Report No.155
Publication No.31
LABORATORIUM VOOR
SCHEEPSBOUWKUNDE
TECHNISCHE HOGESCHOOL DELFT
THE DRIFTING FORCE AND tIOÏIENT ON A SHIP
IN OBLIQUE REGULAR WAVESby
A. Ogawa
L
Abstract.
THE DPIFTING FO10E AND MOMENT QN A SHIP IN OBLIQUE EGULA WAVES.
*
A. Ogaw.a
Two-dimensional hydrodynamic theory is applied in order to obtain the drifting force per unit length on a two-dimensional body floating in oblique regular waves with the object. of applying it to ship forms by a strip method.. A simple resultant formula is obtained.
In order to confirm the foregoing theory, some experiments were carried out on a model of a Series 60, block.coefficient .70 ship t. compare the measured drifting forces and moments with the results calculated by the the.ry. The agreement between theory and experiment
is very satisfactory. .
Naval Architect, Ship Dynamics Division, Ship esearch institute, Tokyo. At Deif t Shipbuilding Laboratory on research assigñment.
Introduction.
A ship floating in waves receives forces and moments from them. These forces and moments consiste, of oscillatory and steady components. The steady forces and moments are generally small quantities of second order and are often neglected in the study of oscillatory ship motions. in some cases, however, they can have, a significant effect on ship motions. For instance, the course-keeping problem of a ship in waves cannot be solved without consideration of these steady or mean later-al forces and moments, the so-clater-alled "drifting" forces and moments, whióh act on the ship.
A theoretical analysis of drifting forces has been presented by Maruo(1)in both the two- and three-dimensional cases. However, no drifting moment is obtained from his theory. Newman(2)has given gener-al angener-alyticgener-al equations for the drifting forces'and moments on an
arbi-trary body and specific formulae for a slender ship. While these
theo--ries-are derived from--rigorous mathematica. treatment's, -t-here---are-some
difficulties in applying them to prahtical hp- forms in oblique waves. In the present pape-r the àuthor, considering the recent develop-ment and practicability of the s-trip method in calculating ship motions attempted to derive the drifting force 'on a two-dimensional body floating in oblique regular waves according to Maruo's theory With the object of applying it to the calculation of t-he drifting force and moment on a
ship.
-Two important results were obtained by this study. First, the drifting force on a two-dimensional section in oblique waves ispropor-tional to D0sin2» ,, where D0 is the drifting force in beam waves =
900), and is the angIe between the direction of the- incident waves and the axisof the body. Añother important conclusion is that D is
composed of six component forces, in general, which are the forces generated by relative héaving, swaying and rolling motions and their coupling terms.
Finally the results of some experiments on a model of a Series 60, block' coefficient .70 ship are shown. These experiments were planned and carried out especially to confirm the foregoing theory by comparing the measured drifting forces and moments with the results calculated by the theory. The agreement between theory and experiment is found to be very satisfactory.'
2
1. The Drifting Force on a 'Two-Dimensional Body in Oblique Regular Waves.
1.1. Momentum theorr.
First,, consider a t,wo-diniehsional horizontal cylinder floating in regular waves. The origin of the right-handed rectangular cdordinate system is taken in the urface of the fluid at rest, the x-axis being parallel to the generatrix of the body and the z-axis vertically up-wards..The depth of the fluid is assumed to be infinite and the waves are progressing inthe direction of the positive y-axis with the angle
,u
(o <,,u<ir )
to the positive x-ax-is (Figure 1). The mean position of the body is assumed to be fixed. The flui& is assumed to be inviscid, incompressible and .irrotational.The presence of the body leads to disturbances of the regular waves. These disturbances generate systems of regular waves which advance in
two directions at a great distance from the body, as will be described
lter
As control surfaces for the application of the momentum theorem con-sider two vertical planes
00 and in Fig. i.) o,f infinite extent
par-allel to the x-axis located at a great distance on each side of the cyl-inder. The change of linear momentum in the. y-direction of the fluid con-fined between the two planes per unit length in the x-direction is:
dM 2
- = Pi
y dz + I p dz dt ,,co
ico
"-00
"-00
rL
2(coo
-f'J
v(,, dz-J
p00 dz - F -00 00 where:y = y-component of the fluid velocity,
F = y-component of the force per unit length experienced by the
cylinder.,
= surface elevation of the fluid, and
suffixes .- andooshow the values at y=-aoand
y=oorespective-ly.
