The cross-flow drag on a manoeuvring ship

Pergamon

THE CROSS-FLOW DRAG ON A MANOEUVRING SHIP

J . P . H O O F T

M A R I N , Wageningen, The Netherlands

Abstract—In this paper, an analysis is given of the experimentally derived local lateral force on a manoeuvring ship as a reaction to the ship's lateral velocity. The tests were performed with models consisting of several segments. Special attention is paid to the longitudinal distribution of the non-linear component of the lateral force, the so-called cross-flow drag. This aspect is of utmost importance when the non-hnear contribution becomes dominant as will occur in a tight turn during which the ship's drift velocity becomes relatively large compared to the ahead speed.

N O M E N C L A T U R E

AP aft perpendicular C, lift coefficient

drag coefficient

C^j corrected drag coefficient

Fn Froude number = U/(g*Lpp)°-^ FP forward perpendicular

G ship's center of gravity g acceleration due to gravity

L lift on the ship = force perpendicular to the resultant speed U of the ship Lpp ship's length

/„ length of the nth segment

niy added mass in lateral direction per unit length r ship's rate of turning

T ship's draught

t„ draught of the nth segment

ll ship's speed component in the longitudinal direction U resultant speed of ship = ^M^ + v^)°-^

V ship's speed component in the lateral direction

X longitudinal direction in the ship fixed system of coordinates

x^ longitudinal direction in the Earth fixed system of coordinates lateral direction in the Earth fixed system of coordinates Y lateral force to the ship

yawing moment around G

N„ yawing moment around the midship section (3 drift angle = atn(v/M)

p mass density of the water.

1. I N T R O D U C T I O N

FOR T H E prediction of the ship's manoeuvrabiUty, use l:an be made of computer simulations. For this approach mathematical models must be available by which the relevant hydrodynamic forces o n the manoeuvring ship are described.

O f t e n these mathematical descriptions of the hydrodynamic phenomena have been derived f r o m a regression analysis of the results of captive m o d e l tests f o r the ship considered [see H o o f t (1986)]. H o w e v e r , such mathematical descriptions are not always

330 J . P. HOOFT

satisfactory because of tlie lack of a pliysical meaning of the descriptions. This imperfec-tion may lead to the f o l l o w i n g problems:

• The impossibility of comparing the hydrodynamic coefficients of two different ships. This problem may be serious when designing the ship f o r o p t i m u m manoeuvrability. • A l i m i t e d accuracy of the .manoeuvring predictions because of the fact that the

simulation results w i l l depend on the accuracy by which the coefficients have been derived by the regression analysis.

For the prediction o f the ship's manoeuvrability i n the initial design stage, no model test results are usually available f o r the determination o f the hydrodynamic coefficients. One can then use computer simulations only i f the hydrodynamic coefficients can be established otherwise.

Nowadays i t is not yet possible to determine the hydrodynamic manoeuvring coef-ficients by means of theoretical methods. Therefore, empirical methods have been developed by means o f which, i n the initial design stage, the hydrodynamic coefficients can be estimated as a f u n c t i o n of the ship's main dimensions; see, f o r example, Inoue

et al. (1981) and K i j i m a et al. (1990).

I n empirical methods the hnear hydrodynamic coefficients are described rather accu-rately as a f u n c t i o n o f only a f e w aspects of the ship's dimensions. T w o reasons can be given f o r the achieved accuracy:

1. The linear hydrodynamic coefficients are most likely rather independent of local parameters o f the ship's hull f o r m (at the bow and/or the stern).

2. For a wide range of ships the linear hydrodynamic coefficients have been determined experimentally. This means that an acceptable level o f confidence has been achieved in the description o f the coefficients as a f u n c t i o n of the ship's main parameters. I t appears that the non-hnear contribution of the hydrodynamic characteristics can be estimated only roughly by empirical methods. This is very inconvenient when tight turns have t o be predicted by means o f computer simulations. I n such manoeuvres the non-linear contributions play a significant role. Three reasons are mostly given f o r this unfavourable aspect:

1. The non-Hnear hydrodynamic coefficients are rather sensitive to the local f o r m parameters of the ship. This means that much more i n f o r m a t i o n (data) is required as a f u n c t i o n of the larger number o f parameters.

