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OCR for page 643

Calculation of Free-Surface Flow
around a Ship in ShaBow Water by Rankine Source Method
H. Yasukawa
Mitsubishi Heavy Industries
Nagasaki, Japan
Abstract
Rankine Source Method was applied to the
shallow water and channel problems, and cal-
culations were made of free-surface flow
around a ship moving in calm water and her
wave-making resistance. The calculated
results were compared with experiments and
calculations by conventional linearized
analytical method. It was shown that the
Rankine Source Method promises an improvement
in predicting free-surface flow around a ship
in shallow water than the conventional
linearized analytical method.
1. Introduction
Flow around a ship traveling in shallow and
restricted water is much different from that
in deep and unrestricted water due to the in-
fluence of sea bottom and bank walls. There-
fore many studies have been carried out for
investigation of performance and safe naviga-
tion of the ship in the shallow and restricted
water such as river, coast, harbor, channel
and so on.
The theory on ship waves and wave-making
resistance in shallow water was presented by
Havelock in 1922 ~13. Kinoshita and Inui cor-
rected the Havelock's theory so as to satisfy
the boundary condition of sea bottom exactly
[2~. Further Inui extended its theory to the
channel problems [34. By these studies the
hydrodynamic characteristics of waves and
wave-making resistances near the critical
speed (F. h = l.0, where Fnh means Froude num-
ber base] on water depth h) were made clear.
Kirsch carried out calculations of wave-making
resistance in various water depth and channel
width for mathematical ship hull form and dis-
cussed the shallow and channel effect t43.
Bai calculated wave-making resistances in
shallow channel by localized Finite Element
Method t5~. Mueller compared experiments with
calculations by conventional linear theory in
shallow water, and indicated that the conven-
tional linear theory could not predict free-
surface elevation and pressure distributions
with sufficient accuracy [63. Recently Mel
and Choi presented the higher order theory on
643
a slender ship moving in channel t73.
The other hand, there is Rankine Source
Method developed by Gadd t8] and Dawson ~9], a
kind of numerical method for solving steady
free-surface potential flow in deep water. In
this method the free-surface condition based
on double body flow is employed, and the ac-
curacy of predicting wave-making resistance is
generally better than the conventional linear
calculations. Further detailed informations
for wave elevations, pressure distributions
and velocity fields around ships can be ob-
tained easily.
In this paper the Rankine Source Method was
applied to the shallow water and channel
problems. Calculations of free-surface flows
and wave-making resistance were made for Inuid
S-201 ship hull form [10], and the calculated
results were compared with experiments and
calculations by conventional linearized
analytical method. It was shown that the
Rankine Source Method promises an improvement
in predicting free-surface flow around a ship
in shallow water than the conventional
linearized analytical method.
2. Formulation of Shallow Water and Channel
Problems by Rankine Source Method
Let us consider a ship moving in center of
channel in calm water. We assume that sec-
tional shape of the channel is uniform in
lengthwise direction. The coordinate system
is defined as shown in Fig.1. x-axis coin-
cides with the direction of steady uniform
stream whose velocity U is identical with the
ship speed. The origin of the axis is at the
point of intersection of still water surface,
midship section and center plane of the ship.
y-axis is horizontal and normal to x-axis, and
z-axis directing vertically upwards.
Supposing a ship is in an inviscid, irrota-
tional, incompressible fluid, the velocity
potential ¢, which represents flow around the
ship and satisfies Laplace's equation V2¢ = 0,
is introduced. ~ has to satisfy the followin&
boundary conditions :
~xCx + ~y~y ~ ~z = 0 on z = `, ( 1 )

