Wave Resistance of Ships in Low Speed
Eiichi Baba*
From the flow measUrement arouAd the bow of a slOwly moving ship it was found that, except in the thin boundary layer near the free surface, velocity components agree well with the calculated values from the double body potential. In tt.e thin free surface layer the velocity components change depth wise very rapidly. Based on this experimental result an asymptotic formula to calculate wave resistance of s/ow ships is derived by assuming that the flow around the floating body can be expressed as a sum of two parts. One is the double body flow and the other is an additional thin free surface layer superposed on the double body flow The formula includes a local nonlinearity around the body and gives better agreement with experiment than the conventional linearized wave
resistance theories. Appliëations of the prèseAt theory to the hull form design are cOnsidered.
1. Introduction
In this paper the contents of the previous two papers by the present author (1975 a, b) are consOlidated with detailed derivations of the wave resistance formulas and
with a recent result of theoretical and experimental studies.
It is the objective of the present paper to introduce a wave resistance theory for slow ships. The theory was primarily developed in order to get better understanding
of the mechanisth Of wave breaking phenomenon observed
around the bow of full forms. It has turned out, however
that the theory is also used to predict the wave resistance of slow ships.
In Sec. 2 a brief review of the studies of slow ships is given. In Sec. 3 a result of flow measurement around the bow of a blunt form is presented. From the flow measure-ment it was found that except in the thin boundary layer
near the free surface, velocity components agree well with the calculated values by the use of the double body velocity
potential. In the thin free surface layer, on the other hand, the velocity components differ from the double body solutions and change depthwise very rapidly until they
reach the value of the double body solutions. Based on this experimental result .a wave resistance theory for slow ships
is developed in Sec. 4. In the theory the double body
solution is used as the zero-order solution and an additional
velocity potential is then derived in such a way that the sum of two velocity potentials satisfies the free surface
conditions.
In Sec. 5 the wave resistance of simple forms such as a vertical circular cylinder is calculated for a qualitative examination of the theory. In Sec. 6 the calculated wave resistance of a Wigley's paraboliô form is compared with experimental values which are determined by means of wave pattern analysis. In the low speed range (F < 0.25)
a good quantitative agreement is observed. In Sec. 7
appli-catiOns of the present theory to the hull form design are
considered.
* Dr Eng, Resistance and Propulsion Research Laboratory,
2. A brief review of the studies of slow ships
In 1968 Ogilvie reported the study on a two-dimensional
problem Of the thin free surface layer on the non-uniform flow which is determined by the double body soFution. Ogilvie assumed that the basic non-unifOrm flow varies slowly with space variables while in the thin free surface
layer physical variables such as velocity and wave height are
assumed to vary very rapidly. In a paper by Hermans (1974) a wave resistance problem in low speed is treated in the similar manner as Ogilvie. A theory to analyze the
free surface flow around a semisübmerged horizontal
cylin-der, perpendicular to the incoming flow is developed. Timman (1974) extended Hermans' theory further to study the free sUrface flow around a semisubmerged three-axial
ellipsoid with its middle axis horizontally on the still water
plane so that the free surface meets the bow over a relative-ly broad part of the front. The problem is then transformed
into tWo-dimensional one in the plane perpendicular to
the middle axis. Keller (1974) studied wave patterns of full
forms in low speed by a different manner from those men-tioned above, but based on the similar assumption on the free surface flow. Keller assumed that for small Froude number, the flow consists of the double body flow plus an oscillatory flow which represents the wave motion.
Dagan (1975) discussed the non-uniformity of the solution of thin ship expansions in low speed free surface problems.
In a recent paper by Newman (1976) a boundary value problem fOr slow ships has been developed. The free surface
condition obtained by Newman coincides with the free surface condition which was derived independently by the present author. The present study is a direct extension of Ogilvie's t'heory to the three-dimensional case. Originally Ogilvie did not intend to apply his theory to a surface piercing body. Here, however, a theory for conventional
ship forms in lOw speed limit is developed, since the results
of the flow measurements around a blunt form motivated
MTB 109 August 1976
piercing bodies.
3. Flow rneaJrement around the bow ota slow speed ship For the flow measurement by use of a 5-hole pitot tube, a simple hull form M.2201 as shown in Fig. 1 was used. The principal particulars of the ship model are shown in Table 1.
The flow measurements were conducted in two different load conditions. Fig. 2 and Fig. 3 show the flow patterns around the bow of shallow draft (200 mm) and deep draft (400 mm) respectively at speed U=1.089 m/sec (U/../L 0.1420). With an increase of ship speed short waves observed in Figs. 2 and 3 are transformed into rather
con-fused 'disturbed flow (breaking waves) as shown in Figs. 4
and 5 (U1.284 rn/sec. U/ ,/E=0.1674).
Since there are rather confused short waves on the free
surface, only the mean elevation of' the free surface from
the still water level was measured first (Fig. 6). Then the velocity components (u, v, w) were measured up to the free surface by use of a 5-hole spherical pitot tube of
7 mm diameter. The measured results are shOwn in Figs. 7 through 10 comparing with the velocity components
calcu-lated numerically from the double body velocity potential which satisfies the rigid-wall free surface condition. In those figures z 0 corresponds to the free surface. The rigid-wall solutions are extrapo!ated up to the free surface from the values on the still water level, since they change
depthwise very little.
From the comparison it is found that except in the thin
boundary layer near the free surface the measured velocity components agree well with the calculated values based on
'the rigid-wall solution. In the thin free surface layer mea
Fig. 1 Lines of M.2201
Table '1 Principal particulars of M.2201 FP
sured velocity components show rapid changes with an increase of depth while the calculated values show very
little change. 100 mm
EIH
Fig. 3 U 1 .089 rn/sec in deep draft Fig. 5 U =1.284 rn/sec in deep draft- -ShaIIo draft Deep,draft
10
-500 300 0
(F.p.)
Fig. 6 Measured wave heights at U =1.089 rn/sec
uIu, v,u,w/u
-Free surface- o-a-'-
100
0 - I 0 $ IOi
I-
ID.
t
l00.i.
rneoid cuIcuiatedj
0.5 4 LÔ2 Still -W?evlevel 100mmFig. 7 Velocity components at x 350rnm, y 0
Load ' Deep Shallow
Lpp('m) 6.000 6.000 B' (ml 1.000 1.000 d (mm) 400.00 200.00 n(kg) 2004.7 958.68
L/B
6OO 6.00 Bid 2.50 5.00 Cb 0.8353 0.7989 C, 0.8359 0.7999 Cm , 0.9993 - 0.9987load U/U V/u W/U deep o Shallow
I
£Fig. 2 Flow pattern around
bow of a full fOrm at U =1.089 rn/sec
(U//7J
=01420)iii shallow draft
-LL
Fig.
4 U
1.284 rn/sec(U//T=0.1 674)
a
Fig. 8 Velocity components at x -175mm, y 0
U/a, V/U,W/u Taking the rectangular coordinate system fixed on the
0.5 L00 body with the origin on thestill water plane weset x-axis
directing to the uniform flow U and z-axis directing
up-wards as shown in Fig. 11. Supposing a ship floating on an inviscid irrotational incompressible fluid we consider that the velocity potential fOr the free surface problem is - expressed as a sum of two parts:
-100
(x,y,z) = ø,(X,Y,Z) +(x,y,z),
(1)N
where 0,. (x,y, z) is the potential fOr the rigi&wall problem, and 0 (x y z) is an additional potential to ,. (x y z) so that the sum satisfies the free surface conditions. Fig. ii -200 shows the modeled scheme of the present problem Though
05 10 we are considering short waves appeared aroühd a slow
ship, the surface tension effects on the wave formation is
neglected for the simplicity of treatment.
The boundary valuC problem for the present study is
written as follows:
0= 'I+ .I, +
(2)U2=gHx,y)+-[+cI+1
onz'H(x,y),(3)
u/U, V/U,W/U9---
as
as
u/U,v/U,w/Uas
100 1.0200 E EFig. 9 Velocity components at x O y 40Omrn
.00 where ,.(x, y) is the wave height due to the double body
potential, i.e.
100
N
1.0200
Fig. 10 Velocity components at x1 75mm, y=400mm
4! Theoretmal study
4.1 Bouncb.rv value problem
From the results of flow measurements we may assume
that the rigid-wall solution is quite accurate everywhere
except in the thin layer near the free surface.
o (//x2+y2)
for x <O,where g is the acceleration of gravity. The last ohe is the condition insuring that waves only follow the ship.
According to Ogilvie, we assume that the wave height is expressed as the sum of two parts:
r(X,y)[U2_0x(x,y,0)_Oy(X,y,0)],
(7) since,(x,y,0) = 0.
