ARCH EF
MANOEUVRABILITY OF SHIP IN CONFINED WATE
Katsuro Kijima , Department of Naval Architecture, Kyushu University , Japan
INTRODUCTION
Sfety of navigation is of importance, especially in restrictèd waters such as harbour,. bay or canal, and the necessity of marine
traffic contol is gradually gaining importance. For safety of navigation in these waters, it is necessary to know the precise manoeuvring characteristics of ships including the effects of water depth, channèl wall or the another ships in the proximity of them. On trie other hands, it will be hoped that the prediction of ship nanoeuvring performance at the initial stage of design is established. In the past, many method relating to this prediction.have been proposed, but it seems that much effort will be required to get the manoeuvring characteristics of ship under any condition.
In 1MO the inter-in guidelines for stïmating manoeuvring performance.in ship design and it's assessment have been discussèd. Especially being required high manoeuvring performance, we have to carefully consider the effects of chanñel wall and the hydrodynátnic. interaction bEtween ships in narrow waterways. Relating to these hydrodynamic interactions, NewnanU., Yeung2 and Dand' have shown the calculation method on the hydrodynamic inter-action force between ships, however.véry little was investigated on the unsteady hydrodynamic
interaction of ships in narrow water .chánnel. The author has shown the hydrodynamic behavior of two ships during meeting and overtaking in narrow water channel by applying Tan's methód in reference 6)
In this paper a numerical calculatión nethod has been proposed for predicting ship nanoeuvring characteristics in narrow water channel, taking into consideration the effects of channel wall, interaction forces betweeñ ships, and ship marioeuvring charactéristics for the overtaking condition in narrow water-w4ys.
MATHEMATICAL MODEL FOR PREDICTION OF SHIP MANOEUVRING MOTIOÑ
-In general, the equation for maneouvring motion is shown as follows by using coordinate systems in Fig.l
cos ß snft)±On'+m)r' sth ft = X'
(m'+m)(L,/C/)((Ù/TJ) sin. fi+ó cos ft)4(m'4.m)r' COSfi
=
= N'
I-Lab. v
Schepsbouwkunde
Tec.hische Hogeschool
Deift
The superscript ' refers to the non-dimension-alized quantities which are as follows.
m', m, m = m, m,, m,/j ßL'd.
Ï,, i, =
i, /.L
pL'd. X', r = X, Y /.. pLdU'.N' = N/.. pL1d(J', r' = rL/U
where L: ship length, d: draught, U: ship speed, m,mx,my: ship's mass, added mass of x,y axis.,
ß : drift angle, r : angular velocity, X,Y : external force of x,y axis respectively, N : yaw moment about center of gravity of ship.
o
Yo Y
Fig.l Coordinate systems
Now let the subscript " H T
symbolize hull, " P " propeller, " R " rudder and " E " interaction or external force etc.
The individual component of thé force and moment, duE to propeller, rudder and
interact-ions between .ships or ship and channel wall, in the right hand side of the equation (2.1) can be written as follows.
X'.
X +.X
+ Ç +
Y'
N' = N
+ N +
For the longitudinal component of forces, the following equations re assumed.
XX,r'sInfi+% cos!fi
z; = (i_r,)
n'DI.Kr(Jp)/(+ LdU')
(2.3)
IC,.(J,) = C,+CsI,+C,J
.1, = Ucoa
fi(1wp)/(n.D,)
where t: thrust reduction coefficient, n : propellEr revolution, D : propeller diam-eter,
Vp :
effective wake fraction coefficient at propeller location, J: advance coefficient, u: longitudinal component of U, C11C2,C3:constant. X. = aX'/.D etc..
X0
(2.2)
QoC)
J
For the lateral force and yaw moment acting on hull, we have the following expressions
which
the author has proposed.
= Y'.fi+Y',.r'+Y,'j il +Yrlr'I +(Y',,j+Y,.r')lr'
= N,'fl+N,'t'+NI,fi .81 +N,r'Ir'I ±(N'j+NI,,r')ftr' j(2.4)
Once the body shape of the ship is known, it is then possible to estimate the lateral force and yaw moment shown in equation (2.4) by using the hydrodynathic derivatives obtained by
theoretical or experimental methods.
For estimation of rudder force and yaw moment, various methods have been proposed, however in this paper an extension from Hirano's expression has been used.
= -<1t,)(1wa)CN am 8(A,/Ld)
= (1+a,)(1w,)2C,, cos 6(A,/Ld) N = .(xl+aßx,)(1ws)'CN cos 8(A,/Ld)
CN 6.13 k5/(k,+2. 25). C1+C,(s)) eLaa ¡(s) =
K = O.6(lwp)/(1w,) ,
= W50.W,/Wpo= ö(ft+r')
q = s 1. O(1w,)U ces ft/nP w, =w,,.exp(-4.O(ft+r'/2)')i.