OCR for page 643

Now Al is expressed as :
¢1 = OF + tH + (C,
where
OF : velocity potential representing free-
surface,
tH : added velocity potential representing
ship hull due to the presence of free-
surface,
tC : added velocity potential representing
channel due to the presence of free-
surface.
(F. tH and tC are represented by Rankine
sources which are distributed on undisturbed
free-surface SF, ship hull SH and channel sur-
face Sc as follows :
IF(P) = J.lsFCJF(Q)GF(P, Q)dx'dy', (8)
tH(P) D.SHOH(Q )GH(P, Q')dSH, (9)
Fig.1 Coordinate system tC( ) ~Sc3C(Q )GC(P' Q")dSc, (10)
1(~2+¢y+~2_U2) + go = ~ on z = i, (2,
tnH = 0 on ship hull surface, (3)
ARC = 0 on channel surface, (4)
where ~ is elevation of free-surface, g the
acceleration of gravity. nH and no mean the
outward normal directions on ship full and
channel surface respectively. Eqs.(l) and (2)
represent the exact free-surface conditions.
Eqs.~3) and (4) represent the boundary condi-
tions of ship hull and channel surface. Here
in order to simplify the calculation, the ex-
act free-surface conditions (1) and (2) are
linearized as :
is expressed by following form : where
(5)
where
¢0 : velocity potential for double body flow,
¢~ : velocity potential for steady wavy flow.
It is assumed that the double body flow is
dominant in the flow field, and we neglect
higher order terms with respect to ¢~ for
eqs.(l) and (2~. Then the linearized free-
surface condition based on double body flow is
derived [9] as :
tOS¢ISS + 2~0S¢OSS¢IS + g¢IZ = -¢OS¢OSS
on z = 0, (6)
where the subscript S means partial derivative
along the streamline of double body flow on
still water surface. In this paper eq.(6) is
employed for the free-surface condition.
Double body potential ¢0 can be obtained by
Hess and Smith's method t114. Therefore our
concern becomes to solve the ¢~ so as to
satisfy the boundary conditions.
respectively. Here oF, oH and oC are strength
of source distributions for free-surface, ship
hull and channel respectively. P is field
point (x, y, z). Q. Q' and Q " are source
points for free-surface, ship hull and channel
respectively, and are defined as follows :
Q = (x', y', O ),
Q = (xH, YH, ZH),
Q" = (xc, Yc, ZC)
GF is represented as :
GF(P, Q) = 1/RF' (ll)
RF = /(x-x' )2 + (y_y')2 + Z2.
GH and GC are representing the double body
flows for ship hull and channel as :
GH(P, Q' ) = 1/RH + 1/RH', (12)
Gc(P, Q " ) = 1/RC + 1/Rc', (13)
where
644
RH = i(x-xH) + (Y-YH) + (Z-ZH) ~
RH' = /(x-xH) + (Y-Y8) + (Z+ZH) '
RC = /(X-xc) + (Y-YC) + (Z-ZC) '
RC' = i(X-xc) + (Y-YC) + (Z+zC)

