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OCR for page 593
Ship Wave-Resistance Computations
G. Jensen
Ingenieurkontor Lubeck, Germany
V. Bertram and H. Soding
Ibstitut Bur Schif~bau Hamburg, Germany
Abstract
A method for the numerical determination of the po-
tential flow around a ship moving steadily at the free
surface of an ideal fluid solves iteratively for the non-
linear boundary condition at the free surface, the body
boundary conditions and the equilibrium floating posi-
tion. The method can also be applied to shallow water
and hulls with transom stern. The radiation condi-
tion and the open boundary condition are enforced by
a special arrangement of the collocation points at the
discretisation boundaries. This paper also describes a
new simple and flexible panel method for satisfying the
body boundary condition; the method could be used
for other potential flow problems as well.
1. Symbols
F
B breadth of the hull
D draught
Do draught at rest
frictional resistance coefficient
wave resistance coefficient
unit vector pointing in the direction of the tow-
ing force
pressure force on the body
f panel area
FA additional force on the body
En = U2/ I, Froude number
9 acceleration of gravity
G weight
G = (O,O,G)
G(p, ~ potential at point q due to unit source at p
593
n
p
p
-
q
r
R
Ru
S
So
s,t
T
T
A
U
vn
H water depth
k point on tangential sphere
lPp ship length between perpendiculars
M source strength
unit normal pointing into body
point
projection of a point on the body surface onto
the tangential sphere
pressure
point
ratio between the area of the projection and
that of the original surface element
radius of tangential sphere
wave resistance
wetted or panelized body surface
wetted surface at rest
vectors tangential to the body surface
moment due to pressure on body
additional moment on body
free stream velocity
velocity component normal to body surface
vs. v' components of velocity tangential to body sur-
face
point on free surface
x,y,z right-handed coordinate system; the x and y
axes are on the undisturbed free surface, x
points upstream, z vertically downward
point of action of towing force
center of gravity of ship's mass
towing force
xz
XG
z
do
-
p
submergence of dipole
moment of dipole
trim angle (bow down trim is positive)
trim angle at rest
velocity potential
approximation of
= ~ - Us
correction of ~
density of water
nondimensional sinkage at midship section; see
(26)
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z
F
K
nondimensional trim; see (26)
z-coordinate of free surface
approximation of
suffices:
1,2,3 vector component in direction of the :~-, y- or
z-ax~s resp.
refers to point or source at the free surface
refers to point
n
p
q
refers to point or source on the body surface
component in direction of the normal n
refers to point p
refers to point q
x, y, z partial derivatives
2. Introduction
For the design of a ship hull and its power require-
ment it is of great practical interest to know the flow
and the resulting forces due to the steady motion at the
surface of a calm ocean. Experimental techniques try
to separate viscous from potential-flow forces to scale
measured force coefficients from a model to the full-
scale ship. Due to the difficulty of computing viscous
flow forces for high Reynolds numbers, the same sepa-
ration is used in numerical predictions as well. Serious
efforts to compute the potential flow and the accompa-
nied wave resistance of a ship started with the pioneer-
ing work of Michell t1] in 1898. In spite of the great
emphasis placed on this problem, numerical solutions
are only now at the threshold of practical applicabil-
ity. They neglect breaking waves, spray, bow vortices
and other details of the flow, which may then be de-
scribed by a potential satisfying Laplace's equation and
boundary conditions at all fluid boundaries.
The major difficulty in this problem is the nonlin-
ear boundary condition at the unknown location of the
free surface. To circumvent this difficulty, most known
methods use, apart from the physical simplifications
stated above, additional mathematical simplifications:
Almost exclusively a linearized free-surface boundary
condition is implied at the location of the undisturbed
free surface, and in most cases the solution is described
by a superposition of complicated singularities that
meet this linearized free-surface condition identically.
All linear methods show good agreement with mea-
surements only for special hull forms or high Froude
numbers En = U2/~, where U is ship speed, g
acceleration of gravity, and Lpp the ship length be-
tween perpendiculars.
