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OCR for page 375
Adequacy of Free Surface Conditions
for the Wave Resistance Problem
H. Raven (Maritime Research Institute Netherlands, The Netherlands)
ABSTRACT
This paper compares three different lin-
earized free surface conditions for steady
potential flow around a ship, viz. the Kelvin
condition and the slow-ship conditions of
Dawson and of Eggers. All are implemented in a
Rankine-source method of the type proposed by
Dawson. The comparisons concern the predicted
wave resistance and wave patterns, the magni-
tude of the nonlinear terms neglected in the
FSC, and the remaining errors in the dynamic
and kinematic conditions at the predicted free
surface. Rather substantial errors are found
for all linearized methods. Dawson's condition
does not perform any better than the Kelvin
condition except for full hull forms. The
occasional prediction of a negative wave re-
sistance for full ships is explained in terms
of an energy flux through the free surface.
The condition of Eggers, though theoretically
preferable, is found to lead to non-con-
vergence of the basic perturbation expansion
near the bow in one of the cases and so to
give worse results than Dawson's.
NOMENCLATURE
D downstream plane up to y=h
Do downstream plane up to y=0
E energy flux
En Froude number
FS free surface Y=h
FSo undisturbed free surface y=0
g gravity acceleration
n
n
n
linearized wave elevation
double-body wave elevation (23)
h-nr
wave elevation including nonlinear
* terms (18,25)
wave elevation including nonlinear and
transfer terms (19,26,28)
p fluid density
total velocity potential
~double-body potential
§' f-.
1. INTRODUCTION
intersection of plane D with free
surface
normal vector
n longitudinal component of n
px pressure
Rw wave resistance
Rwf wave resistance according to (7)
Rwp wave resistance according to (9)
U ship speed
Vn normal velocity component of control
surface
x,y,z Cartesian coordinate system; x astern,
y upward, z to port; origin at ~ L un
less stated otherwise
H.C. Raven, MARIN, P.O. Box 28, 6700 AA Wageningen, The Netherlands
375
One of the oldest and most extensively
studied problems in ship hydrodynamics is the
determination of the wave pattern and wave
resistance of a ship sailing in still water.
This is partly related with its visible
importance for the efficiency of a ship and
the fact that it is the primary quantity
derived from towing tank tests; but also the
rather obvious formulation of an appropriate
mathematical model for the potential flow with
free surface boundary conditions has played a
role. Particularly in the sixties and seven-
ties much work has been done to simplify this
basic model to a method, simple and yet accu-
rate enough in practical applications. The
most important result of all this activity was
probably the formulation of the slow-ship
theory, based on linearization with respect to
the flow about the hull at zero Froude number,
i.e. the "double-body flow". At the Second
International Conference on Numerical Ship
Hydrodynamics two papers applying this condi-
tion, by Baba [1] and by Dawson [2], attracted
much attention due to the realism of the pre-
dictions shown. Additionally Dawson's method
raised much interest owing to its very origi-
nal and apparently straight-forward implemen-
tation differing substantially from current
methods of that time.
Since then, Dawson's method or a very
similar one has been implemented by many
others. One member of this family is our pro-
gram DAWSON, which follows the same basic
procedure but differs in several details from
OCR for page 376
the original method. More about it can be
found in [3]. This program is being used
extensively as a design tool at MARIN, giving
detailed information on the pressure and
streamline direction on the hull, the wave
pattern, the resistance and other forces, etc.
With careful interpretation this allows to
optimize a design before model tests are being
performed. As a matter of fact this has led to
a few quite successful designs. It appears
that in some respects the ideal role of
Numerical Ship Hydrodynamics integrated in the
design procedure is being approached here al-
ready.
Still, in practical applications to
current ship forms certain problems and doubts
on the reliability of the predictions may
arise. In studying these it appears that some
aspects of slow-ship theory have not yet been
settled entirely. The principle of Dawson's
method is generally taken for granted, as is
illustrated by the fact that even the alge-
braic mistake [3] in the derivation of the
free surface condition is copied without
comment in most papers! Current work on wave
resistance calculations is mostly concerned
either with further extensions of Dawson's
method to new applications, or with methods
solving the nonlinear problem, and does not
address the basics of the methods so widely
used now.
In a certain sense this paper makes a
plea for a renewed critical look at the wave
resistance calculation methods. It questions
the assumed superiority of the slow-ship con-
dition by showing that for a large class of
ships the predictions are nothing better than
if the Kelvin condition is used instead. The
paradoxical occurrence of negative wave re-
sistance predictions, perhaps more widely
known but ignored, is studied in Section 3.
The validity of the linearization assumptions
is investigated by estimating the nonlinear
terms and by directly evaluating the velocity
field at the predicted free surface. Some of
the conclusions motivate and direct the de-
velopment of a method to satisfy the nonlinear
boundary conditions; Section 6 briefly dis-
cusses the prospects for such work.
2. WAVE RESISTANCE PREDICTIONS
2.1. Background
The mathematical model pertinent to cal-
culation of the wave pattern and wave re-
sistance of a ship is that of a potential flow
subject to kinematic and dynamic free surface
conditions (FSC's). Solving this problem in
its exact form is quite complicated, not only
owing to the nonlinearity of the dynamic con-
dition but also since both conditions must be
applied at the unknown free surface. The
general approach is therefore to linearize the
FSC. This linearization can be performed with
respect to either the undisturbed uniform flow
or the double-body flow. In the former case,
the Kelvin FSC is obtained:
fax by O
(1)
which, if combined with the exact form of the
hull boundary condition gives rise to a
"Neumann-Kelvin" problem. The other approach
results in the slow-ship FSC, which can e.g.
have the form chosen by Dawson:
Fn2~ t~ _ + Ott a ]~`,2 + t2)
(.x ax + (z az) (.x fx + (zfz ~ ax ~ {z) +
+ by = 0
(2)
where ~ is the double-body potential and ~ is
the total potential. Present methods also
impose that the perturbation potential ¢'=~-.
satisfies the exact hull boundary condition.
Since the appearance of the slow-ship FSC
this has become by far the most popular
boundary condition in wave resistance calcula-
tion methods. Its assumed superiority is
probably based to a great extent on the
results of the First Workshop on Ship Wave
Resistance Calculations in 1979 [4]. Here many
different methods were applied to common test
cases, viz. the Wigley hull, an Inuid form,
the Series 60 block 60 hull, a fast naval
vessel and a tanker model. For all these cases
Dawson's predictions were consistent and
acceptably close to the experimental values.
On the other hand, Fig. 1 illustrates the
results of methods solving the Neumann-Kelvin
problem. The only conclusion to be drawn from
this figure is that large numerical errors
must be present in most of them, since solu-
tions of the same problem predict resistances
differing by a factor of 2 in some cases. The
magnitude of these errors forbids all conclu-
sions on the relative merits of the free
surface conditions used, and the results are
certainly no reason to do away with the
Neumann-Kelvin approach! Still this is what
happened: from about 1980 onwards, the majori-
ty of the effort in the wave resistance field
concerned Dawson's method including the FSC
(2). It seems that the motivation for the
general preference for the slow-ship condition
is not very sound.
Actually a comparison of predictions by
Neumann-Kelvin methods and slow-ship methods
is only allowed if one is very careful on the
point of numerical accuracy. This is because
these two classes of methods use an entirely
different approach and contain therefore
numerical errors of a different origin.
Neumann-Kelvin methods generally exploit the
known form of the Green function for the
problem. So-called Kelvin source panels are
distributed over the hull and along the water
376
OCR for page 377
51 ~
3
2
O_
RESIDUAL RESISTANCE IHUANG & von KERCZEK, 1972)
RESIDUAL RESISTANCE (TODD, 1963)
WAVE RESISTANCE (LONG. CUT, TSAI & LANDWEBER, 1975)
--- WAVE RESISTANCE (XY-METHOD, WARD, 1964)
~ ~ ~ WAVE RESISTANCE (MODEL FIXED AT ZERO TRIM AND SINKAGE, CAl ISAL, l9B0)
1
~ X
_ _1'~
) art ~
. '? A//'
. /'
in'/
~ _
0.15 0.20 0.25 0.30 a_
FROUDE NUMBER
Fig. 1 Wave resistance coefficient for Series
60 model. Symbols indicate results of
various Neumann-Kelvin methods [4]
line; the potential induced by these panels
automatically satisfies the Kelvin free sur-
face condition. Numerical errors are present
in the hull panelling, the treatment of the
singularities of the Green function, the
numerical integrations over the panels and the
waterline integral.