(x,y) is a superposed wave on.-.-i-e. U
4(x,y,z)
rigid-wall olutiOn
Fig. ii Scheme Of tilemodeled Phenomenon 1(X,y,z)SoIurion of thin surface layer
MTB 109 August 1976
flr
ttt
21
I 2-oI
IIO.I
15
'U
i-t
I I I I IA______
0 I I I I I 0 0 ,n is a normal-
Ux =+Hj-
on zHx,z),
vector on the body surface,O(h,41x2+y2)
forx>0,
as
x2+y2-°°
(4)
(5)
MTB1O9 August 1976
In the low speed limit we introduce the following
as-sumptions about orders of magnitude. The detailed reason-ing of the assumptions is found in Ogilvie's paper
Ø(x,y,z) = 0(U),
,(X,y) =0(U2),
a a a
-'
, -= 0 (1) when operating on4)(x,y,z) or
ax ay az
Cd)
4)(x,y,z)=0(U5),
ix,y)0(U4),
p,
__.., --
0(U-2) when operating on 4)(x,y,z,)ax ay az
orix,y).
These assumptions are interpreted physically as follows.
To an observer at the free surface around a slow ship, it appears that there is a uniform stream below him which has a speed equal to the local value (cbrx, øry). This
Ob-server can not see the free surface disturbance represented by ,(x,y), for it varies on a length scale which is too large. Only the small, superposed waves, represented by
(x,y), will be visible to him.
Under the above assumptions about orders of magni-tude, the lowest order terms of the free surface boundary conditiOns (3) and (4) are written respectively (see
Ap-pendixA):
-gix,y)+ 4)rx(X,9,0) Ø(x,y,z)+ 4)(x,y,0)
x cb(X,Y,Z) =0 on Z = r(X,Y), (8)
4) (x,y,z)
-jx, Y) 4),-(x,y, 0)-
x,y) 4)(x,y,0)
= -
[r(x,y)ct)rx,y,0)] +[r(x,y)4)ry(X,y,O)],
ax ay
onz,(x,y).
(9) Eliminating (x,y) from (8) and (9), and taking the lowest Order terms, which are of 0(U3), we have the followingfree surface condition (see Appendix A):
[4),-., (x,y,0)--+ 4),. (x,y, 0)-_-] 2 4)(x,y,z)
+4)(x,y,z)
D(x,y),
on z = r(X,Y) (10)where
ar
-
D(x,y) = [,(x,y)4),(x,y,0)
ax+ftr(X,Y)4)ry(X,Y,O)].
(11)In general, we have to solve Eq. (10) on z = ,(xy).
Here, however, for the simplicity of treatment we introduce a following non-conformal transformation of coordinãtès.
x' = x,y' =y, z' = z -,(xy).
The Laplace equation fOr4) (x',y',z') is written as
0a2Øa24)
a2ax'2 ay'2 az'2 order
[UI
-2--
a2+0(U5).
ay ayaz
[U3]
Thus, to leading order, 4) satisfies the usual Laplace
equa-tion in terms of the new variables x', y', z'. Taking the lowest order terms and dropping the primes on the new
variables,.our boundary value problem is written as
(12)
[ø,x(x,y,0)+4)ry(x,y,0)f]20(x,y,z)
+Ø(x,y,z)=D(x,y) onz0,
(13) 0(1/s./x2+y2)forx>0,
çb(x,y,z)=
asx2+y2-o
(14) o(h/Jx2+y2)forx<0
+øry(x,y,0)4)y(x,y,0)}
(15)In addition to those conditions the body boundary
condi-tion must be added. Here, however, this conditiOn is omit-ted, since Ø(x, y, z) is considered as the solution of the first
step of an iteration cycle.
In a recent paper by Newman (1976) a free surface condition which is identical to Eq. (13) has been derived independently., In hiS derivation the double body
poten-tial is also used as the zero-order solution for the slow ships.
When droppWig the terms including cbry, we have a free surface condition which coincides with the Ogilvie's two-dimensional expression. Ogilvie obtained an explicit solu-tion by the use of complex funcsolu-tions. In our three dimen-sional problem, Fourier's method is used with a slight modification.
4.2 Solution of the present boundary value problem
First we introduce a double layer potential 'I' (xy,z) which equals to D(x,y) at z 0 as
'4'(x,y,z)
_4LfkdkJ'dO
X1fJddflD(,).
(16) We then assumethe form of velocity potential 4)(x,y,z) asØ(x,y,z) .=
_f kdkf
do ekz+ixc0s05mn0)
xF(x,y,k,O),
(17)where
F(x,y,k,O) is
determined in such a way that Ø(x,y, z) satisfies Laplace equation and the free surface condition(13). From (17) we may consider that
F(x,y,k,O)
isrelating to the wave amplitude which depends on the in-tensity of disturbance acting on the free surface. In our problem D(x,y), the right hand side of (13), is considered as a disturbance acting on the free sürfacè. Then we may
a ,. a2
ax' ax'az'
assume intuitively thatF(x,y,k, 0) is expressed in terms of
the double body potential. This assumption means that the operation a/ax or
a/ay
onF
does not change the order of magnitude due to the assumption (C). (x,y,z) is already assumed to be of0(U5)
in the thin boundary layer near the free surface wherex,y,z=0(U2).
Therefore, from (17), we may deduce the following orders fork andF(x,y,
k, 0):
k0(U2),F(x,y,k.0) 0(U9).
Then,
4 (x,y, z)=
kdkf
do e kz+ilc(xcosO+iisin8)[_k2cos2OF+2ikcosO.f+.L
ax ax2
[U5]
[U7]
Eu9]
The second and third terms in the above bracket are of
higher order than the first term. The same rule is applied to and When taking the lowest order terms, we see
first that 0 satisfies Laplace equation. Then, substituting
them into the free surface condition (13),
F(x,y,k,0)
is determined as follows.k0(x,y,0)
F(x,y,k,O)
lko(x,y,0)k k
xffdd7iD(n)e
-1kcosO+sinO) whereko(x,y,0)g/fr,b(x,y,0)cos0 +Ø,(x,y,0)sinO
2 (20) We see thatF(x,y,k,0)
is expressed in terms of rigid-wallsolution. Thus
F
satisfies the previously assumedcondi-tions,
e. aF/ax, aF/ay=o(F).
Substituting (19) into (17), and taking the radiation condition (14) into account, we have the following expres-sion of our velocity potential:
dO k0(x,y,0)
xpvf
dkk0(x,y,0) k
+ ,.1T/2XJ
ko(x,y,0)ekoOsin k0(x,y,0)&i}dO.
(21) whereW= (x)cos0 + (y) sin0.
The wave height due to the potential cb(x,y,z) is
deter-mined from (15) when taking the lowest order terms of derivatives with respect to x and y. Here, it should be noted that a/ax,
a/ay=o(1)
when operating onk0(x,y,O):17/2 x
f
(x,y,0)cosO+0ry(X,y,0)sinO xpvJ'
k(
0)k
ffddnD(,17)
(/2
cos1k0(x,y,0)}dO
f0 (x,y,b)cos0 +Ory (x,y,0)sin0 In the far downstream where cb,., -+
height is expressed as
(x,y) dO [S(0) sin -- sec20 (xcos0 +ysinO) 7/2
+ C(0)cos _sec20(xcosO +ysinO)} 1, (23)
where
C(0) +iS(0)
= ---sec0
irU3
i__sec2O(cosO+tsriO)
j2
Then the wave making resistance R is expressed as
RwlrpU2J
IC(0)+zS(0) cos30dO, (25)0
where p is the density of water.
In principle, we can calculate wave making resistance of
any body shape, since D(xy) is determined from the rigid-wall solutions which are obtainable numerically or
analytically.
5. Waveresistance of simple forms at lowspeed
5.1 Wave resistance of a vertical circular cylinder and a semisubmerged sphere
Calculation of wave pattern around a ship by the use of
the present theory makes it possible to compare directly with the observation. However, this calculation needs lengthy numerical work even for a geometrically simple form. For the preliminary stage to examine the validity of the present theory, it may be enough to calculate wave
making resistance of simple forms.
For the convenience of analytical work, a vertical
infinite circular cylinder and a semisubmerged sphere are
considered. The double body velocity potentials are written as follows.
0(x,y,z)=0(x,y).=U(x+
x2+y2/22Xfor a vertical circular cylinder of radius a, (26)
a3x
3/)
2(x2+y2+z2)
for a semisubmerged sphere. (27)
From (11) we have the following non-dimensional expres
sion forD(x,y):
dO MTB 109 August 1976 (22) U, cb 0, the wave (24)MTB 109 August 1976
L_i_P(s)cosl3+J_Q(s)cos313 s 1 (28)
where the origin is fixed at the center of the cross plane of the cylinder or the sphere and the still water plane, and.