OCR for page 643

3. Numerical Procedure 4. Wave Height and Wave-Making Resistance
Eqs.(3), (4) and (6) can be discretized by
following procedures:
(a) A finite area of the still water is
divided into ME rectangular panels IF
(j = 1 ~ ME), the hull surface into MH
panels THE (A = 1 ~ MH) and the channel
surface into MC panels ~ (n = 1 ~ Mc).
(b) It is assumed that variance on each
panel is represented by value on the
panel, and the source strength is con-
stant in each panel.
The system of simultaneous equations with
respect to the source strength oFJ (j = 1
MF), (7HdQ (A = 1 ~ MH) and C7Cn (n = 1 ~ Mc) are
compose as follows :
ij]~oF j: + ~BiQ]~GHQ) ~ tCin]~GCn} = {COii
[Dkj ~ l3Fj ~ + [EkQ] l3HQ: + [Fkn] i()Cni
[Gm; ] t(JFj ~ + [HmQ] {C>HQ: + [Imn] iC>Cn:
where
Aij = [ha (Aoi ;~2+Boi A,; )dx dy ]P=P
—1~ ,]
- 2~6ij ~
siQ = [JI~HQ(AOi~+sOi aS )dSH]P=
C;n = [JI:~c (AOi~+Boi as )dsc]P=
Dkj = rr7:F j anHdx dy ]P=Pk
Eke = [ TITHE a nHdSH]P=Pk
Fkn = [TI~cnanHdSc ] P=Pk '
Gmj = Errs j ancdx dy ]P=Pm ~
HmQ = t~r7HtanCdSHl P=Pm '
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
~ - circa anCdSC ] P=Pm
Aoi = f osi/g
(23)
Boi = 2¢ositossi/g | (24)
C0i = -~2sitossi/g J
Si. is the Kroenecker delta function.
For the calculations of a 2/3 S2 terms in
eqs.(15) - (17), the 4 points upstream dif-
ferencing was used so as to satisfy the radia-
tion condition of free-surface numericaly t93.
The C>Fj (j = 1 ~ ME), o~Q (A = 1 ~ Mp) and
oCn (n = 1 ~ Mr) are obtained by solving the
matrix of eq.(14).
645
Wave height ~ is represented by a
linearized form of eq.(2) with respect to ¢
as:
<; = 2t (U2-tOx~¢oy
~2¢ox¢~x~24'oy¢~y)~z=O' (25)
Pressure p is represented by a linearized
form of Bernoulli's equation as :
P = 2p(U2-~2X_~2 _~2
-2~0xl~x-2loy¢~y~2¢oz¢~z), (26)
where p is water density.
Wave-making resistance Rw is evaluated by
integration of pressure on ship hull as:
Rw -5sHP nHx dSH, (27)
where nHX is x-component of directional
cosines of the ship hull surface.
In this paper, p and Rw are represented by
the following non-dimensional expressions :
C p/lpU2 (28)
Cw = RW/2PSO U2, (29)
where S. is wetted surface area of ship hull
in still water.
5. Results and Discussions
For evaluation of the present method, cal-
culations in deep and shallow water were
carried out for Inuid S-201 ship hull form
t10] and compared with experimental data
t64~103. Further, calculations were made of
wave-making resistance of the ship in channel.
5.1 Results in deep water
First, calculations in deep and un-
restricted water were carried out. Fig.2
shows panel arrangements for ship hull sur-
face. We used 2 types of panels for valida-
tion. Panel H-1 has 220 panels and panel H-2
has 440 panels. Fig.3 shows panel arrange-
ments for free-surface. Panel F-1 has swept
back [9], and panel F-2 has not. When we cal-
culated the flow by using the free-surface
panels with rapid change of panel width in
lengthwise, oscillations occurred in the cal-
culated results for free-surface source
strength. So the free-surface panels with
smooth change of the panel width were used to
prevent the oscillations. Fig.4 shows varia-
tion of calculated wave-making resistance with
respect to the size of free-surface panel
region. Wave-making resistance coefficient Cw
converges around BFp/L = 0.4, where BFp is
lateral width of free-surface panel region and
L the ship length.

