,
(2)
where U is the magnitude of the free-stream velocity, and
a, v, and u' are the components of the perturbation velocity
in the x-, y-, and z-directions, respectively.
While the free surface generality exhibits energetic wave
breaking at some distance aft of the transom, it is assumed
here that the effects of these breaking waves do not propa-
gate upsteam any significant distance. Therefore, there will
exist a region aft of the ship and forward of the breaking
stern waves over which the kinematic and dynamic free-
surface boundary conditions are valid. Furthermore, if the
effects of these breaking waves do not extend forward to
the hull itself, then our solution in the region immediately
surrounding the ship will be unaffected by an application
of the kinematic and dynamic free-surface boundary condi-
tions over the entire free surface. On the free surface Z. the
non-linear kinematic boundary condition is written as
~-U + ¢~)Z~ + MAZY = oz. on z = Zip, y). (3)
The Bernoulli equation is applied on the free surface to give
us the dynamic free-surface boundary condition,
gZ + ~ ~v~2 = ~ u2, on z = Zip, y). (4)
It is assumed that both Equations (3) and (4) hold over
the entire free surface, including the region directly astern
of the transom. In addition, it is necessary to impose a
radiation condition to ensure that the free-surface waves
vanish upstream of the disturbance.
The boundary condition to be applied on the hull sur-
face SB, excluding the transom, is simply the zero normal
flow condition
V · n = 0, on y = SB(X, I; U),
(5)
where n is the hull normal vector (directed into the fluid
domain), and we note that the sunk and trimmed position
of the hull surface depends upon the forward speed. It is
assumed that the flow separates cleanly at the transom and
that the velocity vector at the trailing edge is tangent to
the hull.
Since there can exist no discontinuities in the pressure
within the fluid domain, the pressure on the hull just for-
ward of the transom must be equal to the pressure on the
free surface just aft of the transom. Therefore, we have
an additional boundary condition to be applied on the hull
surface at the transom:
2 EVE = 2U2-gal, on y = SB((X!,ZI; U), (6)
where x' and z' denote points at the intersection of the sunk
and trimmed hull surface and the transom. Since Equa-
tion (5) requires that the normal flow on the hull be zero,
Equation (6) can be viewed as a restriction upon the magni-
tude of the tangential component of the flow at the transom.
We note since we are requiring that the pressure go to zero
at the intersection of the hull surface with the transom, the
present requirements are actually more restrictive than the
Kutta condition applied at the trailing edge of an airfoil, for
which we require only that the flow be finite and tangent to
the foil, and that the pressure be continuous.
E`ree-Stream Linearization
If we assume that the potential and its derivatives are
small relative to the free-stream velocity U. we can replace
the non-linear free-surface boundary conditions with their
linearized counterparts. Combining Equations (3) and (4)
to remove the explicit variable Z. and retaining only the first
order terms in ¢, we can write the linearized free-surface
boundary condition as
(7)
where ho = g/U2 is the fundamenatal wavenumber. The
boundary condition is now applied on the position of the
undisturbed free surface.
The hull boundary condition remains unchanged with
the exception that it will now be applied at the position of
the hull surface with zero sinkage and trim, SB(X,Z;U =
O). Once the velocity potential has been determined, the
sinkage force and the trim moment can be calculated and
an improved estimate of the position on the hull surface
SB(X,Z;U) can be used to solve the problem iteratively.
However, for the validation cases presented in the following
sections, the sinkage and trim were known from experimen-
tal measurements and the hull boundary condition could be
applied at the sunk and trimmed position of the hull surface
on the first iteration.
After dropping the non-linear terms, the pressure con-
dition to be applied on the hull surface at the intersection
with the transom, Equation (6), becomes simply
¢2~= + kook = 0, on z = 0,
U¢2 = gal, on y = SB(Xt, Z[; U). (8)
The linearized problem addressed here is very similar to
the Neumann-Kelvin (N-K) problem: the potential must
satisfy Laplace's equation, Equation (1); throughout the
fluid domain subject to a zero normal flow boundary con-
dition, Equation (5); on the hull surface and subject to
a linearized free-surface boundary condition, Equation (7).
However, in the present problem we have the additional re-
quirement that the pressure on the hull must go to zero
at the stern, Equation (8). This additional independent
209
OCR for page 210
boundary condition, a restriction on the longitudinal com
ponent of the velocity which does not occur in the formula
tion of the Neumann-Kelvin problem for ships which do not
have immersed transoms, indicates the need for additional The function ~ is given by
unknowns in our numerical approach to the problem.
We solve this boundary-value problem by employing
two different approaches, both of which are capable of mod
eling vortex sheets in the presence of the free surface. One
approach uses Havelock sources and dipoles which are dis
tributed over the hull surface and in the wake. The lin
earized free surface boundary condition is implicitly satis
fied by the use of Havelock singularities, and therefore no
singularities are required on the free surface. The other
approach uses Rankine sources and dipoles which are dis
tributed over the hull surface, the free surface, and in the
wake. The details of the two approaches follow.
Havelock Singularity Formulation
Our first approach to the numerical solution to this
problem distributes Havelock singularities over the hull sur
face and along a trailing wake sheet. The hull surface and
the wake sheet are divided into discrete panels comprised of
coincident sources and normal dipoles of uniform singularity
density. The Havelock singularity is a Green function for the
problem which satisfies the linearized free-surface boundary
condition on the mean free surface. The unknown singular
ity densities are determined by imposing the hull boundary
conditions at the centriods of each panel.
