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OCR for page 145
Numerical Evaluation of a Ship's Steady Wave Spectrum
F. Noblesse
David Taylor Research Center, Bethesda, USA
W. M. Lin
Science Applications International, Annapolis, USA
R. Mellish
David Taylor Research Center, Bethesda, USA
ABSTRACT
The study presents a modified mathemati-
cal expression for the wave-spectrum function
in the Fourier representation of the wave
pattern of a ship advancing at constant speed
in calm water. This new expression is
obtained from the well-known usual expression
via several applications of Stokes' theorem
resulting in the combining of the integrals
along the waterline and over the hull surface
of the ship. The modified expression for the
wave-spectrum function is considerably better
suited than the usual expression for accurate
numerical evaluation because the significant
numerical cancellations which occur between
the waterline and hull integrals in the usual
expression are automatically and exactly
accounted for in the modified expression, as
is demonstrated mathematically and confirmed
numerically.
INTRODUCTION
Near-field potential-flow calculations
about ships advancing at constant speeds in
calm water are routinely required for evalua-
ting their hydrodynamic characteristics, both
in calm water and in waves, and for determin-
ing the required propulsion and control
devices. Calculations of far-field ship wave
patterns are also important in connection
with wave-resistance predictions and remote
sensing of ship wakes. In particular, the
latter practical application requires the
capability to determine the short divergent
waves in the wave spectrum having wavelengths
between 5 cm and 40 cm associated with Bragg
scattering of the electromagnetic waves in
typical SAR systems used in remote sensing of
ship wakes. No meaningful predictions of
such short waves can be obtained on the basis
of currently available numerical methods.
More generally, numerical predictions of the
steady wave pattern at large and moderate
distances behind a ship are notoriously dif-
ficult and unreliable, as was recently made
clear at the Workshop on Kelvin Wake
Computations [1]. Ship wave-resistance cal-
culatio-ns are also known to be unreliable.
145
Several alternative numerical methods
have been developed for evaluating near-field
flow about a ship, that is, flow at the hull
surface and in its vicinity. These include
finite-difference methods, e.g. Coleman [2]
and Miyata and Nishimura [3], and the more
widely used boundary-integral-equation
methods, also known as panel methods. The
latter methods can be divided into two main
groups, according to the Green function that
is used. These two groups of methods are the
Rankine-source method and the Neumann-Kelvin
method, which are based on the simple Rankine
(free-space) fundamental solution and the
more complex Green function satisfying the
linearized free-surface boundary condition,
respectively.
The Rankine-source method was initiated
by Gadd [4], Dawson [5] and Daube [6], and
has since been adopted by many authors. The
Neumann-Kelvin approach has a long history.
A survey of recent numerical predictions
obtained by a number of authors on the basis
of the Neumann-~elvin method may be found in
Baar [7]. This study and that by Andrew,
Baar and Price [8] also contain extensive
comparisons of the authors' own Neumann-
Kelvin numerical predictions with experimen-
tal data. An approximate solution, defined
explicitly in terms of the value of the
Froude number and the hull shape, to the
Neumann-Y~elvin problem was proposed in
Noblesse [9]. This slender-ship approximation
was recently used by Scragg et al. [10] and
compared to both Neumann-Kelvin predictions
and experimental data in [7] and [8] and to
experimental data in [1] and [11].
The aforementioned alternative numerical
methods for predicting flow in the vicinity
of a ship are not all directly suitable for
predicting the far-field wave pattern of a
ship. More precisely, the finite-difference
method and the Rankine-source panel method
require truncating the flow domain at some
relatively-small distance away from the ship
and therefore can only be used for near-field
f low calculations. (However, these near-field
f low predict ions can be used as input to the
far-field Neumann-Kelvin flow representation
considered in this study. ) On the other hand,
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Representative terms from entire chapter:
waterline integral
the Neumann-Kelvin theoretical framework is
equally suitable for near-field and far-field
flow predictions. Indeed, the far-field
Neumann-Kelvin flow representation is a
simplified particular case of the correspond-
ing near-field representation.
The problem considered in this study is
that of evaluating the steady wave spectrum
and the wave pattern of a ship at moderate
and large distances behind it in terms of the
near-field flow on the hull surface. The
near-field flow thus is assumed known for the
purpose of the present study, which is con-
cerned with the prediction of the steady wave
spectrum and the wave potential behind a ship
stern within the Neumann-Kelvin theoretical
framework as was just noted.
