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OCR for page 629
Stability and Accuracy of a NonLinear Mode!
for the Wave Resistance Problem
A. J. Musker
Admiralty Research Establishment
Haslar, England
Abstract
A nonlinear Rankinesource method for
predicting the wave resistance of a surface
ship in calm water has recently been
published by the author. In the present
paper, an investigation into the stability
and accuracy of the method is presented. The
method is shown to be quite insensitive to
changes in the various parameters which are
necessarily associated with the numerical
procedure but which do not feature in the
mathematical formulation of the problem. A
stable parameter regime is identified which
has been found to give excellent agreement
with towing tank data for the Series 60 hull.
1. Introduction
During the last decade there has been a
great deal of activity in the field of
numerical ship hydrodynamics. New methods
for simulating a wide range of phenomena have
been proposed; such methods invariably
involve the generation of a grid covering
either the boundaries of the domain (as in a
panel method) or the complete interior of the
domain (as in an Euler or a NavierStokes
code).
Although it is usual to compare any
numerical predictions with available
experiment data (or, where appropriate,
analytic data) it is somewhat rare to find
examples of the effect of systematic
variations in grid geometry on the accuracy
of the solution. Equally, there may be
factors other than the grid geometry which
affect the outcome of a simulation and these
too are often overlooked. In addition to the
accuracy of a method, its stability also
needs to be assessed. This may be achieved
with the aid of theoretical treatment but
ultimately the degree of stability of a
method can usually only be gauged by its
actual behaviour in trial calculations.
This paper addresses both the accuracy and
stability aspects of a panel method due to
629
Musker [1] designed to calculate the
potential flow past a ship hull in steady
translation in calm water. The method
employs nonlinear forms of the freesurface
boundary conditions and on the basis of some
initial tentative calculations it has been
found to give good agreement with experiment
data for both the Wigley hull and the Series
60 hull (see Musker [2]).
Although the above problem can be
formulated in terms of an integral of a
source density distributed on the hull and
the free surface, in practice the domain
boundaries are made discrete by representing
them using a finite number of panels. In
addition, the free surface is not known a
priori because it is part of the solution.
Consequently panels must be associated either
with the calm freesurface (with the actual
freesurface represented by means of a
Maclaurin expansion of the prevailing
boundary conditions) or with the approximate
freesurface relating to the most recent
iterate in a convergent sequence of surfaces.
The present method employs the first of these
two options and clearly falls into the
category of socalled Rankine source methods.
Rankine source methods have been the
subject of intense development in recent
years in connection with a particular
formulation of the NeumannKelvin problem
(see, for example, Raven [3]). The
NeumannKelvin (NK) problem models the
wavemaking of a hull by a solution of
Laplace's equation subject to a linearised
form of the freesurface condition and an
exact form of the kinematic condition on the
hull surface. Despite the much disputed
problems of uniqueness of solution for the NK
problem a large number of computer codes have
been prepared based on this theory. The
beauty of the Rankine source method lies in
its inherent ability to provide engineering
solutions to the NK problem (usually in a
modified form using doublebody
linearization) whilst at the same time
providing the potential for addressing the
OCR for page 629
nonlinear problem. On the other hand, the
more conventional approach to solving the NK
problem, using a Havelock or Kelvin
wavemaking source distribution, has the
important advantage of requiring fewer panels
which automatically satisfy the important
radiation condition that waves should travel
downstream.
Staying with the linear problem for the
moment, it is interesting to note the great
diversity of solutions quoted in the
literature. Figure l shows a collection of
computer predictions of wave resistance for
the Series 60 hull (CB = 0.6, model fixed)
drawn from the two Workshops on Ship
WaveResistance Computations held at the
David Taylor Research Center (see references
[4] and [5]). All these predictions use
linearised theory for treating the free
surface but differ in their method of
solution (NK using Rankine source technique,
NK using Havelock source technique, thin ship
theory  in which the hull boundary condition
is applied at the centreplane etc). It is
interesting to note that a mean curve drawn
through the data would be significantly above
the tank results. More recently, Chen and
Noblesse [6] have remarked on the
considerable scatter evident in published
predictions of wave resistance based on
similar theory. As noted by Bat [4] if no
algebraic or computer truncation errors are
incurred then results of numerical
computations based on the same mathematical
formulation should be the same. Clearly this
is not the case.