There are several different methods for evaluating the
potential ~ due to a distributed Havelock source of uniform
density a,
~ = (T,/~; dSG(x,y,z;<,r1,()~
where G is the Havelock Green function and s' is the panel
surface. We use a form of the Green function which allows
us to analytically calculate a wave spectral function by in-
terchanging the order of integration:
G=-1+ 1
T T'
~ 2 Loo Loo ek(Z+~)+ik=(=-()+iky(y_~)
+ BRIM-ho J dry Jo dk~ kid-kok
-i 2° /. day /3(ky) ek(Z+~)+ik' (~-~)+iky(y-77) )
-oo ~
where
and
r = /(x _ ()2 + (y _ 9)2 + (Z ~ ()2, (10)
T = j(X _ ()2 + (y _ 9)2 + (Z + ()2
k = \/~.
The variables kin and ky are the longitudinal and lateral
wave numbers of the free-surface waves, respectively. In
the single integral, the longitudinal wave number is not an
independent variable, and is related to ky by
210
k' = [1 (ho + A)] ~/2 .
2 (ho + I/;)
~ = ..
to + 4ky
The first two terms in the Green function correspond to a
Rankine source and its negative image above the free sur-
face; their contributions to the potential are computed using
standard techniques. The free-surface wave contributions
to the potential are contained in the two integral terms of
Equation (9~. To compute these terms we first define a wave
spectral function C(k=, by ),
C(k=,ky) = |/ dSek/:-it-inky (11)
sit
where the integration over the panel surface is performed
analytically. The wave contribution to the potential can
then be written as
2 ~ ~ ~ ~ ek3+ik~ =+iky y
Flu, = a-ho J dLyt dkxC(k='kY) k2 _ kok
-Hi ko / day is, C(k', ky)ekZ+iks2+ikYY,
where it is understood that we are taking the real part of
the expression. In practice, all of the boundary conditions
involve derivatives of the potential rather than the potential
itself, but the spatial derivatives can be calculated simply by
multiplying the spectral function by the appropriate wave
number prior to the integration.
For a surface distribution of Havelock dipoles directed
normal to the surface, the potential can be written as
p = p J i dS On G(x y z; ( a/ ()
where ,u is the dipole density. When we substitute the def-
inition of the Green function "Equation (9)], into this ex-
pression, we find that the potential contains contributions
from a Rankine dipole and its negative image above the
free surface, and contributions from the wave terms. The
wave spectral function for a Havelock dipole distribution,
Cd(k2:, ky), can be analytically integrated over a flat panel,
and is related to the wave spectral function for a Havelock
(9) source, Equation (11), by
Cd (k=, ky) = (nzk-invoke-inyLy)C(k=, By),
where (ins, my, no) are the components of the unit vector n,
normal to the dipole panel. The wave contributions to the
potential can then be determined by integrating over kX and
k
Y
I2 °° t°° ekz+ik=~+ikyy
4'w = ~-ho dry J dk=Cd(k='ky) k2 _ kok
-Pi 2 | day /3, Cd(k',ky)ekz+ik'=+ikyy
OCR for page 211
Since both sources and dipoles are distributed over the
panels on the hull surface, there exist two unknowns for
each panel and we have some flexibility over the manner in
which the singularity densities are to be determined. One
could use a slender ship approximation to set the source
densities a priori,
a = Un~/4,r,
and then determine the dipole densities by imposing the hull
boundary conditions at the centroid of each panel. Alterna-
tively, the source strengths could be determined initially by
solving the Neumann-Kelvin problem in the absence of the
pressure condition, and then solving for the dipole strengths
which satisfy both the zero normal flow condition and the
pressure condition.
The zero pressure boundary condition, Equation (8), is
highly dependent upon the tangential flow at the centroid
of the last dipole panel on the hull surface. There exists
a discontinuity in the tangential flow across a continuous
distribution of surface dipoles which is proportional to the
tangential gradient of the dipole strength. Unfortunately,
this important term is lost when the continuous surface of
the body is approximated by discrete panels of constant
dipole strength. In order to include this term, we employ
a finite differencing scheme for the panels immediately up-
stream and downstream of the last panel on the hull surface.
If the pressure boundary condition is to be applied at the
ith panel, we denote the immediate upstream panel as i + 1
and the adjoining wake panel as i-1. Then the tangential
gradient over the ith panel can be approximated by
/,~ ~ (Ri+1-pi-! )
pan
where Lpan is the length of the ith panel. The tangential
component of the perturbation velocity Vt due to this dipole
gradient is
At = +2~pi,
where the negative sign corresponds to the outboard side of
the panel and the positive sign corresponds to the inboard
side. This tangential velocity must be added to the veloc-
ity calculated at the centroid of the uniform dipole panel
only for those panels on which the zero pressure boundary
condition is to be applied.
If we panel the wake with M dipole panels, and the
hull with N panels on which are distributed both sources
and dipoles, then the zero normal flow boundary conditions
applied at the centroids of the N panels on the hull surface
can be written as
~ RjVni (pi) = Un ~`-~ ajVn; (*i), i = 1, . . ., N.
~ ~ .
where IN is the normal component of the perturbation ve-
locity. The additional M boundary conditions required are
the zero pressure conditions applied at the centroids of the
last M panels on the hull just forward of the trailing wake
panels:
-2,r^pi + ~ ~jU(Xi)
= U at,-~ ajU(Xi), i = 1, . . ., M.