This theory expresses the wave potential
in terms of a Fourier representation, as is
well known and is specifically indicated by
Eq. (3) in this study. The wave-spectrum (or
wave-amplitude) function in this Fourier
representation is defined by the sum of an
integral along the mean waterline and an
integral over the mean wetted-hull surface.
This expression for the wave-spectrum func-
tion is ill suited for accurate numerical
evaluation because the waterline integral and
the hull integral largely cancel out, as is
shown further on in this study. Errors which
inevitably occur in the numerical evaluation
of the waterline and hull integrals cause
imperfect numerical cancellations between
these integrals and corresponding large
errors in their sum. This fundamental diffi-
culty was recognized in [9] and in [12],
where attempts to remedy it were presented.
However, these ad hoc approximate numerical
remedies, based upon combining the waterline
integral with the contribution to the hull
integral stemming from the upper part of the
hull surface are not satisfactory.
A conceptually simpler and numerically
more effective remedy is presented in this
study, in which a modified mathematical
expression for the wave-spectrum function is
obtained via several applications of Stokes'
theorem resulting in the combining of the
waterline integral and the bull integral.
This new expression for the wave-spectrum
function is considerably better suited than
the usual expression for accurate numerical
evaluation because the significant numerical
cancellations which occur between the water-
line and hull integrals in the usual expres-
sion are automatically and exactly accounted
for, via a mathematical transformation, in
the modified expression obtained in this
study. The fundamental advantage of the new
expression over the usual one may readily be
appreciated from Fig. 4.
Another interesting feature of the modi-
fied expression for the wave-spectrum func-
tio-n is that it only requires the tangential
velocity at the hull, not the potential,
whereas the usual expression requires the
values of both the velocity potential and its
gradient at the hull. The modified expres-
sion thus defines the wave-spectrum function
in terms of the speed and the size of the
ship, the shape of the mean wetted-hull sur-
face and the tangential velocity at the mean
bull surface. This expression is suitable
for use in conjunction with any near-field
flow-calculation method, including boundary-
integral-equation methods based on source
distributions and other numerical methods in
which the velocity vector is determined
directly on the mean hull surface rather than
derived from the potential. It provides a
practical and reliable method for coupling a
far-field Neumann-Kelvin flow representation
with any near-field flow-calculation method,
including methods based on the use of Rankine
sources or finite differences.
NEUMANN - ELVIN REPRESENTATION FOR
THE WAVE POTENTIAL
As already noted, this study considers
steady potential flow about a ship advancing
at constant speed in calm water of infinite
depth and lateral extent. Nondimensional
coordinates and flow variables are defined in
terms of the length L and the speed U of the
ship and the water density p. The undisturbed
sea surface is chosen as the plane z = 0,
with the z-axis pointing upwards, and the x-
axis is taken in the ship centerplane (port-
and starboard-symmetry is assumed) and point-
ing toward the bow, as is depicted in Fig. 1.
The Froude number and its inverse are denoted
by F and v, respectively, and are given by
F = U/(gL) = 1/v , (l)
where g is the acceleration of gravity.
Within the so-called Neumann-
More precisely, the problem considered
this study is that of evaluating the wave
potential () at any point (5, n , ~ < 0)
behind the ship stern. It is shown in [9]
and [13] that the wave potential may be
defined in terms of the following Fourier
integral representation:
W(~) = (2/~) |o exp(v2`p2) cos(v2npt)
Im exp(iv (p) K(t) dt , (3)
where p is defined in terms of the Fourier
variable t by the relation
p = (1 + t2) 1/2
(4)
and K(t) represents the wave-spectrum func-
tion defined further on in this study. The
wave potential () in Eq. (3) is expressed
in terms of a familiar Fourier superposition
of elementary plane waves propagating at
angles ~ from the ship track given by
tang = t .
(5)
The amplitudes of these elementary plane-wave
components are essentially given by tile func-
tion K(t), which may thus be referred to as
the far-field wave-amplitude function or as
the free-wave spectrum function. The wave-
spectrum function K(t) contains essential
information directly relevant to a ship's
wave resistance and wave pattern. In partic-
ular, the wave resistance, R say, experienced
by the ship is defined in terms of the wave-
spectrum function by means of the well-known
Havelock formula
R/(pU L ) = |o [K(t)] p dt · (6)
The wave-spectrum function K(t) in Eqs.