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Panel H-1
~11..1 1 ~ ~ ~ ~ ~ ~ ~ ~ ~ 1 111 -
Fig.2 Arrangements of hull surface panels
~ Panel F-1
Fig.3 Arrangements of free-surface panels
Fig.5 shows comparison of wave-making
resistance curves in deep water. In this
figure, 'fixed coed.' means to take no account
of sinkage and trim into the computations.
The calculated results by using panel H-1 and
F-1 show good agreement with the results by
panel H-2 and F-1, so the panel H-1 seems to
be sufficient for the number of hull surface
panels. For the hump and hollow in Cw curve
the results for panel H-1 and F-1 show better
agreement with Inui's experiments [10] than
the results for panel H-1 and F-2. There is a
discrepancy between the present calculation
and the Dawson's one [123. The reason for the
discrepancy may be due to the difference of
number of the panels. (Dawson used total 512
panels [123.)
5.2 Results in shallow water
Next, calculations in shallow and un-
restricted water with horizontal sea bottom
were carried out. We calculated in hid =
2.389 and 3.413, where hid means the ratio of
water depth and ship draft. For this case
Mueller has made detailed experiments t63.
Panel H-2, F-1 and the sea bottom panels as
shown in Fig.6 were used for the computations.
Fig.7 shows variation of calculated wave-
making resistance for the free-surface and sea
bottom panel regions in shallow water. The
sea bottom panel region is larger than the
free-surface region by 10% of ship length. It
seems that the convergence does not achieve
yet at BFp/L = 0.65. Thus we need larger
0.007
3
Panel H-1 & F-1
Deep water ~ | BFP ~
Fn=0.310 / ~\ /
Ship hull Free-surface
panels
0.006
f-o~o
1 1 1 1 1 1
0.2 0.3 0.4 0.5 0.6 0.7
BFP/L
Fig.4 Variation of calculated wave-making
resistance for free-surface panel
region in deep water
panel region than that in deep water.
In the present calculations, however, the
free-surface panels with BFp/L = 0.55 (BBp/L =
0.65, where BBp is lateral width of bottom
surface panel region) were employed for saving
computation time.
Figs.8 and 9 show comparison of wave-making
resistance curves in shallow water. The
present calculations (in 'fixed cond.') are a
little larger than Mueller's experiments t6]
in low speed range, and the tendency of hump
and hollow in Cw curve shows good agreement
with the experiments. However, Froude number
at the calculated maximum Cw is different from
the experiment. The tendency of hump and hol-
low in calculated C curve by conventional
linear theory (Have~ock's integral) [6] is a
little different from the experiments. How-
ever the Cw values show good agreement with
the experiments as well as the present
calculations.
Now, let us compare Cw curves in Figs.5, 8
and 9. In the present calculations, with
decrease of water depth Cw near the critical
speed increase and Froude number at the maxi-
mum Cw becomes smaller. These tendencies
agree with the experimental results. However,
the Froude number at the maximum Cw is larger
than the experiment and the differece of this
Froude number becomes larger with decrease of
water depth. The present method of calcula-
tion does not take into account the effect of
sinkage and trim. For reference, therefore,
attempts were made of calculations of wave-
making resistance by use of sinkage and trim
estimated from Mueller's experiments [63. In
this computation hull surface panels were
rearranged. In Figs.8 and 9 'free coed.'
means the calculation made by this way.
Froude number at maximum Cw in 'free coed.'
becomes smaller than that in 'fixed coed.' and
comes closer to the experiment. Thus, it is
suggested that for the improvement of the
present method the effect of sinkage and trim
should be included.
646

OCR for page 643

0.025
0.020
Cal
$~N 'it
total pane' numbers
0.015
0.010
0.005
- 2< Cal. (panel H-1 & F-l ) 640 ~
- C1 Cal. (panelH-1 &F-2) 650 !
——~——Cal. ( panel H-2 & F- 1) 860 |
x Cal. by C.W.Dawson (12) 512 J
O Exp. by T.Inui (10)
Pa
~ fixed coed.
Six
~-
f~
,( x JO
max. Cw
Exp.~ Cal. (fixed)
OO00L' ~ ~ ~ ~ ~ ~ I I ~ I
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65
En
Fig.5 Comparison of wave-making resistance curves in deep water
Fig.6 Arrangement of ship hull, free-surface
and sea bottom panels (in/d = 2.389)
647