F = U//, x/a= scos3,y/a
=ssinl3, s= /2+y2/a,P(s)=2s4s6,Q(s)=s2
for a vertical circular cylinder,Q(s)=--s
fora semisubmerged sphere. Substituting D(s,13) into (24), we have
C(0)+iS(0)=i4asec3U
{cosOfP(s)Ji()ds
_cos3OfQ(s)f3(S20)ds
where the cross area of a body and the still water plane iS
excluded from the. integral range, and J1, J3 are Bessel
functions.
The asymptotic expression of the amplitude function in the low speed is derived based on the stationary phase
method:
C(0) + iS(0) = i-1--F[P(1)cosO +Q(1)cos3O]
x cos(J_sec2O___) + O(F5),
2F2 4
where P(1)1, Q(1)=-1 for a vertical circular cylinder, and P(1)=75/128 Q(1)=-27/128 for a semisubmerged sphere.
The wave making resistance coefficient C in low speed limit is expressed as follows.
C
Rw/PU2(2a)2FO(Fn)
for a vertical circular cylinder,for a semisubmerged sphere.
C values are shown in Fig. 12 comparing With experimen-tàl values of a simple ship model M.2201 used for the flow
measurement in Sec. 3 and of a conventional full form of Cb=O.84. It is found that the order of magnitude. of
theo-retical values agrees with that of experimental values which are determined by Hughes method (CwCtotai
- (1+k)c1
Hughes' k is the form factor). It is also shown that the trend of theoretical C curves with respect toFroude number resembles that of experimental curves. This result, encourages us to apply the present theory to slow ships. -4 4u10 - - - Experiment - Theory ypu'if
L Lead wateillae far M.2201 and Conventional form
L 2a for Cylinder and Seft,
,-sjbmerdhme-_0
-0.10 0.11 0.12
Simpl, form M.220i Brarde theory deep draft> //ciroular
Sünpform Shallow draft Present theory
(5erd
a- Corwaimonal _-.- full form CBO.84 0.13 ..0.140.15 0.16 u4grFig. 12 Comparison of wave resistance
5.2 Comparison of the present theory with conventional theories
For the comparison with other theories we calculate wave making resistance of a vertical circular cylinder by two other methods The first method is to calculate wave making resistance due to the surface source distribution
on the cylinder which
is determined by the.. so-called Zero-Froude-number approximation by Kotik and Morgan (1969). This' is the conventional method Of calculatingwave making resistance. The second method is the Brard's
one which includes the contribution of the line singularity around the intersectiOn of the, body and the still water plane In addition to the above surface source distribution
(Brard 1972).
In the conventional method the amplitude function due to the surface source distribution for a vertical, circular cylinder (b/a= e - 1 'in (39) of Brards paper'together'with c=L1) is written as
B (0) + iA(0) = -
i4aJ1 (sec20).
(32)The asymptotic form of Cat low speed limit is'expressed
as
zi
It should be noted that the first term cancels out the amplitude function due 'to the surface source distribution. The asymptotic fOrm of the sum of both amplitude
func-tions (32) and' (34) in low speed is written as
flmory
(Circular Cfilnder)
-
8Fsin(-1-+--)+ O(F2). (33)
The order of magnitude of C is different from (30) by F Therefore it is evident that this formula gives practi
cally unacceptable high values in low speed.
In the second method, the amplitude fUnction due to ('31) the line singularity is derived as follows for a vertical
circUlarcylinder [ c - 1 in (52) Of Brãrd's paper
B3(0) + i&4(0) i4aJ1 (_ijsec20)
+ i8a.F,seo9cOs30J0(__t2sec2O)
B(0) + iA.(0)
+6B3(0) + iöA (0)cos30cos (_!_sec20__IL)
+ O(F,).
(35)2F, 4
This is quite similar to the amplitude function (29) derived by the present theory. The order of magnitude of both
expressions agrees with each other i e O(F) The asymp totic form of C at low speed limit is expressed as
C =
+ 32V7F,sin(+---) +O(F,),
(36)which is identical to the wave resistance of a vertical cir-cular cylinder derived independently by Guevel et al.
(1974) in the frame work of the linearized free surface condition. C values expressed by (36). are also shown in
Fig. 12. C curve by Brard's theory is rather oscillatory. compared with that by the present theory where the oscillatory term is disappeared as a higher order quantity for the case of a vertical circular cylinder. The reason for the strong oscillatory property is due to the linearization
of the free surface conditions in Brard's theory as explained in the followings.
D(x, y)
of a vertical circular cylinder is rewritten from(11):
iU
2D(x,y)=
(U+rx)pxx(px+(Pry)I
+ c0ry cPrxy -
(i'
+ } , (37) where p is the perturbation velocity potential of the double body approximation:p(X,y)=
Ø,(x,y) - Ux=
Uax
for a vertical circular cylinder.
Neglecting formally any products of derivatives of the
perturbation potential, we have
U2 .
2U3a2 x3+3xy2
g (x2+y2)3
The, non-dimensibnál expression of D(x,y) in the polar
coordinate system is written as D(s,13)
. S3cos33 (38)
4UF,
This expressiOn coincides with the second term of the right
hand side of (28). Therefore the asymptotic form of the
amplitude function in the low speed limit is expressed from (29) as
C(0)+S(0)
(39)
which IS exactly the same as Brard's amplitude function (35) which includes the contribution from the line singular ity. This theoretical result indicates that the reason for the
MTB1O9 August 1976
strong oscillatory property in Brard's theory is due to the
linearization of the free surface conditions. Recently Nakatake and Yamazaki (1976) proved that the expression
of the velocity potential, which ihcludès the line integral
term in addition to the Havelock wave source term, can be
derived from the velocity potential (21) of the present theory after the formal linearization.
In the lOw speed, however, the linearized free surface
condition becomes less and less accurate as Brard suspected.
Then we may say that the present higher order theory is one of the ways to overcome the lack of accuracy arising in the low speed problem with the linearized free surface
conditions.
In Appendix B wave resistance formulas of a
sèmisub-merged three-axial ellipsoid and a vertical infinite strut of
arbitrary across section shape in low speed limit have been derived.
6. Examination of the present theory by experiment For a direct comparison of the wave resistance obtained
by the present slow speed theory with experiment, a mathematical hull form defined by the following formula
is considered.
y = 2Bf(1---)11
(L)21L/B= 10.0,d/L
= 0.0625. (40)This form has been known as Wigley's parabolic fOrm. For this form it is expected that the viscous effect on the wave generation is very small compared with that of blUnt forms such as a vertical circUlar cylinder and a
semisub-merged sphere.
By the use of the expressions (B-6), (B7) and (B-8) in Appendix B, the amplitude function of the Wigloy's
para-bolic form and the wave resistance is calculated
numerical-ly. In this computation the velocity components due to the double body potential around the load water line are obtained first by Hess and Sniith method with 26 x 10
source panels lengthwise and dèpthwise respectively. Then the funätion F(L3) expressed by (B-8) is calculated, where the numerical differentiations are Used fOr a17/a13,
and The integration with respect to j3 in (B6) is carried out carefully by searching the zero points of the
integrand. The wave resistance values are then obtained by
the use of the expression (25), where the upper limit .of the integration with respect to 0 is about 750 The validity of the procedure of the numerical computation has been confirmed by applying the computer program to the case
of a sernisubmerged sphere fOr which the asymptotic value of the wave resistance is known analytically by the. expres-sion (31).
Fig. 13 shows the computed results comparing with :Michell's theory and the experimental results of Wigley's form of 5 meters long. In the range of Froude number less than 0.25 a remarkable attenuation of wave resistance
MTB 109 August 1976 .4 *10 5 Vt{A(e)}cos30/L2
FnU/(jt0.2O
N
. Wove pOttern analysisMichell \ / (longitudinal Cut)
\
'I
iuJA A'ikA
20° 30° 40° 50°
e
Fig. 14 Comparison of wave spectra Of Wigley's parabolic model
curve with respect to Froude number is attained by the-present author's theory. A quantitative agreement is also
observed between the present theory and the experimental values determined by means of wave pattern analysis and by the conventiOnal Hughes method.
A comparison of wave spectra at Frouie number 0.20
is shown in Fig. 14. When comparing with Michell's
linear-ized theory, it is a characteristic of the préséht theory that
the wave spectrum corresponding to the transverse waves (small 0-values) is considerably reduced and approximately
agrees with the value détérmined -by the wave pattern
analysis.