(3) and (6) may be expressed as the sum of
two terms [9], as follows:
it(t) = Ko(t) + K¢(t) , (7)
where Ko represents the (zeroth-order)
slender-ship approximation and Rip the
Neumann-Kelvin correction term in the
Neumann-Relvin approximation Ray + K¢. More
precisely, the function JO + Kid corresponds
to the usual linearized Neumann-Kelvln
approximation, in which the nonlinear terms
in the free-surface boundary condition are
neglected. These nonlinear terms yield an
additional term in the expression for the
spectrum function K, defined by an integral
over tile mean free surface [9,13], which is
ignored here. The slender-ship approximation
Ko is defined explicitly in terms of the
value of the Froude number and the hull
shape, whereas the Neumann-T~elvin correction
K~ also depends on the value of the potential
at the mean bull surface. The functions X~
and Kit are considered in turn, beginning with
the stender-ship approximation Do.
in THE SLENDER-SHIP APPROXIMATION
The slender-ship approximation Ko(t) to
the wave-spectrum function K(t) is defined in
[9] as the sum of a line integral Kw(t) along
the ship's mean waterline w and a surface
integral Rh(t) over the ship's mean wetted-
hull surface h, as follows:
Ko(t) = KW(t) + Kh(t) , (8)
where the waterline and hull integrals are
given by
K = J (E++E )nx ty dQ , (9a)
Kh = v |h exp(v p z)(E++E )n da . (9b)
In these expressions, E+ represent the
trigonometric functions defined as
E+ = exp[-iv p (ux + vy)] , (10)
where u and v are given by
u = 1/p and v = tip ; (lla,b)
it may then be seen from Eq. (4) that we have
1 > u > 0 and O < v < 1 (12a b)
_ _ _ _ ,
for O < t < ~ , with
u + V2 = 1
(13)
Furthermore, w and h represent the positive
halves of the mean waterline and of the mean
wetted-hull surface, respectively, as is
depicted in Fig. 1 where h = s + b with s -
hull side and b = hull bottom. Also, do is
the differential element of arc length of w
and da the differential element of area of h.
Finally, n = (ox, ny, no) is the unit vector
normal to h and pointing outside the ship,
and t = (tx, ty, tz = 0) is the unit vector
tangent to ~ and pointing toward the bow, as
is shown in Fig. 1.
The hull bottom of a typical ship is a
nearly horizontal surface, so that we have
nx ~ O on b, but nx is usually significant on
the hull side in the bow and stern regions.
However, the bull side of a typical ship is a
nearly vertical surface, i.e. we have nz ~ O
on s. It is therefore convenient to express
the hull integral as the sum of integrals
over the hull side and the hull bottom, and
to modify the bull-side integral into a form
involving tile source density no instead of
nx by using Stokes' theorem in tile manner
sI~own in [13]. The slender-ship approxi~a-
tion Ko(t) may then be expressed in the form
Ko(t) = K *(t) + K ,(t) + Kh*(t), (14)
where tile functions K *(t), KW,(t) and K~l*(t)
are det ined as
low* = is (E +E ) (nx -u )ty di , ( 1 5a )
147
K , = u2 J , exp(v2p2z)(E++E )t do ,
Kh* = -iv u J exp(v p z)(E++E )n da
1 ~2 ~ ~ 2 2~= 1~ >~ AN
(15b)
I v Jb =~`v ~ ~~+ =- MANX ~~= e (15c)
In the foregoing modified expression for
the slender-ship approximation Ro(t), the
function Kw*(t) represents a modified water-
line integral, with source density (nX2-u2)ty
instead of nX2ty in Eq. (9a). Furthermore,
the function Kw'(t) corresponds to a line
integral along the waterline-like curve w'
separating the hull side s and the hull
bottom b, as is shown in Fig. 1; the unit
tangent vector t = (tx, ty, tz) to the lower
waterline w' points toward the bow. Finally,
Rh*(t) represents a modified hull integral
consisting of the sum of an integral over the
hull bottom b and the hull side s, with
source densities given by nx and -iunz ,
respectively. The latter source density is
null for a wall-sided ship and, more
generally, vanishes in the limit t ~ ~ , as
may be seen from Eqs. (4) and (lla). The
hull-side integral therefore is generally
less important in the modified expression
(14) than in the original expression (8). In
particular, the hull-side and hull-bottom
integrals in Eq. (15c) are null for a wall-
sided ship with a flat horizontal bottom
(i.e., a strut-like form), for which Eq.(14)
thus expresses the slender-ship approximation
Ro(t) as the sum of two line integrals. For
large values of v2p2 = (sec26)/F2 , the trig-
onometric functions E+ defined by Eq. (10)
oscillate rapidly. The dominant contribution
to the modified waterline integral Rw* in Eq.
