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OCR for page 657
A Hybrid Mode} for Calculating Wave-Making Resistance
V. Aanesland
Norwegian Marine Technology Research Institute
Trondheim, Norway
Abstract
A hybrid method for calculating the wave-making
resistance of ships has been developed. The method
is based on three-dimensional potential theory, and the
flow is assumed to be steady. The fluid is divided into an
inner and outer domain by two vertical control surfaces.
These surfaces are parallel to the free stream, extending
to infinity and one at each side of the body. The inter-
nal flow is matched to the external flow on the control
surfaces in order to satisfy the radiation condition.
A three-dimensional source-sink method is adopted
in the numerical treatment of the problem, using a source
function for an infinite fluid. A distribution of source
density on the wetted part of the body, on a local part
of the free surface and on the control surfaces has to
satisfy the boundary conditions in the inner domain.
The wave resistance can be calculated by two differ-
ent methods. One is to calculate the pressure distribu-
tion on the wetted hull surface and to integrate for the
force in longitudinal direction. The other is to use the
inner solution combined with a control surface integra-
tion. The fluid velocity and wave elevation along the
control surface are needed.
Introduction
For a ship in service it is important to have as low to-
tal resistance as possible in realistic sea conditions. Both
model tests and calculations have been used extensively
in order to be able to estimate the resistance at the de-
sign stage. Methods of varying sophistication have been
developed, and many restrictions and assumptions are
made in each case.
A large number of contributions on the wave-making
problem have been published. Concentrating on theo-
retical and numerical solution methods which are inde-
pendent of model tests, an important contribution was
given by J. H. Michell, t1], who introduced thin-ship
657
theory where the resistance is given by the geometry of
the hull. T. H. Havelock, [2], used a Green's function
method instead of the Fourier-integral method used by
Michell and confirmed the results of Michell. The reason
for referring to these two authors is not only because they
where among the first to attempt to solve the problem
theoretically, but also because their results will be used
in the present solution method. For a comprehensive
review on the wave resistance problem, see for exam-
ple Wehausen, [3~. Two other contributions will be of
importance when relating the present work to already ex-
isting theories. G. E. Gadd, t4i, and C. W. Dawson, [5i,
introduced the idea of distributing discrete source panels
on both the ship hull and the mean free surface. The
wave-making part of the velocity potential is calculated
as a perturbation to the double-body flow. Gadd used a
nonlinear free-surface boundary condition, while Dawson
linearized with respect to the double-body flow. In the
former case an iterative procedure was adopted because
the position of the free surface is unknown.
It can be argued that the methods of Gadd and Daw-
son need a considerable amount of computer power and
time. If preliminary results are needed quickly, a number
of thin ship and slender ship methods are available. On
the other hand, the restrictions set by these methods can
be troublesome. For the present author, the flexibility of
the method is of greater importance than a fast run-
ning program on a medium-size computer. The capacity
of new computers is increasing rapidly, and the corre-
sponding cost is decreasing. Indeed, the use of super-
computers minimize this problem. The present program
has been run on both VAX785 and CRAY X-MP. On
the latter computer a speed up factor of about 100 is
typical. In other words a computation of about 1 hour
on the VAX machine is finished in half a minute on the
CRAY.
The present method incorporates some ideas from
Gadd and Dawson, the thin ship theory and some new
ideas. Results are presented for a single point source,
the Wigley parabolic model and the Series 60 block 0.60
ship.
OCR for page 658
The boundary value problem (vi) radiation conditions.
The steady-state wave-making problem is formulated
; n a Cartesia n coordi nate-system x, y, z movi ng with the
ship velocity. The x - y plane describes the undisturbed
free surface with the x-coordinate positive towards the
stern, see figure 1. Another coordinate-system x~,y~,z~
is fixed in the ship and coincides with x,y,z when the
ship is in its equilibrium position and with no forward
velocity. The fluid is assumed ideal, (i.e. inviscid, in-
compressible and homogeneous), and its motion is irro-
tational. Surface tension is neglected.