The trailing wake sheets are extended straight aft from the
transom for a distance of half a ship length. The effects
of terminating the dipole sheets are therefore confined to a
region well aft of the hull.
Rankine Singularity Formulation
The Rankine singularity approach involves seeking the
perturbation potential 0 of Equation (2) as the solution to
the integral equation
2~ = ~1 dS¢~3nt. r-//s~ dS 0nt r
+ //SF dS¢0n~ r I/SF ins r
+ //SW /1 [3nE r '
(12)
obtained from Green's second identity. The field point may
lie on the hull boundary SB or the mean free-surface level
SF. The variable T. as it is in Equation (10), is the distance
between the field point x and the singular point (, and no is
the unit normal vector directed into the fluid domain. The
hull surface is taken as the surface with either zero or known
sinkage and trim. According to whether the field point x
is on SB or SF, one of the integrals involving <~/0n: is a
principal-value integral. The surface Sw represents a dipole
sheet in the wake across which there is a jump ,~` in the
potential.
Using the integral equation as a basis of a solution tech-
nique, we panel the hull, a portion of the mean free-surface
level near the hull, and the wake. The panels in the wake are
grouped in longitudinal strips within which all the panels
have the same normal dipole moment. Each of the panels
is flat and is assumed to have constant source and normal
dipole distribution. These approximations allow one to dis-
cretize the integral equation.
The integrals over the hull surface and the mean free-
surface level involve ~ and its normal derivative. On both
surfaces, the boundary conditions can be used to reduce the
number of unknowns at a boundary. On the hull surface the
normal derivative is given by the zero normal flow condi-
lion, Equation (5~. On the mean free-surface level, the lin-
earized free-surface boundary condition, Equation (7), can
be used to express the unknown normal derivative in terms
of longitudinal derivatives of ¢, i.e. ¢~1:- An upsteam finite-
differencing scheme involving the values of 4> at the centroids
of free-surface panels is used to approximate X2; at a point
x on the mean free-surface. Therefore, on the boundaries
SB and SF only the function ~ at discrete points remains
unknown.
Since free-surface panels are arranged in longitudinal
strips with their centroids lying on curved lines, the finite-
differencing scheme for determining ¢~= must include panel
centroids from several strips. In most cases an 11-point
scheme is used: five points from the longitudinal strip of
panels in which x lies and three points from each of two
adjacent strips. At the upsteam ends of the strips of free-
surface panels, we assume that ¢= and ¢~= are both zero
(Sclavounos and Nakos 1988~.
Aft of the transom, there are strips of free-surface pan-
els that originate at the transom. On the panel closest to
the transom, ¢~= can be approximated from the transom
depth and the hull shape. At the next panel downstream,
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OCR for page 212
a differencing scheme involving ~ on that panel and ¢, A=,
and ¢~= on the panel nearest the transom is used. Approxi-
mations for ¢= and ¢~= on the free-surface panel nearest the
transom are obtained from the linearized Bernoulli equation
and the fact that the position of the free surface at the tran-
som is known. In particular, we use Equation (8) and the
partial derivative with respect to x of Equation (8~. On the
third panel downstream of the transom in these free-surface
strips, a lower order upstream differencing scheme involv-
ing only ¢' at panel centroids is used. For all other panels
in these strips, the 11-point upstream differencing scheme
is used.
On the transom, although it is unwetted, a boundary
condition must be specified because we have formulated the
boundary value problem in terms of an integral equation.
That integral equation involves an integral over a closed
boundary which incudes the hull, the mean free-surface
level, and possibly a vortex wake across which ~ may be
discontinuous.
Several treatments of the transom boundary are pos-
sible. We may ignore the boundary which is equivalent
to specifying that both ~ and ~= vanish on the transom, a
possible overspecification of boundary conditions. However,
if the perturbation potential and its gradient are small, this
approximation may not be bad. Another option is to treat
panels on the transom in the same manner as other panels
on the hull are treated. In this case, ¢= on the transon is
known and is set equal to Un=. Without a surface of dis-
continuity extending downstream from the periphery of the
transom, we should expect the flow to turn around the cor-
ner of the transom as it streams past the hull. The resulting
calculated flow field will not satisfy the criterion that the
fluid leaves the hull tangentially, and it will violate our lin-
earization assumptions. Treating transom panels like other
hull panels thus seems to require a lifting surface extending
downstream. A third option is to set ¢~ equal to a value
determined by Equation (8), the linearized Bernoulli equa-
tion. The three options are complicated by the fact that we
have already decided to set ¢= and ¢~= at the free-surface
panels immediately aft of the transom equal to those val-
ues required by Bernoulli's equation. Thus, for the sake of
continuity in the boundary conditions, we choose the third
option and set ¢= equal to the value required by Equa-
tion (8~.
There remains the task of determining it, the strength
of the normal dipole moment on the trailing wake. Two op-
tions are readily apparent. We may, of course, set ~ to zero
and thereby ignore the possibility or requirement of having
a discontinuity of ¢' in the wake. Otherwise, we are required
to determine ~ from Bernoulli's equation. This is done by
considering the linearized form given by Equation (8~. For
each wake strip, two points are chosen. One point is at the
centroid of the hull panel nearest the transom. The other
point is slightly aft of the transom on the side of the strip of
dipole panels facing the fluid domain. A difference equation
for ~ is obtained by discretizing Equation (8) using these
two points.