On the other hand, most slow-ship methods
use a distribution of Rankine sources on both
the hull and a part of the free surface sur-
rounding the hull. Discretization errors are
made in the panelling of both surfaces, but
also in the difference scheme used to imple-
ment the velocity derivatives in the free
surface condition. Additionally, the trunca-
tion of the free surface domain and the
"numerical" imposition of the radiation condi-
tion may introduce errors.
However, a fairer comparison of different
FSC's is possible. The flexibility of Dawson's
numerical implementation allows to treat all
sorts of linearized FSC's in basically the
same way. Instead of using Kelvin sources to
solve the Neumann-Kelvin problem, we now use a
distribution of Rankine sources on the hull
and a part of the free surface for both FSC's;
and the velocity derivative is implemented by
a difference scheme. Actually, the only change
needed in the program DAWSON is to replace the
double-body flow terms in the free surface
condition by a uniform flow, as appears from
the formulations given above.
Due to the very similar implementations
of the FSC's the numerical errors will now be
of comparable magnitude; and using the current
experience on the required discretization we
can make sure that these errors have little
influence on the predictions. It is true that
near the stagnation points, in the limit for
zero panel size, numerical errors could
locally again dominate the comparison due to
singular behaviour. But in practice these
singularities are always "discretized away".
Thus the comparison tells us how important the
double-body flow contribution to the FSC is
for practical discretizations, which have been
checked to be adequate.
2.2. Results
This methodology has been applied to a
number of ships. First the standard test cases
were attempted. For the Wigley hull, a discre-
tization with 20*6 hull panels and 10*38 free
surface panels was used in the calculation for
Fn=0.40. As expected in view of the near-
uniformity of the double-body flow for this
slender hull, the difference between the
Neumann-Kelvin and Dawson resistance predic-
tions amounted to only 1%.
A more realistic test case is the Series
60, block 60 hull. 24*20 hull panels and
10*128 free surface panels were used. Fig. 2
displays the predicted resistance curves,
compared with some of the experimental data.
Here again the differences between both
methods are negligible, except perhaps above
Fn=0.32. This contradicts the results of the
Workshop mentioned before, where the Neumann-
Kelvin predictions were, on the average, sub-
stantially higher. Which FSC is more accurate
cannot be deduced from comparison with experi-
mental data, not only because of the small
differences but also due to the disappointing
scatter of all available measurements.
6
5
0.20 0.25
FROUDE NUMBER
, I _ I ~
- RESIDUAL RESISTANCE (HUANG & von KERCZEK, 1972)
RESIDUAL RESISTANCE (TODD, ig63)
WAVE RESISTANCE (LONG. CUT, TSAI & LANDWEBER, 1975)
--- WAVE RESISTANCE (XY-METHOD, WARD, 1964)
3' *( ~ WAVE RESISTANCE (MODEL FIXED AT ZERO TRIM AND SINKAGE, CALISAL, 1980)
Calculated (Dawson FSC) I I
___ Calculated (Kelvin FSC)
~,y,/',/'
,~.. .
a,,'
__ A_ ._
/
,// 1
//
i/
//
//
1
0.35
Fig. 2 Wave resistance coefficient for Series
60 model, predicted with Kelvin and
Dawson's FSC
377
OCR for page 378
Fig. 3 compares the predicted wave
profiles for both methods. Both are in close
agreement with each other and with the data.
Generally, Dawson's FSC leads to a small
forward shift of the bow wave and a somewhat
more pointed wave shape. The largest differ-
ences are found aft of the ship's stern: The
Neumann-Kelvin predictions often have a sub-
stantially larger wave amplitude here. This
can probably be explained by the damping
effect of the base flow acceleration in the
slow-ship FSC.
The same sort of comparison has been made
for several practical cases. The fact that
there is remarkably little difference between
the Dawson and Kelvin predictions is a re-
current feature for all ships with a block
coefficient up to about 0.60 or 0.70. This is
in marked contrast with the general convic-
tion' Since in commercial projects the present
use of the predicted wave resistance values is
generally qualitative or comparative rather
than quantitative, we can safely state that
the Kelvin FSC performs just as well as the
slow-ship FSC.
unto
Fig. 3 Wave profiles along the hull predicted Fig.
with Kelvin and Dawson's FSC; Series
60, Fn = .35 and Fn = .2Z
- Dawson
- Kelvin
There are, however, exceptions. Since the
difference between both FSC's increases with
the nonuniformity of the double-body flow,
there will be a limit on the hull fullness for
which the above statement is valid. Fig. 4
shows what happens for very full hull forms,
in this case a 55000 tdw tanker with a block
coefficient of 0.82. At higher speeds (in
excess of the service speed) both predictions
are similar (though not identical) again, but
for decreasing speed quite substantial dif-
ferences aDDear. The K~1 v; n Per ~;11 yields a
experi-
predicts
a rapid decrease of the resistance. Even so
the predicted wave patterns are very much
alike (Fig. 5) and give no indication of such
drastic resistance differences. Again Dawson's
condition typically results in a forward shift
of the bow wave. Due to the large curvature of
the waterlines of such full hull forms this
brings about a large resistance difference,
concentrated at the forebody.
large resistance far exceeding the
mental value, while the slow-ship Fort
For the same ship at ballast draught the
resistance predicted using Dawson's condition
is in good agreement with the experimental
data, while the Kelvin resistance exceeds this
by a factor of 3. At full draught the Kelvin
result is 4 times as high as the experimental
result at the service speed, but
Dawson's condition yields a negative re
sistance! This physically unacceptable
behaviour has been found for several slow,
full-formed ships.
11
10
q
8
6
378
calculated wave resistance for a tanker model
.
. _. tow Kelvil ~_
Cw Dawson
. . _
. ,
,' ,_
/
/
0 0 0.: 2 0.
I- _
r r ~
o
4 Calculated wave resistance for a tanker
model
0 Experiment
OCR for page 379
Fig. 5a Calculated wave patterns of tanker model, with Dawson's (left) and
Kelvin's FSC (right), at Fn = 0.177 (service speed)
o
at_
N
ID
-
- 1 i.
I I I I
o
O-
to 1
0
o
o
16 32
Summarizing the results of these compari-
sons, for a large class of ships there is no
reason to prefer Dawson 's FSC to the Kelvin
FSC; for practical discretizations the double-
body flow effect is of minor importance. In
extreme cases however, appreciable differences
in resistance may occur, but none of the FSC's
gives an accurate and reliable result.
The next section first tries to resolve
the mystery of negative wave resistance. Then,
we shall assess the adequacy of the FSC's by
other means.
3. THE PARADOX OF NEGATIVE WAVE RESISTANCE
3.1. General
The fact that a negative value of the
wave resistance is sometimes predicted by
Dawson's method is known to more people that
Dawson
Kelvin
1
11 l,\: 1,\. 1/` ~ .1/~6 ^\\ /e
l: 4{ 1`, ~ ~ /° i\ i\~ll2 ''\-/1~ ~144~/lm
Fig. 5b Calculated wave profiles along the hull
379
apply this method to real commercial ship hull
forms. Generally this phenomenon is attributed
to an insufficient resolution of the large
pressure gradients on the hull. As a matter of
fact we often find large differences in the
pressure on adjacent panels, and the simple
pressure integration over the hull could well
be numerically inaccurate.
The accuracy of the pressure integration
and of alternative formulations of the re-
sistance based on Lagally's law has been
discussed in [3]. It has been shown there
that, provided the pressure integration is
corrected for the zero-Froude number pressure
integral, these formulations are all of about
the same level of accuracy, determined by the
accuracy with which the hull boundary condi-
tion is satisfied and so by the density of the
hull panel distribution. Therefore, for the
present test case the hull panelling has been
OCR for page 380
refined, from 545 to 1090, and then to 1415
panels on one half of the hull. Table 1 shows
that this does reduce the zero-Froude number
pressure integral (which has the exact value
zero due to d'Alemberts paradox). However, the
wave resistance (according to all three ex-
pressions) converges to a negative value!
Similarly, increasing the free surface panel
density did not give any substantial change in
the predicted wave resistance. Therefore,
contrary to the general belief discretization
errors can probably be rejected as cause for
the negative resistance, at least in some of
the case investigated.