If the possibility of algebraic or coding
error is dismissed (itself a dangerous
assumption) then the explanation for the
disparity in the predictions most likely lies
in the mechanism whereby truncation errors
appear. The obvious source of such errors is
the manner in which the hull is divided into
facets or panels. The distribution of such
panels around the hull, particularly near the
bow, stern and waterline is bound to have an
effect on the solution. In the case of a
Havelock source the numerical evaluation of
the Green function poses its own problems.
If a Rankine source rather than a Havelock
source is used then perhaps an even greater
problem presents itself: how large an expanse
of the free surface in the vicinity of the
hull needs to be modelled using additional
panels and how should the resolution of this
region be chosen? The manner in which the
source density is distributed algebraically
on each panel will also affect the solution.
Recent work by Ni [7] has shown that the use
of higher order curved panels with linearly
varying source density is computationally
more efficient compared with constant density
panels. Finally, Rankine source methods
invariably employ Dawson's approach [8] to
satisfy the radiation condition that waves
should not travel upstream of the hull. This
approach involves the use of an upwind finite
difference advection scheme for the wave
disturbance and can take many forms.
Clearly the operator of any Rankine source
code has a great deal of freedom in the
manner in which a particular problem is
posed. The purpose of the present paper is
to investigate the effect of making certain
decisions about the geometry of the mesh and
the type of advection scheme used for the
particular case of the method described in
Reference [l]. This method is first outlined
for completeness and the 'degrees of freedom'
at the operator's disposal are identified and
minimised to manageable proportions. These
are then systematically adjusted and the
effect on the accuracy of prediction and the
stability of solution is recorded. All the
calculations were performed for the Series 60
hull (CB = 0.6).
2. }~E~al~L~LE
A Cartesian coordinate system is used in
which the xy plane is in the calm free
surface with the x axis positive upstream and
the z axis positive downwards. The fluid is
assumed to be inviscid, incompressible,
irrotational and infinitely deep.
A scalar velocity potential is defined
such that grad. is the local fluid velocity
vector. This potential satisfies Laplace's
equation everywhere within the fluid subject
to the following boundary conditions.
(grad ¢) · n' = 0
on the hull surface,
[aZ]O ([aZ2]0 dl [dl]o
+ ~ 4: [[82¢ 1 dx + [82¢ 1 dv1 (2)
dl ltaxBz]O dl [Byaz]O dlJ
and
630
r ~ r ~
laxio laxazio
' [ ] [ ''] + [~] [a2~] 1
ay O ayaz O az O aZ2 ol
= 2 [U~  [3xjO [8YjO i8Z]o] (3)
OCR for page 629
on the calm freesurface. In the above
equations, n' (nx, ny, nz) represents the
outward normal vector on the hull surface, 1
represents a distance measured along a
doublebody streamline measured positive in
the locally upstream direction, Up is the
ship speed, g is the acceleration due to
gravity and ~ represents the wave elevation.
The last two boundary conditions are
nonlinear and result from Maclaurin
expansions of the exact boundary conditions
prevailing on the free surface [1]. For the
purpose of comparison, the above set of
boundary conditions reduces to the
NeumannKelvin problem if the spatial
velocity gradients are set to zero. The ship
speed is defined such that
grad ~ = ( U, 0,0) (4)
at ~ except where waves are present.
Dawson's approach [8] is used to ensure that
waves are adverted downstream.
The velocity potential is prescribed by
means of a source density distribution
associated with the hull (and its optical
image in the z = 0 plane) and the free
surface. Constant, planar, source density
panels are used throughout the analysis and
in the case of the free surface panels these
are raised very slightly above the calm
waterplane to improve the modelling of the
spatial velocity gradients [9], [10]. The
latter are calculated analytically and great
care has been taken to avoid singular regions
(in the numerical sense) by employing
suitable asymptotic expressions within the
program. To avoid leakage, an additional
system of vertical panels (referred to
loosely as 'deck panels') Joins the
waterline on the hull to the neighbouring
system of freesurface panels; a Neumann
condition is imposed at the centroids of
these panels.
