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Numerical Solution of the "Dawson"
Frec-Surface Problem Using Havelock Singularities
C. Scragg, I. Talcott (Science Applications International Corporation, USA)
ABSTR ACT
A method of solving the steady free-surface ship wave prob-
lem satisfying "Dawson's" double-body linearization of the free-
surface boundary condition, which employs distributed Havelock
singularities on both the hull surface and on the free surface is pre-
sented. The use of Havelock singularities, as opposed to Rankine
singularities, allows the solution to be extended into the far-field
without difficulties. The present technique combines the superior
aspects of Rankine/Dawson methods in the calculation of near-
field waves and the far-field superiority of the Havelock methods.
~,n~r`~,rn'.~in~ results are presented for two simple hull forms, a
(JUAN ~` ~6. 1445 , ~ in. vat `__ ~ r
submerged body of revolution and a Wigley hull.
NOMENCLATURE
B
C(k=, ky)
Cw
g
G
k$
k
y
ho
L
n
p
Rw
S(z, y)
U
V
(x, y, a)
Z(x, y)
To
INTRODUCTION
Beam
Wave spectral function
Wave resistance coefficient
Gravitational constant
Green function
Draft
Wave number
Longitudinal wave number
Lateral wave number
Characteristic wave number = g/U2
Ship length
Unit normal vector into the fluid
Pressure
Wave resistance
Hull surface
Ship speed
Fluid velocity vector
Ship-fixed coordinate system, with x forward,
y to port, and z upward
Free-surface elevation
Double-body wave elevation
Velocity potential
Double-body velocity potential
Perturbation potential
Fluid density
Havelock source density
In 1977, Dawson [1], introduced a method of linearizing the
free-surface boundary condition using a perturbation about the
zero-Froude number potential. Since then, there has been signif-
icant interest in utilizing zero-Froude number or "double-body"
linearization schemes in the field of wave resistance and in the
prediction of Kelvin waves. Although there are several different
methods of linearizing the free-surface boundary conditions (see
Raven, Hi), we refer to this basic approach as Dawson's method
even though we do not actually use the same version of the lin-
earized free-surface equations given by Dawson in his pioneer-
ing work. Several researchers have developed computer codes
which satisfy the exact hull boundary condition and Dawson's
free-surface condition by distributing Rankine singularities over
the ship's hull and on the free surface. During the 1988 Workshop
on Kelvin Wake Computations (Lindenmuth, et al. [33), it be-
came apparent that the best of these Rankine/Dawson codes were
capable of predicting quite accurately the wave elevations in the
near-field region directly around the ship. However, these codes
encountered difficulties in the prediction of the freely-radiating
far-field Kelvin waves. The solutions exhibited excessive numeri-
cal wave damping and/or wave reflections off the computational
boundaries.
Although Rankine singularities provide a convenient and effi-
cient method for the calculation of the zero-Froude number prob-
lem, they actually introduce some numerical difficulties into the
calculation of the Kelvin wave field. These difficulties are avoided
by solving Dawson's problem with distributed Havelock singular-
ities. Since Rankine sources are symmetrical, it is necessary to
impose some sort of numerical radiation condition to prevent up-
stream radiating waves. This difficulty is not encountered when
Havelock singularities are used since the Havelock singularity in-
herently satisfies the radiation condition. The use of Havelock
singularities also eliminates problems associated with wave re-
flections off the computational boundaries. Since neither Rankine
singularities nor Havelock singularities distributed over the hull
surface alone can satisfy Dawson's free-surface boundary condi-
tion, it is also necessary to panelize some region of the free surface
surrounding the hull. With Rankine singularities, wave reflec-
tions at the edge of the computational domain can create serious
problems, usually solved by the introduction of some numerical
damping scheme. But with Havelock singularities, there is no dif-
ficulty at the edge of the panelized region since Havelock singu-
larities always satisfy the linearized free-surface boundary condi-
tion, and consequently the far-field waves always propagate away
from the hull as linear Kelvin waves. At moderate distances from
the hull, the zero-Froude number potential approaches the undis-
turbed free-stream potential, and consequently, Dawson's free-
surface boundary condition limits to the linearized free-surface
boundary condition satisfied by Havelock singularities. There-
fore, with Havelock singularities distributed on the free surface,
the singularity strength necessary to satisfy the Dawson free-
surface boundary condition will smoothly approach zero as the
double-body flow approaches the free-stream. The computational
domain is defined quite naturally as the limited region of non-
zero singularity strength directly around the hull. Furthermore,
this computational domain in which free-surface panels are re-
quired, is determined by examining the zero-Froude number so-
lution, eliminating the need for elaborate free-surface panelization
schemes.
259
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This approach leads to a solution which satisfies the same
field equations and the same boundary conditions as the Rank-
ine/Dawson codes, and therefore, the near-field solutions are the
same. But since the use of Havelock singularities eliminates the
numerical problems associated with Rankine singularities, this
method leads to solutions which are also valid in the far-field.