2. O n l y f o r a limited number of ships have non-hnear hydrodynamic coefficients been determined experimentally. This means that only a l i m i t e d level of confidence has been achieved i n the description of the coefficients as a f u n c t i o n of the ship's main parameters.

3. O f t e n only the hydrodynamic coefficients have been pubhshed w i t h o u t the actual model test results. Some of the authors use quadratic non-linear coefficients, while others apply tertiary non-linear coefficients. I n this way the validity o f a presented non-linear coefficient is hmited and cannot be compared w i t h the corresponding coefficient f o r another hull f o r m .

I t was thought that a better description of the non-linear component of the lateral force could be achieved i f one could establish the local non-linear force component

instead o f tlie total force on the ship. This local component was defined by the local cross-flow drag coefficient; see, f o r example. H o o f t (1987).

Burcher (1972), Clarke (1972), Matsumoto (1983) and Beukelman (1988) have presented the results of their experiments w i t h segmented models. F r o m these tests the lateral forces o n each of the segments can be derived. I n the present paper a description is given of the analysis of test results w i t h segmented models. Based o n this analysis an empirical method can be derived to estimate the distribution o f the non-linear lateral force component over the ship's length. The intention of this method is:

1. T o achieve a more accurate prediction of the non-linear component o f the lateral force.

2. T o realize a more accurate mathematical description of the hydrodynamic forces f r o m measurements on the t o t a l model.

3. T o estabhsh eventually a simulation program that is capable of predicting the ship's manoeuvrability w i t h sufficient accuracy. Such a manoeuvring prediction method would provide a means to investigate the effect o f the variations o f the ship's h u l l f o r m o n its manoeuvrability.

2. T H E V A R I A T I O N O F T H E L A T E R A L V E L O C I T Y O V E R T H E L E N G T H O F

While considering the ship's manoeuvring performance, three velocity components can be discerned at each instant (see Fig. 1):

• u, the ahead velocity of the centre of gravity G

• V, the lateral velocity o f the centre of gravity

• r, the rate o f turning around the vertical axis.

The combination of d r i f t velocity v and yaw rate r leads to a local lateral velocity v(x) that varies over the length of the ship:

When considering a t u r n by a port (positive) rudder angle then i t is seen that the ship w i n t u r n to port (negative r) at a positive d r i f t velocity v. I n this t u r n the local

T H E S H I P

v(x) = V + x*r . ₍₁₎

xe

yet

332 J . P. HOOFT

d r i f t velocity becomes zero (the turning p o i n t ) somewhere near the bow of the ship

(x>0), while at the stern the lateral velocity wiU become quite large relative to the

ship's ahead speed. A s a consequence of this resuh i t is f o u n d that, i n general, the lateral hydrodynamic reaction force:at the bow w i l l be small, while at the stern quite a large force w i h be experienced.

3. T H E L A T E R A L F O R C E AS A R E A C T I O N T O T H E D R I F T V E L O C I T Y W h e n considering first the lateral hydrodynamic force on the ship as a reaction to the ship's d r i f t velocity v only (at zero rate of t u r n i n g ) , the foUowing description is o f t e n applied:

Y(v) = Y^^*u*v + Y^^*v*/v/ (2)

or, i n a non-dimensional f o r m :

Y O ) ' = y ^ * c o s ( p ) * s i n O ) + y^p*sin((3)*/sin((3)/ (3) i n which the lateral force has been made non-dimensional by dividing by 0.5p Lpp^U'^

or 0.5p LppTU^ w i t h the total velocity U defined by:

The d r i f t angle |3 is defined by:

(3 = a r c t a n ( v / w ) . (5) The total lateral force Y{v) is the resuhant of all lateral forces y „ ( v ) o n each of the

segments i n which the ship model has been subdivided:

Y{v) = E Yniv) (6) while the total yawing moment around the center of gravity amounts to:

N{v)^J,{x^*Y^{v)). (7)

For f u r t h e r analysis the local lateral force on each segment is w r i t t e n i n a non-dimensional f o r m according to:

F „ ( P ) ' = y „ ( v ) / ( 0 . 5 p / A f / " ) (8) w i t h /„ the length of the segment and t„ hs average draught; see, f o r example, Fig. 2.