This numerical study ii,dicates the validity of the
asymp-totic solution (21) of the lowspeed free surface condition (13) which takes into acôoUnt the nonlinearity of the flow
around a shiP
7. Applications to the hull form design
7.1 Characteristic of wave resistance coefficient in low
speed limit
In Sec. 5.1 it was found that the wave resistance
coeffi-cient of full forms with deep draft has the following form
in the low speed limit:
C = inF,,
fU2L2
Wigley's parabolic model Y 2B(X,&Xl_X4..)(u_(Z/d)5) %lO. d/LO.0625 Micheli
/
/L\
/1
/\__/
70°Fig. 13 Comparison of wave resistance coefficients ofWigley's parabolic model
0
Wave resistance
-determined from towing test
(Hugher method)
-Wave Pattern Onólysie
(ion.itudinol cut) 0.25 0.30
F, U/
where m is determined by the body geometry alone. The exponent of the Froude number is different from that derived from Michell s wave resistance theory in the low speed limit, where C is proportional to F.
Hughes (1966) and Prohaska (1966) used the expression = nzF, in determining the form effect on the viscous
resistance from the towing test data in low Speed range. For
instance, Prohaska proposed the following formula for the determination of the form factor k,
C/C', (1+k)+mF,/C'f,
-where C is the total resistance coefficient, and is the
frictional resistance coefficient. At Hydro-Og Aerodynam-isk Laboratorium results of about 200 model experiments - were plotted based on this formula. For Froude numbers between 0.1 and 02 the Cf/C'1values for a great majority of the models did plot on straight lines against Ft/Cf
values. However, for full -forms, say withCb > 0.75, it was
found that the points may plot on concave curves. ihis indicates that either (1+k) or m or both are speed
depen-dent.
-- lnui (1968) studied the exponent of the Froude number for full forms in low speed assuming a form
C/C1= (1+k) + CW/cf. (42)
where c'=mF'1.
From analyses of a number of model experiments of full forms. lnui found that for deep draft condition n6 gives better fitting to C than n=4. In the shallow draft condi-tion, on the other hand, n=4 or n=5 gives better fittings. Fig 15 shows some examples of lnui's plottings Of the experimental data of a full form with Cb=O.8O based on
the fOrmula (42).
Fig. 16 shows another example of fittings. C-values were determined by Taniguchi (1966) from the towing
test data of geometrically similar models of 4.2, 7 and 10 m
long with CbO.BO. lnui obtained the following fittings for C-curves:
C (Full load)
Rw/i_pU2V
= 83i9F,,
(43)Rv
(41) C(Ballast load) 28O7F,
xiÔ5 O4 0.3 0.2 0.1 0.15 0.10 0.20
I.e 1.6 1.4 Ct,cq Full 0
.
.
.
0 0 p 9, 0.0,0.12 0.l4 Ct/Cf (IK)m.F,!/Cf 0.
0 S 016 0.18 0.20 F S S.
S flc 6 S Fn 0.19 0.10 0.14 0.15 0.16 0 0.5 1.0F/CI
l.5z102Fig. 15 lnui's plotting based on Prohaska's proposal
Fig. 16 lnui's fitting of C-curves of a tanker model
-8.88Fcos[S(1+)--+--],
(44)where c7 is the displacement volume, C Le/L, Le is the
length of entrance,
c=p/C, and
F
is the forward shift of bow waves as shown in Fig. 16. For this ship form C =0.26, 0.173. lnui's expression (43) forCs,,, in fullload condition well corresponds to the expression (41)
obtained by the present theory.
The water line form at the ballast condition of this ship form has a point where the curvature equals to zero.
For such a form the present theory suggests that C, in low speed limit is proportional to as explained in
Appendix E (E-5). Therefore lnui's fitting (44)
approxi-mately corresponds to the theoretical prediction.
Thus, the present wave-resistance theory can be used for a reasonable determination of the level of viscous
resistance curve in low speed limit.
7.2 Decomposition of wave resistance into bow wave
resistance and stetn wave resistance
For the sake of convenience, an infinite vertical elliptic cylinder is considered. C is expressed in a form (see
Appendix B)
I' -D
TT2,2_ r'6I.
L -
mLrfl,
where the oscillatory term of O(F,) disappears, and
67T/
316(1+e)
r
2sin4Ocos8O(1+e2tan2O) 'adO, Joe =B/L.
Since mJ, is determined by a geometrical parameter e alone, the calculation of C is easy when mL is known for a givene. Table 2 showsmL values for variousL/B.ratios.
From the table it is found that for a fixed Froude number, the wave resistance coefficient C, is not always getting smaller with increase of L/B ratio. The reason for this tendency is explained as follows. Beyond a certain L/B ratio, say L/B'5, the change of velocity around the elliptic cylinder becomes larger with increase of L/B ratiO when
the ship speed U and the ship length L are fixed, since the velocity which was null at the stagnation pOint has to reach near the ship speed within a very short distance. ThiS means
that the Froude number defined by is very large
so that the disturbance on the free surface around the
bow becomes large and then contributes to an increase of wave resistance.
When introducing ship breadth B as the characteristic length instead of ship length L, the wave resistance
coef-ficient is rewritten as
C
,Rw/PU2B2 = mBFflB,
where mB = emL, and FnB=
U//.
mB values are also listed in Table 2. Fig. 17 showsmB
and mL values versus L/B ratios together with those of a simple full form having semi-circular bow and stern with parallel middle bodies of different lengths. It is observed thatmB values tend to reach a constant value with increase
of L/B ratio while mL values increase. This calculation
indicates that the ship breadth B is an appropriate
charac-Table 2 mLandmB mL
2
MTB 109 August 1976 1.00 0.60 0.40 0.20 0.10 0.05 L/B 1.00 1.667 2.50 5.00 10.00 20.00 mj, 26.01 11.43 9.543 13.38 30.76 92.34mB=mL
26.01 6.858 3.817 2.676 3.076 3.117 Bo.L Lost S CCf 1.8 1.6 .4MTB1O9 August 1976
/
/
30
Simi circuLar bow
m6 a.nd ;tsrn
6 7 8 9 10 Ii 12 13
Fig 17 Comparison of rn-values
teristic length rather than the ship length L for a compara-tive study of wave resistance of full forms which usually
have long parallel bodies.
Figs. 18 and 19 show C values versus F,,. The figures show that C, values tend to reach constant values for a fixed FflB with increase of L/B ratio This implies that the interference effect of potential flow between bow and
stern form in generating waves is negligibly small. In other
words, wave resistance may be separable into two parts, e. one due to the entrance part and the other due to the
run part, whenL/Bratio is greater than, say, 6.
In order to verify this theoretical prediction, towing
test data of several models having the same bow form and
stern form with different length of parallel body are exam med as shown in Fig. 20. There. is no appreciable effects of the difference of L/B ratios on wave resistance C'
except in the higher speed range (F, > O.5) where inter-ference effects between bow and stern may exist. Thus those experimental results support the prediction of the
theory in the low speed range.
Further, Fig. 21 shows other experimental results of models having different stern forms with same bow form and parallel middle body. The effect Of difference ofstern forms is very small. Usually the afterbody of full forms is designed to have a relatively fine form compared with bow forms in order to prevent flow separation which has
unfavorable effects on propulsive performance. In addition
to this
the viscOus boundary layer and wake behind aship may reduce the waves generated by the stern. There
fore wave generation from the run is relatively small com-pared with that from the bow Then we may say that the wave résistánôè characteristics of full forms are mainly
depending on the bow geometry.
In Appendix F a design method of full forms based on
these experimental and theoretical evidences is described.
In summarizing the present section we may conclude
0
a
0
Elliptic cyLinder
L, 1.0
Fig. 18 C-valuesfor elliptic cylinders of variOus
L/B ratios
Fig. 19 C-vaIues for full forms of semi circular bow
and stern
Fig. 20 Comprison of among the 3 ship forms having the seme entrance & run with different parallel body
that the bow form can be treated separately from other parts of the body when L/B> 6 Therefore, we may con-centrate our effort upon the improvement Of bow forms
in order to reduce wave resistance of full forms of
conven-tional type.
For instance, Fig. 17 is considered as a comparison of mB values of two different water line forms, i.e. one is a semi-circular water line form and the other is an elliptic
020 015 - -010_____________________________________
EI
C52
/
infinite cylinder with ómi circular bow
and Stern/ Fe': Fn!U/J -;---.-FnB=WJ
,
,
,
mL , WptiC __' cytlfld.y 0.50 ass 040 04B oso 0.25Fig. 21 Comparisons of C among the models, having the same entrance & different run
water line form. The latter gives smaller mB values. This means the elliptic water line form is better than the semi-cirëular water line form in reducing wave resistance. In
practice we have to determine the optimum water line forms under the given design constraints. When we apply the present theory to general full forms of finite draft, we may obtain information about the effect of not only of water line forms but also frame line forms on wave
resistance.