THEORY
Consider a ship moving with steady forward speed U in the
presence of a free surface. We define a ship-fixed coordinate sys-
tem with the positive x-axis in the direction of travel, the y-axis
directed to port, and the z-axis vertically upward. The origin
is located on the mean free surface. We assume that the fluid
is incompressible and inviscid and that the flow is irrotational.
Consequently, we can define a velocity potential ~ which satisfies
the Laplace equation throughout the fluid domain,
V2~ = 0, (1)
and is related to the fluid velocity vector V by
V = Vie. (2)
On the surface of the body S(x, z), we require that the flow
be tangential to the hull surface,
Vat n = 0, on y = S(x, z), (3)
where n is a unit normal vector directed out of the hull.
On the free surface Zip, y), the velocity potential must sat-
isfy the kinematic free-surface boundary condition,
~2 = USED + Gym, on z = Z~x,y), (4)
and the dynamic free-surface boundary condition,
gZ + 2vq} Vie = 2U2, on z = Ztx,y), (5)
where 9 is the gravitational constant. In addition, we require
that the disturbance created by the body must vanish at points
infinitely far away, and we require that the far-field free-surface
waves generated by the body may not radiate upstream of the
ship.
The free-surface gradients Zen and Zy can be written in terms
of the velocity potential by differentiating equation (5) with re-
spect to x and y. Then by substituting the gradients into equa-
tion (4), we can write a single free-surface boundary condition
which must be satisfied by the potential:
2(V~.V~+2(V~.V.)y~y+g~z = 0, on z = Z(z,y). (6)
The manner in which this non-linear free-surface boundary con-
dition is linearized is what distinguishes the Dawson problem
from the Neumann-Kelvin problem. In both problems, we seek
a solution to the Laplace equation (1) which satisfies an exact
hull boundary condition (2~. In the Neumann-Kelvin problem we
rewrite the potential as the sum of a free-stream potential and
a perturbation potential id', and we assume that the perturba-
tion potential is, in some sense, small relative to the free-stream
potential,
4? = -Up + up'. (7)
If we substitute equation (7) into the free-surface boundary con-
dition (6), and retain only terms which are linear in A', then we
obtain the linearized Kelvin free-surface boundary condition
~+ko~' =0, on z=0, (8)
where ho is the characteristic wave number defined by
o u2
(9)
To show that the boundary condition can be applied at the po-
sition of the mean free surface, one can expand the potential in
a Taylor series about z = 0, and assume that the free-surface
elevation Z is of the same order as the perturbation potential.
In Dawson's approach to the problem, the potential is di-
vided into a double-body potential ~ and a perturbation potential
in,
it = ¢,+~p (10)
and it is assumed that the perturbation potential is small rela-
tive to the double-body potential ¢. The double-body potential
corresponds to the limiting solution as the Froude number goes
to zero (i.e. g >> U ), for which case the free surface acts as a
reflection plane. The double-body potential is a solution to the
Laplace equation at all points outside of the body,
V2¢ = 0, (11)
and satisfies the exact hull boundary condition,
n · V) = 0, on y = S(x, z), (12)
and a reflection boundary condition applied on the position of
the undisturbed free surface,
(: = 0, on z = 0. (13)
The double-body solution can be readily obtained by well-known
panelization methods utilizing Rankine (1/R) singularities, and
it will be assumed throughout the remainder of this discussion
that the double-body potential and its derivatives can be treated
as known quantities.
The perturbation potential must be a solution to the Laplace
equation throughout the fluid domain, and must satisfy the same
hull boundary condition,
n · Via = 0, on y = Sly, z). (14)
To obtain the linearized free-surface boundary condition which
must be satisfied by the perturbation potential, we substitute
equation (10) into the free-surface boundary condition, equa-
tion (6), and retain only first order terms in A,
2(V~ · V¢)sy$ + (V~ · Versus + 2(V~ · V¢)yyy
+ (V~ · Vy~y~y + gyz = -9¢z - 2 (V) verse= (15)
-2 (V~ · V¢)y by, on z = Z(z, y).
In order to apply the free-surface boundary condition at the po-
sition of the undisturbed free surface, it is necessary to expand
equation (15) in a Taylor series about z = 0. By defining a wave
elevation Z0 which depends only upon the double-body potential
Ze = 2} (u2 - V) · V¢), (16)
and assuming that the wave elevation Zip, y) is composed of Z0
plus additional terms which are of the order A, we can obtain the
linearized boundary condition to be satisfied on the mean free
surface:
(¢s)2Yss + 2~¢S¢y~ysy + (¢y,2yyy + DOSES + (y~xy)Ys
+ 2~¢s~xy + Amp Fly, + gyp: = - 9Zo~zz
-Obsess + ~y~xy)¢s ~ hasty + ~y~yy)¢y,
(17)
where we have used the reflection condition, As = 0, to remove
terms involving the vertical component of the double-body flow
on z-0.