In 1978 Dawson t2] published a method using a dis-
tribution of Rankine sources (potential = 1/distance)
on the body surface and on a local part of the free
surface around the body. The flow is imagined to be
superimposed from the double-body flow, i.e. the flow
which would result in case of a rigid free surface, and
a correction A. The source strength distribution is de-
termined from the body boundary condition and a so-
called double-body linearization of the free-surface con-
dition. This linearization neglects nonlinear terms in
~ and is applied at the plane, undisturbed water sur-
face. The radiation condition which states that waves
may occur only behind the ship is enforced by a one-
sided finite-difference operator for the second deriva-
tive of the potential in the direction of the double-body
streamlines appearing in Dawson's free-surface bound-
ary condition. This numerical method of satisfying the
radiation condition has the disadvantage that the sur-
face waves are damped a little and are slightly shorter
than they should be.
Several authors have tried to extend Dawson's
method to an iterative solution of the exact, nonlin-
ear problem, but it was only quite recently that Ni t3]
and Jensen t4] succeeded. Both methods show good
agreement with resistance force measurements for the
few cases compared so far.
Ni's method uses source-panels on the wetted part
of the body surface and on a local part of the free
surface as computed in the previous iteration step.
In each step the body boundary condition and lin-
earized dynamic and kinematic free-surface conditions
are used to derive a system of linear equations for
the unknown source strengths and surface elevations
at control points. After solving this system the body
is repanelized automatically up to the computed wa-
terline. The radiation condition is enforced by means
of a one-sided finite-difference operator as in Dawson's
method.
Our method, on the other hand, employs Rankine
point sources in a layer above the water surface, and
in each iteration step it uses the same panelization of
the body and a mirror image of it above the water
surface. A linearized free-surface boundary condition
for the flow potential is used in each step of the itera-
tion together with the body boundary condition. After
solving the resulting system of linear equations for the
source strengths, the shape of the free surface is deter-
mined from the dynamic free-surface condition. The
radiation condition and the open boundary condition
are enforced by adding an extra row of control points
at the upstream end of the free surface mesh, and an
extra row of source points at the downstream end. The
effectiveness of this simple method was shown already
in [4,5~. For the body boundary condition a new, flexi-
594
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Representative terms from entire chapter:
boundary condition
ble panel method with simple numerical integration of
the influence function is used.
3. Problem
We want to compute the stationary flow of an incom-
pressible, irrotational fluid around a laterally symmet-
ric body at or near the free surface of an otherwise
unbounded fluid. Far upstream the fluid has the uni-
form velocity U opposite to the direction of the x-axis
of our Cartesian x, y, z coordinate system with z point-
ing downward.
~
~=~-~
-
The velocity potential ~ meets Laplace's equation:
/\~= 0 (1)
at points below the free-surface height ~ and outside
of the body and, in case of limited water depth, above
the bottom.
the dynamic free-surface condition; for z = ~ outside.
the body:
I
4. Free surface boundary condition
The unknown surface height ~ can be eliminated
from (4) and (5~; at z = (:
2V˘V (vi)2 _ 9~5z = 0. (13)
This free-surface condition is nonlinear; it is valid at
the unknown location of the water surface. Therefore
we iterate solutions with a linearized condition which
assures that - if convergence is reached - the solution
fulfills (13~.
To derive this condition, let ~ and Z be approxi-
mations for ~ and (. We substitute ~ = 4} + ~ in (13),
neglect terms nonlinear in derivatives of A, and obtain
at z = (:
V4}V (2 (V~2 + V4>V~) (14)
+V~V (2 (Vamp - 9(~z + Liz) = 0
4} and ~ are developed into a Taylor series about Z.
The series is truncated after the linear term. If prod-
ucts of ~—Z with derivatives of ~ are neglected, (14)
becomes
V4iV (2 (V~2 + V4iV~) + V~V (2 (V45~2) (15)
- gaff + Adz) + t2V~V (Vet _ gee] (( _ Z) = 0
at z = Z. The index z designates partial derivatives.