(14) therefore stems from the points, if any,
where the phases v2p2(ux + vy) of the trigo-
nometric functions E+ are stationary. These
points of stationary phase are defined by the
conditions udx + vdy = 0, which yield the
relations
ut + vt = 0 , tx = v , ty = + u ; (16a,b,c)
the latter two relations can be obtained from
Eq. (16a) by using Eq. (13) and the identity
tX2 + ty2 = 1.
The term u2 in the integrand of the modi-
fied waterline integral Kw* defined by Eq.
(15a) stems from the integral on the hull
side in Eq. (8), as may be seen by comparing
the alternative expressions for the function
Ko given by Eqs. (8), (9a,b) and Eqs. (14),
(15a,b,c). We have nx = -t along the top
waterline of a wall-sided ship; Eq. (16c)
therefore shows that the term nX2-u2 in the
integrand of the modified waterline integral
Kw* vanishes at a point of stationary phase
for a wall-sided ship. This result indicates
that the waterline integral and the hull-side
integral in Eq. (8) cancel out in a first
approximation (specifically, within the
stationary-phase approximation) for a wall-
sided ship. The major contributions to these
two integrals thus are combined into the
modified waterline integral Kw* in the modi-
fled expression (14), and the modified bull-
side integral in Eq. (14) is less important
than the original hull-side integral in Eq.
(8) as was already noted.
The modified expression for the slender-
ship approximation Ko(t) defined by Eqs. (14)
and (lSa,b,c) thus is better suited for accu-
rate numerical evaluation than the usual
expression defined by Eqs. (8) and (9a,b) for
large values of v2p2, that is for small
values of the Froude number and/or large
values of t = tans. However, significant can-
cellations may be expected to occur between
the line integrals Kw* and Kw' in Eq.(14) for
relatively large values of the Froude number
and small values of tans. More precisely,
the term -u2(E++E_)ty in the integrand of the
top-waterline integral Kw* defined by Eq.
(15a) and the integrated u2exp(v2p2z)(E++E_)ty
of the lower-waterline integral Kw' defined
by Eq. (lSb) may nearly cancel out if
exp (v2p2z) ~ 1, that is for small values of
v2p2d where d is the ship draft.
It therefore is useful to express Eq.
(14) in the following form:
K~(t) = K*(t) + K'(t) , (17)
where the functions K*(t) and K'(t) are
defined as
lo'* = 1 (E +E )
w ~ _
[nx -u +u exp(v p z)]ty dQ , (18a)
K' = u |w' exp(v p z)(E++E )t do
- u | exp(v p z)(E++E )t dI
- iv u | exp(v p z)(E++E )n da
+ v |b exp(v p z)(E++~ )n da
. (18b)
In the integrals along the top waterline w in
Eqs. (18a,b), z is to be taken equal (or,
more generally, approximately equal) to the
vertical coordinate of the point (x,y,z) on
the lower waterline w1 , in such a way that
the integrals along the lower waterline w'
and the top waterline w in Eq. (18b) nearly
cancel out.
In the simple case of a strut-like hull
form we have nx = 0 on the hull bottom b and
no = 0 on the hull side s. Furthermore, the
lower waterline w' is identical to the top
waterline w except for a vertical translation
equal to the ship draft d, and z in the
integrals along the lower and top waterlines
w' and w in Eq. (lab) is equal to -d. For
such a simple strut-like hull we then have
K'(t) = 0 and F.q. (17) yields Ko(t) = K*(t).
The modified waterline integral K* defined
by Eq. (18a) thus provides an exact expres-
sion for the slender-ship approximation To in
the special case of a strut-like hull form.
For a simple hull in the shape of a strut
the alternative expressions for the slender-
ship approximation TO defined by Eqs. (8) and
(17) become
148
Y`o(t) = K (t) + ~ (t) = Y~*(t) , (19)
where the hull integral Ah in Fq. (8) was
replaced by the hull-side integral Ks since
we have nx = 0 on the horizontal bottom of a
strut. The real and imaginary parts of the
functions Kw(t) , KS(t) and fit) _ K*(t) are
depicted in Fig. 2 for 0 < t = tans < 5
(corresponding to 0 < ~ < 79°) for a specific
strut-like hull form at three values of the
Froude number, namely O.1 (top row), 0.2
(center row) and 0.3 (bottom row). The strut
considered for the calculations presented in
Fig. 2 has beam11ength and draft/length
ratios equal to 0.16 and 0.07, respectively,
and consists of a pointed bow region 0.2 < x
< 0.5 with parabolic waterlines, a straight
middle-body region -0.3 < x < 0.2 and a
rounded stern region -0.5 < x < -0.3 with
elliptic waterlines.