Equation (12) is not the only integral equation that can
be used to obtain ¢. We could replace 1/r by 1/r + 1/r' and
obtain a second integral equation. The main difference is
that in the first equation ~ is expressed in terms of dipoles
and sources on the mean free-surface level, whereas in the
second case only sources appear on the mean free-surface
level.
We originally intended to use double-body linearization
instead of free-stream linearization for the Rankine-source
solution, but we encountered difficulties. In order to pre-
vent the flow from turning the sharp corner formed by the
transom, we used a wedge to extend the hull downstream for
the double-body solution. There is no difficulty in obtain-
ing such a solution. The non-zero Froude number solution,
however, requires paneling on the entire free-surface near
the hull, including the region aft of the transom. Normally,
the double-body potential is itself differenced to build finite-
difference coefficients so that the influence coefficients can
be calculated. In order to proceed in this manner for the
free-surface panels aft of the transom, the double-body po-
tential was continued from the hull extension to the mean
free-surface level by means of a Taylor series expansion.
This was necessary because the fictitious potential inside
the hull is identically zero. Once this was done, differenc-
ing of the analytically contimled double-body potential was
performed for first and second order derivatives in the mean
free-surface level. Differencing for the first derivatives pro-
duced reasonable approximations. However, differencing for
the second derivative resulted in large numerical errors.
A second and related issue with respect to linearization
schemes is that the free-surface elevation computed from
double-body linearization shows evidence of numerical in-
stability, in that numerical noise seems to grow in the strips
of free-surface panels in the downstream direction. This
should be expected in view of the results of Sclavounos and
Nakos {19881. The fact that this numerical instability is
not so pronounced in the free-surface linearization scheme
was not expected.
The differencing schemes we used for double-body and
free-stream linearization are not the same. The double-
body di~erencing scheme involves centroids on the same
free-surface strip because the free-surface paneling is deter-
mined so that the strips of free-surface panels are bounded
by double-body streamlines. For free-stream linearization,
the free-surface paneling cannot line up with the stream-
lines of the free-stream flow because free-surface paneling
must conform to the shape of the hull and not penetrate
into the hull. Thus the required differencing scheme is more
complicated because it must include centroids of panels in
several strips of panels. However, once it has been decided
which centroids to include in the finite difference stencils,
the finite-difference coefficients for approximating ¢~= can
be determined rather easily by eliminating truncation er-
rors. A linear system of equations is set up to eliminate
these errors up to fourth order. The method of singular
value decomposition is used to obtain a set of differencing
coefficients among the possible sets of differencing coeffi-
cients. For free-stream linearization at a particular Froude
number, the coefficients of the derivatives of 0 in the free-
surface boundary condition are known constants. This is
not the case in double-body linearization, in which we are
required to numerically approximate the coefficients of the
derivatives of ¢, which introduces another source of error.
PREDICTIONS
In order to validate the computational methods which
have been developed based on the theory which was just dis-
cussed, predictions have been made for a high-speed tran-
som stern model at two Froude numbers, 0.25 and 0.4136.
The hull form examined for this study is that of DTRC
Model 5415, which was studied as part of the Compara
212
OCR for page 213
Fig 1 Panelization of high speed transom stern ship
(DTRC Model 5415~.
tive Study of Numerical Kelvin Wake Code Predictions, re-
ported by Lindenmuth et al. (1990~. A sample paneling of
Model 5415 is shown in Fig. 1. On this model, Lindenmuth
et al. reported a wave trough at the transom for Fn = 0.25
and a crest at the transom for Fn = 0.4136.
Because of concerns about violating linearization cri-
teria, it was thought best to concentrate developmental ef-
forts on the Havelock singularity code at the lower Froude
number. The Rankine-source code requires paneling the
mean free-surface level near the hull. Because the wave-
lengths associated with a lower Froude number are smaller,
more paneling is required to resolve the same expanse of
the free surface. More paneling requires more computer
time and slower development time. Therefore, for the sake
of efficiency, the development and modification of the Rank-
ine singularity code were concentrated at the higher Froude
number. For these reasons, the results in this section are
presented in a somewhat inconsistent order. For the Have-
lock singularity method, results are presented first for the
low Froude number and the major part of the results corre-
spond to this Froude number. For the Rankine singularity
method, the opposite is true; results are presented first for
the higher Froude number, and then for the lower Froude
number.
Havelock Singularity Results
For the Havelock singularity calculations, Model 5415
was paneled at the measured sinkage and trim, with 320
panels on the hull and 8 trailing dipole panels in the wale.
For the first test case, we set the source strengths using
the slender-ship approximation, and solved for the dipole
strengths. We found that the dipole strengths required to
satisfy the zero normal flow condition on the hull were un-
realistically large, and this approach has been abandoned.
By starting with source strengths obtained from a N-K solu-
tion without any Kutta condition, we are in effect, using the
sources to satisfy the normal flow condition, and using the
dipoles to satisfy the tangential flow condition. This leads
to a better behaved solution. The source strengths on the
hull, from the N-K solution without a Kutta condition, for
Fn = 0.25, are presented in Fig. 2a. Each curve in the fig-
ure corresponds to a row of panels on the hull (although the
panels on the bulb are included in the solution, their source
strengths are not plotted on this graph for reasons of clar-
ity). These source strengths account for the interactions
between the hull and the free-surface waves, but not for the
zero pressure condition at the transom stern. The dipole
tar
o
O
\ o
o a.