7.3 Effect of protruding bow in reducing wave
resist-ance
7.3.1 Problems arising in the case of shallow draft
Discussions on the formula of wave resistance coeff i-cient C have been concerned with cylindrical full forms of deep draft. In this section we consider full forms of
shallow draft.
Calculations of C-values for various Bid ratiOs are car-ried out fOr semi-submerged ellipsoids of
L/B1.Q. As
shown in Fig. 22 ellipsoids of large Bid ratios give high
Cs,-values with strong oscillations with respect to Froude
number.
To find the reason for the increase of C-values for the
case of a large B/d ratio, calculations of the function D(x, y), the right hand side of Eq. (13), which is the dis-turbance induced on the free surface by the double body velocity potential, are carried out. Fig. 23 shows com-parisons of D(x, y)-values on the x-axis for various Bid
ratios.
It is observed that peak values of D(x, y) increase and come c!ose to the body with the increase of B/d ratio. In Fig. 24 calculated wave heights represented by 0)
for various Bid ratios are shown When comparing Figs 23 and 24, it is found that steeper waves give higher peak values of D(x, 0). Those calculations indicate that the reason for the enormous increase of C values in the shal-low draft condition is due to steepening of waves. Further, it is understood that large values of D(x, 0) at x,'a =1.0 contribute to enlarging the oscillatory term of C, , the
third term of formula (B-14), since there are relations:
F(0)=D(a, 0)/UF,, and F(ir)
D(a, O)/UF,.
When waves induced by- the double rnodeI'-potentiaIbecome steeper
it
is anticipated that the assumption of the present theory, i.e. the condition of slowly varying basic flows, is violated and the free surface becomes lessL1
B/d=l0C UFU B/a'4.0 'O.O(d'oo)-
-ao UFig. 23 D(x,0)-values for semi-submerged ellipsoids of variousBidratios
- U
12 4 0 X/a. 0 B/d'00 (doo) -1.2 Lon9itudülaL center pLOne(L/B'l.0)
-20
'jdoo)
Longftudinat center pLane
(LIe' 1.0) MTB1O9 August 1976 '0.0 'd'2.0 B/d'40 BId' 10.0 ,d'0.0 0 20
U__rrnrn
Ce#0.82C_________
-Fig. 22 C-values for semi-submerged ellipsoids of various Fig. 24 rx,0)-vaIues for semi-submerged ellipsoids of
Bidratios variouSBidratiOs
0.25 &50
0.35- 040
ass 050 FnB'Wfj 0.13 0.07 0.00 0.09 0.10 0.11 012 0.14 0.15 0l 0I 050 0MTB 109 August 1976
and less stable, so that even a small perturbation induced by the additional potential (x, y, z) results in breaking
waves.
From (22) the wave number of superposed waves on (x,y) is known as
kO(x,y,O)g/4,(x,y,O)cosO +Ø,.(x,y,0)sinO}2.
(45)This expression indicates that wave length becomes very short near the bow, since the flow becomes stagnant. Furthermore steep local waves enlarge the amplitude of
the superposed short waves, since the amplitude depends
on the intensity of D(x,y).
According to the theoretical studies by Longuet-Higgins and Stewart (1960, 1969), short waves riding on long waves tend to be both shorter and steeper at the crest of the long
waves than they are in the long-wave troughs; If the lông
waves become steeper, the steepening and shortening of the superposed short waves are drastic. This is considered as one of the triggers of appearance of breaking waves (white-caps) on the crest of waves in deep water.
Therefore we may consider that in the vicinity of blunt bows there exist similar free surface phenomena to those
of ocean waves.
-When wave breaking occurs, the present theory is not applicable any more. However, we may consider that the function D(x, y) is used as a measure of wave breaking inception. Although a correlation between D(x,y) and the
incipient breaking of bow waves has not been derived yet, it, is considered that wave breaking phenomena will be suppressed by reducingD(x,y)-valües in front of the bow.
- 7.3.2 Reduction ofwave resistance byprotruding bow From both model experiments and service peformances
of full forms, the effectiveness of the protruding bows in
reducing wave resistance has been confirmed.
To explain this effect a protruding bow is considered separately from the main body and is replaced by a
sub-merged sphere for the sake of simplicity. Then calculations
of D(x, y)-values of the submerged sphere for various
im-mersions are carried out.
Neglecting the effect of induced velocities on the sphere
by its image system above the free surface, the velocity potential for the submerged sphere in the uniform flow is
written approximately as
Ua3 x
ør(x,y,z) = Ux +
x22+(z_f)2
+ x 1, (46)
x2+y2+(z+f)2
where f is the immersion of the sphere.
After substitution of velocity components derived from 0,. into (7) and (11), an expression for D(x, y) on x-axis
at the free surface is obtained as
D(x,O)_6(a3-4af2)
(1j2)
UF, (a2+72)%
i2(3af2-3o3)i
where F=U//, and a=x/a,f =f/a, and
2
1+ 1
(1 3a
(a2+f
2342+j2
Fig. 25 shows D (x, 0) values for different immersions.
It is shown that a negative peak value appears in the front part of the sphere and it becomes larger with a decrease
of immersion.
This simple calculation suggests that a submerged sphere
or a protruding bow works in cancelling D(x, y)-values
induced by a main body in front of the bow, i.e. the
protruding bow is effective in reducing steepness of local bow waves. Further it is suggested that the shallowerim-mersion gives a greater effect of cancellation. Then we may
consider that in the ballast load condition the protruding bow contributes to the reduction of wave breaking resist-ance which is attributed to the steepening of bow waves. In the full load condition, on the other hand, it is
con-sidered that the effect of the protruding bow is relatively small compared with the effect in the ballast load con-dition. This theoretical prediction coincides with our ex-periences about the protruding bulb which are mounted
near the bottom of bow.
For the improvement of resistance characteristics in the full load condition Couch and Moss (1966) developed bow forms with protruding bulbs which are raised up to
the middle part of the stem (R-series). Experiments showed
a marked improvement in the full load condition. This is a good experimental evidence to verify the above prediction
about the bulbs of shallow immersion.
Fig. 25 D(x,0)-values for a submerged Sphere (41)
8. Concluding remarks
From the studies in the previous sections we may expect that the present asymptotic solution of the non-linear free surface problem is applicable to the prediction of wave resistance characteristics of conventional ship forms which have not always been tractable by Michell's
linearized wave-resistance theory.
For instance, the asymptotic expression CmF, may
be used in determining a reasonable level of viscous
resist-ance curves of full forms in low speed limit, and the wave
The author wishes to express his deep appreciation to Mr. K. Tamura, manager of Resistance and Propulsion
Laboratory, Nagasaki Technical Institute and Mr. K. takekuma, senior research engineer of the laboratory, for
their stimulating and encouraging discussions. Thanks are
also due to Mrs. M. Hara and Dr. Y. Kayo for their efforts
Baba, E. and Takeküma, K. (1975a), A Study on Free-Surface Flow around Bow of Slowly Moving Full Forms, J. Soc. Nay. Arch. of Japan, Vol.137, 1-10.
Baba, E. (1975b), Blunt Bow Forms and Wave Breaking, The First STAR Symposium on Ship Technology and Research, The
Society of Naval Architects and Marine Engineers.
Brard, A. (1972), The Representation of a Given Ship Form by Singularity Distributions When the Boundary Condition on the Free Surface is Linearized, Journal of Ship Research, Vol. 16, No.1, 79-92.
Couch, R. B and Moss, J. L. (1966), Application of Large Protrud-ing Bulbs to Ships of High Block Coefficient, Trans. SNAME, Vol. 74,392-441.
Dagan, G. (1975), Waves and wave resistance of thin bodies moving
at low speed: the free-surface nonlinear effect, J. Fluid Mech., Vol.69, part 2,405-416.
Guevel, P., Vaussy, P. and Kobus, J. M. (1974), The Distribution of Singularities Kinematically Equivalent to a Moving Hull in the Presence of a Free Surface, International Shipbuilding Prog.,
Vol.21, 31 1-324.
Hermans, A. J. (1974), A matching principle in nonlinearship wave theory at low Froude-number, Delft Progress Report, Series F., Vol.1.
Hughes, G. (1966), An Analysis of Ship Model Resistance into Viscous and Wave Components, Parts I and II, Trans. of RINA,
289-302.
lnui, T. (1968), Separation of Ship Resistance Components, Symp. on Ship Resistance and Propulsion, Soc. Nay. Architects of
Japan, 39-53.
Keller, J. B. (1974), Wave Patterns of Non-Thin or Full-Bodied
Ships, 10th Symp. on Naval Hydrodynamics.
Kotik, J. and Morgan, R., (1969), The Uniqueness Problem for Wave
Resistance Calculated from Singularity Distributions Which are Exact at Zero Froude Number, Journal of Ship Research, Vol.13, No.1, 61-68.