The top row of Fig. 2, corresponding to
the small value of the Froude number F equal
to 0.1, shows that the function K~ is signif-
icantly smaller than the waterline and hull-
side integrals Kw and Us in Eqs. (19) and
(8). This numerical result is in accordance
with the previously-established theoretical
result that the major contributions to the
integrals Kw and Rs cancel out for small val-
ues of the Froude number. The function Ko ,
especially its read part represented by a
solid Line, is also appreciably smaller than
the functions Kw and Ks in the center row of
Fig. 2 corresponding to F = 0.2 and, to a
reduced degree, in the bottom row correspond-
ing to the fairly large value 0.3 of F.
The integral KW (t) along the lower
waterline w' in Eqs. (14), (15b) and (18b) is
also depicted in Fig. 2. The top row of this
figure, corresponding to F = 0.1, shows that
the lower-waterline integral KW'(t) is negli-
gible in comparison with the function,Ko(t) _
K*(t) for all values of t due to the expo-
nential function exp (v2p2z) in the integrand
of the lower-waterline integral Kw . How-
ever, this integral is significant for small
and moderate values of t = tans in the center
and bottom rows of Fig. 2 corresponding to
F = 0.2 and 0.3, respectively.
For typical hull forms Eq. (17) expresses
the slender-ship approximation Ko(t) as the
sum of the modified waterline integral K*(t)
defined by Eq. (18a) and the remainder ~'(t)
defined by Eq. (18b). The remainder K'(t)
may generally be expected to provide a rela-
tively small correction to the dominant
waterline integral K*(t). In particular, the
integrals along the lower and top waterlines
w' and w and the hull-bottom integral in Eq.
(18b) decay exponentially due to the exponen-
tial function exp(v2p2z) in their integrands.
These three integrals thus are negligible for
sufficiently large values of v2p2 , for which
the major contribution to the remainder R'
stems from the upper part of the hull side in
the hull-side integral in Eq. (18b).
It may thus be useful to divide the bull
side into an upper part -d < z < 0 and a
lower part z < -d , where ~ is solve fraction
of the depth of the hull side s. The upper
hull side can be approximately defined by tile
parametric equations x = a(Q) + z a(Q ) and
y = b(Q) + z 3(Q) for -(A) < z < 0 , where
represents the arc length along the top
waterline w defined by x - a(Q) and y = b(Q),
and a = ax/az and ~ = ay/a' are the slopes of
the hull surface at the waterline. The con-
tribution of the upper part of the hull side
to the hull-side integral in Eq. (18b) can
then be expressed as an integral along the
top waterline w, which can be grouped with
the top-waterline integral K* defined by Eq.
(18a). In this manner the do~lnant waterline
integral it* is modified by including the con-
tribution of the upper part of the hull side
to the hull-side integral in Eq. (18b),
whereas the remainder K' is modified by
restricting the integration in the hull-side
integral to the lower part of the hull side.
This modified remainder K' thus is expo-
nentiaLly small for large values of v2p2 =
(sec20)/F2 and can only be significant for
small and moderate values of v2p2. The hull
bottom b, the lower part of the hull side s
and the lower and top waterlines w' and w in
expression (18b) for the remainder it' may
then be approximated by using a relatively
coarse discretization, whereas a finer
discret izat ion may be used for representing
the top waterline w in expression (18a) for
the dominant waterline integral K* . The
modi f led f orm of the top-waterline integral
( 18a ) including the contribution of the upper
hull side to the hull-side integral in Eq.
(18b) can easily be derived from Eq. (18b).
THE NEUMANN-KELVIN APPROXIMATION
The correction terra K<, in Eq. (7) for the
Neumann-Y~elvin approximation Ko + K¢, to the
wave-spectrum function K is defined by the
sum of a waterline integral and a l~ull-
surface integral [ 9,13]:
K<~(t) = KW(t) ~ KW'(t) + KH'(t), (20)
where Kw(t) and Kw'(t) are the waterline
integrals and KH' (t) the hull-surface
integral def ined as
A Iw (E/E_) (tX4)t+sx~s)ty dQ , (21a)
Kw ' = iv p | (E++E )¢ t do, ( 2 1 b )
KH' = (v p ) |h exp (v p z )
(E+n++l3: n )¢ da . ( 21 c )
In the foregoing equations E+ are the trigo-
nometric functions defined by Eq. (10). The
functions n+ in Eq. (21c) are defined as
n = -n + i(un + vn ) . (22)
+ z x y
In Eqs. (21a-c) and (22), t = (tx, ty, 0) is
the Ullit vector tangent to the waterline w
and pointing toward the bow, as was already
defined, s = (sx, sy, sz ) is a unit vector
149
tangent to the hull surface h and pointing
downwards and n = (ox, ny, nz) is the unit
vector normal to ~ and pointing outside the
ship, as is shown in Fig. 1. Finally, it and
Is in Eq. (21a) represent the components of
the velocity vector V) in the directions of
the unit tangent vectors t and s to h,
respectively.