A negative value of the wave resistance
in the presence of a physically realistic wave
pattern radiated by the ship as shown in Fig.
5 is, of course, paradoxical. Apparently such
a wave pattern need not have the correct
energy budget. The wave pattern represents
radiation of wave energy; since this energy
travels with the group velocity which in
harmonic deep-water waves is half the phase
speed, it lags behind the wave crest propaga-
tion, hence it must be supplied at the wave
origin, i.e. at the ship. But a negative re-
sistance means that instead the ship extracts
energy from the waves. A steady flow can only
exist if some other source of energy is
present.
3.2. Energy Conservation
To investigate what the source of energy
can be we consider the energy fluxes through
the boundaries of a control volume surrounding
the hull and moving with the hull, in a sta-
tionary frame of reference (Fig. 6). We define
the x-axis to point astern, the y-axis ver-
tically upward with y=0 at the undisturbed
water level.
As derived in [5] the general expression
for the energy flux through a surface, ex-
pressed in a general unsteady potential I, is:
E = - rip { P.t( En - Vn) - p Vn}dS (3)
where Vn is the velocity of the surface itself
in the normal direction, and the energy flux
is defined positive in the sense consistent
with that of the normal. For a steady flow,
at = Uf , where now V) is the disturbance of
the uniform flow and U the ship speed.
The energy fluxes out of the control
volume then become:
through the hull surface:
EH = -U l~ P . nx . dS = -U . R (4)
(where n is the inward normal on the hull);
through the upstream plane:
EU = -U l~ pay dS
through the downstream plane:
ED = +U l~ pay dS - ~pU l~ (¢x - by - gads
(5)
(6)
where the lateral and upstream boundaries (but
not the downstream plane) have been assumed to
recede to infinity. It is noted that no energy
flux is present through the free surface,
since the pressure is equal to zero and ~ -V ,
for exact satisfaction of the free surface
conditions.
Table 1 Calculated resistances for tanker model, En = 0.1765
A) influence of hull panel density; Dawson's FSC.
I Sumner or nu' ~ panels per side 1 545 ~ 1090 |
| Zero-Fn pressure integral | .00071 | .00042 |
Wave resist. pressure integration ~ -.00082 ~ -.00068
~ Wave resist. Lagally integr. over FS ~ -.00054 ~ -.00048
! Wave resist. Lagally integr. over hull ~ -.00068 ! - ~ 00054 ~
1 .. 1 _
B) influence of free surface condition; 1090 hull panels
~ Free surface condition
1415
.00020
-.00067
_.00047
-.00052
~ Kelvin ~ Dawson ~ Eggers |
- 1 1 1
Wave resist. pressure integration ~ .00270 ~ -.00068 ~ -.00158
Wave resist. Lagally integr. over FS ~ .00282 ~ -.00048 ~ -.00146
Wave resist. Lagally integr. over hull ~ .00288 ~ -.00054 -.00156
! Waterline integral (10) I .00062 1 .00054 .00015
380
OCR for page 381
f ~ ~
1 1 Rwp = up Or (-< + ~ + 2)dS +
P [l fx by dS (9)
-
t
Energy conservation now demands that
. -EH = U.R
ED wf _. Eu
._
EFso
_ I_
Do ~
EDO U. Rwp ~ Eu
Fig. 6 Control volumes for energy balance;
above: exact case;
below: linearized case
Conservation of energy then requires that
Rwf = Up l~ (~¢x ~ by + (z~dS +
+ Peg ~ n2 dz
T
(7)
The first integral is over a transverse
plane astern of the hull and the second one
along its intersection IT with the free
surface; the latter line integral basically
takes into account the contribution of the
potential energy, as it stems from the pay
term in the pressure. This is a well-known
result that is the basis of wave-pattern
analysis methods.
However, the derivation may not be
entirely relevant to our calculations. The
linearized FSC is imposed on the undisturbed
free surface, not on the actual wave surface.
The fluid domain considered thus extends only
to y=O, and so should our control volume.
Therefore we now redefine the control Volume
by taking not the exact free surface as its
upper boundary but the y=0 plane.
The energy fluxes must then be slightly
modified. An energy flux through the un-
disturbed free surface may now be present,
according to the general expression (3):
O FSo (8)
~ y
It is important to note that here no FSC
has been substituted yet.
It is noted in passing that the energy
supply through the hull surface, and thereby
the resistance found, now only concerns the
y<0 part of the hull. This introduces a dif-
ference that can be approximated by:
= + UPS ~ n nx do (10)
WL
However, this waterline integral is often
ignored, following Dawson [4], since it gener-
ally does not improve the results; its magni-
tude for the present case is included in Table
1 and found to be not negligible, but not
affecting the conclusions on the negative re-
sistance either. Similarly the integral over
the downstream plane extends only to y=O.
In any case, Rwf may be considered as the
resistance that is physically associated with
the generation of the wave pattern that is
predicted in the far field, since (7) has been
derived without making any simplification of
the boundary conditions. On the other hand,
the pressure integration over the hull in our
linearized calculation need not be equal to
Rwf because of the simplifications of the FSC
and the different control volume, but it will
give a result in agreement with Rwp. There-
fore, the origin of the paradox, being a
contradiction of the calculation and our
physical insight, must be sought in the dif-
ference between both expressions.
Substitution of the potential of a free
harmonic wave in Rwf shows that the wave re-
sistance so obtained is positive definite, in
agreement with our physical observation of the
radiated wave pattern. Also its value is
independent of the position of the aft plane,
since in harmonic waves there is a constant
horizontal energy transport and a pure ex-
change between kinetic and potential energy.
In expression (9) however, the horizontal
energy flux is not constant: the potential
energy is absent as the control volume extends
now only to y=O, and it thus cannot compensate
the variations of the kinetic energy flux. But
still the resistance from (9) is independent
of x, since for harmonic waves the variations
of the kinetic energy flux are now compensated
by the flux through the undisturbed free
surface, the integral over FSO.
More generally, we can evaluate (9) by
substituting the Kelvin condition:
381
OCR for page 382
Of'
a
by = unX ; ~ = - g'';
- P fir ox by dS = +Pg l~ hex dS
= Y2pg ~ n2 dz + pug ~ h2 nx do (11)
T WL
where n is now the inward normal on the water-
line. So Rwp and Rwf are equal to leading
order, except for a waterline integral which
is of opposite sign to (10); this difference
in sign has been called "Gadd's paradoxon" and
is dealt with in [6]. Apart from this, if the
Kelvin condition is imposed (7) and (9) give
the same result at least asymptotically; so
the pressure integration over the hull in our
linear problem, which satisfies (9), is equal
to the resistance deduced from the far-field
wave pattern corresponding to (7), hence it is
positive and independent of x if a system of
harmonic waves is present at the aft plane.
Again the energy flux through FSO just takes
into account the variations of the potential
energy. It appears that this correspondence
relies on the precise relation between the
velocity components at the free surface and
the wave elevation, so on the FSC.
However, for free surface conditions
other than Kelvin's this equality may be lost.
E.g. for the slow-ship condition we find from
(9):
Rwp = ~kp [[ (_~2 + f2 + ~z)dS +
- P ·~r (X (~xnX + (ZhZ + fxbrx +
+ ~ZnrZ)dS
(12)
which does not correspond with Rwf generally.
If a realistic system of waves is present
behind the ship, (7) will again give a
positive resistance. But Rwp may be different
as part of the wave energy may have been
supplied through FSO instead of by the ship.
The intuitively expected correspondence
between a radiated wave pattern and a re-
sistance acting on the hull is thus not
verified for all FSC's. In the extreme case
the pressure integration over the hull can
give a negative resistance: the ship rides on
waves generated by the free surface condition,
in this case governed by the double-body flow
nonuniformity.
In order to obtain a more realistic wave
resistance prediction for such extreme cases,
two approaches now suggest themselves. Since
Rwf seems to have the desired property of
positive definiteness, we could try to deduce
the resistance from the far-field wave pat
tern. Otherwise, we could perhaps modify the
FSC so as to eliminate the excess energy flux
through the free surface.