As a result of the assumption of irrotationality the flow field can be described by a velocity potential and Bernoulli's law gives:
z
where the atmospheric, pressure has beenset equal to zero. The magni-tudesof the fluid velocity and the wave height are considered to be small quantities of first order. Then, substituting equation (1.2) in equation (1.1) and retaining terms up to the second order:
F =
-+
fj
[_()
()2
dzirJJ-'()
PJØQ
-f
1 2 2-g(-Taking the average value during one period, the periodical terms dis-appear, and the time-independent drifting force per unit length of the
cylinder is obtained as:
f
2(2
2J_L x -
ay _az
+
F,(
c_dO
-(ç00
3aid-
jpg
Thus the mean drifting force is represented by a quadratic of the first-order small quantities.
1.2. The direction of the scattered wave.
The scattered waves are generated by the relative motions between the body and the fluid, in oblique waves, the relative motions of
sec,
tions of the body located at different axial stations have a phase dif-fèrence which is determined by their relative positions. Consequently the scattered waves from each section have also phase differences, and by the Huygens' principle the scattered waves on the whole have -the same direction in the positive y-direction and the reversed in the nega-tive y-direction to the incident wave. The conception of the principle is the same as the snáke-type wave generator and is shown, in Fig. 2.-5-+(f)2
2+(.f)2
]
ax
øo - ayco az1.3. Wave velocity potentials,
According to the, relations of surface wave motion of small ampli-tude, the kinematic aurface' condition is:
()
at;
-
azand the first-order Bernóulli equation is:.
)
z=o
The velocity potèntial 4, is written as:
1. C' iw't
,=
RejØe
Eliminating from (1o5) and (1.6) and introducing (1.7), there results:
where
2
K
g
with the wave length:
21t
XK
- KØ = O
at z=0
and the wave velocity:
w
C
=
The 'incident wave potential.is expressed by:
0
=c
ew a
where:
5
Kz-iKxcos.i -iKysinj.t
= incident wave amplitude,.
and the scattered wave potential's as shown before by:
(y _-(y
-t'
+o)
(1.5) (1.6) (1.8) (1.7)where A and are scattered wave amplitudes in complex quantities at -ou and oo respectively.
-6.-01 =
cA_eK1CO5J
Ky
SiIiL
D
=
f
[_I.J2
D=
sin2p.[J+
where A is the conjugate complex of A
i.k. Energy relation.
Since the body is not allowed, by assumption, t.o drift from its mean position, the average work done by the wave on the body is zero. Accordingly the net energy f ux through the control surfaces Sco and
S must be eqiai, to zero.' The energy flux through a fixed plane o-thogonal to the y-axis is given by:
= -j;f P
}d
x dz . (1.17)Applying the relations (ia?),, (1.9), (1.10), (1.11) and (1.1k) the average rate of the energy flux is given by:
()
=_gc8in,LL,Ç+A4
y=+oo 2 -JA[
6 dE=pgcs1n)A
y=-2 'ay co az(1.16)
.18.) (1.19)'-7-.
+[iøi
-iøi
(i 15)
The resultant velòcity potential is then:
0
= 0
+ 01
(1.12)Considering the following relations from èqs.
(i6)
an:d (1.7):¿;
2)
= j.
he eq.
(lok)
is written as:Equating them:
2 2
- a
(A+A)=
IAl
+ AJ .
(1.20)Putting equation (1.20) into eq'iation Éi..i6), the drifting force per unit length is;
'.=pgsin2
JA 12.. (1.21)Equation (1.21) reduces to the result obtained by ?laruo in the case of beam waves for which /h= 7c/2, and shows a' similar trend with p. as the experiments on a s-hip model b.y Lalangas(3).
1.5. Construction of the scattered waves.
In the treatment of the theory, there is no restriction on the
ori-gin of--the --scattered--waves- except- that- -they -result -from--disturbances
-due---to the presence of the body floating on the surface of the fluid. That is to say, the amplitude A in equation (1.21) can be expressed as
fol-lows:
le
je
A e + e + e (i 22)
where
H' s and are the amplitudes of the scattered waves caused
by the heaving, sWaying and rolling motions of the body relative to the motion of the fluid, respectively, and 11 and are the phases of thescattered waves. Then:
1A1
s2++2
RH
(1.23)Thus the total scattered wave is replaced by the componen,t scattered waves and their cross-coupling terms. Applying equation (1.23) to
equa-tion (1.21), the drifting force can be written as:
D=sin2p.(DH+Ds
+DR+DHS + + DRs) (1 .2k)where the terms DH, D5,e.tc. represent the component forces generated -by the respective component. scattered waves.