Numerical evaluation of the waterline and
hull integrals in Eq. (20) is a seemingly
simple task, given the value of the potential
on the mean hull surface h + w ; in partic-
ular, the integrands of the integrals defined
by Eqs. (21a-c) are continuous functions.
Nevertheless, accurate and efficient numeri-
cal evaluation of these integrals requires
careful analysis because the trigonometric
functions E+ defined by Eq. (10) oscillate
rapidly for large values of v2p2, as is the
case for typical values of the Froude number
F = 1/v and of the Fourier variable p2 = 1 +
t2 = sec2O, and because the potential ~ in
the integrands of the waterline and hull
integrals Kit' and KH' in Eqs. (21b,c) is mul-
tiplied by tile large numbers v2p and (v2p)2,
respectively. More precisely, we have 102 <
v4 10 for
> 72°; values of (v2p)2 as large as 105
thus are possible. The waterline and hull
integrals Kw' and KH' in Eq. (20) may then be
expected to be dominant and to largely cancel
out, as is shown in Fig. 3.
More precisely, Fig. 3 depicts the func-
tions Kid , KW + Kw' and KH' for O < t < 10
(corresponding to 0 < ~ < 85°) for the simple
bull form considered previously in Fig. 2
with an assumed simple mathematical expres-
sion for the value of the velocity potential
at the hull surface. Specifically, the
potential in Eqs. (21a-c) is taken as ~ - F2
exp(v2z) cos[v2(x-1/2)-3~/8], which corre-
sponds to an elementary plane progressive
wave. This simple hull form and assumed sim-
ple expression for the potential at the hull
surface are used for tile calculations pre-
sented in Fig. 3 because they permit accurate
numerical calculations (the required integra-
tions can be partially performed analytical-
ly) and they are adequate for the purpose of
numerically illustrating the essential prop-
erties of the several alternative mathemati-
cal expressions for tile Neumann-Kelvin
correction Kid examined in this study. Figure
3 corresponds to a value of the Froude number
equal to 0.15. It may be seen from Fig. 3
that the function ~¢ is considerably smaller
than the waterline and hull integrals Kit +
Kw' and KH' . In particular, the waterline
and hull integrals rK~ + Kw' and KH' do not
appear to vanish in the limit t ~ ~ (D ~
90°). Significant cancellations therefore
occur between the waterline and hull
integrals in Eq. (20). These significant
cancellations occur for all values of ~ but
are especially notable for large values of 3,
corresponding to the snort divergent waves in
the spectrum.
The errors which inevitably occur in the
numerical evaluation of the integrals Kw +
Kw' and KH' cause imperfect numerical cancel-
lations between these components and corre-
spending large errors in their sum. Numerical
errors in the sum Kid can be especially diffi-
cult to control because the errors associated
with the numerical evaluation of the hull
integral KH' and the waterline integral Kw +
Kw' are not necessarily comparable (due to
differences in the errors associated with
numerical integration over hull panels and
waterline segments). The usual expression
(20) for the Neumann-Kelvin correction Rip in
Eq. (7) thus is ill suited for accurate
numerical evaluation. A modified mathemati-
cal expression in which the cancellations
between the waterline and hull integrals KW +
Kw' and KH' depicted in Fig. 3 are automati-
cally and exactly accounted for, via a mathe-
matical transformation, is presented below.
