Analysis for the effect of shallow water upon turning

-ÄRCHL

Lab. y. Scheepbouwkunde

B S R A Translatio'i No 2068 Tehnische Hogeschool

Deft

ANALYSIS FOR THE EFECT OF SHALLOW WATER UPON TtJRNINQ

by

StL rum

.

Makoto Kan, member _{vor Scheeps.} -. Nvgate

-. SthecpSbOU

ari a

Tatsuro Hanaoka, member

J. Soc. Nay. Arch.. Japan, 115 (1964), p..49 (June)

niary

A rz.ethod is develped which predicts the turning ability of a.ship ir sha11or water.

The shallow water efft

oefficients of the lateral fôrce and yawing momént actiig on a turning ship were evaluated by means of low aspect ratio wing theory. Some expressions for the shaalow water effect on the. turning ability were deñvèd by the use Of the coefficients. This method is compared with expemental data. The deviations between theory and experiment are not small, but an important resuitthis method is that it provides a means of extrapolating the results of tests deep water. This is a valuable asset for design work when test facilities are limited.

Itroduction

This report describes a simple theory and

lculation

results on the èffect of shallow water on turning.

Though saJly the shallOw water efft is considered

together with the wave-making phenomenon, t1e wave-thaking phenomenon is neglected he re not only bemuse if it is brought in the theoretical calculation becomes very complicated and

difficult to solve, but also because no

_{al poblerns arise except n}

the case of especially high-speed slips. The main purpose, of this

report is to investigate roughly what hydraulic thotion would bng a.out tue shallow water effect on a ship!s turxng ability in cases were the wave-making phenomeo can be eliminated, that is, when the ship's speed is very slow.

Since the effect of shallow water on tuing has been studied cOjunction with tests on controlability in canals, most_tests

ve hee made in the longitudinal basin [i], [z], [3]. The

test [4 J on the shallow water effcct caïe.d Out recently b Ozeki

model ship aíid à. wide bain. _{However, such tests cán only rarely} be carried out because the turning test. requires a very wié basin.. Therefore future study will _{still be caied out in thé longItudinal} basin _{The authors believe tnat this report will be a useful}

reference in such cases.

1. _{Shallow Water Coefficient of the lateral force and}

mOment due to obliqu

_saiing.

Taking he x-axis to be in _{the dire ctioi of the ship} length, the change of speed in the x-direction is considered to be less than the. speed change _{in the y ¿nd z-directions beca.use} a ships shape is long and

_{narrow. Theefore, two-dimensional}

hydrauli6 analysis is quite _{adequate. Assumiig that the ship is} sailing at a speed r with _{an angle pf yaw of and regarding} the free surface as a solid wall ad the ships body _{as a thin blade,} the fo.lowing bounday _{conditions can be obtained.:}

at the ships body _(Oø/Oy)0V. at the water surface and

at the basin bottom

where k is the water depth.

The two-thmensiona1 hydraulic flow conditioh which will

satisfy the above boundary conditions is equivalent to thê

:riydrauic condition of flat plate.s with width 2d. (where d is the

ship?s draft) placed infinitely at intervals of, 2h in the direction of the z-axis perpendicular to the direction of uniform flow, _in

a uniform flow of velocity VLí. _{In this case the velocity potential}

is _{ritten in terms of a complex functiòn as follows}

w

cos(rzd/2h) Where = z + yj

When the water is infinitely

_{.dep, then h}

is obtained as follows:

[1.3].

Therefore Eqn. [1. 4)

fl..5J The suffixes r and .1. mean the starboard and the _{port sides respectively.} Introducing the _{acceleration potential}

_c,

_{it gives the relationshIp}

between _{the pressure}

_{p ad the fluid density p;}

pip. Leting

the Velocity _{potential be}

.

= V (/o x) for the. coodnate.

wVç'2

_[1.4].