E to
rat
.~
In
~1
0.5 0.4 0.3 0.2 0.1 0.0 -0. 1 -0.2 -0.3 -0.4 -0.5
Conget~udunaL posutcon / shop length
(a) Source strengths on the hull.
. , . , . , . , . -, . . . . . . . .
n.s 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0 5
borage tud~r~aL pose talon / shop Length
(b) Dipole strengths on the hull.
., ., ., ., ., .,,,,,,,, 1
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
l~at~eraL pose talon / transom beam
/
o
o
o~
~D
o
.
=) o
E o
o
o
o
JO
o
,0~`
\_
\
(c) Dipole strengths across the wake.
Fig 2 Singularity strengths on the hull, En = 0.25. Each
curve represents a row of panels at a different
depth.
strengths required to satisfy the Kutta condition, Fig. 2b,
are driven entirely by the zero pressure condition, in the
sense that the dipole strengths would go to zero in the ab
213
OCR for page 214
sence of this boundary condition. Apparently the ejects of
imposing the Kutta condition on the hull at the transom,
as represented by the dipole distribution, do not extend
very far forward of the transom. The dipole strengths are
negligible over most of the length of the hull, but as the
flow accelerates to satisfy the zero pressure condition, the
dipole strengths drop quickly and smoothly to a minimum
value at the transom. The dipole strengths are held con-
stant on the trailing wake panels. The dipole distribution
across the wake is shown in Fig. 2c. The similarities be-
tween this distribution and the spanwise distribution on an
airfoil are striking. The significance of this trailing vortic-
ity can be appreciated by noting that the magnitude of the
dipole strengths is comparable to the strengths calculated
on an airfoil of equal span, operating at a negative angle
of attack of approximately five degrees. The sense of the
vorticity will result in an upwelling flow on the centerline
of the wake, and a diverging free-surface current across the
wake, and may well be the physical mechanism responsi-
ble for such flows observed in the centerline wake region of
transom stern ships.
Contour plots of the contributions to the near-field
wave elevations due to the sources alone and due to the
dipoles alone are presented in Figs. 3a and 3b (note that the
contour interval is reduced in Fig. 3b). The elevations have
been non-dimensionalized by the fundamental wavenumber
ho. The changes in the wave field due to the dipoles are lim-
ited to a small region directly aft of the transom and a group
of diverging waves which occur along a cusp line which orig-
inates near the stern. The total near-field wave elevations
which result from the combined distribution of sources and
dipoles on the hull are presented in Fig. 3c. The correspond-
ing experimental results are shown in Fig. 3d. Including the
Kutta condition at the transom leads to a solution which
exhibits a very steep rise in the wave elevation aft of the
transom, which will likely lead to wave breaking.
Wavecuts along y/L = 0.324 are presented in Fig. 4.
The measured wavecut is presented in Fig. 4a, while that
predicted from a N-K solution without a Kutta condition
is shown in Fig. 4b. In Fig. 4c we note that the transverse
waves due to the dipoles are very small relative to those
generated by the sources. The most significant far-field ef-
fect of the Kutta condition is the same group of diverging
waves noted in the near-field contour plots. The combined
wave fields are given in Fig. 4d. The inclusion of the dipoles
results in a free-wave amplitude spectrum which is in good
agreement with that obtained from the experimental wave-
cuts, Fig. 5. The predicted wave resistance, Cw, of 0.00042
compares well with the experimental wave pattern resis-
tance, Cwp, of 0.00037.
Fig. 6 presents contour plots of wave elevation for Fit =
0.4136. Fig. 6a presents the results for the combined dis-
tribution of sources and dipoles. The near-field waves gen-
erated by the dipoles alone are shown in Fig. fib. The cor-
responding experimental results are shown in Fig. 6c. As
before, the effects on the wave field of imposing the Kutta
condition at the transom are relatively limited. The dipole
distribution across the wake is shown in Fig. 7. The vor-
ticity is somewhat stronger and has the same sense as be-
fore, leading to an upwelling along the centerline and a di-
verging flow across the wake. The measured wavecut along
y/L = 0.324 is presented in Fig. 8a. The predicted wavecut
due to the combination of source and dipole panels is shown
in Fig. 8b. Although the predictions are slightly higher than
/`koZ = 0.02
. ~
(a) Due to Havelock sources alone.
`koZ = 0.01
`koZ = 0.02
(b) Due to Havelock dipoles alone.
~'1 ,~ ~7~.5~
(c) Due to combination of Havelock sources and dipoles.
I
~ 1
/`koZ = 0.02
/`koZ = 0.02
,,. :~-
O.
(d) Measured. (from Lindenmuth et al. 1990)
O
rat ~
U1
-
o
-
11~\
(:)
to
~,,,. ,,,~,,' '>?>', ~
:~)'J~
, ~
-o.s o.o o.s 1.0 1.s 2.0 2.s 3.0 3.s ·.o
(e) Due to Rankine singularies. (Transom panels, no wake
panels)
Fig 3 Non-dimensional near-field wave elevation contours
for Model 5415 at F., = 0.25.
214
OCR for page 215
lo -
o.o to.o 20.0 30.0 40.0 so.o 60.0 70.0 80.0 90.0 ~oo.o
X ~ HOVENUMBER
(a) Measured. (from Lindenmuth et al. 1990)
........ .
.o 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0
X ~ W8VENUMBER
(b) Due to Havelock sources alone.