OCR for page 643

Free-surface panels
0008:
0.007 _
0.006 _
L
\
/ | FP | / 1
I '//i~/~ ' I ~ 1
7 ~
Ship hull Sea bottom
Panel H-2 & F-1 panels
h/d = 2.389, En = 0.310
I I I I I I
0.2 0.3 0.4 0.5 0.6 0.7
BFP / L
l ~
0.3 0.4 0.5 0.6 0.7 0.8
REP/L
Fig.7 Variation of calculated wave-making
resistance for free-surface and
sea bottom panel regions in shallow
water
Fig.10 shows comparisons of wave profiles
in shallow water (h/d=2.389~. Agreement be-
tween the present calculations and Mueller's
experiments is good as a whole. However, it
can be pointed out that calculated wave height
at the fore part is lower and tendency of wave
profiles at the aft part is a little different
from the experiments. The reason for this
difference seems to be the nonlinear effect of
free-surface condition and the viscous effect
which are neglected in the present formula-
tions. The conventional linear calculations
are less satisfactory than the present calcu-
lations. The difference of the calculated
results in between 'fixed coed.' and 'free
coed.' in small.
Fig.ll shows comparisons of pressure dis-
tributions on sea (tank) bottom in shallow
water (h/d=2.389~. The present calculations
show fairly good agreement with Mueller's ex-
periments at the tank bottom below the hull
surface y0/L = 0.0 where yO is lateral dis-
tance from center line of ship. The conven-
tional linear calculations show good agreement
with the experiments also. The pressure coef-
ficient at y0/L = 0.1667 is smaller than the
experimental one near the negative peak value.
The conventional linear calculations are also
smaller than the experiments near the negative
peak value. The difference of the calculated
results in between 'fixed coed.' and 'free
coed.' is small.
Figs.12 and 13 show comparisons of prospect
view of wave-pattern and wave contour around a
ship for different water depth. Froude number
in the computations is En = 0.410 (Fnh = 0.849
for shallow water case) and at its Froude num-
ber the wave-making resistance increases
remarkably in shallow water. From the figure
it is found that the waves go down near the
midship and much swell behind the ship hull in
shallow water. Thus the effect of shallow
water on the ship waves appears more
remarkably near the stern part than at the
fore part of the ship.
It was shown that the present Rankine
Source Method made an improvement of predict-
ing the wave-pattern around a ship (wave
profile), and we can predict the change of
wave-pattern for various water depth as shown
in Fig.12. However, the accuracy of predict-
ing the wave-making resistance by the present
method was same order as that by the conven-
tional linear calculations. Thus, improvement
of the present method may be required for bet-
ter prediction.
5.3 Results in channel
Finally, calculations in shallow channel
with rectangular section were carried out.
We calculated in hid = 2.048 and W/L = 2.0,
where W/L means the ratio of channel width and
ship length. For this case Inui has made the
calculations by conventional linear theory
[103. Panel H-2, F-1 and the channel panels
as shown in Fig.14 were used for the
computations.
Fig.15 shows comparison of wave-making
resistance curves in the channel. The conven-
tional linear calculations by Inui [10] are
larger than the present calculations as a
whole, and particularly increase of the wave-
making resistance due to channel effect is
much larger near the critical speed. Further
the shift of hump and hollow in Cw curve can
be seen between both calculations. The dis-
continuity of Cw curve at Fnh = 1.0 occurs in
the conventional linear calculation, but it
does not in the present calculation. Thus, it
was showed that the present calculation of
wave-making resistance was different from the
conventional linear calculation.
The Rankine Source Method was applied to
the shallow water and channel problems. Cal-
culations were made of wave-making resistance,
wave-pattern around a ship (wave profile) and
pressure distributions on sea bottom. The
calculated results were compared with experi-
ments and calculations by conventional linear
analytical method. It was shown that the
Rankine Source Method made an improvement of
predicting the wave-pattern around a ship in
shallow water. However, the accuracy of pre-
dicting the wave-making resistance by the
present method was same order as that by the
conventional linear theory. Improvement of
the present method may be required for better
prediction. For example, the effect of
sinkage and trim should be considered exactly.
648

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0.025
0.020
3 0.01 5
0.010
0.005
I I,- Expel |Cal.(fixed) l
0 0000.15 0.20 0.25 0.30 0.35 0.40 o.i5 0.50 0.55 0.60 0.65
O Cal. ~ fixed coed.
~ Cal. ~ free coed. ~
-- - - - - Exp. by E.Mueller (6)
x Conventional linear theory ,'
~ Havelock's integral ~ (6) ,'
h /d = 2.389, h /L= 0.233 ,'
, i/
,* ~ /
/l'/;
,'~ ~~
if/
/ ~
~ max. Cw
\ ~
\
\
0.4 0.5 0.6 0.7 0.8
Fnh
0.9 1.0 1.1 1.2 1.3
Fig.8 Comparison of wave-making resistance curves in shallow water (in/d = 2. 389)
u.u'~
0.020
Cal
O Cal. ~ fixed coed.
Cal. ~ free coed. ~
Exp. by E.Mueller C63
0.015
0.010
0.005
— x Conventional linear theory ,f/,/
~ Havelock's integral ~ (6) ~_;, ~ ;/
,, `,//
h/d=3.413, h/L=0.333
\
max. Cw
\
Exp. l | Cal; ~ fixed ~
v vvv
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65
En
0.3 0.4 0.5 0.6 0.7
0.8 0.9 1.0 1.1
Fnh
Fig.9 Comparison of wave-making resistance curves in shallow water (in/d = 3.413)
649