The lateral, force acting on the yawing ship is expressed

=

--3-.

which moves at a constánt _{speed V in the minus dirction along} the x-axis. _{Therefore, Eqn. [i. 5 J is written as follows:}

pV J"LJ2Jd\ Xra 0x1

Because _{- = O forward of the ship and}

= - aft of

the ship, assuming d is ¿ons_{tant in the direction of the}

_ships

length, Eqn. [1.6) is

_{written as follows:}

Fs=PVJ ør(L12)dz

_a

(L/2) is the velocity potential on the _{surface of the ship's} body and is derived from _{Eqn. [i. 3 J as fo1lows}

ør(L/2) =.-cosh'cos(7z/2h) cos(rd/2h)

In the case of infinitely

_{deep water, itis derived from Eqn. [1.4]}

as follows:

ø(L/2)=Vç. s'd_z2

£1.9)

Inserting the above into _{Eqn. [1. 7 J and using}

_{suffis h an}

co

for shallow water and infinitely deep _{water respectively, the forces} can be obtained as follows:

V2c6d2f' cosh1COS(nß12

7r

i

cos(ir/2ô)

F=-LpV22

2

where = h/d and ß zld. _{Introducing F}

_IF

¡h

'co

following is obtained:

r' h_1cos(7ß/2)

-

_{j_j}, cos _cos(r/2o)- dß_k,.

Fig. 1 shows the _{values of kF.}

_{This k, value is}

equal to the ratio of the apparent _{mass of the flat plates}

which. are placed. intlie.

direction perpendicujar to the uniform flow in a canal to the apparant mass p77d2 of flat plates placed in the infinitely _{wide basin.}

The moment Mip._{about-the centre of gravity due to the force}

F/J _{is written as follows:} PV Zia d_f

-[1.6) [1.7] [1.8) [1.10] £1.11) the

t1. 12)

[1.13) Partial integration

_{gives MJi= -FD.L/2.}

_{Then the relationship}

M, IM

= F

also obtained for the _moments.

2. Shallow Water coefficient of the lateral force and rriornent due to

turning motion

If the turning Yadius is far greater than the ships length, the lateral force due to the turning motion can be calculated

by the same method as described in Section 1. However, the boundary

conditions given by Eqn. [1. 1] should be replaced by

_(Ç/

_{y)0 =}

.C2.xVJL, whereflis the non-dimensional expression of the

angular velocity of turning. As the calculation method is eact1y the same as given in Section 1, the calculated results only are given here. The lateral force Fn. due to the turning motion is written

as follows:

Foh=-_.PVC1d2f1 cosh' cos(r.&/2ô)d

-1 cos(r/2)

Fç =rp VÇ1d/4

The moment M2 about the ¿entre of gravity is M

_{F. L/Z.}

Therefore, the shallow water coefficient can be written in the same way as in the previous case; M.C.h/N1= Fh/F2 .=

3. _{Shallow Water Coefficient of Rudder}

Usually the ratio of height to width of a rudder is 1. 5 to 2. 0. However, assuming the free surface is a solid wall by

neglecting the wave-making phenomenon so _{as to simplify calculation,}

it is sufficient to consider the flow field of

a blade having a ruer

ratio \ of 3 to 4. Lettingo be the geometrical angle of incidence of the blade, the effective value ofc changes due to the vortex flow caused by the blade and its rLection from the bottom of the basin and the free surface. _{Expressing this effective angle of incident by} i, the following is obtained [6]

[3.1] where çis approximately givenas follows:

log

[3. la] From Eqn. [3. 1] and the equation CL = Zirki, the lift coefficient

of the blade is given as follows:

2 hr

[3.2] In the case of infinitely deep water 0 = 0, and then the effect of

If

_{4, then r}

_{0. 13. Because k 1, k}

can be calculated from

Eqn. _{[3. 3] and shown in Fig. 1 together with the values of} kF.

It is much smaller _{than the coefficient kF for the ship's body.}

4. _{Shallow Water Coefficient}

of the angular velocity of turning and

angle of yaw.

The linear kinetic equations for constant turning of the ship are written as follows:

where C.,, C

and C _{are values obtained by dividing}

F J, FJfl and òF

Io (where

F is the lateral force

due to the rudder) by 1/2. pV Ld respectively, while

and CM are values

_{obtained by dividingMJ}

_MIfl

and Vj/3c (where

_{M is the}

_{moment about the centre}

of

gravity due to the

_{lateral force F) by lJ2.p V2L2d.}

Cm is also

obtained by dividing the _{ship's mass rn including the}

apparant mass along the ship length direction by lJZ.pL2d:

From Eqn. [4. 1] and [4. 2], fl-and_{Ware obtained as follows:}

l'y- =

CM.ç±CvQ.r2+C.o

T Cfç.CM.. CpQ±C.CM.,,

The differential _{factors CM etc. on the right hand side}

of the above

equation are the values in the case of deep water. The ratio

kt i

_{regard to te angle}

_{of yaw is written}

as follows: 1 F(CCpaCp,'CJJQ)Cy,C(, kF CMCpCpCMQCfl'C,,, [4.3] [4.53 [4.6] [4.1]