O
T
o.o 10.0 20.0 30.0 40.0 so.o 60.0 70.0 80.0 90.0 100.0
X ~ WRVENUMBER
(c) Due to Havelock dipoles alone.
Fig 4 Wavecut at y/L = 0.324 for Model 5415 at F,l
0.25.
the data, the qualitative agreement between the measured
and predicted far-field wave cuts is very good. The free-
wave amplitude spectra are compared in Fig. 9.
215
. . . . . . . . . . .
.o 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0
X 34 WRVENUMBER
(d) Due to combination of Havelock sources & dipoles.
Fig 4 (Cont.) Wavecut at y/L = 0.324 for Model 5415 at
Fn = 0.25 .
Preducted
li1ec~sured
o.o t.o 2.0 3.0 4.0 5.0 6.0 i.o 8.0 9.0 10.0
LateroL Wave Number ( Ky )
Fig 5 Comparison of measured (dashed line) and pre-
dicted (solid line) free-wave spectra for Fn = 0.25.
(measured from Lindenmuth et al. 1990)
Rankine Singularity Results
Since the flow configuration is assumed to be symmetric
about the center plane of the ship, only half of the hull, free-
surface, and wake are paneled. In either case the hull was
paneled with 324 panels. The transom was paneled with 8
panels and the wake with 8 strips of panels. The free surface
was paneled with 780 panels for the high Froude number
case: 10 strips of 62 panels to the side and 8 strips of 20
panels aft of the stern. For the low Froude number case,
more panels were deemed necessary because of the smaller
wavelengths. Therefore the same expanse of the mean free-
surface level was paneled in 10 strips of 80 panels to the side
and with 8 strips of 23 panels aft of the stern. The paneling
for the high Froude number case was allowed to be finer
near the bow and stern whereas the paneling for the low
Froude number case was nearly uniform in the longitudinal
direction. Fig. 10 depicts the free-surface paneling for Fn =
0.4136; the hull paneling is similar to that shown in Fig. 1.
Figs. 6d and be present contour plots of the wave ele-
vation predicted using the Rankine source method for Fn =
0.4136. Fig. 6d corresponds to the case where wake panels
OCR for page 216
/`koZ = 0.02
O ~
(a) Due to combination of Havelock sources and dipoles.
AknZ = 0.01
(b) Due to Havelock dipoles alone.
lo
AkoZ = 0.02
i=-~o~(:
(c)-Measured. (from Lindenmuth et al. 1990)
lo
,~, . _ .
AkoZ = 0.02
°mL~1-~
(d) Due to Rankine singularies. (No transom panels, wake
panels)
AkoZ = 0.02
of_:
-0.5 o.o 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
(e) Due to Rankine singularies. (Transom panels' no wake
panels)
Fig 6 Non-dimensional near-field wave elevation contours
for Model 5415 at En = 0.4136.
-0.3 -0.2 -0. 1 0.0 0.1 0.2 0.3 0.4 0.5
Lateral posctcon / transom beam
Fig 7 Dipole strengths across the wake for
o
Fir = 0.4136.
~ . . . · -
. D.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0
X ~ hlRVENUMBER
(a) Measured. (from Lindenmuth et al. 1990)
i
0 ., ., ., ., ., ., . ',,,, 1, _
0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 10 .0
X ~ HRVENUMBER
(b) Predicted due to combination of sources and dipoles.
Fig 8- Wavecut at y/L = 0.324 for Model 5415 at En =
0.4136.
are present and the panels on the transom are neglected.
The normal dipole strengths of the wake panels were de-
termined from the linearized Bernoulli equation. Fig. be
corresponds to the case in which transom panels were in-
cluded, but the normal dipole strengths in the wake were
set to zero. There is very little difference between these two
sets of results. A third attempt at finding a solution was
made for which both the wake and the transom panels were
216
OCR for page 217
-O .8
o.o 1.0 2.0 3.0 to 5.0 6.0 7.0 8.0 9.0 10.0
Lateral Wave Number 1 Ky )
-Comparison of measured (dashed line) and pre
dicted (solid line) free-wave spectra for Fn =
0.4136. (measured from Lindenmuth et al.
1990)
0.75
-
Hull Depth
| Wave Elevation
x/L 1.0 (Transom) 1.25
(a) Fn = 0.4136.
1.5
Fig 10-Free-surface panelization for Fn = 0.4136.
present. The third attempt failed because the resulting lin-
ear system of equations was too ill-conditioned to obtain a
solution in 32-bit arithmetic. We are using direct solvers
here because the iterative schemes we have been using do
not converge for the problems being considered here.
Because of the presence of dipoles on the free surface,
we are not able to separate the trailing vorticity due to
the presence of the transom from that normally on the free
surface. Therefore, there is no figure corresponding to Figs.
2c or 7.
Comparing the predicted wave elevations in Figs. 6d
and be with the wave height obtained from experiments,
Fig. 6c shows that the bow wave and the mid-ship trough
are underpredicted. The discrepancy at the bow may be
due to the paneling not being fine enough. In the stern
area the predicted wave height does match the hull depth
at the transom, as is indicated in Fig. lla. This figure shows
hull depth versus longitudinal position in the left half and
predicted wave height versus longitudinal position in the
right half. The hull depth and wave height are plotted for
the eight longitudinal strips of panels on the main hull and
the eight longitudinal strips of free-surface panels extending
from the stern downstream. The wave height and the hull
depth match at the stern. We may thus conclude that the
pressure condition, Equation (8), is indeed satisfied. The
predicted wave height rises from the transom depth to a
peak of koZ = 0.14 before falling again. There are no ex-
perimental data in this area to determine the accuracy of
these predictions. To the side and aft of the stern both the
experimental data and the predictions show a wave height
Outer Transverse Location
Hull Depth
Wave Elevation
0.75 x/L 1.0 (liallsom) 1.25
(b) Fn = 0.25.