It is important to note that since the double-body potential
tends to the free-stream potential as we move away from the body,
the Dawson free-surface boundary condition limits to
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Representative terms from entire chapter:
boundary condition
source panels over the hull surface and M panels over the undis-
turbed free surface, the pertubation potential can be written as
U2~cc+gyz = 0, as ~ ~ oo. (18)
This is identical to the linear Kelvin free-surface boundary con
dition written in equation (8~. Consequently, the differences be
tween the Dawson solution and the Neumann-Kelvin solution
must be due to the differences between their respective free
surface boundary conditions, equations (17) and (8), which occur
in a relatively limited region around the hull.
We propose to represent the double-body perturbation po
tential y by a distribution of Havelock singularities on the hull
surface and on the free surface in a region immediately surround
ing the hull. At points farther away from the hull, where the Daw
son free-surface condition tends to the Kelvin condition, the Have
lock source strength necessary to satisfy the free-surface boundary
condition goes to zero. In this approach it will be necessary to
add the double-body potential ¢, which is represented by a dim
tribution of Rankine singularities, to the perturbation potential
A, represented by Havelock singularities, in order to calculate the
total flow field around the hull.
Alternatively, one could represent both the double-body po
tential and the perturbation potential by distributions of Have- where
lock singularities on the hull and the free surface. In this ap
proach, we seek the total potential A, which must be a solution to
the Laplace equation subject to the hull boundary condition (3~.
To obtain the linearized boundary condition which must be sat
isfied by ~ on the mean free surface, we substitute equation (10) and
into equation (17~:
(<;5 )2
retained the free-surface integration, no contour integral around
the waterline occurs.
The first two terms in the Green function correspond to a
Rankine source and its negative image above the free surface.
These terms can be evaluated using standard techniques. The
contribution to the potential due to free-surface waves is con-
tained within the two integral terms in equation (21~. To calcu-
late this wave potential we first define a wave spectral function
C(k=, by) which can be evaluated analytically for a flat panel of
uniform source density:
C(k=,ky) = ~ deem-ik=6-ityn
3i
(23)
Then the contribution to the potential from the two integral terms
in the Green function is
2 l°° /~°° ek=+ik=~+ikuy
=a-ho J dLyt dk~ C(k='ky) k2 _ kok (24)
-i,, ° ,/ day /3 C(k', ky~etz+ik=~+ilyy,
-oo ~
The boundary conditions on the hull and the free surface involve
first and second derivatives of the potential which can be cal-
culated by multiplying the spectral function by the appropriate
wave number prior to performing the integration.
The calculation of the Kelvin waves generated by a discrete
Havelock singularity at zero depth can be particularly trouble-
some. However, in the present formulation of the problem, the
singularities are distributed uniformly over flat free-surface panels
of finite dimensions. By first performing the spatial integration
over the panel, we effectively filter out much of the high wave
number content of the Havelock source and we can obtain well
behaved spectral functions for an arbitrary panel, even one at zero
depth. Some of the higher order derivatives of the potential which
occur in the Dawson free-surface boundary condition can lead to
an integrand which contains sufficient high wave-number content
to present numerical difficulties. However, these high wave num-
bers correspond to waves which are not properly resolved by the
free-surface panelization and which should be removed from the
spectral function to prevent aliasing. In the results presented in
the following sections, a cosine squared filter has been applied
to the spectral functions with the filtering length set at twice
the length of the free-surface panels. The spectral functions have
been cut-off at half the panel dimension.
It is interesting to examine the limiting behavior of a distri-
bution of Havelock singularities as both the depth of the panel
and its collocation point go to zero. If the panel is located at
an infinitesimal depth ~ < 0, then the collocation point should
be interpreted as being locating at the limit as the field point
approaches the panel from below, z = c~. The linearized free-
surface boundary condition is actually satisfied on the other side
of the panel, at z = 0, and there will exist a 4'T discontinuity in At
across the panel due to the first Rankine term in equation (21~.
Consequently, a free-surface distribution of Havelock singulari-
ties will not satisfy the Kelvin free-surface boundary condition
at points located on the panel itself, although it will satisfy the
boundary condition at all other points on the free surface. There-
fore, on panels located at points away from the hull, where the
Dawson free-surface condition approaches the Kelvin free-surface
condition, the Havelock source density must go to zero, for oth-
erwise the free-surface boundary condition could not be satisfied
at the collocation point.
The N unknown source strengths on the hull surface panels
and the M unknown source strengths on the free-surface pan-
els are obtained by solving a set of independent linear equations
composed of the N hull boundary conditions and the M free-
surface boundary condition. Consequently, we have eliminated
the finite differencing schemes usually employed in the solution
to the Dawson problem.
RESULTS - SUBMERGED BODY OF REVOLUTION
To investigate our numerical approach, we initially examined
a fully submerged body, since this would avoid any difficulties
associated with the intersection of the hull with the free surface.