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1.c
ore
0.6
0.4
0.2
0.0
-0.2
-0.4
1.2
~ '3 ~—Ace,
Fn = 0.289, Fnh = 0~599
-0.6 -0.4 -0.2 0.0
Fore x / ~ Aft
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-06 ~ 1 1 1 1
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
~ ~ Fn = 0.332, Fnh = 0.688
~ w~
l
Fore x / ~ Aft
O Cal. ~ fixed coed.
Cal. ~ free coed. ~
Exp. by E.Mueller (6)
0.2 0.4 0.6 - - Conventional linear
theory t6)
h/d=2.389, h/~=0.233
Fig.10 Comparison of wave-profiles in shallow water (in/d = 2.389)
0.3
0.2
0.1
0.2
0.1
0.0
-0.1
Fn = 0.332, Fnh = 0.688
Yo/L=0.0
o.o~? , ~1
-O 1
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Fore x /
0.~!
Yo/~=0.1667
Aft
_? .,
1 1 1 1 1
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Fore x / ~ Aft
Y Measured line
on tankbottom
Ship hull
MU ~
—L/2 0
. Yo
. ~ ~ ' x
L/2
h/d=2.389, h/L=0.233
Fig.ll Comparison of pressure distributions on sea bottom in shallow water (in/d = 2.389)
650

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Fn-0.410 ~
~1~
shal low water Aid,
( had =2.389
/
Fn =0.41 0
1.00f
0.80t
0.60t
0.40~
~ on
n Act
~ o. of
Coo
-0.20
-n 4n
.
-0.60 _
-0.80 _
-1 .00 _
it,
o.o
deep water ~
, ~
g.l2 Comparison of prospect view of wave-
pattern around a ship for different
water depth
(factor of wave height is 1.5)
2~/L~ ~ / ~
~ :~-o.o_,-
2.SO
shal low water -°°
( hid =2.389 )
Fig.13 Comparison of wave contour around a ship for different water depth
(contour interval A: is 10.0)
651

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Fig.14 Arrangement of ship hull, free-surface
and channel panels
(in/d = 2.048, W/L = 2.0)
o.osa
0.040
3 0.030
0.02a
o.o10
-—~—- Shallow
O Channel '
h/~=2.048, h/~=0.2
- W/~=2.0
water ~ Present Cal.
/
water l Conventional
- Channel J linear theory ~ 1 o) /
,, ~
~ // .,,,,
// /
" '4
~ -I
0 oca
0.15 0.20 0.25 0.30 0.35 0.40 0.45
0.50 0.55 0.60 0.65
En
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 13 1.4
Fnh
Fig.15 Comparison of wave-making resistance curves in channel
652