3.3. Far-Field Resistance Calculation
Evaluation of (7) is not a straight-
forward matter in a Rankine-source method,
since it requires to calculate the velocities
in a dense grid of field points in a trans-
verse plane astern of the ship. The integral
over D thus found must then be supplemented
by the Line integral and a contribution of the
difference D-Do. An alternative would be to
apply one of the formula's used in wave
pattern analysis methods, which are based on
(7).
Before undertaking such a study we better
first consider its chance of success. Since we
want to get a resistance independent of the
x-position of the transverse plane, this plane
should be located outside the region where
there is still an excess energy flux through
the free surface. We know that for vanishing
double-body flow disturbance the slow-ship
condition boils down to the Kelvin condition,
for which we have shown the absence of
leading-order energy flux. Therefore we must
choose the downstream plane far enough from
the hull to avoid the double-body flow
disturbance. In practice a logical choice
would be a position at or beyond the aft edge
of the free surface panel distribution.
However, since no singularities are
located aft of this plane, d'Alemberts paradox
is valid for the 'body' generated by the
collection of sources on the double body and
free surface: the momentum flux through the
transverse plane will be always zero! The wave
resistance is an internal force inside this
virtual body and is compensated by an opposite
force exerted by the hull on the free surface
panels.
and
Therefore the integral over Do is zero,
Rwp = ~P l~ fix By dS (13)
FSo
which is precisely the expression obtained
from Lagally's law. Similarly the first term
of (7) is dominated by this effect and looses
the properties it has in harmonic waves; thus
Rwf cannot be supposed to be positive definite
and independent of x any more. This must be
caused by the truncation of the free surface
domain locally distorting the potential field
corresponding with harmonic waves.
Rw (1) will only give a reliable result
if the free surface panelling is continued a
large distance beyond the aft plane D, which
means a considerable extra expense in computer
time.
382
OCR for page 383
In view of their relation with (7) and
the assumptions on the potential field, the
same is likely to be true for wave pattern
analysis methods. Besides, these could be
subject to errors caused by the inherent
damping of the numerical approximation. The
experiences of Maisonneuve [7] with such a
resistance evaluation do show that this is not
a good alternative for the pressure integra-
tion.
3.4. Energy Flux Through The Free Surface
Another, more fundamental remedy for the
occurrence of a negative wave resistance could
be to modify the FSC so as to eliminate the
excess energy flux through the free surface.
Whatever the FSC imposed this energy flux into
the fluid domain can be written as
FS l~ (P U (x EVn - U · Ap · nx)dS (14)
where the integration is over the predicted
wave surface, AVn is the remaining normal
velocity and Ap the pressure prevailing there.
To eliminate the energy flux at every point of
the free surface both AV and Ap should be
zero, so we have to satisfy the exact non-
linear FSC's; the best that can be achieved in
a linearized method is a reduction of the
energy flux to higher order in the perturba-
tion parameter.
For the slow-ship condition this param-
eter is the square of the Froude number Fn.
Since the resistance coefficient is su~posed-
to be of O(Fn ), dE/dt should be O(Fn ). fix
and ~ contain the double-body disturbances
which are 0(1), hence both AV and Ap must be
O(Fn ). Now Dawson's FSC does contain terms up
to O(Fn ) and ought to satisfy this require-
ment; but as has been pointed out in [3] and
will be shown in the next section, Dawson's
FSC is inconsistent due to the absence of
terms incorporating the transfer of the veloc-
ities from the actual to the undisturbed free
surface. Including these terms, ~ ~ + I'
, yields an FSC that can be writienY{s:
YY
Fn {~(+xx + fizz + (x ax + Liz a - z) (.x + Liz) +
(.xx + fizz + ax ax + it a-z) (.x tx + If +
~ fix ~ {Z)) BY (15)
This in fact reduces AV and Ap to O(Fn6) and
thus should suit our purpose.
Now this is exactly what has been derived
in a different way by Eggers [8], in a paper
that seems to have been given little atten-
tion. From the analogous requirement that the
far-field resistance found from a slow-ship
calculation should be invariant to leading
order for the choice of control volume, he
derived this same FSC (15). Although in his
paper this is suggested to form an additional
proof of the correctness of the order assump-
tions made in the linearization, the above
consideration shows that actually it is
subject to these same assumptions.
Therefore, Eggers's FSC does provide
equality of Rwf and Rwp up to higher order in
Fn, just like the Kelvin condition does up to
higher order in the wave steepness. This could
solve the problem of negative wave resis-
tances. The FSC (15) has therefore been imple-
mented in our program and applied to a few
cases. For the Wigley hull the predicted re-
sistance differed by 3% from the Dawson pre-
diction, and the wave profile was almost iden-
tical. For the Series 60 model the resistance
curve is included in Fig. 7 and turns out to
be similar in shape but consistently lower
than those obtained with the two other FSC's.
Surprisingly however, the predicted Rw
for the tanker form (Table 1) is even more
negative than that found with Dawson's FSC.
The wave profile along the hull is hardly
different; the wave pattern (Fig. 8) shows a
remarkable reduction of the Kelvin angle, but
is otherwise similar to Dawson's. These re-
sults suggest that only more energy has been
supplied through FS. Moreover, the resistance
curve goes down for increasing speed, and at
somewhat higher speeds flow reversal at the
free surface is predicted! The results were
found to be extremely sensitive to details of
the implementation. It thus seems that for
large double-body flow disturbances this FSC
is not adequate, notwithstanding its theore-
tical consistency. An explanation of this will
be given in Section 5.
6
.
. ~t ~
.35 0 .40
Fig. 7 Predicted wave resistance coefficient
using Kelvin, Dawson's and Eggers's FSC
Kelvin
- Dawson
- - Eggers
383
OCR for page 384
Fig. 8 Calculated wave pattern for tanker
model, with FSC of Eggers
3~5. Summary
Summarizing some of the findings of this
section, the paradoxical occurrence of nega-
tive wave resistance in the presence of a
plausible radiated wave pattern can be ex-
plained by the possibility of an energy flux
through the undisturbed free surface differing
from the amount needed to represent the poten-
tial energy variations. This energy flux can
be expressed in the remaining normal velocity
and pressure at the calculated free surface.
Reduction to higher order of both quantities
leads to the FSC derived by Eggers. Although
this gives reasonable predictions for slender
ships, for the tanker hull an erroneous flow
field is sometimes obtained and the resistance
is again negative. Therefore the higher order
contributions to the energy flux are probably
significant here. The next section attempts to
shed some light on the magnitude and origin of
these higher order terms in general.
4. HIGHER-ORDER TERMS
4. lo Approach
In the derivation of the linearized free
surface conditions several linear and nonlin-
ear terms have been dropped based on the fact
that they are of higher order in the pertur-
bation parameter adopted. Now the notion of
higher order just means that these terms be-
come insignificant asymptotically for a van-
ishing value of the parameter; no information
is available a priori on the actual magnitude
of these terms in practical cases. Therefore
the range of applicability of a linearized
method can in general only be determined in
practice.
It is clear, however, that agreement with
experimental data is not a very suitable yard-
stick for the accuracy of wave resistance cal-
culations and has therefore not been used in
Section 2. The residual resistance is often
affected by uncertainty on the viscous rem
Distance, and wave pattern measurement tech-
niques give results consistently lower than
the residual resistance and with a consider-
able amount of scatter. Therefore a method to
assess the magnitude of the neglected terms
directly could give a much better idea of the
adequacy of FSC's. In addition it might pro-
vide directions for setting up a method to
solve the exact nonlinear problem.
Evaluation of the nonlinear terms would
in principle be possible from velocity mea-
surements at a very dense grid of points near
the free surface; but no such measurements
seem to be available. A better way would be to
compute these terms from solutions of the non-
linear problem; but these exist only for un-
realistic test cases, or in the form of
numerical solutions of insufficient accuracy.
For these reasons another approach has
been chosen here: to evaluate the nonlinear
terms 'a posterior)', from the flow field cal-
culated by a linear method. Of course this
technique does have its own restrictions: How
the predictions would change if the nonlinear
terms were included in the FSC is not clear;
nor is there any certainty on how the terms
themselves would change in magnitude.
Additionally, the singular behaviour at the
stagnation points will cause some of the
higher-order contributions to blow up. Hence
also the comparison of the higher-order terms
is limited to practical discretizations,
although, as we have found, within fairly wide
margins: approaching the stagnation points
magnifies the relative magnitude of the non-
linear terms but does not immediately affect
the comparison of FSC's.