To decrease the number of unknowns, ~ is substituted
by expressions involving Z. 4~(Z) and ~(Z) only. To
this end (4) is developed into a Taylor series as well
and linearized:
= 29 [(Vim - U2]z=`
2i [(v~2 + 2V~V~ - Up
21 [˘V~2 + 2V~V(p
+2VIV4>z (( - Z) - U2] z .
This gives
(16)
~ [(v4~2 + 2V4?V,o - U2] - gZ (17)
with ~ and ~ being evaluated at z = Z.
Inserting (17) into (15) and substituting ~ by ~—4>,
we obtain the linearized free surface condition
596
V~V [- (Vet + VIVA] + -Vow (Vet _ 9˘Z
+ t2v~v ~vq>~2 - god] (18)
2 [- (V~2 + 2V~V~ - U2] - 9Z O
9 - V~V~z
at z = Z outside of the body. The denominator in the
last term is zero if the vertical component of the parti-
cle acceleration VIVID is equal to the acceleration of
gravity g; this is the stability limit of the approximate
flow Hi.
5. Radiation and open boundary condition
~ is approximated as the sum of the potentials of
the uniform stream, a regular mesh of point sources
in a layer above the free surface and the potential of
singularities on the body surface. The point sources
are generally located vertically above the collocation
points. Only at the upstream end of the grid there
is an additional row of collocation points, and at the
downstream end there is an additional row of point
sources.
To validate this simple scheme to enforce the radi-
ation condition and the open boundary condition, the
flow around a submerged dipole with Kelvin condition
at the free surface was computed and compared to an-
alytical solutions given by Nakatake A. Figs. 1 and
2 show the resulting free surface deformation from the
analytical and the numerical solution for different sub-
mergence speed parameters F = gt/U2. ~ is the sub-
mergence of the dipole. The numbers at the contour
lines indicate values of (go<3/~4,rU,u), where ,u is the
moment of the dipole. The agreement is excellent.
6. Body boundary condition
The Neumann condition (2) at the body surface is
fulfilled using a new panel method [4]. The commonly
used method of Hess and Smith [7] uses plane quadri-
lateral panels with constant source strength and control
points in the (suitably defined) centre of each panel.
The integral over the derivatives of the source poten-
tial can be evaluated analytically for each panel. This
results in complicated expressions containing transcen-
dental functions. Such expressions are expensive to
evaluate numerically.
lath ~ a 4~ `' ( -—~ ~
\~\v~/} ~ ~~-~--` ~
\ it ~ ~ ~ \
~ J
I\
_ _ ~ \
\
— _
Fig.1. Contour lines of surface elevation due to a submerged dipole for F = 0.5
The upper half shows the analytical solution due to Nakatake t6],
the lower half shows the numerical solution
I / ~ / ~
5~ =' / ,- ~ / /; /, ~
,,~:~411~
hi' A\;;'
OCR for page 598
Here we show a possibility to evaluate these inte-
grals with simple numerical integration by eliminating
the singularity of the integrand at the control point on
the panel:
f (id) = ~ M(fi) G(`p, ~ dSp, (~19)
s
˘~ is the potential at a point q induced by a source
distribution M on the body surface S; G(p,~ is the
potential of the unit source at p: G = - (dip—-.
The normal velocity on the body surface S is
vat id) = no Vq˘(~) = (20)
( M(O no) VQG(p, A) dSp—9 My)
s
If the normal velocity is prescribed by the boundary
condition, the velocity on S in two different tangential
directions t is to be determined only:
vt = toy) Vq˘~) = ~ M(
Table 1: Error depending on discretization
N number of elements
I N 11 1 1 4 1 16 1 64 1 256 1 1024
1 error 11 13.5 1 5 62 1 3 19 1 1 58 1 0 78 1 0 39 1
This shows that the method (like that of
Hess&Smith) is of first order; the error decreases pro-
portional to the mesh spacing.