We chose the submerged prolate spheroid for which Neumann-
Kelvin results have been presented by Doctors and Beck [53. The
ellipsoid of revolution can be defined by
r = ~0 L [1 - (2x/L)2]
(25)
where r is the radius of the body and L is the length. The hull cen-
terline was located at a depth of z = 0.16L. The body was pan-
elized with 240 panels on the half-body (symmetry about y = 0
is assumed), using 8 rows of 30 panels with cosine spacing in the
longitudinal direction. For this body of revolution, the hull sur-
face can be panelized with flat quadrilaterals, and panel warpage
is not an issue. The free surface was panelized using 240 square
panels (on the half-space y > 0) with dimensions of L/10 on each
side. The free-surface panels were arranged on a rectangular grid
centered over the body, with 30 panels longitudinally and 8 pan-
els laterally. The panelization of both the free surface and the
submerged spheroid is shown in Figure 1. This panelization was
chosen to cover the entire region of the free surface over which
the double-body velocity magnitude differed from the free-stream
by more than 1.0%. For this submerged body, the maximum dif-
ference between the double-body velocity magnitude and the free
stream velocity is ~ 9%. The magnitude of the double-body
velocity on z = 0 is plotted in Figure 2, where the solid lines
represent 1% contours for which the magnitude is greater than
the free-stream and the dashed lines represent contours less than
the free-stream velocity. We expect that the Dawson free-surface
condition differs from the Kelvin condition only over this limited
region, and consequently, the Havelock singularity density should
go to zero on the panels near the edges of our panelized domain.
Approach I- Perturbation Potential
At a Froude number of 0.4, the solution of the perturbation
potential, id, resulted in the distribution of Havelock singulari-
ties on the free surface shown in Figure 3 (positive contours are
shown as solid lines and negative contours are dashed). As ex-
pected, the source density is greatest near the body and goes
smoothly to zero at the edges of the domain. The greatest source
density which occurs on any edge panel is less than 2% of the
peak value (-0.012), which occurs directly over the body. The
corresponding Havelock singularity densities on the hull surface
are shown in Figure 4. Each curve represents a row of panels
at one circumferential angle on the body. The peak singularity
densities found on the hull occur on the row of panels nearest
the free surface, and are about twice the value of the peak free-
surface sources. Contrast this singularity distribution with that
obtained using our Neumann-Kelvin solution technique, Figure 5,
which of course, does not have any singularities on the free sur-
face. The Havelock/Dawson solution has peak singularity densi-
ties which are about one fourth of the peak values obtained in
the Neumann-Kelvin solution, but more importantly, the Have-
lock/Dawson solution is significantly smoother, indicating that
the geometric approximation of flat panels of constant source
strength is more accurate for an equal number of hull surface
panels. We also note that the Havelock/Dawson solution yields
singularity densities which go smoothly to zero at both the bow
and stern.
The near-field free-surface elevations are calculated from the
pertubation potential using a linearized version of the Bernoulli
equation which is consistent with the linearized Dawson free-
surface boundary condition:
p = 2pU2 - pgz - 2pV) · Vie - pV) · Vie, (26)
262
where p is the pressure and p is the density of the fluid. The
free-surface elevation calculated over the panelized domain is pre-
sented in Figure 6 (non-dimensionalized by the characteristic
wave number ko). As expected, the edges of the computational
domain do not create any wave attenuation or reflections. The
extension of this solution to the far-field is accomplished by us-
ing equation (23) to calculate the free-wave spectrum C(k', by)
associated with the Havelock singularities on both the hull and
the free surface, and then calculating the far-field potential from
the far-field limit of equation (24~:
he ~-~ ho ~ day t~ C(k' ~ by jEkZ+ik=~+ityy, (27)
-Go ~
In Figure 7 is shown a comparison of the far-field Kelvin waves
calculated from the Havelock/Dawson solution (solid line) and
from the Neumann-Kelvin solution (dotted line). The similarity
is remarkable. The components of the far-field waves generated
by the singularities on the hull and on the free surface can be
calculated separately, and this result is shown in Figure 8. Al-
though the singularity strengths on the hull are twice as large as
those on the free surface, the far-field waves generated by these
singularities (solid line in Figure 8) are significantly smaller due
to the exponential attenuation with depth.
Approach II- Total Potential
The second numerical approach we examined was the solu-
tion of the total potential ~ by distributed Havelock singularities
on the hull and free surface. The free-surface singularities ob-
tained are shown in Figure 9. Again we see the rapid reduction
in the singularity density as we move away from the body. At the
outermost panels, the densities (less than 3% of the peak values)
are similar in absolute value to those obtained in the previous ap-
proach, and may represent some measure of the numerical noise.
The peak values (-0.007) are less and the distribution of Have-
lock singularities seems to be limited to an even smaller region
than was observed in the solution to the perturbation potential.
Unlike the pertubation potential, the singularities do not seem
to go monotonically to zero as we move laterally away from the
body, but exhibit a slight oscillatory behavior, suggesting that
this approach may require a finer free-surface panelization. The
singularity densities distributed over the hull surface are shown
in Figure 10. Both the qualitative and quantitative similarities
between these source densities and the corresponding Neumann-
Kelvin results (Figure 5) are striking. The strong singularities at
the bow and stern are an order of magnitude greater than the
peak values found on the free-surface panels.