4.2. Derivation Of Higher-Order FSC's
To define the neglected terms we must
first of all expand the FSC's to higher order.
The combination of the kinematic and dynamic
free surface conditions that is actually im-
posed demands that the flow at the undisturbed
free surface has a direction parallel to the
isobar planes. Only afterwards the free sur-
face elevation is retrieved by using the dy-
namic condition. In view of this, in the fol-
lowing I have chosen the approach to start
from the kinematic condition and to derive the
higher order terms in it.
Now the prime difficulty here is that the
exact FSC should be applied at Y=h, not at the
undisturbed free surface y=O. To derive the
error in the FSC would require the calculated
velocity field at Y=n; but this cannot be eva-
luated since for To, the singularities gener-
ating the flow lie inside the fluid domain.
Therefore we have to resort to Taylor expan-
signs to express flow quantities at Y=h in
those at y=O. But precisely the validity of
these expansions has been a point of debate in
the derivation of the slow-ship FSC. The wave
384
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Representative terms from entire chapter:
wave resistance
elevation including the double-body contribu-
tion is O(Fn ); it has been argued that the
perturbation I' has a wavy character with wave
number O(Fn ), so all terms in the Taylor
expansion for the transfer would be of the
same order; neither the analytic continuation
of the flow field, nor the truncation of the
expansions would then be permitted. We have,
however, adopted the basic order assumptions
of Eggers [8] as discussed in [3] and conse-
quently ignored the possible order reduction
by differentiation.
The nonlinear terms can be formulated
consistently, including only the contributions
of next-higher order, or inconsistently in an
attempt to approximate the exact FSC as close-
ly as possible. There is a quantitative dif-
ference between both forms but the conclusions
are the same for most cases.
The best estimate of the wave elevation
is that which includes the nonlinear terms and
the transfer terms; it is noted n below. For
the Kelvin condition, the perturbation param
eter is the wave steepness ~ = ~ . The ve-
locities at Y=n are: Fn
Ax = (+x)y=o + n fxy + 0(~3), etc. (16)
Substituting this into the dynamic FSC gives
the following expressions for the wave height
n
linear:
n = -Fn fix + 0(£2)
nonlinear:
n = n-~l2 Fn2 :¢X2 + (,2 + +,2] + o(~2) (18
including transfer:
n* = -n- Fn2 n fxy + 0(~3) (19)
The consistent decomposition of the kinematic
FSC is then:
by= Ox+ (§xbX+ ~znZ) + (nx- Ax) +
term 1 term 3 term 4
(nX - -ax) - n my + 0(~3) (20)
term S term 6
while an inconsistent form is:
n* = -n- Fn2 n (+x fxy + Ty~yy + Adz) (21)
(y = Ax + (¢xnX + ~znZ)+~lx aaX + by aaZ](n-n
term 1 term 3 term 4
(¢x ax + at az](n --n) +
term 5
* * *
n ( lynx + ~yznZ - EYE) (22)
term 6
For the consistent form of the slow-ship
FSC the wave height expressions are:
double-body:
or = Y2 Fn (1 ~ (x - (z) (23)
linear:
n = Y2 Fn2 (1 - (x - t2z _ 2.Xfx - 2tzfz) (24)
nonlinear:
n = h - Y2 Fn2 (+,2 + ¢,2 + ¢,2) (25)
including transfer:
n* = -n- Fn2
where
~ = ~ - En hr(.xfxy + (z¢YZ) (28)
and the inconsistent form is identical to
(22).
Actual flow
Doubln-bodv flow
l
Fig. 9 Transfer terms for double-body flow may
reduce the accuracy
In all above expressions, the notation of
the terms is as follows. Terms 1 and 2 are
linear and are included in the linearized FSC.
Terms 5 and 6 result from the transfer of the
FSC to the undisturbed free surface, while
terms 3 and 4 are the other nonlinear con-
tributions to the FSC. Term 7 is a linear
transfer term for the vertical velocity. If,
here, we would not neglect the terms connected
with the double-body flow transfer, its form
would be
Arc yy - ~ tyy = 0(Fn2) (29)
which we shall denote by 'term 8' in the fol-
lowing. This is the transfer term alluded to
in the previous section; it is neglected in
Dawson's FSC but is present in that of Eggers.
It may be noted that for the double-body
flow replaced by a uniform flow, the in-
consistent expressions become equal, but the
consistent odes (19,20) and (27,28) do not for
term 5 and ~ .
4.3. Results
Evaluating all these terms was restricted
mainly to the Wigley hull, the Series 60 block
.60 model and the tanker form dealt with
before. For clearness of presentation we shall
sum up the main conclusions drawn, grouped
under the test case concerned.
The figures illustrating these results
are set up as follows: the quantity plotted is
indicated in the caption; as all terms are
contributions to the vertical velocity, they
are non-dimensionalized by the ship speed and
compared with either the vertical velocity
itself or the dominant contribution to it
(term 1), as indicated. The abscissa is the
panel number on the longitudinal strip of free
surface panels along the hull and the centre
line. The panel lengths are uniform. The loca-
tions of bow (left) and stern(right) need no
further indication.
1. Series 60 block .60 at Fn = 0.35 and 0.22
* Of the linear terms included in the FSC,
only term 1 is significant (Fig. 10). Only
at bow and stern term 2 may give a modest
contribution, but due to the considerable
phase shift between ~ and Or this contribu-
tion is often of the wrong sign. Probably
term 2 could well be deleted from the FSC
but computationally this gives no simplifi-
cation at all.
* The linear term 8 neglected in Dawson's FSC
is generally larger than term 2 and is most-
ly concentrated at the bow (Fig. 10). For
the Series 60 hull its maximum contribution
is about 20%, at Fn=0.35. Including this
term reduces the wave resistance by 20% in
this case.
* The nonlinear terms (i.e. the error in the
linearized FSC) are dominated by the trans-
fer terms. The effect of imposing the FSC at
the correct location is often far more
important than to include all squares and
cross-products.
* The sum of the higher-order terms is shown
in Figs. 11 and 12 for two Froude numbers.
It appears that at Fn=0.35 it is of quite
substantial magnitude, locally as large as
the linear terms. That still the predictions
are acceptable must be due to some sort of
386
o.2l
O . 1
I
-O. 1 -
-0.2
Fig. 10 Linear terms in free surface con
dition; Dawson's FSC, Series 60,
Fn = .35
Term 1
------ Term 2
- - Term 8
error cancelling, or to the fact that the
evaluation a posterior) is too pessimistic. Y 0
At Fn=0.22 all methods have fairly small
error terms, confirming that linearization
is adequate in this case.
The sum of the neglected terms (terms 3 to 8
for Kelvin and Dawson, 3 to 6 for Eggers),
which indicates the adequacy of the
linearized FSC, is of comparable magnitude
for all FSC's; all are oscillating with
similar amplitude and mean value. No signi
ficant improvement for the slow-ship condi
tions is observed.
* Eggers's FSC gives results closely corres
ponding with those of Dawson's; although
term 8 is now included in the FSC, the sum
of neglected terms is not substantially less
than for Dawson's FSC, since term 8 has a
much smaller wave number than the other
terms; therefore the mean value of the error
is changed but the result is not visibly
improved in general.
z. Tanker model, Fn - 0.1765
Similar conclusions were drawn for this
test case, except for the following:
* The nonlinear terms for the slow-ship FSC's *
are of moderate magnitude here, so the
linearization is not particularly inadmis
sible (Fig. 13).
387
n 1
Y o
Y o-
TV
~ ,1~\ ~ 1
\ He |,/ I if
~! IN ,'. ,~,,t,ilkil~l
~ Al V ',3iJI ~I~~N`l T
-0.1- 1) \~J
0.1 - Eggers ~
Jilt
-G.l
~ (Vl/
Fig. It Nonlinear terms in free surface
conditions; Series 60, Fn = .35
--- Term 1
_ Terms 3 + 4 + 5 + 6
1160
1'
1\
~0
~1
~0
The transfer term 8 is now quite large at
the bow, about 70% of the vertical velocity
(Fig. 14). It is understandable that this
causes the substantial difference between
the predictions using Dawson's and Eggers's
condition.
The nonlinear terms are significantly larger
for the Kelvin FSC than for both slow-ship
FSC's now (Fig. 13).