According to Equation ( 3 ) one now describes the non-dimensional local lateral force Yni^y by means of a hnear coefficient Cy„ and a non-linear coefficient Cdn.

y „ ( p ) ' = C y „ * s i n O ) * c o s ( p ) - C r f „ * s i n ( p ) * / s i n ( ( 3 ) / (9) i n which the hnear coefficient Cy^ corresponds to the derivative Y'„^ and the drag

coefficient Crf„ corresponds to - Y n p p .

I t can be shown that i n E q u a t i o n (2) the linear term is linearly dependent on the longhudinal velocity component u and not o n the total velocity U. This means that f o r zero f o r w a r d speed {u = 0) no linear contribution i n the lateral force exists. This

Todd 7 0 m o d e l ; Lpp/T = 17 .5; z e r o t r i m . 5 t h segment f r o m t h e bow; Fn = 0 . 15. 0 0 2 - ₊ • } • 0 0 1 -0 1 - -t-0 • ] 1 \ 5 1 0 2 0 0 0 1 • 0 . 0 2 - ₊

FIG. 2. Example of the measured lateral force on a segment as a function of the drift angle (3; obtained from Beukelman (1988).

holds not only f o r the lateral force on the total ship but also f o r the local lateral force on each of the segments o f a ship model.

I f one considers the l i f t L to be a lateral force that is perpendicular to the total undisturbed incoming flow U, then the coefficient Cy„ is related to the local l i f t coefficient Cl^(n) as foUows:

C y „ = - a p ( n ) * c o s O ) . (10) Inserting this result into E q u a t i o n (9) leads to:

y „ ( p ) ' = - a p ( n ) * s i n ( p ) * c o s ( p ) 2 - C4H=sin(p)*/sin((3)/. (11) I n the application of E q u a t i o n (11), use is made of the assumption that the l i f t

coefficient C/p(n) is independent of the d r i f t angle p. This means that i t is accepted that the local cross-flow drag coefficient is a f u n c t i o n of the d r i f t angle p . Various physical arguments can be given to prove this assumption to be correct. I t should be borne i n m i n d that both the l i f t coefficient and the cross-flow drag coefficient vary over the length of the ship.

F r o m the test results o n each segment as presented i n Fig. 2 one determines the local l i f t coefficient Cl^(n) at the range of small d r i f t angles p . W i t h this value of C/p(«) one determines f o r each segment the cross-flow drag coefficient f r o m the measurements at higher d r i f t angles:

r , rn, - - Y n i ^ y - a p ( » ) * c o s ( p ) ^ . s i i > ( p )

"^"""^^^ s i n ( p ) * / s i n ( p ) / ~ • ^^^^ F r o m this equation one finds, i n Table 1, the values of the drag coefficients Cd^ f o r

334 J. P . HOOFT

TABLE 1. EXAMPLE OF THE DERIVATION OF THE CROSS-FLOW DRAG COEFFICIENTS Cd„ FROM THE MEASUREMENTS

Drift angle p (in degrees)

Measured dimensionless latitudinal force Y'^

Drag coefficient Cd^ with C/p (5) = -0.287 4 +0.01478 1.057 8 -H0.01906 1.038 12 -1-0.00806 1.134 16 -0.00398 1.015 20 -0.02001 0.912

Todd 70 model (7 segments); LppIT = 17.5; zero trim; 5th segment from the bow; Fn = 0.15. 4 . T H E L O C A L L I F T C O E F F I C I E N T Cl^{x)

According to Jones (1946), one finds tliat the local l i f t per unit length is determined by the instantaneous apparent acceleration o f the local lateral added mass of water alongside the ship:

Lg = v*w*m^g (13) w i t h THy being the lateral added mass of water per unit length of the ship and ^ being

the distance o f the cross-section f r o m the f o r w a r d perpendicular. I n Equation (13) the derivatives Lg and m^g are defined by:

