A physical mathematical model on hydrodynamic forces and moment acting on a hull during large drifting and turning motion under the conditions of slow speed

A PHYSICAL-MATHEMATICAL MODEL OF HYDRODYNAMIC FORCES

AND MOMENT ACTING ON A HULL DURING LARGE DRIFTING AND TURNING MOTION UNDER THE ONDITIONS OF SLOW SPEED

Keiichj Karasuno Faculty f Fisheries,

Hokkaido University Kazuyuki Igarashi

ABSTRACT

The mathematical model of

hydro dynamic forces that _{occur in}

association with ship turning motion while drifting has been derived

subsequently from the improved physical-mathematjcaj model for oblique motïon,whjch was derived by the author (Karasuno et al. 1990). The hydrodynamic forces in the model

cons ist of six _{elementary forces of}

fluids. They are as follows 1) ideal forces, 2) viscous lift due to viscosity

3) induced drag , 4) cross flow drag, ₅₎

cross flow lift due to the _{asymmetry of}

cross flow fore and aft , and 6)

frictional drag et al. in longitudinal direction. The viscous lift, induced drag, and cross flow lift are assumed to occur.mainly along the trailing and leading edges of the ship's hull, and to

connect with drifting angles at both edges. Furthermore, it is assumed that

the coefficient of cross flow drag is C0 CD9OI sin ß3

K'p

_oes2

). f2(u,v,r,xjj,

£2(u,v,r,x ,L)=(u2+( . r)9",_{(u2+v2+fr. »2,11I2} Experimental data generateu from a

turning ship model of a pure car carrier,

TECHNISCHE IJNIVERSITEjT

Laborato,jum ,ioor

Scheepehydromechanjc

'thIef

Mek&weg 2,2628 CD De7ft

were incorporated within the

mathematical model andO resulted in good agreements.

INTRODUCTION

Some mathematical models. can describe the hydrpdynainic forces acting on a hull during large drifting and turning _motion. There is, however, _{no mathematical} model which Consists of terms having the plain meanings of the physics on hydrodynamics, especially concerning non-linear terms and the X component of the mathematical model. In the view of

the physics or _{hydrodynamics, this}

paper describes the model consisting of

the six elementary hydrodynamic _forces.

Therefore ; 1) each term of the model

has hydrodynamic meanings, 2) few

coefficients of the _hydrodynamic

characteristics exist in this model, 3)

many hydrodynamic derivatives of X,Y and N components in the model are

derived from few er coefficients of the fluid -dynamic ch aracteris tics mentioned

before, and 4) such detailed studies _on

elementary hydro dynamic forces and the scale effects will be capable of

estimating the forces of a full-scaled

ship.

PMRS!

'93

UITWIATIOJIAL CONIERENCIOXMARINE

In the prospect mentioned above, _the mathematical model of turning motions is based on the model of oblique

motions with large drift composed of the six elementary forces.

In addition, the viscous lift, induced drag and cross-flow lift_{are modeled} respectively as two separate groups of

forces acting along the trailing and leading edges of the ship. This is

because of the previously mentioned assumptions.

ELF,MFNTAR.Y TI YDRODYNAMIC

FORCES

The ideal hy.drodynamic force is concerned with the added mass.

Accordingly, a hull moving obliquely in an ideal fluid has no lift and drag.

However, in actual fluid, viscous phenomena exists and makes some

additional effects on the hull. The following forces., are made from the

yiscous effect, i.e. the viscous lift, the induced drag, the cross-flow drag, the cross-flow lift, and the frictional drag including the residual. resistance

(Figs. 1,2,3). 4 N,r r XA

y..

.Y,v Fig. 1. Coordinate System 1) ideal fluid forces

If there are no viscous effects and _no free surface effects, hydrodynarnic

>.x

.1 . D. a ,t N1: -Y_C (m - m

)..v

Fig. 2 Schematic model of hydrodynamic forces : Y V.. Obliqu e motion D._3,1 Turning.iirotion V U,

Fig. 3 Viscous lift and induced drag

forces and moment in steady ship _motion are described by equations using added masses.

2) viscous liLt

If there are any viscous effects in _the fluid, we must modify the ideal forces

mentioned above by adding the _viscous

lift due to the Kutta-condition along _the trailing edge and'so on. Therefore, _this viscous lift is different from the

conventional lift, because the

conventional lift is the sum of the ideal force and Viscous lift.