[4.2]

C, _{CFÇL.CJÇ. CFQ--C C. [4.4]}

When the rudder _{angle c is the same for the cases of}

shallow water and deep water,

_{the ratio kft in}

regard to the angular velocity of turning is written as follows:

f

Whenkis calculatedirom

_{[4.5], CCFYf_}

CMf.CFandCC?J are so nearly equal that a large error

may be brought into the calculation. To avoid this error, the calculation should rather be carried out as follows: Considering a casé such that the rudder agle is chosen, so as to make the tWO valués of l2for shallow water and deep water equal, thé kinetic

equaiors are exprssed asEqns. [4:7] and [4. 8], lett-ig the

rudder angle ad the angle of yaw beh and

'h respectively:

kFCFçbft±kp.Cpfl±kCpØ/=C7T.

.CMP '/j+ l'p C.2+kC,,.ce4= O

Eliminating U/h and ut(.i) froxi the above eqùations and.

Eqns. [4. 1] and [4. 2] and applyi,ng the linear relationship betweenftand , the. folloWin equation can be derived:

[47]

[4.8]

From this, kj

can be calculated.

5. Shallow Water Coefficient of Increment of Trim & the Dipping of the ships body düe to turning.

It requiresa complicated procedure to calculate the shallow water coeffient of the increment of the dipping of the ships body due to turning. Assuming the body to be a single cylinder, the increment / of the dipping of the ships body is Obtained here just to investigate

its. general tendency. In this case, two-dimensional flow is considered.

Considering th cond.iti that there is uniform flowin the direction of minus y at the speed of U as shown in Fig. Z and

letting p0 be the atmospheric pressure and p and q be the pressure and the flow velocity at apoint (a cos O, - a sin e), the force per unit, length acting downwards due to.the flow U is expressed as follows:,

(psp) sin Od o=.-af (q2_Ü2) sin OdO

In the case of ifiitely deep water q = ZU six 8, and in thé case of shallow water it is multiplied by the factor k(e). _{The above}

equation is reduced to the following,:

=-

PaU2f' f4i(0)2sin2O-1)sinûdO'

[5.2]

lkp fl

C C(çfCFç

-7-The ship will dip _{so as to balance the downward}

force with increased buoyancy. _{In order to calculate more accurately}

ofthe increment of buoyancy, it is required to examine thehId _{= h/a) due to the dipping of the ship.} change

However, this

change is not _{examined here and thc first approximation} only is obtained. _{Letting d be the}

average increment of the dipping o1

the ship, the _{increment of buoyancy due to}

d is 2,PgLad, which

should be equal to _{the integral of over the ship.}

Then the following equation is _obtained:

4d=-._Lf_{i:: r U(4 (0)xinO-1) sin Odo}

dx

[5.3] In the case of constant turning, U = V(y+.(2JL), the above will be reduced _{to the following equation:}

V2(12ç._O))

4d=-_________f

.4&(0)sin2O_1tsjflOdO

[5.4] Using the shallow wate r cofficients k _{and in}

regard tOfl.and z1' which _{were obtained in Section 4}

and neglecting

the change of velocity, the shallow water coefficient of the mean increment of the dipping of the ship is written as follows:

= 4dh/4d,. = K(12 k±sk) /(12+2)

where K is as follows:

K=f4

_{(0)2sino_1))51 OdO/fg}

(4 s1n20-_1)sjn_OdO [5.6]

and s =

/i/,

_{which can be}

obtained from the turning test in

deep water, s _{3 is quite reasonable.}

The

the aft trim.

L2a1cgt/6 be

the following

increment of trim _{t due to turning tends to increase} Letting the moment of t about th centre of gravity equal to the _{moment of f about the centre of gravity,}

is obtained:

[7]

2g

j

4ì«O)sin2Ol)sinOdO

Therefore, its shallow _{wate r coefficient is written as follows:}

k(e) which is _{necessary for the above calculation is the} factor of the _{side wall effect of the tangential flow}

on a cylindrical

Surface which is _{placed in the centre of the uniform flow canal.}

[5.. 5]

[5.7]

The method of calculating k(ê) is described in the following

section.