Successive application of the boundary
conditions at a large number of control
points on the hull and calm waterplane leads
to a system of nonlinear simultaneous
algebraic equations for the unknown source
densities and wave elevations. The
associated Jacobian matrix is calculated
analytically and the equations are solved
iteratively using Newton's method without
relaxation; each associated linear system is
solved directly using a Crout factrisation.
The criterion adopted for complete
convergence is that the root mean square of
the sum of the residuals (comprising the
most recent corrections to the source
densities and wave elevations) is less than
0.002. Although strictly dimensionally
inconsistent, in practice this criterion
relates to a correction in wave elevation
of 1 mm for a hull of length 100 m.
Further details of the nonlinear algorithm
used can be found in Reference [1~.
Having solved the equations, the pressure
at a particular control point on the hull is
found from:
+ 1 (U2  Grad . l 2 )
The wave resistance is found by
integrating the pressure around the hull
taking due account of the hydrostatic
correction associated with the predicted
waterline (note that for consistency with
the applied boundary conditions this
correction is not performed for the linear
case):
ITS
where S is the wetted area. As is often
found in panel methods, a residual resistance
is observed at zero speed because of the
manner in which the hull is made discrete.
Accordingly, the result for the double body
calculation (where no waves are present) is
first subtracted to yield the final wave
resistance.
The free surface grid uses doublebody
streamlines generated by RungeKutta
integration to within a lateral error of 105
of the ship length. These streamlines are
separated laterally by equal intervals at the
upstream edge of the calm freesurface and
for each streamline the control points are
positioned at equal intervals in arclength;
this process is facilitated using cubic
splines. The free surface panels are
constructed automatically such that their
streamwise edge projections onto the z = 0
plane follow the streamlines and have edges
along the waterline which are equal in
length. An additional constraint is imposed
in that the panel aspect ratio away from the
hull is typically unity.
A fourpoint upwind finite difference
scheme is used to advect disturbances in the
downstream direction. The boundary condition
imposed on the first three control points at
the upstream end of a given freesurface
streamline is different from equations 2
and 3. Instead, the vertical fluid velocity
in this region is forced to be zero. In
addition, the last three control points at
the downstream end are treated differently.
Here, the coefficients in the advection
scheme are reduced gradually to 25 per cent
of their normal value to dampen the waves at
the edge of the domain.
631
OCR for page 629
It should be noted that the method does not
employ any clustering of panels in the
vicinity of the bow and stern  in the
author's opinion such clustering provides
too many extra degrees of freedom to allow a
sensible study to be undertaken. In
addition, the highly variable steplength for
the finite difference advection operator in
such a scheme will almost certainly affect
the stability of the solution algorithm.
The hull panels are arranged to mesh
exactly with the free surface nodes along the
waterline. Below the waterline, panels are
constructed automatically in one of two ways.
Firstly, they can be constructed such that an
equal number of quadrilateral panels is used
at each longitudinal station ('singlepatch
method'  see [1]). Secondly, a mixture of
quadrilateral and triangular panels is used
to allow the number of panels at a particular
section to be reduced in the ratio of local
section depth to keel depth. This is done in
such a way as to maintain a reasonably
constant panel area and aspect ratio (which
are thought intuitively to be important from
the point of view of numerical stability)
whilst at the same time lifting the
restriction imposed by the singlepatch
method that the hull panels below the
waterline must be generated on the basis of
strictly vertical stations. This is known as
the 'multipatch' method and was developed to
extend the capability of the panel code to
hulls with more complicated geometry 
particularly in the bow and stern regions.
The BLINKS computeraided design system [11]
is used for interpolation purposes.
3. Stability and Accuracy Study
The following list defines the most
important factors which need to be addressed
before a calculation can be performed for a
particular hull geometry:
a. Length of domain.
b. Halfwidth of domain (ie excluding
image in xz plane).
c. Distance between the bow and the
leading edge of the domain.
d. Freesurface grid resolution.
e. Hull grid resolution.
f. Freesurface panel elevation (above
z = 0 plane).
g. Type of advection scheme.