As before, the near-field free-surface elevations are calculated
from a consistent linearized version of the Bernoulli equation. To
obtain the pressure equation in terms of the total potential ~
rather then the perturbation potential id, we substitute equa-
tion (10) into equation (26), which leads to
p = 2pU2 - pgz + 2pV) · V) - pV) · Vat . (28)
The non-dimensional free-surface wave elevations are given in Fig-
ure 11. Although the wave heights obtained here are slightly
higher than those obtained from the perturbation potential (Fig-
ure 6), the overall agreement is quite good. The far-field wave
elevations obtained from the free-wave spectrum are compared
to the Neumann-Kelvin results in Figure 12. The present results
(solid line in Figure 12) are somewhat higher than those obtained
from the perturbation solution (Figure 7) and consequently, the
quantitative agreement with the Neumann-Kelvin results (dotted
line) is not quite as good, although qualitatively, the results are
similar. The contribution to the far-field waves from the hull sur-
face singularities dominates the solution, as shown in Figure 13.
The waves generated by singularities on the hull are shown by
the solid line and those generated by free-surface singularities are
shown by the dotted line.
Wave Resistance Calculation
The wave resistance can be calculated either by integrating
the linearized Bernoulli pressure, equation (26) or (28), over the
hull surface, or by calculating the energy in the far-field Kelvin
waves. Pressure integration is the standard technique employed in
Dawson's method, while most Neumann-Kelvin solvers calculate
the wave resistance from the energy in the free-wave spectrum.
Using equation (23) to calculate the free-wave spectrum from the
distribution of Havelock singularities on both the hull and free
surface, we can obtain the wave resistance from
RW = 27rpho Idly ~ Ok', ky)~2
(29)
The non-dimensional wave resistance coefficient Cw is defined as
C RW
-DSU2 '
(30)
cat
where S is the total wetted surface area of the hull. We will
use Cd to designate the comparable resistance coefficient ob-
tained from pressure integration. It is probably more reasonable
to compare C,l, with a drag coefficient obtained by integrating
the full non-linear Bernoulli pressure over the hull surface rather
than the linearized version contained in equations (26) and (28~.
The results obtained from the free-wave spectrum and from pres-
sure integration over the hull are presented in Table I for both
Havelock/Dawson approaches and compared with the Neumann-
Kelvin result. The Neumann-Kelvin solution given by Doctors
and Beck [5; is within 1% of the comparable result presented
here.
Table I. Wave Resistance Coefficient for Submerged Body
C(~) c(2) Coo
H/D Soln: ~ 0.0132 0.0143 0.0150
H/D Soln: ~ 0.0148 0.0162 0.0190
N-K Soln. 0.0124 0.0124
C(~): Obtained by integrating the linearized Bernoulli pressure.
C(2~: Obtained by integrating the non-linear Bernoulli pressure.
C'`,: Obtained from free-wave spectrum.
The steep gradients found in the source densities directly
over the body suggest that the solutions would benefit from the
use of smaller panels in this region. The calculations were re-
peated using quarter size panels on an inboard region defined by
-0.6 < z/L < 0.6 and-0.2 < y/L < 0.2, with the outboard
region using the larger panels shown in Figure 1. The solutions
had more well deRned singularity densities but the resulting wave-
fields and wave resistances did not change significantly.
RESULTS- WIGLEY HULL
Our initial attempts to solve the Dawson problem using dis-
tributed Havelock singularities employed the submerged body de-
scribed above. In this section we report on the results obtained
when the same numerical approach was applied to a surface pierc-
ing body, specifically the Wigley hull form defined by
Y = 2B [1-(2X/L) ] [1-(Z/H) ~ , (31)
where B = beam = L/10, and H = draft = L/16. The pan-
elized free surface, Figure 14, contains 300 square panels (L/10
on each side) in the half-space y > 0. The hull surface paneliza-
tion contained only 50 quadrilateral panels, each panel having a
263
length of L/10 and a depth of H/5. This panelization is much
coarser than we would usually use for a comparable Neumann-
Kelvin solution, but we wanted to match panel size on the hull
with adjoining free-surface panels. The fact that the hull surface
panels are flat quadrilaterals means that there will exist some
gaps between the panel edges.
The double body velocity magnitudes are shown in Figure 15.
Due to the high length/beam ratio of the Wigley hull and the rel-
atively large free-surface panels, the double-body velocity magni-
tudes calculated at the centers of the panels never differ from the
free stream velocity magnitude by more than 3%. On the out-
ermost panels, the double-body velocities are reduced by more
than one order of magnitude. Consequently, we would not ex-
pect the Havelock/Dawson solution to differ significantly from
the Neumann-Kelvin solution. The slight fore/aft asymmetry
which can be seen in the contours is one effect of the gaps in the
hull surface created by the use of flat quadrilateral panels.