Acknowledgments
References
MTB 109 August 1976
resistance formula developed in the text is used for the development of full bow forms of least wave resistance by
taking into account the effect of protruding bows.
However, when the waves are broken, the present theory is not applicable to a study of the free surface flows around ships. Then a special analytical treatment to analyze
turbu-lent vortical flows around the bow has to be developed.
Before an accomplishment of such analytical means,
detailed investigations into the. free surface flow around
the slow ships are required in the future works.
in developing the computer program for the calculation of wave resistance of arbitrary body shape. The author also wishes to express his appreciation to all members of. Nagasaki Experimental Tank for their cooperation in
car-rying out this investigation.
Longuet-Higgins, M. S.. and Stewart, R. W. (1960), Changes in the form of short gravity waves on long waves and tidal currents, J. Fluid Mech.. Vol.8, 565-583.
Longuet-Higgins, M. S. (1969), A nonlinear mechanism for the
generation of sea waves, Proc. Roy. Soc. A.311, 371 -389.
Milne-Thomson, L. M. (1960), Theoretical Hydrodynamics, 4th Edition, 506-511.
Moriya, T. (1941), A Theory of an Arbitrary Wing Section, J. of Soc. Aeronautical Science of Nippon, Vol.8, No.78,1054-1060. Nakatake, K. and Yamazaki, R. (1976), DiscussiOn on the line integral of wave resistance theory, The Proceedings of
Interna-tional Seminar on Wave Resistance, Tokyo, 445-446.
Newman, J. N. (1976), Linearized Wave Resistance Theory, The Proceedings of International Seminar on Wave Resistance, Tokyo, 31-43.
Ogilvie, T. F. (1968), Wave Resistance: The Low Speed Limit, Univ. of Michigan, Naval Architecture and Marine Engineering, No.002.
Prohaska, C. W. (1966), A Simple Method for the Evaluation of the
FOrm Factor and the Low Speed Wàvë Resistance, Proc. of 11th ITTC, 65-66.
Taniguchi, K. (1966a), Study on Scale Effect of Propulsive
Per-formance by Use of Geosims of a Tanker, J. Soc. Nay. Architects of Japan, Vol.120, 19-3.5.
Taniguchi, K., Watanabe, K. and Tamura, K. (1966b), On a New Method of Designing Hull Form of Large Full Ship, based on the Separability Principle of Ship Form, J. Soc. Nay. Architects of Japan, Vol.120,36-45.
Taniguchi, K., Tamura, K. and Babe, E. (1971, 1972), Reduction of Wave-Breaking Resistance by "MHI.Bow", Mitsubishi Juko Giho (Japanese), Vol.8, No. 1, or Mitsubishi Technical Review (English), Vol.9, No.1.
Timman, R. (1974), Small Parameter Expansives in Ship Hydro-dynamics, 10th Symp. on Naval Hydrodynamics.
MTB 109 August 1976
The dynamic and kinematic free surface conditions are
written respectively:
_!_1j2=
gH(x,y)
+ onz =H(x,y),
(A-i) o
42(x,y,z) onzH(x,y),
(A-2)where zH(x,y) represents the free surface.
In our problem we assume that the total potential
'(x,y,z)
and the wave height
H(x,y)
are expressed as follows:c1(x,y,z)= ø(x,y,z)+ø(x,y,z),
(A-3)H(x,y)
=r(X,Y} + ix,y),
(A4)where
x,y) =__[U2_Ø, (x,y,
0)Ø,,(x,y, 0)].
By the substitution of (A-3) and (A-4) into (A-i) and (A-2),
the free surface conditions are written as:
U2
(x, y,0)
Ø(x,
y, 0)1 +gix,y)24rx(X,y,Z)øx(X,Y,Z)
+q(x,y, z)+ ci(x,y, z)-i- z}cb(x,y.z)
+
ø,z(X,Y,Z)+(x,Y,Z) +
2ct2(x,y,z)Ø(x,y, z)],(A-5)
0
[rx(X,YHx(X,Y)]
[Ø,(x,y,z)+Ø(x,y,z)]+ (X,y) + y(X,y)] [øry (X,y, z) + ø(X, y,Z)]
,z(X,Y,Z) ø(X,Y,Z). (A-6)
Based on the Ogilvie's assumptions on the order of
magni-tude (a) through (f) in the text, the following Tayler expan-sions at z=O are derived for ØPX(X,Y,Z), øry(X,Y,Z) and
rz(X,Y, z):
rx(X,Y,Z)=ørx(X,Y,0)+(r+irxz(X,Y,0)
+_(r+)2rxzz(x,y, 0)
= 'P,x,Y,
0) 0)+[U]
[U5]P(x,y,z}=
øpy(x,y,0) +t(x,y)t1ryzz(X,y,
0)4.
[U] [U5]
z)= r(x,y)ctrzz(x,y, 0)
[U3]
+ (x,y)cb,,(x,y,
0) +...
[U5]where the following relations are used
ørz(X,Y, 0) = 0, 0)= 0, ryz(X,Y,0)= 0.
On the other hand, the Taylor expansions at
z=,(x,y)
are derived for cb (x,y, z), Ø,, (x,y, z) and ØZ(X,Y, z):
øx(X,Y,2)x(X,Y,r)(X,Y)øxz(X,Y,r)" [U3] [U5]
Appendix A. Derivation of the free surface condition (10)
øy(X,Y,Z)y(X,y,r)+(X,y)cbyz(X,y.,)+."
[U3] [U5]
Ø2(x,y, z)= z(X,Y,
+ (x,y)gzz(x,y, r)
[U3] [U5]
By substituting those expansions into (A-5) and (A-6),
and taking the lowest order terms, we have
0=
gix,y)
+ ØJ.X(X,Y, 0)Ø(x,y, ,)[U4] [U4]
+ ø,y(x,Y, 0)ø(x,y, ,), (A7)
[U4]
0=
ø(x,Y, ,) +
(x,y)t1(x,y,
0)[U3] [U3]
+y(X,Y)øry(X,y, 0)
+(x,y)ø,(X,y,
0)EU3]
+ry(X,Y)çbry(X,Y,0) r(X,y)ørzz(X,Y, 0).(A-8)
[U3] [U3]
By the use of the relation
ørzz = ørxx The equation (A-B) can be rewritten:
ø(-,Y, ,) -
x(X,y)ørx(X,y, 0)(x,y)çb,(x,y,
0)[r(X,Y)rx(X,Y, 0)] +---[,(x,y)cb(x,y, 0)] .(A-9)
In order to eliminate (x, y) from the equations (A-7) and
(A-9), we take first the derivatives of (A-7):
0= g(x,y) +crx(x,y, O)4xx(X,Y,r)+øry(X,Y, 0)çbyx(X,y,r) [U2) [U2] [U2]
+ ø,xx(X,y,
0)çt(x,y,
+ 0)cb(x,y, ,.)[U4]
+Ø,(X,Y, O)(x,y,
r)+cbry(X,Y,0)øyz(X,Y.r)f[U4] [U4]
[U2] [U2]
[U2]
+ rxy(X,Y, 0)øx(X,y,r) +Øryy(X,Y, 0)(x,y,
[U4] [U4]
+ ørx(X,Y,0) z(x,y, ,)
--+ çb,(c,y,
0) yz(x,y,
r)[U4] ' [Un]
By neglecting the terms of
0(U4)
and substituting x(X,Y) and y(X,Y) into (A-9), we finally have_!_[,(x,y,O) ---+Ø(x,y,0)
] 4(x,y,)
+ctz(x,y,r)
D(x,y),
(A-b)
where
D(x,y)
=---[Ø(x,y, o)(x,y)]
ax+_[ct(x,y, 0),(x,y)].
The amplitude function of the wave represented by 1X, y) in the far downstream is expressed by (24) in the text:
C(0)+iS(0) sec3OffdxdyD(x,y)
x exp[i_2jsec20(xcos0+ysin0)] (B-i)
By the partial integration with respect to x, an asymptotic expression of the amplitude function in the low speed limit is expressed as a line integral around the intersection of
the body and the still water surface:
C(0)+iS(0)=__sec30,fdy
[(-1)
e0lx0
irUn0
pfll
x sec20,fdyD(xo,yo)eY0+Px0+O(U6)
[U4] (B-2) where D(n) ax p=i-2---sec0 q=i__.sec20sin9 U2 U2and (x0, y0) is a point on the intersection of the body and
the still water surface.