Fig 11 Hull depth and predicted wave height verses longi-
tudinal position near the stern at eight transverse
locations.
koZ = 0.08. A pressure integration over the portion of
the hull beneath the mean free-surface level except for the
transom was performed to obtain the wave resistance and
induced drag as the force in the longitudinal direction, Cw.
The resulting value for Cw was 0.00243, which is close to
the Cwp of 0.0024.
Fig. 3e shows the predicted wave height contours for
Fn = 0.25; these results should be compared with the ex-
perimental measurements of Fig. ad. The computations cor-
respond to the case in which the transom has panels and
there are no dipole panels in the wake. In this case the
predicted results are not as good as at the higher Froude
number because of the difficulty in resolving the finer de-
tails of the flow even with the finer free-surface paneling.
In the stern area, Fig. fib shows that the fluid leaves the
hull tangentially just as it did for the higher Froude num-
ber case. This is a good indication that the atmospheric-
pressure boundary condition on the hull at the transom is
being satisfied. There are more data points from measure-
ments for this case. The experimental data indicate a max-
imum wave height in the stern area of about koZ = 0.10
217
OCR for page 218
and a minimum wave height of about koZ = -0.06. The
predictions have a peak value of about koZ = 0.14 and a
trough with depth of about koZ = - 0.10. In this case the
wave resistance, Cw, is predicted to be 0.00053, while the
value from wave pattern analysis, Cwp, is 0.00037.
CONCLUSIONS
The authors have developed two methods which are
capable of modeling the flow about a ship with a transom
stern, using vortex sheets to model the lift effects. Both
methods implement a Kutta condition based on the fact
that the pressure on the hull at the transom must be zero.
This Kutta condition is used to determine the strength of
trailing vorticity or the strengths of dipoles on the tran-
som. The first method employs Havelock singularities which
can be distributed over the hull surface and in the wake.
The linearized free-surface boundary condition (obtained by
free-stream linearization) is implicitly satisfied by the use of
Havelock singularities, and therefore no singularities are re-
quired on the free surface. The second method employs
Rankine singularities which can be distributed over the hull
surface and in the wake. The technique requires that Rank-
ine sources also be distributed over the free surface, and in
this way a linearized free-surface boundary condition (lin-
earization here is also about the uniform free stream) is
satisfied. Computations based on the two methods have
been compared with each other for flow about a high-speed
transom stern ship (DTRC Model 54154. The results show
encouraging agreement with one another and with experi-
ments for the flow configurations considered.
The Havelock singularity method uses Havelock sources
to satisfy the normal flow boundary conditions on the hull,
and Havelock dipoles to satisfy tangential flow conditions.
This leads to well-behaved solutions to the boundary-value
problem. Imposing a zero pressure condition on the hull at
the transom appears to have little effect upon the predicted
far-field Kelvin wave. The effects in the near field are much
more significant. The present results show a wave trough
just behind the ship, and the wave elevation now matches
the depth of the transom. There is a rapid rise to a wave
peak immediately downstream of this trough. The waves
generated by the Havelock dipole exhibit a distinct cusp
line emanating from the corners of the transom. The pre-
dictions are consistent with observed wave fields generated
by transom stern ships.
Perhaps more significant than the predicted wave field
is the prediction of the vorticity shed by a transom stern
ship. The flow is shown to accelerate toward the transom to
satisfy the zero pressure condition; the result is an increase
in the downward dynamic force on the hull. The shed vor-
ticity is roughly equivalent to that shed by a hydrofoil with
a span equal to the transom beam, operating as a negative
angle of attack of approximately 5 degrees. The sense of
the vorticity is such that there will be an upwelling flow
on the centerline of the wake, and a diverging free surface
current across the wake. Similar flow fields are frequently
observed behind transom stern ships. To our knowledge,
this is the first numerical result which offers an explanation
for the source of this vorticity.
The Rankine singularity approach shows little differ-
ence in predicted wave resistance for two forms of problem
formulation: first, when the transom is paneled and wake
panels are absent and, secondly, when the wake is present
and the transom is neglected. A third alternative corre
sponding to paneling both the wake and the transom proved
to be numerically ill conditioned. The ill conditioning very
likely arises from the fact that the equations required to
determine the strengths of the trailing vorticity essentially
duplicate those set up to determine dipole strengths on the
transom. At any rate, from the results we have seen, it
seems best to neglect the wake panels altogether and to
place dipole panels on the transom for this type of lineariza-
tion.
When we attempted to formulate a Rankine singularity
method based on a Dawson type double-body linearization,
we had difficulties. To solve the double body problem, we
extended the hull with a solid surface aft of the transom
in order to obtain a double-body flow that did not turn
a corner at the transom. We then attempted to apply a
Dawson type free-surface condition on this extension of the
transom which appeared not to work. Perhaps the solid
hull extension should be replaced with a wake composed of
a sheet of dipoles extending downstream from the perimeter
of the transom to infinity.