As in most panel methods a smooth body surface
is required. Ships often have sharp stems or sterns.
The body boundary condition will always be violated
near such a corner. The panel method may still be
applicable for practical purposes if the overall solution
is not disturbed. Figure 3 validates this property by
showing the computed deviation from the parallel flow
for a Wigley double model with different panelizations.
Fig.3. View of the forward lower half of a Wigley double model in direction of the Taxis.
The arrows show the deviation from the parallel flow for 8, 32, 128 und 512 panels
on one eighths of the hull surface and for a grid with 208 panels,
which is the 128 panel grid with local refinement near the stem.
I;
599
7. Body at the free surface
For determining the stationary potential flow about
a body sailing steadily at a free surface of an ocean of
finite depth, the methods described in the preceding
sections are combined as follows. Source panels are
used
· on the wetted starboard hull surface, and
.
.
.
on a part of the above-water surface up to a plane
above the uppermost point of the assumed water-
line at the hull.
Further source panels of equal source strength as
those below this plane are arranged on a mirror
image of the hull surface above this plane. The
reason for this is that, on one hand, our special
panel method requires the source panels to con-
stitute a closed surface, and, on the other hand,
this mirror image reduces the necessary width of
the grid of further sources arranged along the free
water surface. Here it is sufficient to use point
sources above the free water surface instead of
source panels on the free surface. We use point
sources located one grid spacing above the free
surface.
· Further, all these sources are mirrored both at
the symmetry plane of the ship and - in case
of shallow water - at the bottom to satisfy the
respective boundary conditions.
The source strengths are determined from the following
conditions:
· On the ship surface up to the plane somewhat
above the water surface, the condition of vanish-
ing flow velocity normal to the surface is satisfied.
For the part of the hull surface above the water,
this condition is applied for two reasons: On one
hand, we do not know in advance where the ac-
tual water surface is, and, on the other hand,
the continuous source strength along the contin-
uous ship surface and its mirror image above the
horizontal plane helps to obtain a smooth poten-
tial at the intersection between body and water-
line. The same argument could and has been
proposed for a continuation of the free-surface
source panels into the body. However, because
we use sources not at, but above the water sur-
face, at the free surface the potential due to these
sources is smooth anyway.
.
The transom stern is also covered by source pan-
els. However, if the water flow is assumed to
separate smoothly at the transom such that the
transom face is not wetted, the usual condition
of vanishing normal velocity is not applied on the
transom. Instead, at the collocation points on
the transom we assume a normal velocity through
the transom of a size which satisfies the dynamic
(pressure) condition at the boundary between the
transom and the rest of the ship surface, assum-
ing the normal velocity to be constant over the
whole area of the panel. This results in a nor-
mal velocity greater than the ship's speed U if
the edge of the transom is below the undisturbed
water surface, and a normal velocity smaller than
U if the transom edge is above the undisturbed
water surface. Only if the edge of the transom
is at or above the stagnation height level, the
condition of vanishing normal velocity is applied.
Whether the transom is wetted or not, has to be
guessed in advance.
· At the free water surface the linearized free sur-
face condition (18) is imposed.
These boundary conditions are satisfied at, typi-
cally, 1000 collocation points on the body and on the
free surface. When the source strengths have been de-
termined, the free surface elevation is computed from
the dynamic boundary condition. The collocation
points on the free surface are moved to this height.
Then the process is repeated to get an iterative solu-
tion of the nonlinear free surface conditions.
On the water surface, the collocation points form
a regular grid outside of the body up to a distance
which was determined by test calculations and which
depends on the Froude number. Usually, point sources
are located about one grid length above each colloca-
tion point. However, if the distance between source and
collocation point is smaller than about 3 grid lengths,
instead of one collocation point a pattern of 4 points is
used, and the average error of the boundary condition
at those 4 points is considered.