Experience with the code suggests that
there are significant differences in
predicted resistance between large and small
domains. Not surprisingly, however, at any
particular Froude number and for a fixed grid
resolution, the resistance approaches an
asymptote as the size of the domain
increases. Accordingly, in order to decrease
the number of degrees of freedom for the
present study, a large domain was used
throughout the investigation. This was
to 3 ship lengths in the longitudinal
direction and extended to 1.5 ship lengths
transversely measured from the centreplane.
The distance between the bow and the leading
edge of the domain was set to one half of the
shiplength; this was based on previous
experience with the method.
The hull and freesurface grids are
related in the sense that the waterline
nodes are forced to coincide prior to the
construction of the vertical deck panels. In
addition, an optional facility exists to
position extra hull panels between waterline
stations defined by the freesurface nodes so
that the hull panel density (defined as the
number of streamwise hull panels per
freesurface panel) can assume any integer
value. If this option is invoked then the
grid generator positions these extra panels
such that their streamwise edges form equal
divisions along the waterline. In this way
the grid generating code can automatically
calculate the mesh on the hull once the hull
panel density and freesurface resolution
have been set since only an integer number of
panels is allowed along the waterline (in
practice the latter condition is met within
the computer program by a very minor
adjustment to the absolute length of the
domain).
It can be seen, therefore, that there are
now only four degrees of freedom which need
to be considered: the grid resolution, the
hull panel density, the freesurface panel
elevation and the type of advection scheme.
It should be noted that no other operator
intervention is called for once these
factors have been set.
4. Numerical Experiment
The Series 60 (CB = 0.6) hull was used for
all the computer runs described in this
investigation. The hull was fixed and no
allowance was made for sinkage or trim (in
the author's opinion this in no way detracts
from the usefulness of the study). Unless
otherwise stated, the following default
conditions were used for all the runs:
hull panel density = 1
and
632
panel elevation = 15 per cent of
average panel
diagonal
OCR for page 629
The latter figure was based on previous
experience with the code and careful analysis
of the ability of constant source density
panels to model spatial gradients of fluid
velocity [10].
Four numerical experiments were performed
with the aim of determining the effect of
varying any one of the abovementioned
degrees of freedom. The runs were performed
at the following Froude numbers:
0.239, 0.271, 0.287, 0.303, 0.319, 0.335
and 0.351
based on shiplength at the waterline.
arithmetic was performed using 64 bit
precision. The experiments are labelled A,
B. C and D and are described below.
Experiment A (effect of freesurface grid
resolution)
All
This was performed in two stages. Firstly
(experiment Al), three free surface grids
were generated each of which was fixed, in
relation to the shiplength, for all Froude
numbers (henceforth such grids will be termed
'mode 1' grids). They are referred to as
coarse (1166 freesurface panels), medium
(1904 freesurface panels) and fine (2673
freesurface panels). The number of
freesurface panels distributed along the
waterline of the hull was 17, 23 and 27
respectively. Singlepatch hull meshes were
used for this experiment.
Secondly (experiment A2), grids were
generated such that their resolution related
not to the shiplength but rather to the
wavelength of the transverse waves expected
to be shed by the hull. In this instance the
grid resolution was described by a parameter,
n, where n is the number of panels per
transverse wavelength. The total number of
panels positioned along the length of the
freesurface domain is thus:
n x domainlength
2~Fr2 x shiplength
This meant that for each Froude number a
new grid (henceforth termed a 'mode 2' grid)
had to be calculated to conform to the chosen
value of n. This exercise was carried out
for values of n ranging from 4 to 12 in
intervals of 2 (it was not anticipated that a
value of n as low as 4 would perform very
well but for the sake of completeness it was
included in the experiment). Multipatch
hull meshes were used for this experiment.
Experiment ~ (effect of hull panel density)
This experiment compared the performance
of the method using two different values of
the hull panel density; the values used were
1 and 3. Mode 2, multipatch grids were
employed throughout, with n set to 10.