The free-surface singularity distribution corresponding to the
solution of the perturbation potential at a Froude number of 0.40
is shown in Figure 16. The singularity densities go smoothly to
zero at a very short distance from the hull. Since the free-surface
panelization does not seem to adequately resolve the distribution
of singularities near the hull, it would appear that smaller panels
distributed over a more limited free-surface domain would yield a
better result for the same number of panels. The corresponding
hull surface singularity distribution is given in Figure 17. Each of
the 5 curves in the figure represents a row of panels at the same
depth on the hull. The singularity densities are comparable in
peak values to the densities on the free surface, although adjoin-
ing hull and free-surface panels can have quite different densities,
as evidenced on panels near both the bow and stern. The den-
sities on the free surface reach their peak values near the ends
of the hull while the hull-surface singularities seem to be tending
smoothly to zero at the ends. Comparing these singularities with
the Neumann-Kelvin solution shown in Figure 18, we note that
(like the similar comparison for the submerged hull) the Have-
lock/Dawson solution results in much smaller peak values and
much smoother trends.
Calculated non-dimensionalized near-field free-surface eleva-
tions are given in Figure 19. As before, the computations are not
affected by the edge of the panelized domain, and can be readily
extended into the far-field. The far-field wave elevations are com-
pared to the Neumann-Kelvin result in Figure 20, where the solid
line represents the Havelock/Dawson result and the dotted line
represents the Neumann-Kelvin result. There is more high wave
number energy in the Neumann-Kelvin result, as evidenced by the
steep diverging waves which occur at non-dimensional distances
of-10 to-12, but otherwise the results are quite similar. The
generation of the far-field waves is dominated by the free-surface
singularities, as can be seen in Figure 21.
For this particular case, the wave resistance coefficient ob-
tained from the Havelock/Dawson solution Cw = 0.00204 com-
pares very well with the value obtained from the Neumann-Kelvin
solution (Cw = 0.00212~. The drag coefficient obtained by in-
tegrating the linearized Bernoulli pressure is somewhat higher
ICE = 0.00258) while the drag coefficient obtained from the full
non-linear Bernoulli pressure is 0.00248. However, these results
may not be too meaningful given the coarseness of the paneliza-
tion on the hull surface.
The second approach to the Havelock/Dawson problem, solv-
ing for the total potential rather than the perturbation potential
yielded a disappointing result. The free-surface singularities, Fig-
ure 22, did not tend smoothly to zero at the edges of the panelized
domain, but exhibited regular oscillations as we moved laterally
away from the stern. The singularity densities shown in the fig-
ure are quite small, one tenth the magnitude of the hull-surface
singularities (which are quite similar to the Neumann-Kelvin sin-
gularities), but they generate a comparable wave field due to their
free-surface location. In fact, the far-field wave elevations shown
in Figure 23 show that a remarkable degree of cancellation will
occur between the waves generated by the free-surface sing~lar-
ities and the hull-surface singularities. We speculate that the
solution is suffering from a loss in numerical accuracy caused by
this cancellation. It would appear that this problem is unique to
this particular approach. In the first approach, the solution for
the perturbation potential, the free-surface waves are dominated
by the singularities distributed over the free-surface panels, and
the loss of accuracy does not occur.
. . .
SUMMARY AND CONCLUSIONS
Our original objective in this effort was to develop a method
ot combining the superior near-field predictions of the Rank-
ine/Dawson codes with the superior far-field predictions of the
Havelock codes. We set out to demonstrate that the use of Have-
lock singularities distributed over the free surface as well as on
the hull surface, rather than Rankine singularities, could result
in a solution to the Dawson problem which was free of the wave
reflections often caused by the boundaries of the computational
domain and free of the wave attenuations introduced by numer-
ical damping schemes. This Havelock/Dawson solution could be
extended to an arbitrary distance in the far-field.
By examining two simple geometries, a submerged spheroid
and a Wigley hull, we have demonstrated the feasibility of the
method. The free-surface singularity densities go to zero at the
outer edges of the panelized domain and both the near-field and
far-field wave elevations are well behaved, with no evidence of
wave reflections or artificial damping. The far-field wave eleva-
tions have been shown to be comparable to those obtained from
a solution to the Neumann-Kelvin problem.
We have presented results obtained from two different nu-
merical approaches. In the first approach, we solve for a pertur-
bation potential represented by distributions of Havelock singu-
larities, to which must be added a double-body potential which
is obtained in the usual manner from distributions of Rankine
singularities on the hull. In the second approach, we solve for
the total potential represented by distributions of Havelock sin-
gularities. The two approaches should yield comparable results,
but we found that with the first approach we did not experience
the degree of numerical difficulties encountered with the second
approach.
The singularity distributions obtained when solving for the
perturbation potential were exceptionally well behaved. The peak
-
values of the singularity densities were comparable on the hull
and on the free surface, and were significantly smaller in peak
value and noticeably smoother than the singularity distributions
obtained from the Neumann-Kelvin solution. For the two simple
geometries examined in this study, the hull-surface singularities
tended toward zero at both the bow and the stern. The wave
fields were dominated by the contributions from the free-surface
singularities.
The second approach, solving for the total potential, yielded
acceptable results only for the submerged body. For both the sub-
merged spheroid and the Wigley hull, the hull-surface singularity
distributions obtained were remarkably similar to those obtained
with our Neumann-Kelvin solver. The free-surface singularity
distributions were significantly smaller than the hull-surface dis-
tributions, but their location on the free surface resulted in wave
fields which were comparable to those generated by the singular-
ities on the hull. For the Wigley hull, it appeared that cancel-
lation between the waves generated by the hull singularities and
the free-surface singularities resulted in a loss of accuracy.