When introducing the stream line coordinate system along the intersection, the value of D(x0, Yo) is rewritten as: D(x0 ,Yo) = [ Ø,(x, y, 0) øry(X,Y, o)
j
r(X, Y) ax ay r(x,Y)ct)rzz(X,Y, O)] y=yo= - sgn (yoh/Ø+ Ø,'- ,(x0, Yo)øzz(Xo, Yo 0),
(B3)where 2 is taken along the intersection. Since
sgn(y)Øry(x,y, 0)
dy = - dQ
O)+(x,y,0)
the amplitude function (B-2) is written:
C(0)+iS(0) =
irU J
aQJu+v
x exp[i-
sec20xo(Q)cos0+yo(Q)sin0 } 1. (B-4)where u = px(Xo,Yo, O),v øry(Xo,Yo, 0), = ørz.XO,YO, 0).
Appendix B. Wave resistance formula of a semisubmerged
three-axial ellipsoid and a vertical infinite cylinder of
arbitrary crose section shape
x
a0 = ah
r
(sin2j3+ 2cos2f3)2
MTB 109 August 1976
-J0 (á2+X)s,/(a2+A)(b2+X)(c2+X)
and a, b, care radii, = b/a, 8 = c/a.
ii) Vertical infinite cylinder of arbitrary crosssection shape (see AppendiA D) F(13)
sinj3 {nBcosf3} i +2nB,
+ nBsinn13}G(f3)1PU3)}_3, (B-la) where G(j3)--cosj3+
n2Bcosn13P(13) +nBsinnflQ(I3),
P(f3) =+sih,213 +{nBcosnj3}2,- EnB, cosnf3}{En2Bsinnf3.
When the equation of the intersection is expressed by Fourier expansion through the relations:
--=---cos13, -'--=BsinnI3,
(B5)L 2
L'
where L is the length of a body, and BA are the Fourier
coefficients, the amplitilde function is rewritten as
C(0)1S(0)_
1 sec OF2f dj3F(13)
x
exp{i0 ç(i(13,0)}
(B-6)where F =U/.../j Froude number, and
1i (13,0) cos13cos0 + (2EBsinni3)sin0, (B-7)
F(13) =
[--(1 U2-52)
(1 j22)
J__sin2i3+( nBcosn13)2+ I ,(B-8)
.Jj2+.2 sgn(sin13)
F(j3) is determined by the double body potential and is
depending on the body geometry alone:
I) Semisubmerged three-axial ellipsoid (see Appendix C)
F(j3) - 2
2 2 2
[(2
)2
2a0 sin 13+e cos13
2a0
2 cos2j3 si n2j3
+jcos2fl li(
2ao' sfr2
05213
(B-9)where
MTB 109 August 1.976
+
2-s/F,'F( 0) F(ir)
1+21EnB2+2
(-1Y'nB2
sin(--+j-)
The velocity potential for a translatory motion of an ellipsoid with velocity U in an infinite fluid is given as
Ux
a0
Appendix C. DeriVation of F(a) for a semisobmerged ellip-soid
integration with respect to a in the expression (B-6). In this case the lowest order term of C, is proportional to F7 instead of F as shown in Appendix E.
The following are the expressions of C for simple
forms in the low speed limit:
Semisubmerged three-axial ellipsoid
c,=4F6fJFL
d0+(
2Jo /1+etanO
,'TT 2-a0
x (B-16) since = 0(n 2),a2 =a1 Hp"(a1,o)I= whereF(0) = 2 )3 2 cos4O sin2O (1 + e2tan2O)
2 a0 e
2
+---(
)cos 0 ii - (
) sin 05
2-an
2-a0
It should be noted that the second term of (B-16), the
interference term, decreases with an increase of draftS. Semisubmerged sphere
CWrF6
++\/F,;7sin(-j-+---)+O(F,),
(B-17)which is obtained by substituting e = 1, 5 = 1 and a0 = 2/3
in (B-16). Th.is expression coincides with (31) which is
derived in a different manner as shown in the text.
Vertical elliptical cylinder
C=16F,
(1-I-c)6xJ'sin40cos80(1 +e2tan2O) hdo
+O(F,),
(B-18)which is obtained by substituting 5 = 00, a =2e/(1+e) in
(B-i 6).
Vertical circular cylinder
lily2
= i6F,26 j sin4o cos5Od0 + O(F8)
0
8192 L'6 1)IE'8 315
'
which is obtained by substituting = 1 in (B-18), and co-incides with (30) derived in a different manner as shown in
the text
where
dX
x abci
, (C-i)(a2+ Xh/(a2+ A)(b2+ X)(c2+ X)
- .
(C-2) + X)(b2+X)(c2+ X)
0 (a2+XhJ( Semisubmerged sphere
Substituting e = 1, = 1 and a = 2/3 in F(j3) of the
semisubmerged ellipsoid, we have
F(j3) __cos2f3(_ sin2j3 + 1).
(B-il)
Vertical elliptic cylinder
Substituting = 00 and ao
2e/(1+e) in F() of the
semisubmerged ellipsoid, we haveF(13)- 2 2 2 3 2e3(1+e)3cos2l3sin2fl
(sin 13+e cosj3) (B-12)
which is also obtained from (B-b) by the substitution of
B1=--e,B=O(n2).
Vertical circular cylinder
F(j3) = i6cos2j3in23 (B-13)
The wave resistance of those forms are obtained by means of stationary phase method when carrying out the
integration with respect to a and 0 as shown in Appendix E.
A general form of wave reSistance coefficient C, is ex-pressed as cw = pu2L2
6 r
FW1) dO+ 2F6 {F(132)}2cos0dO'J0
lii"(j3, 0)1
J0hp"(3,O)
(B-14) where j3, a2 are the solutions of the equation-sinj3cosO + (2nB cosnj3) sinO = 0,
for
0j3
: aft body,a2-ir :
forebody,
and
"ia,o) =-cosj3cosO-(2n2Bsinnj3)sinO. (B-15)
The expression (B-14) can be applied to full forms havingconvex water lines, such as semisubmerged ellipsoids. When the load water line has a concave part, there exists a point where 0) =0. At this point the third derivative of
a,b, care radii of the ellipsoid.
There are relations between A and (x, y, z):
a
y
az_
zax
2(a2+A)'ax
2(b2+x)'.ax 2(c2+X)
Further, the following relations can be derived
(Milne-Thomson):
oX 1 ax
ax_
1 Oyax_
i
a ax h OX'ayh ax' azh
OX' where[1
4
(a2+x)(b2+X)(c2+x)
and (A, , v) are the ellipsoidal coordinates.In order to calculate F(j3) defined by (B-B) for merged ellipsoid, the values [OØr/OX] z = 0, z= 0, and O2Ø/Oz2Jz= 0 are necessary. Those values on the ellipsoid (X0) are obtained:
00, aø,
a,.
ax[1z=o
0xax
Ox Uabcr
dX2-a0j0
(a2+x)/(a2+x)(b2+x)(c2+x) Ux abc 2-a0 (a2+x)v'(a2+X)(b2+x)(c2+A) Ox X02U
2U
b2c2 x2
2-a0
2-a0
a2 pv2U
xy a2c2 Oyz=o.2-a0
a2pv
_abcUx
2b 10z2 Z0
2-a0
ac iiv where iv i6 given: pva2b2+---(
2)x2+(c2 -b2)y2.
aIntroducing a new variable 13 as
x=acosj3, y=bsinj3
together With the definitions =b/a, =c/a,we have
.2
2cosI32-a0
2-a0
sin2I3 + e2cos2f3
When the section shape is expressed by
x 1
y.=
---cosj3,
=-j-=
EB1sinnj3,the values of and (1 -U2 j2) are obtained by the use of
(C-3)
Appendix 0. Derivation of F(13) for a vertical cylinder of
arbitrary section shape
(D-1)
2 eosj3sii13,
2
(C-li)
2.-a0 sin 13+e cos 13=çb,(x,y, 0)
U
2-a0
cc!s13sin2l3 + e2cos213
From those expressions we have
itz2-2 = 1
( 2 )2 sin j3 (C-13)2-an
sin213+2cos213 2._ )2 2esinf3cosl32-a0
(sin2J3+e2cos23)2 sgn(sinj3)sinj3 . 2Jsin2j3 + e2cos213
2-a0
By substituting those expressions into (B-8) and using
B1 --e, B=
0 (n 2), we haveF(j3) for a semisubmerged ellipsoid asF
2 2 2 2cos2l3sin2I3sin2fl +e2cos2j3
2-a0
(sin2f3+e2cos2j3)22
2
+Cs13
1 (_.
)2 sin2-a0
sinj3+eco
By the use of the relations:
.3r= dx
2l.=i.aaj
(a2+x)5/2 3F(j3)for a semisubmerged sphere is obtained as F(f3)= (i.)35213sifl2j3+3COS213
75
27.
= 4cos13[-jjcos3--j--cos3I3].