The main part of this vis cous lift

generates along the trailing edge of the ship and the sub -part_{generates along the} leading edge. The vìscous lift can now be modeled. The viscous lift. along each edge is assumed Lo be concerned with the geometrical inflow along each edge. Therefore, Y components of the viscous lift along the leading edge, Y is

11/2

ju2+(vx1 'ii)2_¡ Here, we define C and C,0 where

C' C+ C11 _{is the total of viscous}

lifts, and C,O(C'L,I_

_{cUI c',}

_{be described} as the distribution factor of the Viscous

uf t.

Therefore, the longitudinal location _of the total viscous lift is

x1.00

_-

_'sgn(u)

due to oblique experiments. 3) iadced drag

The induced drag is oriented by the bound vortices of the ship under the down wash flow due to shedding vortices. Now, the main part of this induced drag generates, _{unlike the} viscous lift, along the leading edge of the ship where the main portion of the bound vortices exists. Furthermore, _the sub-part is generated along the _trailing edge. The induced drag _{can then be} modeled. The induced drag along each edge is assumed to be concerned with the geometrical inflow along each edge. Therefore, Y component of the induced

drag along the leading _edge,

_;iais

u (v+x. .

Y . _. u(v+X.r).

iJl,i IJ1,& £

2

pi Here we define CDL and C1,,

where C'Dl=C»1,1+ CD&,t _{the total of the} induced drags, and c1,.=(c'01,1_ C'DL,J/

may be called the distribution factor of the induced drag.

Therefore, the longitudinal location _of che total induced drag is

1'f0 =/L =X'iC.l.sgn()

due to oblique experiments. 4) cross-flow drag

This cross-flow drag is _{tangential Lo the} cross-flow across the X axis, _{and is} described conventionally by

Y=-

CD1.p.LP,d1V2 The drag coefficient C» should be irrelevant to the drift angle of the ship according to the principle of cross-flow theory. In this paper, however, _this coefficient is assumed to be _dependent to the geometrical inflow angle Px _{at the} ship's section and can be described by the following functIons.

CD =CDgo'fj(X)'f2(u,v,r.x,LI)

CD9O: the drag coefficient on tbe ship's lateral motion

the inflow angle at the section x

f1(f3j: ist modulatefunction due to 3-dimensional flow effect

1()=kn

t3I (i

p

f3)

f2( u,v,r,x,L ): 2nd modulate function due to the curvature of shedding vortices on turning motion

f4 u.v.r.x.L _{) =} (n2+(v+x.r ?)h12

(u2+v2+i(L L112 PP 2)1

cross-flow lift

This cross-flow lift is normal to the cross-flow across the X axis, and is asymmetrically oriented with the cross flow along the fore and aft ends of the ship.

Th en

X. PLd{C(v+xp.r)2_CL.(v+xA.r)2)

-

PL,,d.:{_ à, .(v2+x2.r2)₊ where

XFXA : longitudinal locations of F.P. and A.P.

C'LAS = _{- C'L.Asp}

C'LASAP = C'A + C'LAs frictional resistant force

Under conditions of slow speed motion, the total longitudinal resistance is

mainly composed of the frictional resistance, so we may call it the

frictional resistant force and assume chat this force isonly concerned with

longitudinal velocity u. Then this force is described as follow.

x_

-

_{p s.(1+k).c..0}

_P.L.d.U2.c1.Iu1(.U1

s.tpll effect on forces

As the result of the analysis about X.Y and N forces in the oblique motion, only the hydrodynamic derivatives of X about the viscous lift and the induced drag are shown to be too small as compared with the conventional values of these

derivatives (Table 1). Then we assume that the stall effect works mainly on the

X force and not on the Y force and N moment. This is' because the increase in the induced drag and the decrease of the lift cooperate with each other and likely only to be working in the X direction. Then the stall effect in the.oblique motion is assumed to be as follows (Fig.2).

X

- "(u2+vz)L2+ACD u y

(y'2

-.1. p L .d.U2.(CL.u..v2

-- O

= O

As the stall effect hardly occurs in the range of the small drift angles, it can be

assumed that thern X_ f3 curve is nearly equal to zero in the range of the drift angles between 0° and 30° so that

C'D may b e equal to AC'L or

JxdP may

be zero.

Then we get the relation between óC'1 and that is

1.00 or 1.18

MATHEMATICAL MODEL OF SHIP'S HYDRODYNAMIC FORCES DURING

LQE DRIFTING AND TURNING

MOTION.