Figu re 1: Coordi nate systems
The problem is formulated as a potential flow prob-
lem where the total fluid motion is described by the ve-
locity potential ~(x, y, z). The following conditions have
to be satisfied:
(i) La place's equation in the fluid
V2~=0. (1)
(ii) the dynamic boundary condition on the free surface
~ (~2 + 4?2 + ~2 _ u2) = 0 on y = ((x, y). (2)
where subscripts denotes partial differentiation.
(iii) the kinematic boundary condition on the free surface
~,.~. + I,,,—~ = 0 on y = ((x,y). (3)
(iv) the kinematic boundary condition on the hull
~.,.h.~. ~ ~V + 4) h,, = 0 on SB (4)
where y ~ h(:c~,z~) = 0 defines the hull surface
(v) the kinematic boundary condition on the sea bottom
A,, = 0 on z =—d, (5)
which in the infinite depth case is replaced by
lily V. = U. (6)
658
The hull surface is defined in the body-fixed coordi-
nate system.
y1 = ~h(~1,zl) (7)
The transformation from one coordinate system is
easily performed by using the relations
Hi = ACCOST—(Z—s~sinct
yl = y
(8)
(9)
z] = Using + (z—accost (10)
(11)
where ~ is the sin kage and car is the trim angle.
The hydrodynamic forces in x and z direction and
the trim moment are calculated by integration of the
pressure over the wetted part of the hull.
F'. = - ~ p(:c, y, z~n.,.dS (12)
F =—A p(~:c,y,z)n dS (13)
. so
M`' = / pax, y, z)tn'.(z—z`,) - n (x—xc-')]dS (14)
The Bernoulli equation is used to evaluate the pres-
sure,
p = - Pt(V~2 - U24 - pgz (15)
and the normal unit vector n is positive into the fluid.
Solution method
The fluid is divided into an inner and outer domain
as shown in figure 2. The inner domain is bounded by
the body surface, SB, the free surface, SF, two vertical
control surfaces, So, and a surface at infinity, S.,. The
outer domain consists of the rest of the fluid domain,
which is two segments of a sphere, marked by the dotted
lines in the figure. The outer domain is mainly used
in order to find a boundary condition on the matching
surfaces, while the main task is to find a solution in
the inner domain. The use of Green's second identity
in the outer domain gives the boundary condition. The
derivation is described in appendix A.
The reason for introducing the vertical control sur-
faces is partly to restrict the computational domain and
partly to obtain well defined radiation conditions. The
numerical scheme will show that a disadvantage is that
panels have to be distributed on a restricted part of the
OCR for page 659
control surfaces in addition to the hull surface and the
free surface. On the other hand many elements can be
excluded from the free surface compared to the method
of Dawson and Gadd. It is assumed that the wave effect
will decrease rapidly with depth and a large part of the
control surface can be truncated.
In addition to the two above-mentioned reasons for
using two domains, the scheme also suggests to linearize
the free-surface condition differently in the outer and in-
ner domain. A low-speed linearization is adopted in the
inner domain (similar to the one used by Dawson) and a
free-stream linearization in the outer domain (similar to
Kelvin's thin ship formulation). Assuming that the free
stream linearization is satisfactory in the outer domain,
other linear or nonlinear boundary conditions on the free
surface in the inner domain may be used. Another pos-
sibility is to use a totally different numerical solution
method in the inner domain. A finite difference scheme
may be of interest when a local flow phenomenon is
studied, and the fluid can be described by more complex
eq cations.
In the present case the appropriate boundary condi-
tion on the free surface is found by linearizing the equa-
tions (2) to (3~. On the free surface in the inner domain,
the double-body flow is used as the main flow upon which
the wavy perturbation flow is superimposed. The total
potential is divided into three parts.
= Ux+¢
= ¢~+~]
= Ux+¢O+¢>~ (16)
where
Up = free-stream potential
¢,' = double-body potential
¢0 = disturbance potential in double-
body theory
distu rba nce potential i n low-
speed theory
¢> = disturbance potential in thin-
ship theory.