It has yet to be demonstrated that the Havelock/Dawson
approach can reproduce, for realistic hull forms, the superior
near-field results of the Rankine/Dawson methods which were
reported by Lindenmuth et al. [33. The only geometries investi-
gated to date have been so simple that one would not expect sig-
nificant differences between Neumann-Kelvin results and Rank-
ine/Dawson results. However, these encouraging results indicate
that the Havelock/Dawson method may well give us the ability
264
to combine the best aspects of both the Rankine/Dawson codes
and the Havelock codes.
ACKNOWLEDGMENTS
This work was supported by the Applied Hydromechanics
Research program of the Applied Research Division of the Of-
fice of Naval Reseacrh, and administered by the David Taylor
Research Center.
REFERENCES
[1] Dawson, C.W. "A Practical Computer Method for Solving
Ship-Wave Problems". The Proceedings of the Second Interna-
tional Conference on Numerical Ship Hydrodynamics, Berkeley,
California, 1977.
[2] Raven, H.C. "Variations on a Theme by Dawson". Seven-
teenth Symposium on Naval Hydrodynamics, The Hague, The
Netherlands, 1988.
[3] Lindenmuth, W.T., T.J. Ratcliffe and A.M. Reed. Com-
parative Accuracy of Numerical Kelvin Wake Code Predictions -
"Wake-Off". David Taylor Research Center Report DTRC/SHD-
1260-01, May 1988.
[4] Wehausen, J.V. and Laitone, E.V. Surface Waves. "Encyclo-
pedia of Physics," Vol. IX. Springer-Verlag, Berlin, 1960.
t5] Doctors, L.J. and Beck, R.F. "Convergence Properties of the
Neumann-Kelvin Problem for a Submerged Body". Journal of
Ship Research, Vol. 31, No. 4, December 1987.
265
PRNEL I ZRT I ON
SOURCE DENSITY
Contour cntervaL - O. 0005
o.s o.o o.s '.o 1.5
X / ShLp Length
Figure 1. Hull panels on a submerged body of revolution and
corresponding free-surface panels.
DOUBLE BODY TOTRL VELOCITY
-
In
o
1
1.~5 o.oo
Figure 2. Double-body velocity magnitude on the free surface.
Solid lines are contours greater than 1, dashed lines are less
than 1.
Figure 3. Source density on the free-surface resulting from a
solution to the perturbation potential at Fn=0.40. Solid lines are
positive contours, dashed lines are negative.
SOURCE STRENGTHS
Hull SLnguLarLtLes ~ Perturbation Potential
~_..
0.5 0.4
0.3 0.2 o. I o.o -o. ~ -0.2 -0.3 -0.4 -0.5
tongutudLnat posLtLon / shLp Length
Figure 4. Source strengths on hull surface panels resulting from
a solution to the perturbation potential. Each curve represents
panels along one circumferential angle.
266
u]
o
o
lo
- ~
o-
o i
lo
lo
o o
cn
(n
SOURCE STRENGTHS
Neumann-KeLv~n Sol utcon
0 . ~ l
0 , ., ..., .....
.5 0.4 0.3 0.2 0.1 O.o -0. 1 -0.2 -0.3 -0.4 - .s
CongutudenaL positron / strep length
Figure 5. Source strengths on hull surface panels resulting from
a solution to the Neumann-Kelvin problem at Fn=0.40.
NEVE ELEVRT I ON
Contour Interval ~ 0. 050
o
lo
u,
1.45 O.OC
Figure 6. Free-surface elevations resulting from a solution to the
perturbation potential at Fn=0.40. Solid lines are positive con-
tours, dashed lines are negative.
_
-2.0 -3.0
WRVE ELEVRT I ON
a, Froude Number ~ O .40 YO ~ 2. 00
lo
cot O
At:
cat
_ 0
o
is
lo
I_
G
~ _
ha
r~
o
-4.0 -5.0 -6.0 -7.0 -8.0 -9.0 -10.0 -li .o -12.0
LONG I TUD I NRL D I STRNCE / SH I P LENGTH
Figure 7. Comparison of far-field wave elevations obtained from
the solution to the perturbation potential Dawson problem (solid
line) and the Neumann-Kelvin problem (dashed line).
WRVE ELEVRT I ON
Froude Number - 0.40 YO - 2. 00
.,
Figure 8. Contributions to the far-field wave elevations from the
hull-surface panels (solid line) and from the free-surface panels
(dashed line), corresponding to the solution to the perturbation
potential.
267
SOURCE DENS I TY
Contour Intervals - O. COOS
WRVE ELEVRT I ON
Contour Intervals - O. 050
-
1.45 0.00
Figure 9. Source density on the free-surface resulting from a
solution to the total potential at Fn=0.40.