It may be questioned whether our numerical meth-
ods applied to solve the potential-flow problem are cor-
rect for derivatives up to 3rd order. Without theoret-
ically investigating this matter, we simply found that
the method usually converges to a solution in which
the errors of the nonlinear free-surface conditions are
indeed extremely small at all collocation points, usually
below 1/lOOOth of the ship speed if they are expressed
in form of a velocity. Due to the smoothness of the
potential, errors are relatively small also between col-
location points. To obtain such an accuracy, it was
necessary to apply a normalization of the equations
600
which led to approximately equal values on the main
diagonal of the coefficient matrix.
To decrease the cases of divergence, we found it
helpful to determine the maximum error of the free-
surface condition at the newly determined positions of
the free surface both with the previous and with the
current source strength, and to use intermediate source
strengths if the maximum error was larger with the new
than with the old source strengths.
In the iteration to solve the nonlinear free-surface
condition the pressure distribution is integrated on the
actually wetted part of the hull to obtain the resulting
forces and moments. Together with corrections for the
pitch moment of the viscous resistance and the towing
or propeller force, a correction of the equilibrium float-
ing position is determined, and the panel grid of the
body is shifted and rotated correspondingly before the
next iteration step is performed.
The linear system of equations obtained during each
iteration step is solved by a combination of elimination
steps with a Gaul3-Seidel iteration: At first, only those
elements below the main diagonal the absolute value
of which exceeds a certain limit are eliminated to im-
prove the condition number of the matrix. If the fol-
lowing Gaul3-Seidel iteration indicates no convergence,
further incomplete eliminations using a smaller limit
are performed. This method constitutes a completely
safe and quick solution scheme.
8. Applications
8.1 Series 60 with CB = 0.6 in deep water
The Series-60 hull shape with a block coefficient
CB = 0.6 was chosen because extensive and careful
experiments and resistance evaluations have been per-
formed for this form by Ogiwara A. The principle
relations of his model are:
breadth B = 7P5;
draught at rest Do = ~~P75,
wetted hull area at rest So = 5 86~5.
The hull surface was panelized up to a height
0.3125Do above the floatation line at rest. There were
453 panels used on one half of the body. As in Ogi-
wara's experiments the horizontal towing force was
applied at 0.485lPP from the aft perpendicular and
0.461Do under the floatation line at rest. The vis-
cous resistance coefficient CF = Rf /~0.5pU2So) was es-
timated to be 3.5-10-3 for all speeds. For the speed
range investigated here it had only a small influence
on the trim. The height of the centre of gravity which
is not given by Ogiwara was assumed in the floatation
line at rest.
5.0j
2.5~
1O3cw
0 linear
. non-linear
—measured
.>
~ °
-
o
0.25
0.35
Fig.4. Computed wave resistance coefficients for
Series-60 with CB = 0.60 and values measured
by Ogiwara t8]
In Fig. 4 our computed wave resistance coefficients
are compared to Ogiwara's measurement evaluations.
For comparison also the results of the Neumann-Kelvin
problem are included. The results were obtained with
the same program (1. iteration step) and the same
grids; trim and sinkage were suppressed and the still-
water line used as an integration limit for the pressure
integration. Especially for higher Froude numbers non-
linear results agree much better with experiments.
Figures 5 and 6 show the nondimensional sinkage
and trim defined by
2 Do—D
Or = . ~ _
F=2 L
' —F2 (A—8) (26)
n
as functions of the Froude number For sinkage the
agreement is good. The curve for the computed trim
is similar to the corresponding measurements, but ap-
pears to be shifted somewhat to aft trim.
0.2 0.25 0.3
0.0
. computed
—measured
Fn
Fig.5. Computed and measured t8] nondimensional
sinkage ~ at midship section of a Series-60
model with CB = 0.60
601
0.1~
o.o-
. computed . /
—measured · / Fn
.,,,,,t,,,,,.,~r,,,,~
0.25 _~ · ~ 0.4
· · ~ ~
Fig.6. Computed and measured [9] nondimensional
trim ~ for a Series-60 model with CB = 0.60
/
Aim
\: ~r:
/:J': r:3J ~
/ ~/~L ! ~ ~ / ~~/~-:
/ Ad/ V ////~-
d'
Fig. 7 shows the computed elevation of the free
surface for the solution obtained with the linearized
free-surface condition (Kelvin condition)
U2~2 - 9˘z = 0 at z = 0 (27)
and the solution with the exact nonlinear condition
(13) for F7, = 0.25.