Experiment C (effect of panel elevation)
This experiment was designed to
investigate the effect of elevating the
freesurface panels above the calm
waterplane. As for Experiment B. mode 2,
multipatch grids were employed throughout,
with n set to 10. Three different elevations
were tested: zero, 15 and 30 per cent of the
average panel diagonal.
Experiment D (effect of choice of advection
scheme)
The above three experiments were performed
with two different upwind advection schemes
for the streamwise waveslope:
Taylor Scheme: Slopei =  ~` x
[1.667
OCR for page 629
Experiment A
A comparison of the coarse and fine grids
is shown in Figure 2 and a summary of the
results is presented in Figure 3. The method
converged for all speeds using the coarse
grid. For the medium grid the method failed
at a Froude number of 0.335 whilst for the
fine grid the method failed for the three
highest Proude numbers within the Range used.
It should be recalled here that Experiment
Al was designed around the original method
described in Reference [1] employing mode 1
freesurface grids and singlepatch hull
grids. It is perhaps surprising that the
coarse grid should have led to such good
agreement with experiment data particularly
at the lower Froude numbers where shorter
wavelengths are prevalent. It is similarly
curious to observe the closer agreement
between the coarse and fine meshes at the
lower Froude numbers compared with the medium
mesh results.
The results for experiment A2 (mode 2,
multipatch hulls) are summarised in
Figure 4. It can be seen that all the runs
converged successfully for values of n from 4
to 10. Par n = 12, however, the method
diverged for Froude numbers of 0.319 and
0.335 (note that the lowest Froude number
case was not processed because it exceeded
the capacity of the computer  this was the
only run in the whole investigation which
could not be processed because of size
limitations). Not surprisingly, values of
n = 4 or 6 appear to be quite inadequate to
resolve the flowfield and hence the wave
resistance. The data for n = 8 define the
shape of the curve well but slightly
underestimate the resistance for the higher
Froude numbers. Excellent agreement is
obtained for n = 10 and no improvement in
accuracy is achieved for the stable runs when
n = 12.
Experiment B
In this experiment mode 2 freesurface
grids and multipatch hull grids were used
with a hull panel density of 3 (see
Section 3). Figure 5 shows a perspective
view of one of the meshes from a point below
the waterline looking towards the hull. The
effect of the increased resolution of the
hull can be seen in Figure 6 to be quite
weak. At the higher Froude numbers the
increased hull panel density consistently
lowers the resistance by a very small amount
whilst little or no effect is observed at the
lower Froude numbers. All the runs performed
were completely stable.
Experiment C
Figure 7 illustrates the effect of
elevating the freesurface panels above the
calm waterplane. The absence of data points
for the case of zero panel elevation
indicates the instability inherent in
introducing spatial derivatives of fluid
velocity calculated using panels of constant
source density if the control point lies in
the plane of the panel [10]. With the
possible exception of the two runs at a
Froude number of 0.271 it can be seen that
there is little difference between the
results for 15 per cent and 30 per cent panel
elevations. All the runs performed with
elevated freesurface panels were stable.
Experiment D
Attention here will be focussed on three
results of significance. In the first two
cases the data refer to mode 2, multipatch
grids with n = 10. Figure 8 shows the very
small effect the spline advection scheme has
on the accuracy of the predicted wave
resistance compared with the Taylor scheme;
the results are very nearly identical.
Figure 9 shows the data for Experiment ~ with
zero panel elevation replotted using the
results for the spline advection scheme. The
results are compared with the single Taylor
result from Figure 7. Two important
conclusions emerge from these data. Firstly,
the effect on the accuracy of using a spline
scheme is minimal. Secondly, the results for
zero panel elevation are not only inherently
unstable using the Taylor scheme, as observed
above, but also inherently inaccurate when
they can be made stable by employing a spline
scheme. This again reinforces the findings
of Reference t10].
Further evidence of the more stable nature
of the spline scheme is offered in Figure 10,
which shows the spline results for Experiment
A2 with n = 12, compared with the Taylor
results with n = 10. The results for Froude
numbers 0.319 and 0.335 when n = 12 are now
stable  thus allowing a comparison in
accuracy between n = 10 and n = 12 to be made
throughout the whole speed range. It can be
seen that there is little advantage to be
gained by increasing n above 10.