SOURCE STRENGTHS
HuLl' SunguLarctces ~ Totals Potentcal
' - r.l ~ " -a
0.5 o.4 0.3 0.2 o. 1 o.o -o. t -0.2 -0.3 -0.4 -o.s
l~ongctudenal" positron / shop Length
Figure 10. Source strengths on hull surface panels resulting from
a solution to the perturbation potential at Fn=0.40.
Figure 11. Free-surface elevations resulting from a solution to the
total potential at Fn=0.40.
WRVE ELEV8T I ON
Froude Number - O.40 YO - 2.00
-2.0 -3.0 -i.o -s.o -6.0 9-i.o -8.0 -9.0 -to.o -~i.o -~2.0
LONG I TUD I NRL D I STANCE / SH I P LENGTH
Figure 12. Comparison of far-field wave elevations obtained from
the solution to the total potential Dawson problem (solid line)
and the Neumann-Kelvin problem (dashed line).
268
WRVE ELEVRT I ON
N Froude Number ~ 0.40 YO ~ 2. 00
0
WOK
o, ~
. , . , . , , , . · , ,
-2.0 -3.0 -4.0 -s.o -6.0 -7.0 -8.0 -9.0 -10.0 -1 1 .o -12.0
LONG I TUD I NRL D I STRNCE / SH I P LENGTH
Figure 13. Contributions to the far-field wave elevations from the
hull-surface panels (solid line) and from the free-surface panels
(dashed line), corresponding to the solution to the total potential.
PRNEL I ZRT I ON
.
DOUBLE BODY TOTRL VELOC I TY
Contour Interpol ~ 0. 005
o~
8
a
o
1.45
-
0.00 -1 .45
Figure 15. Double-body velocity magnitude on the free-surface
for the Wigley hull.
SOURCE DENS I TY
. . . . .
-1.5 -1.0 -0.s o.o o.s
X / Shep Length
1.0 ~.5
Figure 14. Hull panels on a Wigley hull and corresponding free-
surface panels.
Figure 16. Source density on the free-surface resulting from a
solution to the perturbation potential for the Wigley hull at
Fn=0.40.
269
SOURCE STRENGTHS
Hulls Sungul~aruti~es - Perturbatton PotentcaL
-
r~
A
a
-
o
o
2 0
cn
. ~
an
-
o
.
o
N
o
o
o
A
_~
0.5 0.4
0.3 0.2 0.1 0.0 -0. 1 -0.2 -0.3
tong~tudunal~ positron / shop Length
^o. ~-o.s
Figure 17. Source strengths on the hull-surface panels of Wigley
hull resulting from a solution to the perturbation potential. Each
curve represents a row of panels at a constant depth.
SOURCE STRENGTHS
~Neumann-Kel~v~n Solution
o
o
on
o
-
Gl
N~ WN
. . .
0.5 0.4 0.3 0.2 0.1 0.0 -0. 1 -0.2 -0.3 -0.4 - .
LongetudunaL position / sheep Length
Figure 18. Source strengths on hull-surface panels of Wigley
hull resulting from a solution to the Neumann-Kelvin problem
at Fn=0.40.
ARVE ELEVRT I ON
Figure 19. Free-surface elevations resulting from a solution to the
perturbation potential for Wigley hull at Fn=0.40.
URVE ELEVRT I ON
Froude Number ~ O .40 TO - 2.00
. , . , . , . , . i!. ,'. , . , , , , , . ,., . 1
.
-2.0 -3.0 -~.0 -5.0 -6.0 -7.0 -B.0 -9.0 -10.0 -11 .0 -12.0 -13.0 -14.0 -15.0
LONG I TUD I NRL D I STRNCE / SH I P LENGTH
Figure 20. Comparison of far-field wave elevations obtained from
the solution to the perturbation potential Dawson problem (solid
line) and the Neumann-Kelvin problem (dashed line).
270
WOVL LLEV8TION
~ ~ - O ~ - 2~
WHVE ELEVOTION
~ ~ - ~ - 2~
-2.0 -3.0 -4.0 -5.0 -6.0 -7.0 -8.0 -9.0 -10.0 -11.0 -12.0 -13.0 -14.0 -15.0
LONG I TUQ I NAL O I STONCE / SH I F LENOTH
Fi~re 21. Contributions to tbe ~-Reld w~e elev~ions hom tbe
bulLsur~ce p~ek tsobd Une) ~d hom tbe he~sur~ce paneb
Q^d 1~), ~d~g ~ t~ ~^ ~ t~ pet~b~-
pote~i~.
SOURCE DENSITY
~- ~1
1.45
Fi~re 22. Source denshy on tbe he~sur~ce resuldug hom
~lution to tbe tots1 p~enti~ ~r tbe Wig~y bull ~ Fn=0.40.
.o -i.o -~.o -s.o -d.o -) o -a.o -g.o -lo.o -ll.o -12.0 -13.0 -14.0 -15.0
LONG I IUD I NAL D I STONCE / SH I F LENGTH
Fi~re 23. Conthbutions to tbe ~-Reld w~e elev~ions Q~ tbe
. bulLsur~ce p~eL (scud bue) ~d hom tbe he~sur~ce p~s
(d~bed line), corresponding to the solution to tbe tota1 potential.
271