Fig. 7. Contour lines of deformed water surface for a Series-60 models CB = 0.60' at F7, = 0.25.
Left side: linear solution; right side: nonlinear solution.
The vertical distance between contour lines is 10-3Lpp.
602
Table 3: Influence of grid spacing at water surface
8.2 Containership in shallow water
The following results apply to a containership the
principal data of which are shown in the following table.
Table 2: Principal data of investigated containership
length between perpendiculars 161.44
breadth 28.40
draught at U=0 10.00
trim at U=0 0
block coefficient 0.68
designed load waterline coefficient 0.82
Dynamic sinkage and trim of ships on shallow water
are of interest not only because of possible grounding.
Forward trim usually encountered on shallow water can
reduce yaw stability so severely that ships may loose
their ability to keep course.
Figure 8 shows results for the draught at the for-
ward perpendicular divided by the stagnation height
U2/2g depending on depth Froude number F72h and
length Froude number An. The theoretically most in-
teresting region around depth Froude numbers of 1 can
be investigated only in case of relatively large length
Froude numbers because otherwise the ship touches
the ground. Limits of ground-touching are indicated
in Figure 9. This is the reason for investigating also
high length Froude numbers which are unrealistic for a
containership. As can be seen, near to a depth Froude
number of 1 large changes of the squat are experienced.
This is known also from model experiments and small
boats.
To determine the accuracy of these results, model
experiments are being performed but not yet finished.
Instead, Figure 10 shows results of approximation for-
mulae according to Barras t9] which were established
on the basis of model experiments and measurements
aboard ships, and results of the slender-body theory of
Tuck t104 applied to our ship. The nearly exact coin-
cidence with Tuck's formula is striking; however, near
to the critical depth Froude number 1 this formula is
not applicable.
Possible errors of our method were investigated also
on the basis of numerical experiments with different
grids on the body and on the free surface. In the fol-
lowing tables ~ denotes the difference in sinkage to the
most accurate computation in To of sinkage at forward
perpendicular FP.
The results indicate that for the small length
Froude number of 0.15 (6 m/s) our results are doubtful.
U
(mls)
6
6
6
10
10
H _
(m) _
14
14
14
14
14 _
., .
,grla spawn,
4.5m x 4.5m
5m x 5m
em x 6m
6m x 6m
8m x 8m
_ at AP
-20%
To
To
atFP |
5 To
16~o
-3%
m
m
m Table 4: Influence of grid size behind aft perpendicular
m AP and sideways
_
U
(mls)
6
6
10
10 _
H
(m)
14
14
14
14
grid size
aft side
-135m 93m
-70m 70m
-19lm 101m
_ -135m 85m
atAP | atFP |
-1% 18~o
2~o -ho
Table 5: Influence of height of source points over still
water plane
U
(mls)
6
6
12
12
12 .
H
(m)
14
14
18
18
18
height
4m
6m
8m
10m
_ 12m
Ł
at AP | at FP
25% -14~o
-2% 0%
-3% 2%
Table 6: Influence of distance limit for taking 4 collo-
cation points instead of 1
U
(m/s)
6
6
6
l H T
(m)
14
14
14
distance
limit
2.0xgrid spacing
3.3 x grid spacing
5.5xgrid spacing
1
atAP | atFP |
-do 1 To
-ho 1 0%
The reason for this is held to be the fact that, due to
limitations of the maximum number of grid points, the
grid length is not small compared to the 23 m length of
the transverse surface waves and the even shorter di-
agonal waves generated at this small Froude number.