J
where, kR: aspect ratio of rudder, AR: rudder area, C : coefficient for starboard and port
rudder, tR: coefficient for additional drag due to steering, WR : wake fraction coefficient at rudder location, a : ratio of additional lateral force due to steering,.
i:
flow straightening coefficient, h rudder height, wD0, w the value of w , wR when advancingsraigt.
By using the above expressions, we can obtain analytically ship manoeuvring charac-teristics for deep water, shallow water and narrow waterways respectively. Althoughthe
above expressions have been originally obtained for deep water, we can apply it for shallow water and narrow waterways by correcting the hydrodynamic derivatives, other coefficients and by including the interaction forces between ships or between ship and channel wall. If the external, forces such as wind or current are considered in the terms of X.. , Y and N , we can also obtain the manoeuvrng caracteristics for wind or current respectively.
3. BASIC FORMULATION FOR INTERACTION FORCES' At the first, we shall estimate the interaction forces between ships in narrow water channel. We now refer to such coordinate
systems as shown in Fig2. Each ship isassumed: to move In a straight line in the channel of üniform water depth " h " and in wàter channel
zith vertical side walls.
If we assume that the free surface is a rigid wall, the.velocity potential (x,y,z:t) must satisfy the following boundary
conditions,-by considering a " double-body " problem.
Fig.2 Coordinate systems in narrow
,th=o
-E8/8?It]BUt(nr)t
[6I8n]c=O
,-.O at
'xt'+y'+zj'-.
where U denotes the speed of the i'th ship, B,: represents the underwater hull surface of
the i'th ship and C the Submerged surface of the water channel. n,:. and n are the unit
25)
ñormal vectors in the normal direction to Band C respectively. (fly)-,: is the component at the unit normal' vector n in the x direction.
It is assumed that the flow field about the ship's body will be made divided into two sub-problems i.é. " inner problem " and outer problem ". Let
. be the slenderness parameter, i.e.
L 0(1), B,: = 0(6), d = 0(6)
where L,:. ,B ,:. and d',. denote the length, breadth and draught of the i'th ship
respec-tively. Ve make the assumption that the water is shallow, and the minimum lateral distance between the i'th ship and j'th ship denoted by SPL.J and between the i'th ship and the
channel wall by 'So , and these parameters havé the following order of magnitude,
h .= 0(6), SPLj = 0(1)
Sj = 0(1)
L= 1,2...,N ,
j.l,2...N,
3.1 Inner Problem
We now refer the inner region fôr the
i'th ship, whichis defined as the rgon which
has the following orders of magnitude.
x,:,= 0(1), y,: , z 0(6),
L. 1,2...,N
If :.. is assumed as the velocity potential in the inner region of the i'th ship, equation (3.1), (3.2) and (3.3) can be replaced by the following, ' $201 850$ owl'
(8/8Njjco= U5(n5)5
COøt/0x53s.±k=O UN Ship N Shp2UI
waterway. (3.1) (3.2) (3.3) (3.4) (3.5)where N is the two-dimensional unit normal vector to the section contour
L(x j. )
of the ith ship body. After some analysises based on the above equations, we obtain the outer limit of the inner region.3.2 Outer Problem
The region in the farfield is assumed as the outer region, and which is as follows.
= 0(1), z = Ou.),
L = 1,2...
The velocity potential in this outer region is expressed as a Taylor expansion, and then neglecting the higher order terms, we get the following expression.
e'
EIx' + Oy'
-This problem is just the. same as the two-dimensional flow about ships in waterS channel.
The outer solution can be represented by a distribution of sources and vortices along the body axis. Thus, we get the inner limit of the outer region.
3.3 Integral Equation and Hydrodynanic Forces
We have obtained the outer limit of the inner solution and the inner limit of the outer solution. Furthermore, the gap between inner and outer solutions must be matched in the
intermediate region << y << 1, i.e. they must satisfy the following condition.
,¡ X,,S) = hrn ,(x y f)
...(3.10).
Using the above expressions, the matching condition is given by the equations, obtained by terms:of similar nature.
From these expressions, we can get the velocity potential which should satisfy the required boundary conditions, and then the lateral force and yaw moment acting on the i'th ship are obtained by using Bernoull's theorem for unsteady flow. ( See reference 6) for more details of the above method.)
4. RESULTS OF NUNERICAL CALCULATION
In this paper, as a first step the intez-action force and moment acting on two ships 'during overtaking in narrow water channel are
examined by the numerical calculations using the method in the previous section.