By the use of the relations: = 00,
a0 =lirn
abf
(b2+
x)'
(1+)'2
2
2r(--)r(i)
2
1_e2
(1_2)I(f)
1+e
We haveF(f3)for an infinite elliptic cylinder asF
(13 -)_.3(1+e)c052135in1213 (sIn2+e2cos213)3 a 0.13 xdx
I5 =-sinl3(-}-+
nB)(nBcosn13)
MTB109 August 1976 (C-12) (C-i 4)(Ci5)
1]
(C-16)conformal mapping method developed by MOriya (1941)
for 2-dimensional wing sections:
1
(X-i4(A-v)
a semisub-[OOr/Oy](C-b)
(C-17) (C-i8)MTB1O9 August 1976
P Po
pU2
The amplitude function expressed by (B-6) can be eval-uated approximately by the use of stationary phase method at small Froude number.
C(0) S(0) sec2O L L
-
2i
x expi
S i]i(13,8)}df3 secO i F(13m) 'II2ti"(13m,O)I 0) + sgn(iV'(37, (E-1) where (13ni3O)'c0si3mcos0+ (2BflsinnI3th)sin0, P"(13m, 0) a2(13, - cos0cOs13 (2En2Bnsinnl3m)sin0, (E-2) áhd f3, 132 are the stationary points which satisfy theequa-tion
aF(13,e)
--a.-
-
sin13cos0+ (2nBcosn13)sin0 = 0. (E3)
By the definitions = Bsinn13, -j-cos13. (E-2) ôañ be
rewritten:
\/"(13m, 0) sin0 d , (E-4)
f2lT
Appendix E. The derivation of wave resistance in low speed
limit
sin /3 + 2nBsinn(3 } Q(/3)
P(f3) =
(i)2 +
r_sin2/3 +
nBcosn$32,+
¶sin/3cos/3
nBcosn/3
n2Bsinn/3LWhen using B1=--e,B1, 0 (n 2) in (0-5), We haveF(/3)
for an elliptic cylinder:
F(j3) 2e3(1+E)3cos2l3sin2/3 2/3 +2cos2/3}3
which ôoincides with the expression (C-18) derived as a
limiting case of a semisubmerged ellipsoid.
(D-6)
the amplitude function ôan be applied to a ship form whose
water line curvature dOes not change its sign, i.e. d2/d2
When there is a point where d2/d2 =0 in the water line, the asymptotic expression of the amplitude function
is evaluated approximately:
C(0)S(0)
cos0
r(4T)./9L L 2ir
x
F(j3)
e1Fi21vO),
(E-5)where I() = 26789385347, 13 m is the stationary point
which satisfies the equations aiJi(j3, 0)
0 82/i(j3, 0)
0
aj3'aj32
In this case L"(/3m, 0)is expressed as
1 . 0
1I (13m.0)=5m 13m5n0
2cos2j3m
sin0-(E-6)
sin/3m
For the sake of simplicity we consider a ship form whose water line has a convex curve such as a semisub
merged ellipsoid For this case the wave resistance coeffi cient iS written as follows by the substitution of the
expres-sion (E-1) into (25) in the text.
2r
2pu2L2
J0 L L
which indicates that the asymptotic expression (E-1) for
Cw1 +C2+C3,
(E7)/1()2 + (_)2},
(D-2)= 1 I72_i2= 1 sinj3 + nB sinnl3)2
/f(i)2
(dfl)2}
(D-3) where p-pô is the static pressure on thecylinder.
Since = 0 for a vertical cylinder we have from (B-8):
F($) = (D-4)
Substituting (D-2) and (D-3) into (D-4), we have
F(13) sinf3 1 + 2 nB cosnf3
x-F sinf3 + EnB sinnj3 G(13)P(13)13 (0-5) where
where
C1
= d ,li"(j3,O)l
=2Ff
)I70
dO 6 (/2 F(131)F(132)cOsOC3-4F,
Jo s/Ls(131,.0)'(132,O)I
x cosE,{I1i(j31,O)-,(f32,O)}
contribution fromthe after body, (E-8)
contribution from
the fore body, (E-9)
+-f(sgn/"(3i3O)-sgnt4i"(j32,O) )]dO,
(E-10)where
C3
is an interaction term between the fore body and the after body For the evaluation of C we can again use the stationary phase method when carrying out theintegration with respect to 0. Let us define first
f(0)=
SeC20{'(13 0)i(3O)}
.. (E-1i) The statiOnary point is the solution of the equationdf(0)_sec3OsinO
P1,O-2,O)
2
sec
cosf3sinO. + (2EBsinn131)cosO
+ cosl32sinO
(2Bsinnf32)cos0 = 0,
(E-12) since there. is the relation between 0 and (j3, 132):a,D(13, O)_ sin13cos0 + (2nBsinnj3)sin0 0
at13 =l3 and 132.
When we put 0 = 0 in the equation (E-13), We have (E-12)
A design method for full fOrms was developed based on
the following three experimental and practical evidences
which were. derived from the analyses of towing test,data of
more than 200 full forms (Taniguchi et al., 1966). It has
been served as a routine method since 1963 in Mitsubishi
Heavy Industries, Ltd.
.i) For a full ship (U//E< 0.20,
Cb>0.80) with a
well designed run part the wave resistance characteristics depend mainly on the geometry of entrance part, and the contri.bution from the parallel part and run part is
negligi-bly small in generating waves.
ii) The propulsion factors depend mainly on the geom-etry of run part Practically there exists. a limitation of fullness for the run part to prevent wOrse propulsive
per-Appendix F. Outhne of a design method of full forms
developed in Nagasaki Experimental Tank
MTB 109 August 1976
j3= 0,
132= r. Thus in the case of 00,isestab-lished. Therefore, we may say that 0 = 0 is the. stationary
.point. Then the approxiinate evaluation of
C3
is given:F(0)F(7r) -cw3 =
Fc
x cos[2,2(O,O)-i(,O)}
+sgn i/i"(0, 0) ---.sgn
)"(ir,.0)}+?Jsgn (g)]
4 dO (E14)Further we can derive the fol'owing relations
d2f1
_i.
1dO2' OO,j3O, 2ir 2F
[2nB, 2
We have then
F(0)F(i}
C3
= '.2ii-F, -. = ='JT+2
F24
Thus we obtain a geneaI expression of C for a ship form
which has a convex load water line.curve:
C= 2F,
r{F1312s0
dO+2F,
(T/I1322c050dO
J0
lii"(j.3,0)I
J0
h/i"(f32,O)I2-s/F
sun(L+2t_).\/1+2nBnJ2+2{(_1)nmnB,j2
F, 4formance and undesirable flow phenomena around the
stern.
iii) A reliable formula to estimate the form factor by
use of given geometric parameters has been developed.
In this method the bow forms are designed so as to reduce wave résistáhce, and stern forms are designed to obtain better propulsive efficiency and the parallel parts are designed to satisfy the required displacement So the
experimental data. such as wave resistance and self propul sion factors are stored-together with geometrical parameters
of each part For instance the parameters for wave resist
ance data are
Froude number:
FflBU/s,/
Breãdth.draft ratio: BidMTB1O9 August 1976
Fineness factor: Cm He/B
where Cm is the midship area coefficient and He is an effective length representing fineness of entrance defined
by
He=(lC'pe)/Le,
where Cpe is the prismatic coefficient of èntränce patt, Le is the length of entrance. When a new ship form is
designed a proper bow form and a stern form are selected
independently from the stored data so as to achieve a
desired propulsive performance.
The design method mentioned above was originally
developed based on the experimental evidences. Its validity
has been confirmed by a number of model experiments.
By the present theoretical study in the text the assumption
of separability of a body into parts has been verified theoretically In addition to this a way of reasonable deter
minatiOn of the viscous résitancè in low speed limit has been provided This may be an important contribution in
Obtaining reliable wave resistance data from model experi-ments.
As far as the bow form design is concerned, the
f011ow-ing method has been Used
A ship like a Oil tanker or bulk carrier usually serves both in full load and ballast lOad conditions with almost same frequency Therefore one must design the optimum entrance not only for full load condition but also for bal
last condition. In the full load condition it. is more econom-ical to Choose shorter entrance under the design constraints.
In the ballast condition one of the measures to quantify the propulsive performance is the difference of service speeds between fUll and ballast lOad conditions with
con-stant horsepower To satisfy the required speed difference
say 1 knot, the entrance length in ballast load condition must be longer than that in full load condition so as to prevent enormous- increase of wave resistance. Thus the Optimum entrance length in ballast !oad condition differs from that in full load condition.
To meet this requirement the MHI Bow was invented The key idea of MHI-Bow is to combine two entrance forms each Of which is optimum in full and ballast load