Fortunately, in practice the squat is hardly relevant at
such small speeds. On the other hand, for Froude num-
bers of 0.25 and more, the table seems to indicate that
errors due to the discretization are small.
603
Computing times are about 10 minutes for each it-
eration step on a VAX 8550 in case of about 1000 collo-
cation points. The necessary number of iteration steps
to obtain an accuracy of 1 cm for the squat ranges typ-
ically from 2 to 5, depending mainly on depth Froude
number.
1.0- _
0.5~
O -
sinkage 0.15
0.2
U2/29 ~
/. ' ° \
i'; i,
Fn=0.15 l ~ ~
O '0.25X
a" 0.2 o
,
~ 0.3
~ ~ . ~ . ~
0.25 0 3
1
I
\
0,.4
\
.
\
,, F7lh
0.5 \1.0-i`9
Fig. 8. Nondimensional sinkage at FP
Dotted lines are limits of ground-touching
em
4m
2m _
\\
. ._ ~.~_ _ .
I-
O lOkn 20kn U
Fig.9. Distance between ship bottom and sea bottom
at FP depending on speed for 3 water depths
10]—
0.5~
O -
sinkage
U2/2g
Barras
Tuck
/
0 15 ~ -0 ~ 0.,S
/ ,~ /
/ /
Fig. 10. Nondimensional sinkage at FP
Fnh
l
according to Barras [9] and Tuck [10]
604
8.3 SWATH ship
For an research SWATH ship of the German navy
we used our method for wave resistance prediction. We
estimated frictional and viscous pressure resistance as
in Salvesen et. al. [11]. The rather ununsual shape
of the cross section caused some difficulties in the non-
linear computation. Therefore, in each iteration step
the free-surface collocation points were not only shifted
in height but also in horizontal direction according to
the current waterline. For a demihull this modifica-
tion ensured rapid convergence. The error in the free
surface was reduced by a factor of 10 in each iteration
step. Each demihull induces a slightly oblique flow at
the other demihull. Due to limitations in time, we did
not incorporate a Kutta condition at the rear of each
strut which would be necessary to take this effect prop-
erly into account. We felt justified in this decision by
the results of Bai et al. t12] who found for another
SWATH ship that including a Kutta condition had no
significant effect on the wave resistance. Bai et al. re-
ported only some differences in the local velocity field
near the trailing edges of the struts. Figure 11 shows a
comparison of computational results with experiments
of the Hamburg Towing Tank HSVA. Unlike the com-
putational model the experimental model was equipped
with rudders, fins and a propeller guard.
1 O3CT O
O ~
5 ~
o
0 ~
Ago
· · ~
0 linear
· non-linear
—measured
Fn
, ~
0.1 0.2 0.3 0.4 0.5 0.6
Fig. 11. Computed total resistance coefficients for
SWATH ship and values measured by HSVA
Only in a few cases a nonlinear solution succeeded.
The agreement with experiments is worse than for
conventional ships. For the medium Froude-number
range we noticed a considerable phase shift between
the waves on the inside and the outside of each demi-
hull before the computation breaks down. The point
with the highest error in the free-surface condition and
also the highest vertical acceleration was at the end
of the strut. This seems to indicate that cross-flow
effects afterall might be important for nonlinear solu-
tions despite Bai's et al. findings. For high Froude
numbers the computation breaks down at a point be-
hind the SWATH ship at the plane of symmetry. Two
wave crests starting from the trailing edges of the struts
superimpose resulting in a splash. This phenomenon
can also be observed in reality. This violates one of
the fundamental assumptions of our method. More
research will be necessary before breaking waves can
be included. Similar agreement with experiments was
found for linkage, t13~. Trim was suppressed both in
computations and model tests.
9. Final remarks
For many practical hull forms the present method
can be used to compute the potential flow with nonlin-
ear free-surface conditions. The wave resistance is pre-
dicted quite well, although we do not yet achieve the
accuracy of experiments. More comparisons to mea-
surements are required to gain experience.