By using Stokes' theorem in the manner
explained in [13] we can combine the water-
line and hull integrals Kw' and KH' defined
by Eqs. (21b,c) into a modified hull integral
~~ , as follows:
KH(t) = KW'(t) + KH'(t) , (23)
where the modified hull integral Kit is given
by
XH = iv P Ah exp(v p z)(E+a++E a ) da (24)
with
a+ = nzal/ax - nxa¢/a~
+ iv(nxa¢/3y - nya¢/3x) . (25)
By substituting Eq. (23) into F.q. (20) we may
then obtain the following modified expression
for the Neumann-Kelvln correction T<¢ (t):
K¢(t) = KW(t) + KH(t) . (26)
The functions Kit' , KH' and KH are depic-
ted in Fig. 3. This figure shows that the
waterline and hull integrals Kw' and KH' are
considerably larger titan the modified hull
integral KH . Although the latter integral
is identical to t'ne sum of the integrals Kw'
and KH' , it clearly is preferable to evalu-
ate XH directly by means of Eqs. (24) and
(25) rather titan as the sum of the integrals
KW' and KH' defined by Eqs. (21b,c). The
modified expression for the Neumann-Kelvin
correction R+(t) given by Eqs. (26), (21a),
(24) and (25) therefore represents a signifi-
cant improvement in comparison with the usual
expression given by Eqs. (20) and (9la-c). It
is shown in [13] that the cancellations
between the waterline integral Kw' and the
hull integral Ad' depicted in Fig. 3 can be
explained mathematically for a wall-sided
ship form by performing an asymptotic analy-
sis in the limit v2p2 ~ ~ similar to that
presented previously in this study for the
slender-ship approximation `~ .
The functions Kw , KH and K~ in the modi-
fied expression (26) for the `,leumann-Relv~n
150
correct ion are depicted in Fig. 3. It may be
seen that the waterline integral KW and the
modified hull integral R}I are appreciably
larger than their sum K`t,, especially for
large values of t. Signif icant cancellations
therefore still occur between the waterline
and hull integrals in Eq. (26 ). Further rnod-
ifications of the expression for the function
K<, defined by Eqs. (26), (21a), (24) and
(25) are then desirable for numerical
calculat ions. These Audi f icat ions are now
presented.
By making use of Stokes' theorem and a
classical formula in vector analysis we can
obtain [ 13] the following alternative expres-
sion for the Neumann=Kelvin correction K
K
,nurnber and the shape of the hull, and the
Neumann-Kelvin correction term Kd>, which
also involves the potential at the hull. The
wave resistance of the ship is def ined in
terms of the wave-spectrum function by means
of the Havelock formula (6 ). The Fourier
variable t in Eqs. (3 ) and (6 ) is related to
the angle ~ of propagat ion of the f ree waves
in the ship wave pattern by the relation
t = tans, as is given by Eq. (5).
The slender-ship approximation TO in Eq.
( 7) is def ined by the usual expression given
by Eqs. (8) and (9a,b), or by the alternative
modified expression defined by Eqs. (17) and
(18a,b). The latter expression defines the
spectrum function Ko as the sum of a modified
integral K* along the ship waterline w and a
remainder K' . In the special case of a
strut-like hull form, the remainder K' is
null and the modified waterline integral K*
provides an exact expression for the slender-
ship approximation Ho.
The Neumann-Kelvin correction R`b in Eq.
( 7) is def ined by the usual expression given
by Eqs. (20) and (21a-c) or by the alterna-
tive modified expression given by Eqs. (27),
(28a,b), (29a) and (34). This alternative
expression involves an arbitrary complex
function C(t), and thus defines a one-
parameter family of mathematically-equivalent
expressions for the Neumann-Kelvin correction
K¢, in F~q. (7). In particular, the first
modif fed expression, def ined by Eqs. (26 ),
(21a), (24) and (25), obtained in this study
is a special case of the general expression
given by Eqs. (27), (28a,b), (29a) and (34)
corresponding to the choice C(t ) = t.
Analytical and numerical considerations led
to the particular expression given by Eq.
(44), which corresponds to the choice C(t) =
uv def ined by Eq. (45) where u and v are
given by Eqs. (lla,b). Figure 4 shows that
the mathematical expression corresponding to
E q. (44 ) is considerably bet ter suited than
the usual expression (20 ) for accurate numer-
ical evaluation because the large cancella-
tions which occur between the waterline
integral Kw + Kw' and the hull integral KH'
in the usual expression (20 ) are automati-
cally and exactly accounted for, via a mathe-
matical transformation, in the new expression
(44 ) involving the modif fed waterline and
hull integrals KW* and KH* ~
Another interesting feature of the new
expression for the Neumann~elvin correction
K~, given by Eqs. (27), (28a,b), (29a) and
(34 ) is that it only requires the tangential
velocity at the hull, not the potential,
whereas the usual expression given by Eqs.