The calculations are carried out on two ships, as shown in Table I, with geometrically similar form and different sizes. Fig.3 shows the coordinate system used for this calculation. The lateral distance between i'th ship and. channel wall is denoted by Spi, and the force and moment coefficients C CF., CM ) are
defined as follows.
3-Table I Màin particulars of ship for numerical calculation
Fig.3 Coordinate systems in overtaknig in narrow water channel
F,
CM1- M,
cF,=
-The channel width W is taken to be 1.25L ship speed ratio u lu 1.25 ( u 4.Ot ),
ship length ratio 1.60, te water depth RId1- 1.20 an Hd2= 1.92.
The results are shown in Fig.4 to Fig.7. Fig.4 and Fig.5 show the lateral force and yaw moment acting on ship 1 for the overtaking condition. We assume that ship-1 is the over-taking ship ( Skt ) and ship-2 as the över-taken ship (4kt ), andthe stagger between ships is non-dimensionalized by the average ship length, i.e. (Li + L )/2 (=SI'), so that -1, 0 and +1 correspond to the bow-stern, midship-midship and stern...)w situations
respectively. The lateraZ force acting on ship 1 increases as Sp/L2incrases,.because of a decrease in the lateral distance between ship-1 and channel wall, ihich eventually leads to an increase in bank suction force.. Furthermore,
by decreasing S/L2 , the lateral force shows
a significaút variation followed by changes in stagger.
On the other hand, as shown in Fig.6 and Fig.7 the lateral force acting on the smaller and overtaken ships depends mostly on the interaction force between the two ships. Especially in this case, we note that the
yaw-ing moment actyaw-ing on the small and overtaken ships becomes larger than that acting on the larger and overtaking ships. Consequently, when a larger and fater ship is overtaking a
smaller and lower speed ships, the smaller ship is greatly influenced by the hydrodynatnic iñteractions between ships.
Ship .1 Ship '2 Length' (n) 160.00 100.00 Bre'adth() 26.84 16.77 Draught(m) 8.98 5.61 Block Coeff. 0.698 . SN SP2 ST (3.9) p Sp' hip1
COo 0S 0.0 -0.5 R PULSION ATÎOACTION -to L oiL 2 1.6 0 UVU2!125 H/d 0: 1.20 H/d 2: 1.92 W /1 ii 1.25 1:0 2.0 Lj/Lo.t60 UVUo125 H/di'1.20 H/d21.92 W/Lo1.25
FÏg.6 Lateral force cóefficient acting on ship-2
As he Second step, the manoeuvring characteristics of ships including the effects of hydrodynamic interactions mentioned above are examined by simulation study used in equations (2.1) to (2.5). This will be considered to be a very important problem from
the viewpoint of safety of navigation in narrow' water channel, and ship design. For the
purpose of numerical calculations, it is assumed that the geometrically Similar ships are ship-1 and ship-2 ( as Shown in Table I ), water channel width W = 200 n (W/L1= 1.25) and water channel depth H 10.78 m (Hid0 1.20) respectively. In this simulation study both ships are always controlled, in order to keep the original course, by rudder. The rudder of the ships are assumed to 'be controlled, as function of' the heading angle and the angular velocity r' within the range of ± 15 degrees as ordinary steering.
-4 =
CPb J0.2 0.1 0.0 -0.1 0.2 Bow ouTwoo Bow INWARD SP/L2:O.8 SP/L7:1.0 SP/L2:0.3 SP/L2:0.5 LVLa1.60 U o/U 2: 1.75 H/di:1.20 H/d7;i.92 W/Li:1.75 SP/L2:0.3 S P/ID: 0 .5 Sp/La:0.8 S P/I 2: 1.0 LVL2:t&0 Uo/U2:t25 H/di:1.20 H/d 2: 1.9 2 W/L:t25 -2.0 -1.0 0.0 1.0-
20Fig.7 Yaw moment. coefficient acting on ship-2
ö =ö,+k1 +k,r'
where' and b0'are the initial rudder angle and heading angle respectively, k, and k2.are gain constants, and in this paper these values are assumed as 5.0 for all condition.
Fig.8. shows the examples of ship trajec-tories as a function of 1ateral clearance between ships in overtaking condition,, in which the speed of ship-i and ship-2 ae set as 5 kt and 4kt respectively. As shown in Fig.8, ship-i and ship-2 start from X/L2= 0.0 nd X/L 2.0 respectively and the initial lateral location of both ships is im symmetrical situation with respect to the center line of the channel width. The loci of ships are plotted about every 100 seconds, and the' initial stagger between ships is asstirned as 200 n.