The free surface condition is simplified as described
by Dawson even though Raven, [11i, has reported that
the formulation is inconsistent. (Anyhow, the difference
in calculated wave resistance is within a few percent.)
The subscript ~ denotes differentiation with respect to
the streamlines of the double-body solution. Neglect-
ing quadratic and higher-order products of ¢~ and its
derivative, the following equation is obtained.
Hi- = - 9 t(~.'~s~s—24~s¢~ss] on z = 0. (17)
6s9
t
Figure 2: Inner and outer domains
This is the same result as Dawson obtained in his equa-
tion (14), [5~.
When matching is performed on the vertical control
surfaces, 0 from the outer solution has to equal ¢0 + At.
Equation (17) can be written in the form
0 + - ffr2l,9¢sS—2~04sO`Iss]
= ~~2~2l~¢,~Sx - (O`~(Ux~s~s] on z = 0. (18)
The problem is solved with respect to source strength
which automatically gives the velocity components i.e.
the first derivatives of the velocity potential. A numerical
operator identical to the one obtained by Dawson, t5i, is
adopted in order to estimate the second derivative with
respect to s.
The vertical control surfaces are assumed to be so
far from the body that the waves will satisfy the linear
free-surface condition. It is then appropriate to use the
Kelvin source function, Go, to describe the outer flow.
It is shown in appendix A that the boundary condition
on the vertical control surfaces is
v= 27r //s, G~;o,7dS on y= sob (19)
and the corresponding boundary condition on the hull is
¢~7 =—U.- on z = 0. (20)
The radiation condition in the inner domain is sat-
isfied by using an upstream differential operator when
satisfying the free-surface condition. The operator in-
sures that waves are only present behind the ship. On
the downstream boundary, an artificial damping is ap-
plied to the free surface condition. The inner solution is
only affected a short distance upstream of the damping
OCR for page 660
Numerical tests of a single point source
F., 1 0.10 1 0.20 1 0.30 1 0.40 1 0.50
1 H,~/L 1 003 1 013 10.29 1 0.511 0.79 1
Table .1: Minimum depth of the vertical control surfaces
as function of Froude number.
area. The calculation of the wave resistance by control
surface integration is obtained from the undisturbed part
of the inner domain. A far-field solution may be applied
instead as a matching condition on a downstream trans-
verse boundary.
The necessary depth of the vertical control surfaces
can be estimated by use of the fundamental wave length,
A, defined by the Froude number, F,i, and the length of
the ship at waterline, L.
Of = 2~F,~L. (21 )
As seen from the formula, the wave length increases
as the square of the Froude number. Using the assump-
tion that an elementary wave disturbs to a depth of half
the wave length, the minimum depth of the control sur-
faces, His, is given by
Hit- = Jo = OFF (22)
Table .1 indicates the necessary depth of the ver-
tical control surfaces. At the higher Froude numbers,
however, the results obtained using depths much smaller
than indicated by the table are good. On the other
hand, it is necessary to have a depth about twice the
draft of the hull, which indicates that the near-field flow
about the hull is dominating the wave making. While
running the program, the contribution to the wave re-
sistance from the first vertical row, the last vertical row
and the bottom horizontal row of panels on the vertical
control surface is checked in order to control the exten-
sion of the surface. The depth of the surface can then be
within the limits of confidence without actually satisfying
the numbers in table .1.
The evaluation of the Kelvin Green's function, Go;,
is only needed on the vertical control surfaces. Until
recently the calculation of this function has been rather
time consuming. In the present work, nondimensional
values have been tabulated and linear interpolation has
been used. New, fast algorithms are now available, for
example Newman [6i, and can easily be included in the
progra m.