The squat of the container ship seems to have been
determined accurately by our panel method for Froude
numbers above 0.20 in case of depth Froude numbers
below 0.9 and perhaps also for critical and over-critical
depth Froude numbers. However, the extremely simple
slender-body formula of Tuck gives the same results
for sub-critical depth Froude number, incuding small
length Froude numbers.
10. References
MICHELL, J. H.: "The Wave Resistance of a
Ship", Phil.Mag. Vol 45, 1898.
t2] DAWSON, C. W.: "A Practical Computer
Method for Solving Ship-Wave Problems", Sec-
ond International Conference on Numerical Ship
Hydrodynamics, University of California, Berke-
ley 1977.
[34 NI, S.-Y.: "Higher Order Panel Methods for Po-
tential Flows with Linear or Nonlinear Free Sur-
face Boundary Conditions", Chalmers University
of Technology, Goteborg, 1987.
- ~~ - ~ '#~ ~ ~
[5]
[6]
[7]
[8]
[9]
[4] JENSEN, G.: "Berechnung der stationaren Po-
tentialstromung urn ein Schiff unter Berucksich-
tigung der nichtlinearen Randbedingung an der
freien Wasseroberflache", Institut fur SchifEbau
Hamburg, Report No. 484, Juli 1988.
JENSEN, G., MI, Z.-X., SODING, H.: "Rank-
ine Source Methods for Numerical Solutions of
the Steady Wave Resistance Problem", Sixteenth
Symposium on Naval Hydrodynamics, University
of Califorinia, Berkeley, 1986.
NAKATAKE, K.: "On the Wave Pattern Created
by Singular Points", Journal of Seibu Zosen Kai
West Japan, No. 31, 1966
HESS, J. L., SMITH, A. M. O.: "Calculation
of Non-Lifting Potential Flow about Arbitrary
Three-Dimensional Bodies", Douglas Aircraft Di-
vision Report No. E.S.40622, 1962.
OGIWARA, S.: "Tank Experiments and Numer-
ical Works on Series 60 Model in IHI Ship Model
Basin", Report to the Cooperative Experiment
Program of 18th ITTC, Kobe, Japan, October
1987.
BARRAS, C.B.: "A Unified Approach to Squat
Calculations for Ships", Bulletin of the PIANC 1,
No. 32, 1979
[10] TUCK, E.O.: "Shallow-water Flow past Slender
Bodies", J. Fluid Mech. 26, Part 1, 1966
t11] SALVESEN, N., von KERCZEK, C.H.,
SCRAGG, C.A., CRESSY, C.P., MEINHOLD,
M.J.: "Hydro-Numeric Design of SWATH-
Ships", SNAME Trans., Vol. 93, 1985
t12] BAI, K.J., KIM, J.W., KIM, J.W.: "The Cross
Flow Effect on the Force and Moment acting on a
SWATH Ship" Seventeenth Symposium on Naval
Hydrodynamics, The Hague, 1988
(13; BERTRAM, V.: "Numerische Widerstandspro-
gnose fur SWATH-Schiffe", Schiffstechnik Vol.
35, No. 3, 1988
605
DISCUSSION
by A. Musker
Have you tried the non-linear calculation
of Fig.4 with the ship f ixed and compared with
the data compiled by ITTC on the experiment of
Kim and Jenkins?
The novel treatment of the radiation
condition deserves f urther study to see how it
behaves with high resolution surface grids.
Author's Reply
We did not try the non-linear calculation
with fixed trim and sinkage for the Series 60.
Fig.4 shows the result for the first and the
final step of the interaction for the same
computation.
We believe that our treatment of the
radiation condition will also work in the
limit of very small grid spacing. The
derivative are taken analytically and the free
surface boundary condition is symmetric.
What is needed to get the desired solution
is a numerical stimulation for an asymmetric
solution.
If the source distribution is in a layer
above the free surface, as in our method, the
vertical distance has to be decreased with the
mesh size.
We performed trial computations with 50
points per wave length for the 2-d case and
did not have any problems.
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