The sum of the neglected terms is somewhat
larger for Dawson's FSC than for that of
Eggers in this case, which is entirely due
to term 8.
We thus conclude that for the Series 60
model even from this evaluation of the non-
linear terms no advantage for the slow-ship
condition over the Kelvin condition can be
observed. Only for very full hulls the slow-
ship condition is again more appropriate than
Kelvins condition for practical free surface
discretizations. The large differences in
neglected terms are largely in agreement with
the wide spread in the predictions for the
tanker case; but the slightly greater accuracy
of the linearization underlying Eggers's con-
dition is not reflected in a better resistance
prediction. Further study of this is therefore
needed and will be performed in the next sec-
t;on
-O.
y
O
-O. 1J
Fig. 12 Nonlinear terms in free surface
conditions; Series 60, Fn = .22
Term 1
- - Terms 3 + 4 + 5 + 6
-
o
in. 13 Terms neglected in FSC; terms 3 to 6
for Kelvin and Eggers, terms 3 to 8
for Dawson; Tanker model, Fn = 0.176C
. .
Fig. 14 Importance of linear transfer term;
Dawson's FSC, Tanker model, Fn = .1765
V
_ Term 8
388
In general it was found that at least for
the more severely nonlinear cases the Taylor
expansions do not converge as quickly as one
might hope, as testified by considerable dif-
ferences between consistent and inconsistent
formulations of some of the nonlinear contri-
butions. Thus any method solving a nonlinear
problem must apply the FSC right at the actual
free surface, otherwise the most important
nonlinear effects are missed or poorly re-
presented. This is a fact not properly
recognized in some of the methods proposed up
to now [9,10]. Another fact learnt from this
exercise is the appearance of double and
triple wave numbers in the higher-order terms
as could be foreseen from the theory. Ac-
cordingly a nonlinear calculation only makes
sense if the discretization is fine enough to
resolve their contributions!
5. DIRECT EVALUATION AT THE FREE SURFACE
Our doubts on the validity and accuracy
of the Taylor expansions partly apply as well
to the calculation of nonlinear terms per-
formed here. The dominance of the transfer,
the difference between consistent and incon-
sistent forms and the oscillatory character of
the higher derivatives of the calculated ve-
locities make the comparison somewhat unsafe.
Therefore another approach has been developed.
As mentioned above, a direct evaluation
of the velocity field at the actual free sur-
face is prohibited in the usual method by the
singularities inside the fluid domain. This
can be avoided by generating the flow field by
singularities not on the undisturbed free sur-
face, but above it at a distance sufficient to
keep clear of the highest waves. In the course
of another study such a method had been devel-
oped. Source panels are located above the un-
disturbed free surface at a fixed distance,
while the collocation points where the FSC is
satisfied remain on the undisturbed free sur-
face. Once the solution has been obtained, it
is an easy matter to compute the velocities
generated by these sources in points on the
calculated free surface. From these, the
residual errors in the exact kinematic and
dynamic FSC can be found.
For the same test cases we then draw the
following conclusions;
1. Series 60 block .60 model, Fn = 0.35 and
0.22
* There are significant differences between
the velocities on the undisturbed free sur-
face and those on the actual free surface,
particularly for the vertical velocity.
Including the transfer terms from Taylor
expansions makes the tangential velocities
fairly accurate except at the stern wave;
but the difference in vertical velocity is
not well represented by the consistent term
8 (Fig. 15).
389
The remaining errors in the dynamic FSC are
represented by the difference between the
"linearized" and the "exact" wave elevation
(Fig. 16), and provide little surprise. The
Kelvin linearization consistently gives
slightly larger errors. ~ , the wave height
approximation employed in the previous sec-
tion, appears to be almost exact in this
case.
The error in the kinematic condition (Figs.
17 and 18) is generally more important than
that in the dynamic FSC. In the case con-
sidered the advantage of Dawson's condition
compared with Kelvin's is now slightly more
pronounced than with the method of Section
4. The error at the bow for Fn=0.35 is 55%
of the vertical velocity for Kelvin and 30%
for Dawson. At lower Fn the difference is
reduced.
1.0
y
0.9
0.1 ~
Vertical
vel oc i ty
i ~
1 ll
{bl
40 1\
80
-0.1
It ~
Fig. 15 Velocities at undisturbed free
surface, with transfer term and
evaluated at actual free surface;
Dawson's FSC, Series 60, Fn = .35
U*, V* (y = 0)
------ U , V (including transfer)
- - Uex, Vex (y = h)
lo2
1.0
.0
n
n
Kel vin
80
Fig. 16 Wave profile along the hull, and error
in dynamic free surface condition;
Series 60, Fn = .35
Olin (linearized)
n ( including transfer)
- ~ (evaluated at Y = Olin)
* The FSC of Eggers leads to slightly smaller
errors than that of Dawson at both speeds.
* It was verified that the evaluation of the
nonlinear terms as made in the previous sec-
tion was qualitatively right and indicative
of the accuracy of the FSC, although unduly
oscillatory.
Thus we find that in this evaluation the
advantage of slow-ship linearization is rather
more pronounced. The reduction of the errors
for decreasing Fn is slow. Although term 8
does not quite well approximate the transfer
effect, including it seems to increase the
accuracy somewhat.
2. Tanker model, Fn = 0.1765
Some remarkable conclusions could be
drawn from these calculations.
1
* Again the error in the dynamic condition is
quite small for the slow-ship approach, And
well represented by the transfer terms (h ).
The Kelvin linearization again leads to a
somewhat larger error in In.
* The error in the kinematic condition (Fig.
19) is very much larger for the Kelvin FSC
than for Dawson's FSC, and amounts to 0.33
times the ship speed, which is 1.57 times
the vertical velocity ahead of the bow. For
Dawson's condition it is 0.152 times the
ship speed. This agrees with the bad wave
resistance prediction obtained with the
Neumann-Kelvin method.
* The same error with Eggers's condition has
sharp positive and negative peaks near the
bow, of +0.178 and -0.192 times the ship
speed, so this method is now suddenly worse
than Dawson's, which matches its bad re-
sistance prediction.
* This large and irregular error is explained
by the excessive transfer effect on v and,
in particular, u (Figs. 20 and 21). There
are extreme differences between the veloc-
ities on y=0 or those including transfer
terms and those evaluated on the free sur-
face, if the condition of Eggers is used.
Even the error in v including term 8 (i.e.
including part of the transfer effect) is
larger than the error in v (without any
transfer correction) with Dawson's FSC.
* A negative u, i.e. flow reversal, was found
at the free surface just ahead of the bow.
The explanation of this lies just in the
fact that the transfer term 8 is included in
the FSC. The FSC can then be written as (15),
and the coefficient of fax is ~ (x ~ 'k which
is zero for (x = ~ = 0 577 In the present
case (x = 0.658, and the coefficient is quite
small. As a result, an excessive value of fax
is not controlled by the FSC. Now
Vex - v = h) + Y2 h2¢ + ... = term 8 +
+ n' any + I/2 h2fyyy + ... (30)
Hex - u = boxy + ok n any + . . . ( 31)
which shows that exactly the consistent inclu-
sion of transfer effects increases the higher
order contributions to the transfer, to the
extent of invalidating the FSC.
This is, therefore the explanation for
the unrealistic results obtained with Eggers's
free surface condition for full hull forms, in
contrast with its theoretical preferability
and acceptable results for less extreme cases.
390
o. ~ ~ :~,'l. ~
v
Dawson
It ~ N iK
_ Eggers
o A ~ ~ ~
to
o
o
~ ~ 'J'-~;f-/~4J:6\4\ '//60 -~1\li/-'1~' ~
Fig. 17 Error in kinematic free surface con-
dition, evaluated at predicted free
surface; Series 60, Fn = .22
V
___ AV
-_ 1
o
8_
o
o
Da
O-
>_ - _
to
o
.
o
8
Kelvin
I\ ~
///~~\\ ~ 1\
~ 16 \4 ~32t-~ 48 _q6
Dawson / \
8
a R- ~R \~\~---~\l t'- ~ --~L 11 '-- ~
o
o
o
-
o
Eggers Jew ~ / ~ \,~
, ~ ~ ~ / I)\ I I ~ ' )(' r '~ ' ~ ~ -~' \' ~ '_ 7f
[5 ~\,~j~ 48 ~ ~7
\ / I
Fig. 18 Error in kinematic free surface con-
dition, evaluated at predicted free
surface; Series 60, Fn = .35
V
- AV
:112\8 ~160
Fig. 19 Error in kinematic free surface con-
dition, evaluated at predicted free
surface; Tanker model, Fn = .1756
It is fair to point out here that the possible
vanishing of the coefficient of ~ has al-
ready been given due attention before.