Verification of a computer program is very important,
t10~. The present code has been tested using single point
sources and single point dipoles situated below the free
surface. The results have been compared with results
obtained by Nakatake [9] in the case where the double-
body linearization in the inner domain is replaced by the
common free stream linearization. Figure 3 shows the
panel distribution in the case of 8 longitudinal rows of
elements on the free surface.
Figure 3: Panel distribution on one fourth of the surfaces
in the case of a single point source situated at F = 1. b
is the distance between the center plane and the vertical
control surfaces.
Different aspects were important to investigate in
these single point tests. The first was to check the ra-
diation condition on the vertical control surfaces. The
closer to the center plane it was possible to position the
surfaces, the better. Then a very limited part of the
free surface was needed to be panelized. The effect of
moving the control surfaces was checked by investigat-
ing the wave elevation along the innermost row of panels
compared to the wave elevation obtained by using the
Greens function definition, equation 25, in appendix A.
The plots presented in figures 4 and 5 show the case
when the nondimensional vertical position of the point
source is F = '`(—z) = g/U2(—z) = 1.0. Figure 4
includes the results as presented by Nakatake.
Secondly, the number of elements needed to dis-
cretize the inner domain was checked. It was found
that a number of about 20 elements pr wave length was
needed to obtain a stable solution. By increasing the
number of elements beyond that limit, nearly no differ-
ence was observed in the case of 8 longitudinal rows of
elements on the free surface. In the case of 3 rows of
elements on the free surface however, even a higher num-
ber of elements would increase the accuracy of the wave
elevation as seen in figure 5 where three different grid
sizes in longitudinal direction is presented. Also included
are the curve when the control surfaces are removed. It
is obvious that this solution is totally wrong both with
660
OCR for page 661
-1 0 2
~ ~~\~\~"o>~°:
Figure 4: Contour plot of the wave elevation due to a
single point source situated at F = 1. The upper part of
the figure is from Nakatake.
Kelvin source
— — 8 rows, Max - Q25
3 rows, Max - 0.40
.- 3rows,6yx=025
— 3rows,6~x=0.15
— - 3 rows, no control surface
respect to the wave length and wave amplitude. In addi-
tion the result for the Kelvin Green's function evaluated
at the center plane is given in figure 5.
A third aspect of importance is the behaviour of the
numerical differential operator. Two-, three- and four-
point upwind operators where tested and the last one was
selected. As can be seen from the curves in figure 5, a
reduction of the first wave length of about 5 percent is
observed. But additional computations have shown that
the wave length is very good further downstream. This
might be caused by the boundary condition applied on
the control surfaces.
Finally, it is important to check to influence of com-
bining a double-body linearization in the inner domain
and a free-stream linearization in the outer domain. New
tests where carried out for the single point source using
these conditions. The wave elevation changed, as ex-
pected, somewhat compared to the tests with only free-
stream linearization in the inner domain, but the effect of
using the vertical control surfaces where similiar. Tests
have also been carried out for single point dipoles which
confirm the results.
The Wigley hull
The hull surface of the Wigley parabolic hull is de-
fined by the equation
Y' = 2 t1—~ L')~1 - ~ L' )~] (23)
where—L/2 ~ xi < L/2 and -H < z, < 0.
The main parameters are given in table .2, and the
body plan in figure 6, which has smooth lines arm fore-
aft symmetry. A lot of numerical and experimental data
Figure 5: Wave elevation along the innermost row of
panels. The results for different positions of the verti- Aloe \Vigley hull
cal control surfaces are compared with results using the
centre plane source function.
it\ \ :7 '''-I -'' TTI- r ,~ TO I Ill
Wigley
B/L,,,, 0.1000
H/L,,,, O. 0625
Series 60 |
0.1333
0.0625
L/L,),, 1.0000 1.0167
Ca 0.444 1 0.600
Cal 0.661 1 0.710
To
ble .2: Main parameters of the hulls
. I:;~-;lt1 ~
~ 1 ~ i' , . T .,
l _~/ / /' ~
The Series 60 hull
Figure 6: Body plans of the different ship hulls.
661