We have found linear solutions to the problem of the
steady flow about a transom stern hull form at realistic
Froude numbers, using models which are capable of prop-
erly including the effects of shed vorticity. Obviously linear
solutions are valid only forward of the region of energetic
wave breaking which occurs in the wake downstream of the
transom. However, this approach allows us to accurately
calculate the flow field near the hull, including the free-
surface wave elevations, the pressure on the hull, the shed
vorticity and the stern-induced resistance. This transom
stern analysis method should allow the design of transom
sterns with lower resistance and lower wale signatures.
ACKNOWLED GMENTS
This work was supported by the Applied Hydrome-
chanics Research program of the Applied Research Division
of the Office of Naval Research, and administered by the
David Taylor Research Center. The efforts of Suzanne Reed
who edited the text and assembled the paper are greatly ap-
preciated.
REFERENCES
Cheng. B. H. 1989. Computations of 3D Transom Stern
Flows. Proc. Fifth International Conference on Nu-
merical Ship Hydrodynamics, National Academy Press:
Washington, DC, pp. 581-92.
Lindenmuth, W. T., T. J. Ratcliffe and A. M. Reed. 1990.
Comparative Accuracy of Numerical Kelvin Wake Code
Predictions "Wake-Off." DTRC Ship Hydromechan-
ics Dept. R & D Report DTRC-90/010, 234+ix p.
Sclavounos, P. D. and D. E. Nakos. 1988. Stability Analysis
of Panel Methods for Free-Surface Flows with Forward
Speed. Proc. Seventeenth Symposium on Naval Hydro-
dynamics, National Academy Press: Washington, DC,
pp. 173-93.
Newman, J. N. 1977. Marine Hydrodynamics. The MIT
Press: Cambridge, MA, 402+xiii p.
Tulin, M. P. and C. C. Hsu. 1986. Theory of High-Speed
Displacement Ships with Transom Sterns. J. Ship Res.,
30~3~:186-93.
218
OCR for page 219
DISCUSSION
Hoyte Raven
Maritime Research Institute Netherlands,~e Netherlands
I have a few questions on this very interesting paper.
1. If no special treatment of the transom stern is applied, a free
surface is predicted that intersects the transom at some point. Since
one expects it to flow off the edge of the transom, one looks for
modifications. But in a linearized method we generally do not care
about the precise location of the intersection as its influence is of
higher order. Is not, then, this method (though a very successful
one) to incorporate in a linearized method something that in principle
prohibits the linearization, vis. the presence of a sharp corner close
to the free surface?
2. You point out the analogy with a Nutta condition. There is
perhaps another analogy with free-streamline theory. This, however,
predicts an infinite curvature of the separating streamline at the
transom edge. Could such a behaviour fit into your method?
3. I was impressed by the fact that the Rankine singularity method
with transom panels but without trailing wake panels was as good as
that which incorporates the value. Does not this contradict the
importance of trailing vorticity?
AUTHORS' REPLY
i. This meeting is indeed an attempt to circumvent a difficulty in
devising a linearization scheme. The objective is to find a basic flow
from which the true flow deviates little. With a dipole sheet
extending to downstream infinity from the edges of the transom, we
are building into the basis flow the fact that fluid moves smoothly
past the transom. The perturbation from the basis flow potential
should then be small.
2. The method of free streamlines might be used to find a hull
extension about which a double-body flow can be calculated. Flows
corresponding to nonzero Froude numbers could be calculated based
on linearizing the free-surface boundary conditions on the mean free-
surface level outside the hull with its extension and on the surface of
the extension.
3. We do not understand this seeming contradiction, but it may be
that we can carry and are in fact carrying vorticity on the free
surface. This matter needs to be studied more.
DISCUSSION
Kazu-hiro Mori
Hiroshima University, Japan
Although the linearized pressure condition (8) is consistent in your
framework, it may not always provide nonjump in pressure. How
was the resulted pressure there? Is the pressure condition satisfied?
AUTHORS' REPLY
For the Havelock singularity method, the dipole strength in the wake
is determined in such a way that the linearized zero-pressure
condition (8) is satisfied. For the Rankine singularity method, the
differencing scheme immediately aft of the stern is set up so that this
condition is satisfied. Consequently, for either method, the pressure
is zero at the intersection of the free surface with the hull. For the
Rankine singularity method, this can be seen in Figs. Ha and fib
where the wave elevation aft of the stern has been calculated based on
the linearized zero-pressure condition (8). The pressure based on the
fully nonlinear Bernoulli equation has not been calculated.
DISCUSSION
Dimitris Nakos
Massachusetts Institute of Technology, USA
The approach followed in the paper appears to be able to w tune.
linear and/or quasilinear numerical solutions of the steady ship wave
problem behind a transom stern, so that they are in better correlation
with reality. My question pertains to one of the most critical, and
potentially most troublesome, assumptions behind the technique
described in the paper...that is the positioning of the line across
which separation occurs. In the case investigated by the authors, the
loading condition of the vessel is such that part of the transom stern
is submerged even in calm water, and the sharp edge of the transom
appears as the safest alternative for Separation line.. In many cases,
however, the transom-like flow may be anticipated. Do the authors
have any suggestions about the positioning of the separation line in
such cases?
AUTHORS' REPLY
The cases in question are nonlinear flows which we are not ready to
handle. As a first approximation, one would not concern himself
with the hull configuration above the level of the undisturbed free
surface. When sinkage and trim is accounted for in subsequent
calculations, it might be appropriate to also take into consideration
this transom-like flow.
219
OCR for page 220
Representative terms from entire chapter:
free surface