The hydrodynamic forces acting on the hull in the large drifting and turning motion are also composed of the six

elementary force s mentioned before which are respectively described with the reasonable mathematical models.

The total hydrodynauijc force is the sum of the six elementary förces considering the stall effects and is divided by the

three X,Y,N _components.

X' -X'1_{+(X'.,,yI+X'Lv+(x}

_{+xJ+'V+x' + X.}

(v'+x'1.r') - r') (u'+fr+x'1.r,_¡ (v'+x '' r

-

oc',).u' '(viz'1 ₎ (u '2+(y+x. r'.2W2_¡

-

(C'DI,

-

&C'.,,1).u'.(v'+.1..)._u'2(v+x.r) ( '2+fr+x. r')2)32

-

(C'DI,

-

AC'1O1). ' '(?+X'1.r') U'2(v'+X'1' r')

W2 u +(viz1'r'rj -C','Iul'u' (i) -C'DI,l'u'(v'+z'I r') (u'2+( +z'11')2 + + xei.r)2P2 C'_DI,t'u'.fr+x' (y' + x"r') -J CD9O'(V' +

(2

+ (y, + z'.,.)2)h12 'i u'2+fr + ,r1} 2 d' dz' N' '.N'3 +(N'Lyj+N'LV.)+(N'DI,l+ND)+N.

(' -

.y + z'1 r') 't' (u'2 +fr+ -Z'1'C','u"(Y + r') U' + _xL'C'DI 'u"fr+z' r'),1 ¡ (u'2 +(v'+ - X'1'C'011'u'.(v' + z'1 r') ' +fr+ (u.2 +fr + Cv' + x' 'r') + 'e

(+c +

In the case of weak turning motion such

as

t

(u.2 _{+(v' +}

il

j

(2)

Table i _{Coefficients of hydrodyna.mic} forces analysed from oblique motions

moL i on _{ahead as tern}

X....

_{C' -dC'}

_{-0.037 -0. 056} X'

_{-C',,.4C',, -0:009}

_{0. 108f}

X'.,., a-c',

-0.035 -0. 054

X'..

.

C'L,

0.023 0. 023

Y'..,

-C'

Y.

Y'...

_C,,g'm',,

-0.342-0.455 -0.672 -0.333 -0. 351 -0. 672

N.,.' -c

r 0.060 -0.059 N

=-C ,,'/

_, N _{Cp,,'m ,}

rn', - ra',

p -0.206 0,003 -0,126 0.077 0.078. 0.003 -0. 126 '1. 140

4C'L

dc'.,

0.379 0. 391 0.446. 0,460

C ,./C

£ 1.332 _1.056

dc i/c

,

dc .,ic

.

4C .,/4C

£ 1.108 0.981' 1.177 0.-174 1,174 1.309 ,, 1. 176 0. 177 0.445 _-0.222 "1+p '2

'2

_{z' d'} dx' ku.2+(v.+z..r2J (3) + Y_{LV,) + ('DI,I +Y1,,) +} -m','u"r' - C'w'u"fr + z'1'r')' U' (u2+fr+xi.r, i i't12 -C'L.l'u'(V' + x'1'r'). .2 11' { +(v'+z'1'r, ¡ u''(v +x'1'r')2 where ₊ = I

t"

t." ....r»

r

I. ."vO

C,_Di,I'-Dj,t

r"

'Di"10 The X,Y,N _{components of}

hydro dynamic forces are as equations (4), (5) and (6) with waving _underlined terms equal to unit.

As the results of _{experiments on the} oblique motion of the ship, _{they show} that the viscous lift along _{çhe trailing} edge is major and along _{the leading edge} is minor, because of i

Furthermore, the experiments show that the induced drag along the leading edge is major and along the trailing edge is minor, because of i>c0>>o . Then, we

may be able to assume that the Viscous lift and induced drag _{during the turning} motion is work along _{one major edge of} each force instead of _{along the two} leading and trailing edges.

Now, in order to _{extend the effective} range of.the mathematical model _{to the} fairly sever turning motion, such as

i

(u2(v*

+:xI.r)2

we take the effective multiplier

+ x.ri)2j

into consideration.