6. Flow around cylinder in Canal

Suppose such a turbulent flow where a cylinder whose

radius is a is in the centre of the uniform flow (velocity U)

in a canal whose width is 2h. This flow pattern is equal to the case of an infinite number of cylinders placed at intervals of 2h in the z-direction. Letting r

_m

be the distance between a point (y, z) on a cylinder placed at the coordinate origin and

a point (y, z') on the m-th cylinder from the origin, the

following equation can be derived from Green's Theorem:

:f

This is integrated with respect to the variables (x', y').

N is a large positive integer. _{Introducing the cylinder coordinates:}

a cos G, z' = a sin 0 2mh, y = a cos e and z a sin O. I

Let the first integral term in Eqn. [6. 1] be Il which has a singular point at (x, y), so that the integral should be done to

avoid this singular point. I is then expressed as follows: IL=rø(0)±fføEiogrnt2dO

[6. 2]

The following is also obtained:

where

From the infinite series,

i sinhA

i

-

2A(coshAcosB)' z+m_C0t

is obtained. Letting the second term in Eqn. [6. 3] change to the form of

_{C/(m)ßI(mi)/(ß)}

and putting N > co

Eqn. [6. 2] is reduced to the following: I1=7rV(0)_7fK(0. 0'YD(O')dO'

[6.5]

(4 nIa52)sr (0'0)12 (4 rn/aò') sin O'

ogr

_{Ä2+(B-2 mir)2 A2±(B-2mrr)2}

A=r(cos0cosû')Jö, B=(sin0sinO')/O, 6=h/a

[6.1]

[6.3]

[6. 3a]

where K is given as follows:

-9-sinh A. sin (O'O)/2sin O' -9-sinh A. cos(O'+O)/2±sinB .sin(O'+0)/2} K(0. O')=

2ô(cosBcoshA)sin(O'+O)/2

Letting the second terni in Eqn. [6. 1] be _{Iz, O/ön}

JÒn = U cos êt independently from the existence of the side wall, the following is obtained:

aU ,

%a=

)j cosO logr-d'

2

--'

[6.7]

where 50 is gilven by'f= et +3' when E2 _{= y2 + (z - 2mh)2 and}

cosy = y/E. Then r2

= E2 + a2

- 2Ea cosf.

Using the following integral equation;

I

Jo

Eqn. [6. 7Jis reduced to the following:

-. 2r

f

cosO'.1ogrdç-[6.6]

[6. 7a]

Then using the infinite series of Eqn. [6. 4], the following is obtained:

aU

sinh(i/3.cosO) I2=7/(0)

=

i1

cosh(r/5 cos O) cos(/O.sin O)

Inserting I and 12 obtained in the above into Eqn. [6. 1], the _integral equation with respect to the velocity potential _{on the cylinderis}

obtained as follows:

0(0)=f(0)f2'°K(O. O')O(O')dO' _[6.9] K(e, OX) and f(G) _{are given by Eqns. [6. 6] and [6. 8]. Solving}

Eqn. [6. 9] to obtain (0), the velocity potential _{around the} cylinder can be obtained from the Third Theorem of Green.

Because K(e, et) has no singular point in the integrated region, it is easy to replace the ,second terni _{on the right hand}

side of Eqn. [6. 93 by an approximate algebraic equation. In

this case the integral is changed into the _{algebraic equation}

by applying Simpson!s law and (0) is obtained by the approximate integral method.

The apparent mass nit of the cylinder is given as follows:

Pa r2' 50

,n'=-- 1

0 do

On

Therefore, inserting Q and OIô

= u cos O obtained above into

Eqn. [6.10], the apparent_{mass can be obtained. Fig.4 shows}

[6.8]

As there is no singular point near the cylinder, tangential velocity on the cylder surface _{can be obtained}

by carrying out directly the nunieical _{integration of, with réspect}

to variable O. _{From this procedure, the values of K given in}

qn. _{[3. 6] are obtained and given in Table 1.}

7. _{Example of Calculation}

For a ship of Series 60 type, a typical high speed tanker, th following values [8] _{have been obtained:}

C1 --0 u0 C-0 C1 _{CL=0 022} CFV=O 290 _{CFÙ=0 067 C=-0 046 C,=0 16} Using these data, ky and k, _{are calculated from Eqns. [4. 6]}

and [4. 9] and given in FiÚ. 3 _{'Ì'hese calculated results show}

a similar tendency to the test. results _{obtained by Ozeki and}

[4]

putting LS] _{s = 2. 7, k}

d and k are calculated from

Ecn _{[5. 5] and [5. 8] and given in Table 1. This is the case}

where the same angle of rudder is taken for deep water and shallow water. If such an angle of rudder is taken as to make

X). or çLi equal in deep _{water and, in shallow water, k}

d and become greater tha. the values giveii in Table L _{If we take} be the same in deep water and in shallow water, the _alue of

d becomes large 'and equal to the values of K listed in

Table 1.