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Computation including the sinkage and trim can
be made by improvement of the present method
as the following iteration procedures:
(a) First, computation of the flow around
the ship fixed is made, and the vertical
forces (linkage force and trim moment)
are calculated.
(b) From the vertical forces, we determine
the amounts of sinkage and trim needed
to hydrostatic balance.
(c) The panels for the ship hull surface are
rearranged to take the sinkage and trim
into account, and the flow is recomputed.
Furthermore, the nonlinear effect of free-
surface condition should be considered.
Acknowledgments
10. Inui, T., "Study on Wave-Making Resist-
ance of Ships", 60th Anniversary Series,
The Society of Naval Architects of Japan,
Vol.2, pp.173-355 (1957~.
11. Hess, J.L. and Smith, A.M.O., "Calcula-
tion of Non-Lifting Potential Flow About
Arbitrary Three-Dimensional Bodies", Report
No.E.S.40622, Dauglas Aircraft Co. Ltd.
(1962~.
12. Dawson, C.W., "Calculations with the XYZ
Free Surface Program for Five Ship Models",
Proceedings of the Workshop on Ship Wave-
Resistance Computations, Vol.2, Bethesda,
Maryland, pp.232-255 (1979~.
Appendix Determination of wave-making
resistance from Mueller's experiments
The author would like to express his sin- Mueller provides total resistance and
cere gratitude to Dr. E. Baba, Manager of Ship residual resistance curves for Inuid S-201 in
Hydrodynamics Laboratory of Nagasaki Research shallow water [63. In this paper, the wave-
and Development Center, MHI, and Dr. T. making resistance was determined from
Nagamatsu, Research Manager of the same Mueller's measured total resistance by follow-
laboratory, for their guidance and valuable ing formula :
discussions.
References
1. Havelock, T.H., "The Effect of Shallow
Water on Wave Resistance", Proceedings of
the Royal Society, A. Vol.100, pp.499-505
(1922).
2. Kinoshita, M. and Inui, T., "Wave-Making
Resistance of a Submerged Spheroid, Ellip-
soid and a Ship in a Shallow Sea", Journal
of the Society of Naval Architects of Japan,
Vol.75, pp.119-135 (1953), (presented in
1944).
3. Inui, T., "Wave-Making Resistance in Shal-
low Sea and in Restricted Water, with Spe-
cial Reference to its Discontinuities",
Journal of the Society of Naval Architects
of Japan, Vol.76, pp.l-10 (1954), (presented
in 1946).
4. Kirsch, M., "Shallow Water and Channel Ef-
fects on Wave Resistance", Journal of Ship
Research, Vol.10, No.4, pp.164-181 (1966~.
5. Bai, K.J., "A Localized Finite-Element
Method for Steady Three-Dimensional Free-
Surface Flow Problems", 2nd International
Conference on Numerical Ship Hydrodynamics,
Berkeley, pp.78-87 (1977).
6. Mueller, E., "Analysis of the Potential
Flow Field and of Ship Resistance in Water
of Finite Depth", International Shipbuilding
Progress, Vol.32, No.376, pp.266-277 (1985~.
7. Mei, C.C. and Choi, H.S., "Forces on a
Slender Ship Advancing Near Critical Speed
in a Shallow Channel", 4th International
Conference on Numerical Ship Hydrodynamics,
Washington D.C. (1985).
8. Gadd, G.E., "A Method of Computing the
Flow and Surface Wave Pattern Around Full
Forms", The Royal Institution of Naval
Architects, Vol.18, pp.207-219 (1976).
9. Dawson, C.W., "A Practical Computer Method
for Solving Ship-Wave Problems", 2nd Inter-
national Conference on Numerical Ship
Hydrodynamics, Berkeley, pp.30-38 (1977).
Cw = Ct ~ Cfo~l+K), (30)
where
Ct : total resistance coefficient,
Cfo : frictional resistance coefficient
corresponding to the flat plat,
K : form factor.
Cfo was calculated by Hughes' formula. Form
factor K was determined as :
K = 0.265 for h/d = 2.389,
K = 0.230 for h/d = 3.413.
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DISCUSSION
by R.C. Ertekin
I wish to make some comments on your
paper. First of all, it is very surprising
that you do not mention the existence of
upstream waves in shallow water. There are 3
papers presented in this conference on the
subject, where you can find all the related
references. Your parameters in shallow water
falls in the range we used in our experiments
(Ertekin, Webster, Wehausen, 1984, 15th ONR
Symp., Hamburg). I am not familiar with
Mueller's paper but if he hadn't observed
these waves the something is wrong with his
observations. In any case, I will seriously
question the results and data for Fnh larger
than, say 0.6 or 0.7 in shallow water because
linear theory is no longer valid, also
steadiness, in general, is not possible.
Even though we have not paid any attention
to Thews and Landweber's (1935) work in the
last 50 years, we now know that in very
shallow water linear and steady results do not
have much meaning.