Then we get the following _{equations (4),}

(5) and (6). X' m','vr' +(x..u...2 +X'.u.v'.r' (u.2 +(v' + 1 _{2 1312} +X'.r + X,.v'2+X'v't' + X'IT.r'2+ X.luj. u' + (4) Y' - - m'1.u'? + ₊ (u.2_+fr+x'1.r'f) +(Y'.12..3 + Ygiiv,rU2V21 + 4. Y' 'u'2'r'3). I (u' +fr+ x1.r)2) ( where X'v.,

C'C',

_X',., X',., _{=(C'L-c').2.x'1.c X',,} C',AsA,.2.X, X',.., _-(C',.c').,' - X',, X' _-(C'01-ac',) x'__ =-ç', X' X' y..,', (v + i1l2 + x't.prd _____________'I (1₊

p'

)f2d dx

'1 v2 + (y'+ x'rf_{J (s)}

N' -(m1_ni'7}u.v. +(N.,,.a.,, +N.u.4.) rIY , x.rjj

-+ N' _{.u4.y'2.1' + N''u4yq}

+ N.u3.)

I 2 1312 lu +(v'+x',.r)21 (V, + CD,Q -J C»o' +IM'

.4.3

- .1u2 ' vu.,, (6) Y,.., _{N'.,, u_XI.Cui.Xi} Y' _{-c'1, N'} Y' _{C'oj.3.X1.C,,. N'} Y' _{u_C'0i.3.XS N'}

y __,,=_c 'C1.

N' _=-x

_(,a+(y +x'r?)t'2 , +vi + I_j,j,.r/2)2)Ili

ANALYSED INGREDTRNTS _OF }IYDRODYNAMIC FORCES DURING

OBLIOUE MOTION

The hydrodynamic forces (Fig.4) measured on a PCC model in small to large drifting motions _(Yoshimura 1989), were analysed by the equations (4), (5) and (6) under the _{condition of} r'=O and

_u2v121

and added masses _{(m'a-m'y)measured by} the constant accerelatjon tests (Obokata et al 1986).

First, we get from the stage of 13=00.900

the frictional d erivatjves _{X' and} cross-flow derivatives x',,,,,

,

_{Y',,,,, N',,,,}

Second, we determine from the stage of_00< P <30° _{the vis cous lift derivatives}

X,_U??'V'

)j*

WV' Wi? .

Third, we get from _{the stage of 30°< 3<90°} the induced drag derivatives

,

Finally, using the stall assumption, the stall effect derivatives _{ac'.,, and ÓC'DI} were derived from

_X'(c'L+c'J

and

Accordingly the derivative

the coefficient p are determined from XWi=_cDL+ACDI) and

+ PY',,,,,(= -C'DI +

The derivatives anaiysed from the

hydrodynamjc forces of _{oblique motions} are tabulated in Table 1.

As the results, the _{ingredients and the} synthesized hydr_{o dynamic forces are} also drawn in Fig. 4.

The synthesized hydrodyiiamic forces are in good agreement with the _measured forces.

ES..IIMAIEDH YDRODYNAMIC

and

FORCES DURING _{TURNING MOTIQj}

LOO I. S 0.50 0. lo 0.05 o -0.05 -0.10

Fig.4 Analised results _{of hydrodynamjc} forces in oblique _{motions of PCC} model ship, and _experimental res ults

From the analysis of the _hydrodynamic forces in oblique motions, we can get all

-ft

180

(d g)

I I

of the hydro dynamic derivatives described in equation (4), (5) and (6) except only one derivative

An.alLzLI.n.gredients of HyjymjQ

Forces During Lateral Turning Motion

0.20 .0.10 -0.10 -0.20 1.00 0.50 o -2.0 q' 0.50 CD90 0.40 0.30 0.211 -0.10 O 0.10 0.211 0.30 0.40 --1.0 CDgofjf2 1.0 ;90.r1.r2 £ i 2.0

r'

Fig. 5 Cross-flow effects on X,Y and N' in turning motions at 13=900

This derivative XÇis _{analyzed from the} hydrodynamic forces X'(Fig.5)

measured on the PCC model during turning motion w ith _{a drift angle p}

(Yoshiniura 1989) using equation _(4).

-0.10 -0.20 -O. 2. O 0.30 .20 0.50 " 0.40 0.30 0.20 0.10 o -0. IO -0.20 -0.30 -0.40 -0. O -1.0 1.0 N0ND.. 2.0 r. Fig. 6 Analysed results of hydrodynamic

forces at (3=00 ¿ .5 CD9O' 1 s. o . -1.0 1.0 2.0 LO LO

Table 2 Measured. and estimated

derivatives ?fhYdrodynamic force/

Comparison Betweenstiniated And Experimental Res ults of Hydro dynamic Forces During Tuning Motioj

Then we can calculate and estimate the hydro dynamic derivatives and forces in

turning motions with drift angles using equation (4), (5) and (6).