Lg = éLlAi • ruy^ = dmy/d^ . (14) F r o m integration o f Lg i n Equation (13) one finds that the l i f t force L over a distance

between 4/ and ^„ ( w i t h ^/ being the closest t o the FP) amounts to: p a

L = v * M * my^di = v*u*(my{^,)-my(^f)) (15)

kf

f r o m which i t is seen that i n theory the total l i f t o n the ship w i l l be zero (paradox o f d ' A l e m b e r t ) because of the fact that in an ideal fluid the added mass per unit length

niy is zero at the bow (^/ = 0) as weh as at the stern (^„ = Lpp).

I n Tables 2 and 3 the local l i f t coefficient Cl^(n) on each of the seven segments o f TABLE 2. COMPARISON BETWEEN MEASURED AND CALCULATED LINEAR LATERAL FORCE COMPONENT ( I N A NON-DIMENSIONAL FORM) ON EACH OF THE SEVEN SEGMENTS OF A MODEL OF THE TODD 70 SERIES, NO TRIM, Fn = 0.15;

SEE BEUKELMAN (1988)

Lpp/T = 22.81 Lpp/T = 17.50 Lpp/T = 14.20

Segment No.

Cip Ctp C^ip C^ip C'i.p C'z.p

Segment No. (measured) (calculated) (measured) (calculated) (measured) (calculated)

1 + 1.233 + 1.134 + 1.387 + 1.440 + 1.577 + 1.655 2 +0.301 +0.312 +0.401 +0.370 +0.487 +0.487 3 +0.172 +0.115 +0.186 +0.142 +0.215 +0.176 4 0 -0.009 -0.040 -0.003 0 -0.001 5 -0.199 -0.259 -0.215 -0.299 -0.242 -0.329 6 -0.172 -0.437 -0.287 -0.606 -0.356 -0.722 7 +0.284 -0.944 +0.143 -0.934 +0.072 -1.109

TABLE 3. COMPARISON BETWEEN MEASURED AND CALCULATED LINEAR LATERAL FORCE COMPONENT ( I N A NON-DIMENSIONAL FORM) ON EACH OF THE SEVEN SEGMENTS OF A MODEL OF THE TODD 70 SERIES, LppIT = 17.5,

Fn = 0.15; SEE BEUKELMAN (1988)

Trim = -3.46° Zero trim Trim = + 3.4°

Segment No.

Ci,p Cz,p Cz.p C'ip

Segment No. _{(measured) (calculated) (measured) (calculated) (measured) (calculated)}

1 + 1.577 + 1.730 + 1.387 + 1.440 + 1.090 +1.034 2 +0.158 +0.125 +0.401 +0.370 +0.659 +0.756 3 -0.229 -0.352 +0.186 +0.142 +0.602 +0.680 4 -0.373 -0.559 -0.040 -0.003 +0.502 +0.548 5 -0.573 -0.826 -0.215 -0.299 +0.201 +0.179 6 -0.502 -0.762 -0.287 -0.606 0 -0.324 7 -0.401 -0.742 +0.143 -0.934 +0.244 -1.197

the ship model is given as derived f r o m the experim.ents described by Beukelman (1988). I n these tables the local l i f t coefficients are also presented, derived f r o m E q u a t i o n (15).

F r o m the comparison between the experimental and theoretical values i n Tables 2 and 3 h is seen that the theoretical values f o r the h u l l f o r m considered agree weh w i t h the measurements f o r the five most f o r w a r d segments.

The theory by Jones (1946) f o r determining the local l i f t force can also be checked in another way while applying Equation (15). For this purpose one reduces f r o m the measurements the distribution of m^(^) by a summation of the l i f t derivative L^^{n) over the segments after E q u a t i o n (15) has been rewritten by:

niyiUn)) = my{Un-l)) + L „ , ( n ) (16) w i t h rriy being zero at the f o r w a r d perpendicular.