8. Conclusion

The authors wanted to find out how the shallow water effect upon turning ability would appear in the actual _behaviour of a ship and they consider that the above calculations

_{are fairly}

satisfactory, though the accuracy is' nöt so goöd.

For exam.ple, _{kF in .Eqn. [i. 12] is not only the ratio of}

to Fi, but also the ratio _{f the apparent mass of a flat plate} placed in narrow canal to the apparent mass of flat plate placed

in infinitely wide _{water. The ratio the apparent thass of}

a

cylinder placed in a narrow canal 'to the apparent mass of' cylinde.

placed in infinitely wide _{water is also shown in Fig. 4. It is'}

' '

seen in this figure 'that the. _{experimental data} cz], [3],

of ' .

the.

h''

lie between the curve of the flat plate and the curve of the cylinder. According to MunkTs linear _{theory, the lateral force} per unit length along the ships length is Vd(mV/)/dx (where m is the apparent mass of the cylinder, _{o that the experimental}

data are plotted in _{very reasonable positions. Because this} test did not pmrent the ship from dipping _{freely, the plotted data} should be shifted to the left in _{order to correspond to the water} depth calculated. _{Therefore, the lateral force acting on he ship} is very close to that in the case assumed for a flat plate.

More accurate values of

_{h1 can be obtained if the apparent}

mass is calcuiated for each section of the ship _{by the method} described in Section 6. _{If the lateral force is calculated from} Munks theory, it is quite useful _{to apply the hypothesis intxoduced}

by Jones f9J and Schmitz

If the shallow water effects of _{turning derivatives are} examined in a wide test tank, _{a longitudinal basin, etc., the effect} of shallow water on the turning _{ability can be known roughly.}

Refe renc e

Baker, G. S.: Steering of Ships in _{Shallow Water and Cals,}

TINA, 66 (1924), 319

Brai-d, R.: Manoeuvring of Ships in Deep Water, in Shallow

Water and in Canal, SNAME, _{59 (1951), 229}

Motora, S. and Couch, R. B.: _{Manoeuvrability of Full Bodied} Ships in Restricted Waters, _{Paper presented before the Fall} Meeting of the Great Lakes and Great River Section of the

Soc. N. A. M. E. Oct. 1961

N. Ozeki & T. Tsuji: Effects _{of Shallow Water on Turning}

Ability, read at the first

_{symposium at the Ship Engineering}

Research Institute, 29th Nov. , 1963.

T. Sasaki: Application of Equi-angle Reflection, Toyama _{Book Co.}

1939 Nov.

Durand, W. F.: Aerodynamic Theory, Vol.11 Julius Springer, _{1934, 251} Cole, A. P[: Destroyer Turning _{Circles, TINA, 80 (1938) 32.}

Eda, H. and Crane, C.L.,

_{Jr.: Research on Ship Controllability,}

Part II, Davidson Laboratory _{Report 923, 1962.}

Jones, R. T.: PTc>perties of _{Low Aspect Ratio Pointed Wings at}

Speeds below and above the Speed of Sound, NACA Tech. Rept.

No. 835 (1946)

Schmitz, F. G.: Application of Slender Body Theory to the dynamic yaw stability and steerability _{of ships. Schiff y.}

Hafen, June 1961, 479.

.2.0 1.0 Table 1. K, k

,, k

_t ;.' 1.4 1.2 1.0 1.0 1.1- 1.3 2.2 2.' s Fig.1

_{k,, k<}

tJ U

;f

f I f f / / / f i / II I Fig. 2 .3

..4 'V--

'

(o-25) .2 O Io Cok,mr -FIc Pkt 1.0 1.4 1.8 .!&:.4

_Frh/Fy

Z2

zeki. & Tsuj± Experiment

1.2 1.4 _1.8 K _{9.93 4.35 2.29} k d 2.36 1.70 1.37 2.43 1.74 1.40 22 1.3 0t. o ° MCtC3 FiIF C?-s)

-

MJV0