Brandsma and Hermans [11] thought this fact a
reason not to trust the order assumptions
underlying the Taylor expansions for the
transfer and proposed an alternative with
imposition of the FSC on the double-body wave
surface. Eggers [6,121 interpreted the change
of sign of the coefficient near the bow as a
change of character of the mathematical
problem and tried to explain this in physical
terms.
But the vanishing coefficient is an
artefact of the Taylor expansion used for
transferring the boundary condition; and as
shown here, this expansion tends to diverge if
the coefficient vanishes. Therefore I believe
that no physical interpretation may be given
to the change of sign, and that it simply
imposes a bound on the hull fullness for which
this free surface condition is applicable (and
perhaps even theoretically preferable).
391
N I
~ 1F 1
a3
a_
l ~ ~ I:
t2~ 128:
\;
Fig. 20 Vertical velocities, at undisturbed
free surface, with transfer term and
evaluated at actual free surface;
Tanker model, En = .1765
V* (y= 0) U*
------ V (including transfer) _____- U
_ VeX (y = A) ___ u
6. PROSPECTS FOR SOLUTION OF THE NONLINEAR
PROBLEM
To my knowledge only very little informa-
tion of this kind on the adequacy of linear-
ized free surface conditions has been
published up to now. Although the consistency
of the FSC of Eggers, as opposed to that of
Dawson, is well-known, it seems as if nobody
had tried yet what results the modification
has. The first results published here seem
already to cast an entirely new light upon
this issue. This suggests that there is still
much more work to do on linearized methods.
On the other hand, all the studies
reported in this paper at best add to our
insight but give no indication on how current
methods can be improved: the widely used FSC
of Dawson has turned out to be not markedly
superior for most ships, but to be a fairly
392
Dawson's FSC ~
l'
ID
I_
to
-1- 1 1
0 16
I i'
-
1
ll
ll
ll
ll
ll
ll
4
96 1 12 128
Fig. 21 Horizontal velocities, at undisturbed
free surface, with transfer term and
evaluated at actual free surface;
Tanker model, Fn = .1765
(Y = 0)
(including transfer)
(Y= If)
reliable choice throughout the range of
practical hull forms. Even the occurrence of
negative wave resistances could not be cured
within the framework of linearized methods.
From the practical point of view, therefore,
this study contributes nothing at all.
Although one may thus disagree on the
question whether or not the linearized methods
are well enough established to make the step
towards a nonlinear method, it is true that
computationally current methods require only
little effort, and the present computer
capacity allows commercial use of far more
complicated programs. Moreover, a nonlinear
method could well resolve some of the problems
met with current methods. The negative re-
sistance predictions dealt with in Section 3
will be eliminated by solving the exact
problem with sufficient numerical accuracy.
Additionally, the magnitude of the nonlinear
terms9 even in cases where the predictions are
generally realistic asks for a method that
includes these terms just for reasons of
reliability of the predictions.
Furthermore, certain physical effects,
known to be important and sometimes forming
the main difference between variations of a
design, are entirely eliminated by the lin-
earization. Examples of this are:
- Slightly or partly submerged bulbous bows;
the phenomena occurring in the vicinity of
the bow are strongly nonlinear and cannot
well be represented by a linear approach. In
addition, the increased submergence due to
the sinkage and the bow wave elevation sub-
stantially alters the effect of the bulb, a
phenomenon poorly represented in a linear
calculation.
- The flow off a very flat stern, as often
found on ferry hulls. In this case the
waterline shape changes considerably due to
the wave elevation; the experience is that
linearized methods overpredict the stern
wave height.
- The gradual transition from the flow off a
stern with flat sections to the flow off a
transom stern; in a linearized approach both
regimes are reasonably well representable,
but one has to choose beforehand which one
is more appropriate. This is not a very
satisfactory approach.
Hence there are several practical
incentives to develop a nonlinear method even
though there may still be more work to be done
on linearized methods. Such a development has
therefore recently been undertaken at MARIN.
Abandoning the linearizations adds
several complications to the problem. The
following aspects should in principle be taken
into account:
- The free surface condition should be applied
at the actual free surface; this is probably
the dominant effect, as concluded from Sec-
tion 4.
- The hull boundary condition must be applied
on the actual wetted surface, instead of on
the surface below the undisturbed waterline.
- All nonlinear terms (squares and cross
products of disturbances) should be in-
cluded.
- The dynamic trim and sinkage of the hull
should be taken into account.
Without restricting oneself from the
outset to a direct extension of current
linearized methods one has a wide variety of
methods to choose from. In the first place,
the nonlinear steady problem could be solved
either by iteration or by a transient ap-
proach. Both methods have already been pro-
posed. The iterative approach may have a
greater efficiency if successful, but meets
problems in the convergence. Secondly, in each
time step or iteration the Laplace equation
for the velocity potential has to be solved
(provided that the potential flow model is
retained). Here again several alternatives are
possible. Panel or boundary integral methods
appear to be the most popular choice.
Whatever the choices made, it is clear
that any advance compared with linearized
methods is only possible if the utmost care is
taken in the numerical solution. The fact that
the nonlinear terms generally have higher wave
number already illustrates this necessity. But
also the behaviour near the hull/free surface
intersection could be far more difficult to
deal with than in a linearized method. All
these aspects deserve separate studies. In any
case most of the nonlinear solutions published
up to now are, in my opinion, not more accu-
rate than linear solutions of state-of-the-art
numerical accuracy.
Even if this can all be solved satis-
factorily, the theory remains limited to
potential flows, without any viscous or wave-
breaking effects. This could prohibit conver-
gence of the solution in the limit of zero
discretization spacing near the waterline, and
additional methods to deal with this region
might become necessary.
Nevertheless I believe that the develop-
ment of a method to solve the problem of
potential flow with nonlinear free surface
boundary conditions is the best next step for
further enhancing the role of Computational
Fluid Dynamics in the optimization of the
wave-making characteristics of ships.
7. CONCLUSIONS
This paper has provided more detailed
information on the adequacy of linearized free
surface conditions for the wave resistance
problem. All of them were implemented in a
Rankine-source method of the type proposed by
Damson. In particular the Neumann-Kelvin
formulation and two free surface conditions
from the slow-ship theory, that of Dawson and
of Eggers, have been compared. These compari-
sons concerned the wave resistance and wave
profile predictions, the magnitude of the
terms neglected in the FSC, and the remaining
errors in the dynamic and kinematic FSC at the
predicted free surface. The main conclusions
are summarized below.
1. For practical discretizations Dawson's FSC
gives results not significantly different
from solutions of the Neumann-Kelvin prob-
lem, for all ships with a block coefficient
not exceeding about 0.60 or 0.70. Also from
the magnitude of the terms neglected in the
linearization no significant advantage for
Dawson's condition is found. This is at
variance with the general preference for
the slow-ship approach.
393
2. For full hull forms the Kelvin FSC predicts REFERENCES
a resistance far exceeding the experimental
value. On the other hand, both slow-ship
FSC's predict, paradoxically, a negative
wave resistance while the predicted wave
pattern is physically plausible. The magni
tude of the neglected terms is much larger
for the Kelvin FSC than for Dawson's FSC
here.
3.
4.
The use of a linearized free surface condi-
tion imposed on the undisturbed free sur-
face changes the energy balance in such a
way that a negative resistance is not ruled
out. Wave energy can locally be supplied
through the free surface which has no coun-
terpart in a wave resistance acting on the
hull.
If the linearized free surface condition is
consistently formulated, the possible nega-
tive contribution from the free surface
energy flux is reduced to a higher order in
the perturbation parameter than the wave
resistance itself. This is true for the
Kelvin condition and for that of Eggers,
but not for Dawson's FSC due to the absence
of transfer terms.