If the amplitudes' and phases of the relative motions of a ship are known, the amplitudes of the scattered waves caused by each motion can
be calculated for each section of the ship, from which the total drifting force and moment can be obtained by stripwise integration.. The basic know-ledge on the scattered waves is available, for instance, from UrselI's paper t'i). At this moment., however, it is very difficult to calculate ev-ery mode of motion of a ship in oblique waves and often it appears to be unnecessary to do so. Tasait.5) has shown the calculated results of the drifting forceà caused by the relative heaving and rolling motions sepa-ratély in tkie case of beam seas,
2. The Drifting Force and Moment on a Fixed Series 60 Model in Beam and Bow Waves.
2.1. Drifting forces on a fixed body.
Because the scattered waves are considered to be generated by the vertical, transverse and rotational motions of the water particles
re-lative to thè ship, the case in which the body is fixed is clearly a
special case of the genera]. theory as given by equation (1.23). Since the body is stationary in this case there is no oscillatory
relative rotational motion, and the drifting force on each section
is:
where:
a2
CH
CsThe amplitudes of the scatteréd waves can be calculated, for instance, by Tasai's methodC6,7] and the phase differenceby a method.described below.
2.2. Phase difference.
From equation
(3)
ofr6)and equation (22) ofC7) , the scatteredwave potentials at y , z = 0 are obtained as follows:
D = jpgsin2).
[C:+.:2 HScos(H_
or, in dimensionless form:
+ 2J7H)?Sc,o(EH._ Es)],
(2.1)
(2.2)
for heave: for sway: gCH'' cos(Ky- cut) irw Sc5'lt
- ;z;
cos(Ky -Cot +) (23)
(2. 10-lows:
heaving motion: z z cos(wt )
H a HM
swaying motion: =
YaC06(0t+ ZM).
lo
On the other hand, each corresponding motion is writtén as fol+
Since the relative swaying motion has a phase lag of with respect tO the relative heaving motin1 each motion can actually be rewritten using the same time base as:
ZH = Z CO5Wt' (2.7)
/ ' 7t
y8 =
yaco5t -
(2.8)where t' denotes the unified time.
Then the corresponding scattered wave potentials (2.3) and (2.'-i)
become: -0H cos(Ky - tot (2.9) gC
0
= -
cos(Ky- tot' + - - SM SooAccordingly, the phase difference between both component waves js.:
2.3.. Calculatin.
The phase of each motion is exressed as follows:
A -1 o
HM=t5fl
PSM
=tan1
6HM T (2.10)(2.11)
(2.12)
(2.13)where A and B are defined iñ
(6)
, and Po and Qin (7)
Assuming the amplitudes of both motions in (2.7) and (2.8) are equal to the root-mean-square of the orbital amplitudes due to the in-cident wave as:
-2KT
-aL..2KT (l-e )j 2
(2.5)
li
where T = draft of the section,
the total drifting force and moment on the fixed model is calculated in the dimensionless form as:
fd dx
(.i,)L
fd. ldx
(2.16)where 1 denotes the distance from the centre of gravity of the ship and the integration is carried out over the length of the ship.
The calculations were carried out fora Series 60, CB= .70 model onthe Telefunken digital computer TP k of the "Wiskundige Dienst" (Com-puter Department) of the '!Technische Hogeschool, Delft".
The results are shown later together with experimental results.
2.k. General description of the tests.
The tests were carried out at the towing tank of the Shipbuilding .Laboratory, Technologiôal University, Delft. The principal dimensions
of the model are given in Table 1.
Table 1. Model.
Length between perpendiculars L 3.01+8 m
Breadth B .1+35 in
12
-Draft T .i74 m
Dis plac ement 1615m3
Block coefficient 700
Prismatic coefficient .71Ö Midship coefficieñt CM 986
Water plane area coefficient C
w .785
12
Two heading angles were shosen for the tests; 9Q0 (beam wave) and 1200 (bow wave). The characteristics of the waves are. shown in Table 2. The general plan of the model.-setting in bow wave case is shown in Fig. 3.
Table 2. Waves,
The model waa fixed beneath a stiff steel girder by means of two strain gauge dynamometers. The outputs from the two dynamomèters were added and subtracted electronically and then integrated over a few
periods to give the res,ultnt mean drift force and. moment, respectively. 1h Fig. + a schematic illustration of the measurement is shown. Two. wave height reco.rders were it of the model as sho!nin
Fig.
3.
Only.the first one Was used for obtaining quantitativ.e wave data.. The second one was used for monitoring the reflected waves and for es-tablishing the time interval during which meaningful force data could be read from the continubus record of the. test, since it was essential to ensure that no undesirable disturbances, such as reflections from the end of the tank, were intercting with the model durin.g the time that the force measurements were made or were distorting the measurement of the incident wave amplitude.Much care was taken not only on the refleôted.waves, but also on
the stiffness of the mechanism,with the result that the natural pe.riod of the whole mechanism was 0.086 sec. Special attention was also given to the acc.uracy and reliability of the electronic instruments, to the directional independence of the sensitivity of thee dynamometer and so on. Du.ring the tests ali instruments performed satisfactorily.