The stability of panel methods for the
linear problem has been analysed recently
using Fourier techniques by Sclavounos and
Nakos [133. They found that the numerical
damping associated with upwind finite
difference schemes decreases as the grid
becomes finer or as the number of upstream
nodes included in the scheme at a particular
control point increases. Although the
present technique is nonlinear this
nevertheless probably explains why the method
has a tendency to diverge for the finest grid
tested (n = 12) using the Taylor scheme.
Their work may also explain the stability
problems experienced by the only other
nonlinear methods which, to the best of the
author's knowledge, have appeared in the
634
OCR for page 629
literature since 1985 (Maroo and OgIwara
[14], Xia [15], Ni [7] and Rong et al [16]).
Marno and Ogiwara used a twopoint formula
but still experienced quite severe
convergence problems, whilst Xia and Ni both
used threepoint formulas. Xia found that
this method diverged for all cases tested if
a 4point scheme was adopted but much better
convergence characteristics were observed
when this was abandoned in favour of a (less
accurate) 3point scheme. Ni employed a
sophisticated higher order panel scheme very
successfully using a threepoint scheme but
again found stability a problem using an
extra upstream node. In the present method,
using mode 2 freesurface grids and
multipatch hull grids, stability only became
a problem at the highest resolution (for the
highest Proude number only) and this was
completely alleviated by utilising a
fourpoint spline rather than a fourpoint
Taylor scheme. In the method of Rang et al a
twopoint scheme was used near the stern
region to provide sufficient damping to
ensure convergence.
Mention should be made of the practice of
elevating the freesurface panels above the
calm waterplane. It might be argued that
the ensuing system of equations will be less
well conditioned in the sense that a control
point on the z = 0 plane would not be so
strongly associated with its corresponding
panel if the panel were raised above it. On
the other hand, the author has shown that the
modelling of the spatial gradients of fluid
velocity which are required in the nonlinear
formulation is inadequate if the control
points are contained in the plane of the
panels. Indeed, it is so inadequate as to
cause the method to diverge.
It should be noted that the amount by which
the panels need to be raised is very small 
typically between 0.8 per cent and 1.5 per
cent of the ship length, depending on the
Froude number, when n = 10. In addition,
Figure 7 demonstrates that the predicted
resistance appears not to be particularly
sensitive to the panel elevation. Xia [15]
also tried raising his freesurface panels
and found improved convergence at some
expense in accuracy. Panel elevations used
by Xia, however, were much larger  typically
between 2.5 per cent and 10 per cent of the
shiplength  and in the author's opinion
these values are too large. Ni [7] has
circumvented the problem altogether by
adopting higher order panels with linearly
varying source density which follow the
calculated free surface as it progresses
towards its final converged solution. The
present work suggests that such
sophistication may not be necessary.
Finally, a synopsis of the published data
for the above nonlinear methods is presented
in Figure 11. The Series 60 data for the
method of Marno and Ogiwara is taken from
Reference [17]. The present results using
the default conditions of 15 per cent panel
elevation (based on average panel diagonal,
not shiplength), fourpoint Taylor advection
scheme, mode 2, multipatch grids with n = 10
are included for comparison.
6. Conclusions
.. . _ _ . ..
A panel method based on a boundary
integral approach to solving the nonlinear
freesurface wavemaking problem has been
thoroughly tested against tank data for the
Series 60 hull. The number of degrees of
freedom at the disposal of the operator of
the computer code has been minimised with the
aim of investigating systematic changes to
the input parameters which may affect the
numerical solution but which do not feature
in the mathematical formulation.
Two types of grids were tested. The first
type, labelled mode 1, singlepatch, uses
panel geometries on the calm free surface
which are independent of Froude number but
which are kept constant with respect to the
shiplength. The hull panels are organised
so that the same number of panels is used at
each station. The second type, labelled mode
2, multipatch, uses panel geometries which
depend on the Froude number in the sense that
a fixed number, n, of freesurface panels per
transverse wave is selected. In addition,
the hull panels are organised into
quadrilateral or triangular panels
automatically such that the size and shape of
panels do not change significantly from
station to station.