(20) and (21a-c) requires the values of both
the velocity potential and its gradient at
the hull. the new expression for K<, obtained
in this study thus def ines the wave-spectrum
function Ro + K
K Kit Ko=K* Kw,
0.04
0.02
0.00
-0.02
—0.04
n na
n nn
—0.04
—0.08
0.12
0.06
0.00
-0.06
-0.12
_
11
O
o
\\
Jl l
It
.,
, ~
dt ~,,.N _
/ ,
\ /%
/'l,j \~_
\ ,
11
O
o
I_
~ O
/ o
CO
l
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
tans tans tans tans
Fig. 2 - Comparison of the usual expression'
0.08
0.00
.08
0.16
n l6
0.16
0.16
0.08
0.00
0.08
0.16
0.16
0.08
0.00
-0.08
—0.16
0.16
0.08
* ~ 0.00
-0.08
—0.16
1 2 3 4 5 6
tans
7 8 9 1 2 3 4 5 6 7 8 9
tans
-
-
0.16
0.08
0.00
.~.
0.08
0.16
0.16
0.08
0.00
0.08
0. 16
0.16
0.08
0.1 6
0.16
0.08
0.00
0.08
0.16
0.16
0.1 6
_ ~
_
-
-
* -4
Fig. 3 - Comparison of the usual expression R¢, =KW +KW' +KH' and the three alternative modified
expressions K,~> = 'Kw + Kit, K,~> = Kw~ + KH" and K,p, = 'POW* + KH* for tile Neumann-Kelvin correction
term K in the expression for tile wave spectrum function K(t) for a simple strut-like bull form
and an assumed simple expression for the potent tat at the hull surface. The real and i Imaginary
parts of the ten functions K¢, 'Kw ~ Kw', Kw', KH', Kw ~ KH ~ KW ~ KH , Kw* and KH are
depicted for O < tang < 10, corresponding to O < ~ < 85°, and for a value of the Froude number
equal to 0. 15.
155
KW+KW KH Kin KW KH
1i,.'
In
1'
-in n4
-n ns
-n 08
n no
—0~24
ll .
\ l -
l ~ ~ l - -
1 2 3 4 1 2 3 4
tans tang
,,
.
,~
1 2 3 4 1 2 3 4 1 2 3 4
tang tans tans
C:
_ . I ~ ~ I
Fig. 4 - Comparison of the usual expression Kid = Kw + Kw' + KH' and the alternative modified
expression Rip = Kw* + KH* for the Neumann-T~elvin correction term Rip in the expression for the
wave-spectrum function it(t) for a simple strut-like hull form and an assumed simple expression
for the potential at the hull surface. The real and imaginary parts of the five functions KW +
Kw' (first column on left), KH' (second column on left), Kid (center column), Kw* (second column
on right) and KH* (first column on right) are depicted for O < tans < 5, corresponding to 0 <
< 79°, and for three values of the Froude number F. namely 0.1 (top row), 0.2 (center row) and
0.3 (bottom row). Large cancellations occur between the waterline integral Kw + Kw' and the
hull integral KH' in the usual expression for Rip which is then ill suited for accurate numerical
calculations, notably for evaluating the short waves in the spectrum corresponding to large
values of tans. The modified waterline integral Kw* and hull integral KH* in the alternative new
expression for Kid are significantly smaller than the usual waterline and hull integrals, and are
comparable to the function Kit . Although the alternative expressions Kid = 'KW + Kw' + KH' and Kid
= Kw* + KH* are mathematically equivalent, the latter expression is considerably better suited
than the former one for accurate numerical evaluation.
REFERENCES
10. Scragg, Carl A., Britton Chance Jr., John
C. Talcott and Donald C. Wyatt, "Analysis of
Wave Resistance in the Design of the 12-Meter
Yacht Stars and Stripes," Marine Technology,
Vol. 24, pp. 286-295 (1987).
11. Noblesse, F., D. Hendrix and A. Barnell,
"The Slender-Ship Approximation: Comparison
Between Experimental Data and Numerical
Predictions," Proceedings of the Deuxiemes
Journees de l'Hydrodynamique, Ecole Nationale
Superieure de Mecanique, Mantes, France,
pp. 175-187 (1989).
156
12. Barnell, A. and F. Noblesse, "Numerical
Evaluation of the Near- and Far-Field Wave
Pattern and Wave Resistance of Arbitrary Ship
Forms," Proceedings of the Fourth Conference
on Numerics Ship Hydrodynamics, Washington
DC, pp. 324-341 (1985).
13. Noblesse, F., and W.M. Lin, "A Modified
Expression for Evaluating the Steady Wave
Pattern of a Ship," David Taylor Research
Center Report No. 88/041 (1988).
14. Noblesse, F., "Alternative Integral
Representations for the Green Function of the
Theory of Ship Wave Resistance," Journal of
Engineering Mathematics, Vol. 15, pp. 241-265
(1981).