In shallow water, the angle that divergent
waves make with the waterline of the ship must
be much wider than the deep water case. Even
the linear theory can predict this. Therefore,
I do not think that your Fig.13 is accurate.
It doesn't look right so it must be wrong.
By the way, considering that you are using
linear theory, could you explain why you need
to distribute source on the sea floor (which
is flat) and on the free surface? Thank you.
Author's Reply
Thank you for your comments. The present
paper deals with steady wave-making problem,
so we did not refer to the papers of unsteady
wave-making problem in detail. The detailed
review on the unsteady wave-making problem in
restricted water is shown in ref.[7].
As expressed in Dr. Ogiwara's reply, we
think that the influence of the solution is
small for unrestricted shallow water.
The present method is a kind of the panel
method where Rankine sources are distributed
on boundary surface, so we need to distribute
the source on the free-surface and sea bottom
surface. In case of flat sea bottom, there is
the method which takes into account infinite
image of the sources distributed on the free-
surface and ship hull surface. However this
method can not apply to non-horizontal sea
bottom generally and has much time for
calculation of the infinite image. For the
above reasons we employed not the infinite
image method but the source distribution
method on the sea bottom.
DISCUSSION
by S. Ogiwara
As referred by your paper, T.H. Havelock
(1922) studied shallow water effect on ship
wave resistance, in which he predicted
significant feature of wave pattern that the
angle of diverging wave increases as the water
depth decreases. Could you simulate the same
feature by the proposed Rankine source method?
Moreover, considering the case of
extremely shallow water, we can find the
soliton which generates periodically forward
upstream from the bow, as pointed out by T.Y.
Wu and others.
The method, you proposed here, is not able
to treat such phenomena, because this method
is involved in the framework of steady state.
How do you think about the limitation of water
depth (or Fnh=U/ ~ ), to which this method is
able to apply?
Finally, Resistance and Flow Committee of
the 19th ITTC is carrying out the evaluation
of shallow water effect on ship resistance and
flow around a hull through the Cooperative
Experimental Program. You are encouraged to
conduct new experiments in order to verify the
effectiveness of your numerical method, and to
contribute to ITTC.
Author's Reply
Thank you for your discussions and
comment. So far as we observed the comparison
of calculated wave-height distributions for
different water depth (see Fig.13), it does
not seems that the remarkable change of
diverging waves appears in the present
calculation. The reason why the change does
not appear may be due to adoption of linear
free-surface condition based on double-body
flow in the present method.
The linear free-surface condition is
employed in the present method, so predicting
accuracy becomes poorer in high speed range.
Actually we can not see that the tendency of
wave-making resistance curve in the present
calculations agrees with that in the
experiments for water depth Froude number
larger than 0.8 (see Figs.8 and 9). However,
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from the view point of practical use, the
present method has sufficient accuracy for the
Froude number smaller than 0.8. Needless to
say, for better prediction non-linear effect
of free-surface condition should be
considered.
It is well known that when a ship moves in
shallow channel near the critical speed,
solution generates periodically forward
upstream from the bow and the flow around the
ship becomes unsteady[Al][A2]. The generation
of the solution is related with blockage
coefficient of the channel (or towing tank)
and the amplitude of the solution decreases
as the blockage coefficient decreasestA3]. All
calculations in the present paper except
Fig.15 are for unrestricted shallow water
taking no account of the channel walls.
[Al] Huang, D.B., Sibul, O.J. and Wehausen,
J.V.: Ships in Very Shallow Water,
Festkolloquium zur Emeritierung von Karl
Wieghardt, Institut fur Schiffbau der
Universitat Hamburg, Bericht Nr.427,
pp.29-49, 1982.
[A2] Wu, D.M. and Wu, T.Y.: Three-dimensional
Nonlinear Long Waves due to Moving
Surface Pressure, Proc. of 14th Symp. on
Naval Hydrodynamics, Ann Arbor, pp.103-
129, 1982.
[A3] Ertekin, R.C., Webster, W.C. and
Wehausen, J.V.: Ship-Generated Solitons,
Proc. of 15th Symp. on Naval Hydro-
dynamics, Hamburg, pp.347-364, 1984.
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