The derivatives are tabulated in Table 2. As a result, the calculated forces

synthesized from estimated ingredients are compared with the hydrodynamic forces measured on the PCC model (Figs.5,6.,7).

The estimated hydrodynamic forces are also in good agreement with the

measured forces, in the range of severe turning motion.

CONCLUSIONS

In this paper, six elementary hydrodynamic forces wçre used to constructs the mathematical mode! of ship turning motion with drifting. Then, each term of the mode! had clear

meanings in physics or hydrodynamics. Furthermore, the mathematical model can describe precisely the experimental results of the hydrodynamic forces acting ana ship's hull that is

experiencing turning motions while drifting. Accordingly, this math ematjcal mode! will be able to estimate the

hydrodynamic derivatives and forces simply by means of a few fluid-dynamic characteristics in the model.

0.10 ?" ____.\-. a.._,_

---

XI, -0.10 -0.20 -0.30 80 120 150. 180 1(d.g)

Fig. 7 Measured and estimated results of hydrodynamic forces at r' =1 .0 oJioad a; ahøad a ta Z' uvv Z' uvr X' irr X uuuvv X' uuuyr i' uunrr X'luju Z' vr X' rr -0.037 0.037 -0. 000 -. 003 -0.000. -0.002 -0.035 0.023 -0,063 0.006 - . 05G -0.056 -0. 014 0.103 -0.106 0.027 0.05t Q.023. -0.063 0.006 T'mw T'mw

r ,,,

T' Uuvyr

r

r

N' iuy N' uuz' H' uuvyy X'UUyyj -0.342 0.060 -0.455 -0.656 -0.051 0.060 -0.086 -0. 206 -0.316 .333 -0.050 -0.351' 0.402 -0.153 0 010 -0.053 -0. 083 0.078 -O.l53

rz o.000 0.009 H' uuvrr -0.164 _0.lOj,

N'y 0.135 0.135 N' unrrr _{-0.028 -0.022} o -0.10 -0.20 -0.30 -0.40 -0.50. -N N, I I 30 60 90 120 150 180 0.50 0.40 0.30 0.20 D. 10 N1(Munk) -- N.

REFTRRNCF.S

Karasuno,K ; J. Matsuno, T. Ito and K. Igarashi, 1991. "The mathematical model of hydrodynainic forces acting on ship moving in an oblique direction with fluid-dynamic concepts, 2nd report ", Jourj.zai of the Kansai Society of na vai ¿rcbit'cts, Japan, No.216, 175-183 Karasuno,K ; J. Matsuno, T. Ito and K. Igarashi, 1992. "A new mathematical model of hydrodynamic forces and moment acting on a hull during

maneuvering motion that occurs under conditions of slow speed and large

turns, 2nd report " , Journ,] of the

¡(ai, svi Society of na va.! architects, Japan, No. 217, 125-135

Obokata, J ; Yasuo Yoshiinura et al 1986. "Measurement of Added mass of Ships with Unconventional Dimensions

,Jouriuzl of the ¡(a.zzsai Society of naval architects, Japan, No.201, 1-6

Yoshimura, Y. 1988. " Mathematical model for the nianoeuvring model at slow forward speed" , Jornal of the

Kaizsai Society a/naval architects, Japan, No.210, 77-84

KARASUNO, KEIECifi, Bachelor of Engineering, 1964 from University of Osaka Prefecture, Master of Engineering 1966, Osaka University, Doctor of Engineering

1972, Osaka University. Presently employed as Professor of Fishing Boat Engineering at the Faculty of Fisheries at

Hokkaido University.

Karasuno,K ; K.'Yoneta and S.Jyanuma, 1990. "Physical- mathematical models of hydro- or aero-dyùamic forces acting on ships moving in an oblique direction ", J'roceed.thgs of MA J? SIA! IC'SM 90 Tokyo, Japan, 393-400

Karasuno,K ; 1992.

"A concept abouta

physical-mathematical model of hydrodynamic forces and moment acting on a hull

during large drifting and turning motion under slow speed conditions",

Proceedings of w or.ksbop on prediction of sJiio .inaizeuverabi7ity, Fukuoka,

Japan, The West-Japan Society of Naval Architects, 133-a59