S. The FSC of Eggers yields a resistance con-
sistently lower than Dawson's, for the
Series 60 model. For the tanker model how-
ever, an even more strongly negative resis-
tance and an erroneous flow field were ob-
tained. The cause of this was found to be
the near-vanishing of the coefficient of
fxx in the FSC near the bow; thus an exc-
essive value of ~ is not controlled by
the FSC. As a result the Taylor expansion
underlying the linearization locally does
not converge, and the reduction of the
energy flux to higher order does not pre-
vent a negative resistance here.
6. The transfer term neglected in Dawson's FSC
and included in that of Eggers was found to 9.
be of substantial magnitude even for the
Series 60 hull.
7. The magnitude of the neglected terms in the
FSC and of the errors in the exact FSC at
the predicted free surface has turned out
to be quite significant even if fair re-
sistance predictions are obtained. E.g. for
the Series 60 model at Fn=0.35, the error
in the vertical velocity using Dawson's FSC
amounted to 307.
8. Although more work would be needed to make
the foundations of linearized methods
sounder, part of the problems and uncer-
tainties might be eliminated by solving the
exact nonlinear problem. In this respect
the present study has shown the importance
of imposing the FSC at the actual free
surface and the higher resolution required
for accurately incorporating nonlinear
effects.
394
1. Baba, E. and Hara, M., "Numerical Evalua-
tion of a Wave-Resistance Theory for Slow
Ships " Proceedings of the Second Int.
Conf. on Numerical Ship Hydrodynamics,
Berkeley, 1977, pp. 17-29.
2. Dawson, C.W., "A Practical Computer Method
for Solving Ship-Wave Problems," Proceed-
ings of the Second Int. Conf. on Numerical
Ship Hydrodynamics, Berkeley, 197i, pp.
30-38.
3. Raven, H.C., "Variations on a Theme by
Dawson", Proceedings of the 17th Symposium
-
on Naval Hydrodynamics, The Hague, Nether-
lands, 1988, pp. 151-172.
Proceedings of the Workshop on Ship Wave
Computations, DTNSRDC, Bethesda, Md., USA,
1979.
5. Wehausen, J.V., and Laitone, E.V., "Sur-
face Waves," Encyclopaedia of Physics,
Vol.IX, Springer Verlag, 1960, pp. 446-
778.
6. Eggers, K., "A Method for Assessing
Numerical Solutions to a Neumann-Kelvin
Problem," Proceedings of the Workshop on
Ship Wave-Resistance Computations, Sup
plementary Papers, DTNSRDC, Bethesda,
Md., USA, 1979, pp. 526-527.
7. Maisonneuve, J.J., "Resolution du Probleme
de la Resistance de Vagues des Navires par
une Methode de Singularites de Rankine,"
Thesis, ENSM, Nantes 1989.
8. Eggers, K., "On the Dispersion Relation
and Exponential Variation of Wave Compo-
nents Satisfying the Slow-Ship Differen-
tial Equation on the Undisturbed Free
Surface," Research Report 1979, Study on
Local Nonlinear Effect in Ship Waves, pp.
43-62. See also: Schiffstechnik Bd.
28, 1981, pp. 223-252.
Maruo, H., and Ogiwara, S., "A Method of
Computation for Steady Ship Waves with
Nonlinear Free Surface Conditions," Pro-
ceedings of the 4th Int. Conf. On Numeri-
cal Ship Hydrodynamics, Washington D.C.,
1985, pp. 218-230.
10. Musker, A.J., "A Panel Method for
Predicting Ship Wave Resistance," Pro-
ceedings of the 17th Symposium on Naval
Hydrodynamics, The Hague, Netherlands,
1988, pp. 143-150.
11. Brandsma, F., and Hermans, A.J., "A Quasi-
Linear Free Surface Condition in Slow-Ship
Theory," Schiffstechnik, Bd. 32, 1985, pp.
25-41.
12. Eggers, K., "A Comment on Free Surface
Conditions for Slow Ship Theory and Ray
Tracing," Schiffstechnik, Bd. 32, 1985,
pp. 42-47.
DISCUSSION
Paul D. Sclavanous
Massachusetts Institute of Technology, USA
I would like to congratulate Dr. Raven for yet another thorough study
on the effect of the free-surface linearization upon the evaluation of
the ship wave resistance. Having read the article, I would like to
offer a conjecture on why the wave resistance from pressure might be
negative and invite the author to discuss it. The potential flow near
the ship bow and stern, subject to either the Neun~ann-Kelvin or a
double-body condition, most likely develops a singular behavior
associated with the finite entry angle of the waterline or from the
singularity of gradients of the double body flow at its stagnation
point. This singular behavior may be sufficiently strong that a
localized contribution to the resistance may arise from the waterline
due to a local singularity of the hydrodynamic pressure. This
contribution would be directly analogous to the leading-edge suction
force in linearized hydrofoil theory, which cannot be captured by
direct pressure integration over the mean chord position. Should such
a singularity exist and be of sufficient strength, it will contribute to
an Of 1 ) component to the resistance arising from the waterline, which
cannot be accounted for by integrating the pressure over the mean
position of the ship hull. This effect, if it exists, may shed more
light into the occurrence of negative wave resistance reported by the
author.
~ .
AUTHORS' REPLY
This is a good point; the possible occurrence of singularities at the
stagnation points should indeed be a matter of concern. However, in
my opinion, the comparison with a leading-edge singularity is not
entirely valid. At a zero-thickness leading-edge, a finite force
contribution arises by the pressure going to minus infinity, while in
the present case a positive resistance contribution could only arise
from a positive pressure which, however, is bounded by the
stagnation pressure. The latter seems to be numerically fairly well-
resolved here. Furthermore, it seems reasonable to assume any force
contribution from the waterline to have a vertical extent scaling with
the wave length or stagnation height. It can then be expected to be
similar to the waterline integral (10), which turns out not to eliminate
the negative resistance in all cases. Hence, a localized force
contribution is expected to be at least of O(Fn4), and to be already
approximately included in the present results.
A more probable effect of the singularity seems to me the occurrence
of a localized energy flux through the free-surface. This, too, may
contribute to the wave generation without being found in the pressure
integration. Recently, I have calculated this energy flux for the
tanker model according to Eq. (14). Its distribution has a large spike
quite close to the bow, strongly suggesting the presence of a
singularity. For the FSC of Eggers, the spike occurs at the point
where the coefficient of ¢,,` vanishes; for the other FSCs it is found
slightly further aft. This spike almost entirely determines the total
energy influx, which, if expressed as a contribution to the wave
resistance coefficient, is of the same order of magnitude as the
pressure integral but may be severely grid-dependent. The localized
energy flux through the free-surface due to this singularity thus
largely explains both the large differences between the three FSCs and
the large negative resistance values even if the energy flux is formally
of higher order. These results therefore support the explanation given
in my paper but stress the possible role of singularities at the bow in
this respect.
DISCUSSION
J. Nicholas Newman
Massachusetts Institute of Technology, USA
One thought is prompted by the case shown in Figure 4 where, for
the lowest Froude number, the Kelvin free-surface condition
overestimates the wave resistance whereas the Dawson condition
yields a negative value. Obviously the average of these two would
be an improvement! This suggestion is not entirely facetious. It is
common in perturbation solutions to find higher-order approximations
oscillating about the correct result and diverging to an increasing
extent. An example is the high-aspect-ratio lifting surface (lifting-
line) theory as described by Van Dyke and reproduced in Figure 5.22
of my book. If this analogy has any relevance, it implies that we
should look for a different asymptotic approximation about the zero-
Proude-number limit and construct a composite approximation in the
manner described by Van Dyke. I admire the spirit of this paper and
look forward to further contributions from the author.
AUTHORS' REPLY
Thank you for pointing out this interesting analogy. Although the
Kelvin condition and the slow-ship FSC are based on different
perturbation parameters, one could loosely regard slow-ship theory as
an approximation to higher order in flow nonuniformity (so, in some
slenderness parameter). Some of my results in fact suggest that for
increasing nonuniformity it diverges (or rather, it produces unrealistic
results) more quickly than the Neumann-Kelvin approach. In the case
of lifting line theory, the similar behavior indicates that systematic
expansions to higher order will not bring us any further, and a
different basic approach is needed to get a higher accuracy. It
appears to me that a strict analogy would imply here that we should
construct a different approximation about the limit for zero flow
nonuniformity rather than for zero Froude number. Alternatively, the
present behavior might indeed suggest the need to revise the zero-
Froude-number limit. Some possibilities for this have been proposed
in the past but seem not to have been pursued.
395