2.5. Test results and discussion..
The test results are shown in Fig,. 5 and 6 together with the
corn-puted results,
Concerning the drifting force, which is the main object of this report, the test results are in general somewhat larger 'than .the
corn '13 corn
-Wave length/model length .50 .65 .80.1.001.25 1.50 1.75 2.00
13
puted results. I.t is clear that one reason for this differenc,e is the approximate method of the calculation., especially in the shorter wave length range. In some cases, the ampltudes of the oscillatory forces were about 30 kg. This means that the measured drifting forces were less than 2% of the amplitudes of the oscillatory force.
Nevertheless, the measured points could always be reproduced within a small region., showing the reliability of the tested points.
Spurious effects of the tank bottom and walls are presnt through-out all of the test results. The bottom effect seems to be another reason for the larger measured values of the drifting force in the case of long beam waves.
Indeed, it is quite probie that the drifting moment is evenmore
strongly affected by the tank wall and bottom. Unfortunately it is
ve-ry difficult to e:stimate these effects quantitatively because the forces and moments themselves are small quantities of second order. Accordingly the results of the drifting moments can be considered as
only a somewhat rough approximation. On this point the author hopes to be able to give some correction to the caiculatïon in a later paper.
In any event, it is clear that the drifting force can be calcu-lated exactly in this way. And since the drifting force plays the major role in the problem of course-keeping quality in waves, as will be shown in a following paper, it can be said that the most important part of' this problem has been solved.
2.6. Conclusions.
i.The drifting force on a ship in obliqùe regular waves can be cil-culated by the strip method.
ii.The drifting forces are very small quantities compared with the corresponding oscillatory forces.
iii.The drifting moments are much smaller quantities than .the drifting forces.
iv.In order to estimate the drifting moment more exactly, a further correction on the calculatioi' and verification tests in a wide tank are needed.
-Acknowiedgèment.
The author expresses his highest gratitude to Prof.ir. J. Gerritsnia Thr his suitable guidance on this study, to Ir. B. de Jong for his very frequent discussion with the author in detailed problems, to all the other members of the Staff of the Shipbuilding Laboratory, Deift Tech-nological University and Mr. R.D. Cooper of the Office of Naval Research for their generous collaborations on this study and experiments.
The author also acknowledges Mr. F. Bertens of the "Wiskundige Dïenst"
(Computer Department) and Nr. W.E. Smith of David Taylor Model Bain for their assistance and cooperation on the practice of programming and cal-culation.
References.
i H. Maruo:
"The Drift of a Body Floating on Waves". J.S.R. Dec.
1960.
2 J.N. Newman:
"The Drift Forces and Moment on Ships in Waves". D.T.M.B. June
1965.
3 P.A. Lalangas:
"Lateral and Vertical Forces and Moments on a Restrained Series
60
Ship Model in Oblique Regular Waves". D.L. Report 920, Oct.1963.
k F. Ursell:
"On the Heaving Motion of a Circular Cylinder on the Surface of a Fluid". Quart. Journ. Mech. and Applied Mathematics, Vol. II, Pt. 2,
19k9.
5 F. Tasai:
"Ship Motions in Beam Seas". Reports of Research Institute' for Applied Mechanics, Vol. XIII, No.
k5, 1965.
6 F. Tasai:
"On the Damping Force and Added Mass of Ships Heaving and Pitching". J. of S.N.A.,J. No.
105, July 1,959.
F. ,Tasai:
"Hydrodynamic Force and Moment Produced by Swaying and Rolling Oscilla-tion of Cylinders onthe Free Surface". Reports of Research Institute for Applied Mechanics, Vol. IX, No.
35, 1961.
sao INCIDENT WAVE
FIG.!.
FIG.2.
SCAT T ERED WAVE SCATTERED WAVEWAVE WAVE..
TANK BREADTH =428m
WATER DEpTH2.5Om
FIG. 3.
WAVE 220m
PERIOD
COUNTERFIG. 4.
DYNAMO-b'
METERS 10m60m
> PRINTER RECORDER RECORDER RECOiA
R D ERF1 +F2
INTEGRATOR FORCE F1 FORCE- INDICATOR
itt>
F1 + F2 INTEGRATOR F2--
INDICATOR GATE PULSE GENERATOR > RECORDERist WAVE RECORDER 2nd WAVE RECORDER
Ist,
o
1.0 0 .10z
.5 oI-z
w
C) LAw-o
C-).05
-I-z
w
o
z
Li-.02
oo
EXPERIMENT 1.0 1.5WAVE LENGTH
/ MODEL LENGTH
FIG.5.
CALCULATION + Ii=900
WAVE2.0
X/L
1.0
o
.05
-.01
1.0 1.5
WAVE LENGTH
/ MODEL
LENGTH,
X/L
2.0
-o
o o o oo
oO,
WAVE LENGTH / MODEL
LENGTH,
X / L
2.0