I n Figs 3 and 4 the experimental values of my are presented as derived by means of Equation (16) using f r o m the model test results presented by Beukelman (1988). I n Figs 3 and 4 the theoretical values of niy are also plotted. F r o m the comparison of the theoretical and experimental values i n these figures it is seen that the theory should be corrected f o r three-dimensional effects on the value o f niy leading to my^.

myXk) = C^{i)*my{i) (17) i n which C^ii) is the correction t e r m as a f u n c t i o n of the longhudinal location. H a v i n g

determined the corrected lateral "added mass" distribution niyj^^) it w i h be possible to determine the corrected derivative

niy^c-niy^M) = niy^Xk) = dmyXi)IAi,. (18) A p p l y i n g the value m^^^ i n E q u a t i o n (15) one now finds .that the t o t a l l i f t o n the ship

amounts to:

L^v*u* my^^d^ = v*u*my^(i = ^^pp). (19) J^FPP

336 J. P . HOOFT calculated measured LPP/T = 22.81 \ LPP/T = 17.50 X LPP/T = 14.20 —, / y

/

/ /

/ /

w

/ /

' /

/

/

/

\

\

FIG. 3. Comparison of the calculated lateral mass niy per unit length with the values derived from the experiments with segmented models for a Todd 70 hull form without trim at ¥n = 0.15; see Beukelman

(1988).

one then finds f o r the t o t a l yawing moment due to the l i f t force distribution over the ship's length:

around the midship section) = v * « * (0. SLpp - Q * my^^dè, Hfpp

calculated measured

LPP/T = 17.50 Tir

1

APP

FIG. 4. Comparison of the calculated lateral mass per unit length with the values derived from the experiments with segmented models for a Todd 70 huU form with Lpp/T = 17.5 at Fn = 0.15; see Beukelman

(1988).

or

f^APP

No = v*u* [-my,(^APp)*Lpp/2 + rriyXO dQ. (20) kppp

338 J. P. HOOFT

5. T H E L O C A L C R O S S - F L O W D R A G C O E F F I C I E N T Cd(^)

A f t e r having determined the l i f t coefficient Cl^{n) on each segment, one then establishes the cross-flow drag coefficient Cd^i3 j o n each segment by means of Equation (12). Some o f the results derived in^this way are presented i n Fig. 5. I n this figure the distribution o f C J „ over the ship length has been plotted f o r various d r i f t angles (3.

The f o l l o w i n g comments on the results i n Fig. 5 can be given:

The cross-flow drag coefficient on the first segment f r o m the bow has most probably been caused by the bow wave. Therefore i t is assumed that this value depends o n the f o r w a r d speed of the ship; see also the findings by Matsumoto (1983).

Aside o f the cross-flow drag coefficient o n the most f o r w a r d segment(s) i t is seen f r o m the resuhs in Fig. 5 that f o r small d r i f t angles (3 the value of Cd w i h increase at larger distances f r o m the bow u n t h some m a x i m u m value is attained after which Cd w i l l decrease a little. A t increasing d r i f t angles p this curve of Cd over the ship's length wih move f o r w a r d .

For large d r i f t angles of approximately 90° the ship sails nearly abeam. I n this condhion the distribution of the cross-flow drag coefficient over the ship's length mainly depends o n the f o r m of the ship's h u h and to a lesser degree on the local Reynolds number. I n Fig. 6 this distribution of Cd is presented f o r a containership and a tanker; the results in this figure f o l l o w f r o m Matsumoto (1983).

The value o f Cd{^) at a given cross-section w i h vary at increasing d r i f t angles f r o m the values shown i n Fig. 5 to the values shown i n Fig. 6.

One now considers the relation between C(i„(p 90°) at an arbitrary d r i f t angle and C(i„(P = 90°) at a d r i f t angle of 90° f o r the various ship types and ship conditions.

For this purpose use is made of the corrected drag coefficient C c J „ f p) which is defined

by:

C 4 P ^ 9 0 )

^^^^^^ ~ C ^ P = 90) ^^^^ leading to the fact that at 90° d r i f t angle the coefficient Ccd equals unity over the

whole length of the ship.

Combining the results i n Figs 5 and 6 according to E q u a t i o n (21) w i h yield the results presented i n Fig. 7. I n Fig. 8 the course o f Ccd over the ship length is sche-matically indicated f o r increasing d r i f t angles.