The following conclusions can be drawn:
a. As expected, the method was
unstable for zero freesurface panel
elevation. The method was invariably
stable, however, for nondimensional
panel elevations (based on average panel
diagonal) of 15 per cent and 30 per
cent. Furthermore, the solution was
observed to be reasonably insensitive to
the choice of nonzero panel elevation.
b. Accurate results were obtained
using a coarse, mode 1, singlepatch
grid for all Froude numbers. This
approach became unstable as the grid
became finer.
c. The mode 2, multipatch grids
exhibited greater stability and were
used for the remainder of the
investigation. The results approached
the tank data asymptotically as n
increased from 4 to 12, with very little
difference in the accuracy between 10
and 12. The n = 12 results diverged at
635
OCR for page 629
two of the higher Froude numbers using 5.
the fourpoint Taylor advection scheme.
d. No case of instability or
significant change in accuracy of
prediction was recorded for any value of
n using the fourpoint spline scheme.
e. A threefold increase in hull panel
density in relation to the freesurface
grid resulted in no significant effect
on the stability or accuracy of the
method.
f. Comparison with other nonlinear
methods described in the recent
literature shows the method to be very
stable and accurate using mode 2,
multipatch grids with n set to 10.
Having identified a stable and accurate
parameter regime, it is hoped to perform a
similar investigation in the near future to
determine the effect of decreasing the extent
of the free surface surrounding the hull.
Tests with different hull geometries are
also planned.
A^Ln~wl ",la.=m~r~t"
This project would never have been
completed without the skill and dedicated
support of Mrs P R Loader of the Admiralty
Research Establishment, Haslar. The author
would like to thank her for her sustained
effort and enthusiasm. Or G Michel of
Aerobel Defence Technology (formerly
Principia Mechanica) provided the interface
software for BLINKS and Mrs C Patis attended
to many of the computer runs. Their help is
gratefully acknowledged.
8. References
11.
12.
1. Musker, A J. "A Panel Method for
Predicting Ship Wave Resistance", 13
Proceedings of the 17th Symposium on
Naval Hydrodynamics, The Hague,
August 1988.
2. Musker, A J. "A Solution of the
NonLinear Wave Resistance Problem",
Proceedings of the International
Symposium on Ship Resistance and
Powering Performance, Shanghai,
April 1989.
3. Raven, H C, "Variations on a Theme by
Dawson", Proceedings of the 17th
Symposium on Naval Hydrodynamics, The
Hague, August 1989.
4. Bai, K J and McCarthy, J H. "Proceedings
of the Workshop on Ship WaveResistance
Computations", DTNSRDC, November 1979.
636
Noblesse, F and McCarthy J H.
"Proceedings of the Second DTNSRDC
Workshop on Ship WaveResistance
Computations", DTNSRDC, November 1983.
6. Chen, C Y and Noblesse, F. "Comparison
between Theoretical Predictions of Wave
Resistance and Experimental Data for the
Wigley Hull", Journal of Ship Research,
Vol 27, No 4, December 1983.
7. Ni, S Y. "Higher Order Panel Methods for
Potential Flows with Linear or
NonLinear Free Surface Boundary
Conditions", Chalmers University of
Technology, 1987.
8. Dawson, C W. "A Practical Computer
Method for Solving ShipWave Problems",
Proceedings of the Second International
Conference on Numerical Ship
Hydrodynamics, Berkeley, 1977.
9. Musker, A J. "A Note on FreeSurface
Flow Prediction", ARE Technical
Memorandum TM(UHR)86306, March 1986.
Musker, A J and Loader, P R. "A Modified
Boundary Element Method for Predicting
Ship Wave Resistance", Proceedings of
the 7th International Conference on
Finite Element Methods in Fluid Flow
Problems, The University of Alabama,
April 1989.
Catley, D, Okan, M B and Whittle C,
"Unique Mathematical Definition of a
Hull Surface, its Manipulation and
Interrogation", WENT, Paris, July 1984.
Kim, Y H and Jenkins, D, "Trim and
Sinkage Effects on Wave Resistance with
Series 60, CB = 0.6", DTNSRDC
Report SPD101301, September 1981.