This figure shows that as S
IL2
decreases from '1.0 to. 0.3,. the possibility of Fig.4 Laterai.force coefficient acting on Fig.5 Yaw moment coefficient ¿cting onship-i ship-1
CMo
-2.0 t0 0.0 1.0 2.0
- ST'
20--20 -1 0 1 Y/L2 (3i pi X/L2 SP/L2:0.5 SP/L2:0.6 20 20 15 10 5 2 O -ì 0 1 Y/U X/L2 0.4
rl
0.0 -0.4Fig.9 TiSte histories of heading angle, drift angle, angular velocity and rudder angle on ship-i lo 2 -1 0 1 Y/Li SP/L2:0.7
-0
Y/U
Fig.8 Ship trajéctories as a function of lateral clearance betveén Ships in overtaking condition f32 ip2 12 - 60 40 52 20
420
o',
o -20-4-20
0.8 r.2 0.0 -0.8-5p/L2:0.8 SP/L2:1.0 p2 f32 150 300 600 TMEtS ECl 52r2
Fig.]O Time histories of heading angle, drift angle, angular velocity and rudder angle on' ahip-2
20 XJL2 ' . 20 X/Li 15 15 10 10 s 5 O o -1 0 1 Y/U -1 0 1
Y/U
Sp/L2=O.8
Fig.11 Ship trajectories when only ship2 is uncontrolled
collision becomes significantly larger. Furthermore, the time histories of heading angle ( ), drift angle (/3;j, angular veloc-ity ( r) and rudder angle ( ) of ship-i and ship-2 for Sp /L2= 0.5 are shown in Fig.9 and Fig.lO. réspectively. In Fig.9 and 10, we can conclude that the ship-2 requires large control force in order to keep her original course, and finally collides with ship-1 because the inter-action forces exceeded the control force.
Fig.11 shows an example of the case when only ship-2 is uncontrolled. In this case, when Sp ÏL2 0.8, the uncontrolled ship-2 collides with ship-i, in spite of that ship-2 can sail in safety when it is controlled.
From these simulation studies, the. eval-uation whether ship-2 collides with ship-i or not s shown in Fig.12. In Fig.l2, black symbols and white symbols represent the case when shlp-2 collides, does not collide with either ship-1 or channel wall respectively. These results clearly show the saf e navigation zone in narrow waterways, and are based on the assumption of the above data. However, this zone will change when changes are brought about in the value of the maximum rudder angle gain constant for rudder control and other related parameters.
-6-5. CONCLUDING REMARKS
This paper shows a method for predicting ship manoeuvring perfOrmance in narrow water channel including
of
ship to ship interaction. The major concluding remarks can be suiierized as follows.(i) During overtaking, the interaction forces such as lateral fOrce and yaw mOment between ships are greatly
affect-ed by the distance between ship and ship, ship and channel wall.
For a certain speed difference there is a possibility of col]ision in the case of smaller and overtaking ships, when the lateral distance between ships is small.
By using the present method, we will be able to estimate the ship manoeuvring performance in narrow water channel.
AcOWLEDGENT.
The author woüld like to thank Mr.Hironori Yasukawa and Mr.Yasuaki Nakiri for their assis-tance in,calculating hydrodynamic interactions and simulation study.
This research was supported by the orant-in Aid fOr Reseàrch of the morant-inistry of Eduôa-tion, and by using FACOM M-382 in the computer center of Icyushu University..
SP/L2 LilL2= 1.6 0 H/d 1:1.2 0 W/L :12 s 1.0
.0000
0 o000
0 0 000 0 0000
0 0 s o s i s 0.5 0 OOs s500
0 0 0.3 o. o. .s-00
0 0 4 5 kt. (Ui) lo 1.5 2.0 U1/U2Fig..12 Safety zone for navigation in narrow waterway on relationship betwéen S/L2 and
REFERENCES
1) J.N.Newman Lateral motion of a slender body between two parallel walls
Jour, of Fluid Mechanics, vol.39,1969 2.) R.W.Yeung : On the interaction of slender
ships in shallow water
Jour, of Flüid Mechanics, vol.85, 1978 3)R.W.Yeung and 1.T.Tan : Hydrodynamic.-.
interactions of ships with fixed obsta-cles ", Jour. of Sh±p Research, vol.24,
.1979 .... .
I.W.Dand " Hydrodynamic aspects of shallow water collisions Trans. Royal Insti-tute of Naval Arch., vol.118, 1976
WT.taÌ : ". Unsteady hydrodynaic interact. ion of ships in the proximity of fixed objects ', Master's.Thesis, Dept. of Ocean Engineering, M.I.T., 1979
K.Kijima and H.Yasukaa Manoeuvrability of ships in narrow waterway Jour of the Society of. Naval Arch, of Japan, vol.156, 1984 ( in Japanese)
M,Hirano : On calulation method of Ship manoeuvring motion at initial design
phase ", JOur, of the Society of Ñavál Arch. of Japan, vol.147, 1980