F r o m the results o f the corrected drag coefficient Ccrf„(p) i n Fig. 7 i t is concluded that this coefficient is stiU dependent o n the longitudinal location and the d r i f t angle, but is no longer dependent on the ship's h u h f o r m .

6 . C O N C L U S I O N S

Experiments have been carried out w i t h segmented models during which the lateral forces on each of the segments were measured. These tests were performed as a f u n c t i o n of the ship's lateral velocity ( i n a towing t a n k ) and as a f u n c t i o n o f the ship's rate of t u r n i n g (under a rotating arm f a c i l i t y ) .

I n this paper the results are presented of the analysis o f the experiments w i t h segmented models as a f u n c t i o n of the ship's lateral velocity. The same k i n d of results were f o u n d by analysing the rotating arm tests as a f u n c t i o n of the ship's rate of turning.

o L P P / T = 22.81 L P P / T = 17.50 L P P / T -= 14.20 1.4' 1.0 Ü 0.6- 0.2- -0.2-APP O . ° « " c <T) > 1 Q O A • A • 0.5 FPP 1.4-1.0 = 12 0,6- 0,2- -0.2-APP ê B D 0 < A ) 0 t a Ï k é UJ a a 0 0 5 FPP 1.4-1.0 0.6- 0.2- -0.2-APP /S = 16° A A m 0 O A D ° * • r A O ) i a • • • fi 0.6 FPP 1.4 |9 = 20 1.0 Ü 0.6- 0.2- -0.2-APP O O a o A O • 0.5 O A FPP

FIG. 5. Distribution of the cross-flow drag coefficient over the ship's length, derived from the experiments with a segmented model for a Todd 7 0 hull form without trim at Fn = 0.15; see Beukelman ( 1 9 8 8 ) .

340 J. P . HOOFT • CONTAWER O TANKER 8 0.8 • O 0 O O 0 0 • 0 D 0 ID n • Q = = = 1 1 APP 0.8 0.6 0.4 0.2 FPP

FIG. 6. Distribution of the cross-flow drag coefficient over the ship's length, derived from the captive drift tests with two segmented model at (3 = 90°; see Matsumoto (1983).

The conclusions obtained i n the present study can be summarized as follows: • Based o n Jones' theory (1946), the distribution of the linear lateral force over the

ship length can be predicted as a f u n c t i o n of the ship's f o r m by means o f empirical formulations.

© A L P P / T = 22.81 L P P / T = 17.50 L P P / T = 14.20 = 8 1.4 1.0 Ü 0.6- 0.2- -0.2-APP • S O J A 0 n o a A • 0.5 = 12° 1.4 1.0 X) 0.2- -0.2-• O s C ) ) 0 • • ê FPP APP 0.6 _FPP 1.4 1.0 = 16° O 0.6. 0.2- -0.2-m • . 1 s n A E D APP •O.S _FPP 1.4 jg = 20° 1.0 •a O 0.6. 0,2 -0.2-1 APP A O 0.5 •

FIG. 7. Distribution of the corrected cross-flow drag coefficient Ccd over the ship's length, derived from the experiments with a segmented model for a Todd 70 hull form without trim at F« = 0 15- see Beukelman

342 J . P. HOOFT

A P P 0 . 5 F P P

g

FIG. 8. Schematic indication of the forward shift of the distribution of Ccd{£,) at increasing p.

• The distribution of the cross-flow drag coefficient can be predicted f o r a lateral d r i f t i n g ship ( p = 90°) as a f u n c t i o n o f the ship's f o r m parameters by means o f empirical formulations.

• W i t h the knowledge of the drag coefficient f o r 90° d r i f t i n g it is possible to predict the cross-flow drag coefficient f o r any d r i f t angle p as a f u n c t i o n o f the ship's f o r m parameters by means of empirical formulations.

• The results presented i n this paper show that i t is possible to develop a manoeuvring prediction program that is based o n a physical theory rather than on regression. Such a program leads to more accurate predictions o f the ship's manoeuvrabihty because the effects o f local h u l l f o r m parameters are taken i n t o account.

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