Sclavounos, P D and Nakos, D E,
"Stability Analysis of Panel Methods for
FreeSurface Flows with Forward Speed",
Proceedings of the 17th Symposium on
Naval Hydrodynamics, The Hague,
August 1989.
14. Marno, H and Ogiwara, S. "A Method of
Computation for Steady ShipWaves with
NonLinear Free Surface Conditions",
Proceedings of the 4th International
Conference on Numerical Ship
Hydrodynamics, Washington DC, 1985.
15.
Xia, F. "Numerical Calculations of Ship
Flows with Special Emphasis on the Free
Surface Potential Flow, Chalmers
University of Technology, 1986.
OCR for page 629
16. Rong, H. Liang, X and Wang, H. "A IOOO
1000 x Cw
4
3
4
1,............
1.8 2.2
Orse
Med~iu'~'
Fire
Exp~~e'~t
Mode 1, Si'~glePatch
._ Panel Elevatior~
Taylor ~dvection Scene
X
X
X/
X W~
_ ~~
i B He 2 al 6 ~ ' ~ · · l ·
Fr
Fig 3 EFFect of Grid Resolution
1000X CW ire=
r~=8
~n=lO
On=12
_ Experdr~,ent
Mode 2, MultiPatch
15% Panel Elevation
Taylor Advection scheme
X10 1
t~ ¢
V + ~
Hi/ ~
Off
, . . . . .
/
V
Fr
F1g ~ Effect of Grid Resolution
638
~ ,
X10 1
8
OCR for page 629
Fig 5 Perspective Ark  w from Men  sty
Hall (Panel D  nudity = 3)
t000 ICE ~ Hull Panel Density = 1
~ Hull Panel Density = 3
4  Experiment
Mode 2, MultiPatch
1~% Panel Elevation
Taylor Advection Scheme
2
.
1.8
/!
xt
Fig 6 Effect of Hull Panel Density
639
3.8
OCR for page 629
\090 of ~
X

Panel Elevat8on
Panel Elevation
Panel Elevation
Experiment
= 0
= 15=
= 30%
Mode 2, MultlPatch
Taylor Advection Scheme
1 ~ X
~ 81.1.18',8~'
]8~8 2#28
to
4

3
2
1
X
,. .~.,.~ ,8~ .
2.9 5.0 3.4 3.8
X10t
F
r
F8g 7 Effect of FreeSurface
Panel Elevat8on
1000~8C ~ Taylor
W X spine
Experiment
Mode 2, MultiPatch
15% Panel Elevat80n
eX/
~_r
/
, ~ , ~ ~
3.0
8r
F"8g 8 Effect of Advect8on Scheme
640
X10 18
OCR for page 629
MOW ~ Spilne Advection Scheme
4.00 0 Taylor Adveictlon Scheme
3.SO
3r 00
2eSO
2.00
\.50
\.00
O.S~
D.00.
\,80
Mode 2, MultiPatch
Zero Panel Elevation
/d
/ ~
/
/
2
I ~ ~ ~ r
2~;0 3~{JO
F
r
Fig 9 Saline Calculation for
Zero Panel Elevation
t000 ~ C ~ Splir~le Schel~le In
~ ~ Taylo'~ Scllerrie In =
4
3
2 ~
to
3.40 3.8U
X10 1
=
Mode 2, MultiPaUch
15~o Panel Elev~t~ion
NO insufficient computer memory to
lowest Denude number when n = 1
2r
my/
3.0 3,4
F."
fig 10 EFFect of Splilne Michelle
For In = 12
641
X10 1
3.e
OCR for page 629
1000 x Cw
4
3
2
1
Ogiwara 1987 t17]
Via 1986 C15]
Hi 1987 C7]
~ hong 1987 C16]
· Present Method
Experiment
Mode 2, MultiPatch
15= Panel Elevation
Taylor Advection Scheme
o
~ a/
,~
Q /`
a _
1.8 2.2 2.6 3.0
F
r
F1g 11 Comparison with Other
NonLinQar methods
Copyright (C) Controller HMSO London 1989
3.4 3.8
X101
642