Physical mathematical models hydro- or aero dynamic forces acting on ships moving in an oblique direction

PHYSICAL-MATHEMATICAL MODELS OF HYDRO- OR

AERO-DYNAMIC FORCES ACTING ON SHIPS

MOVING IN AN OBLIQUE DIRECTION

I. INTRODUCTION

For the sake of vessel safety, It is very important to understand low speed maneuvering cotions of merchant vessels and also fishing boats when operating under windy conditions.

Recently roseach in this field vas conduc-ted using mathematical models describing ship steering cotions at cloy speed while in shallow water. This work vas performed by members of MSS Comit tee In the society of naval architects of .lapan(31. Thoseautbors proposed a new math-esatical model of hydrodynamic lateral forca Y

and yaw moment N for ships coving in oblique directions at slow speed. The model supposes that a coefficient of cross flow drag force varies as C00O50. Isinß I where ß is drift an-gle and C55. is at ß9O'(l.51.

In an attempt to i.prove that mathematical model's ability in fitting experimental results. the mathematical codal of longitudinal forces X was appended to T and N mathematical models (21. In this paper, the authors considered the following elemental forces Influid dynamics. By applying the X,T and N mathematical models to hydro- or aaro-dynamlc forces on ships. good fitting results were obtained.

This mathematical modal Is expressed by the following elements of fluid dynamic forces

(2,41: ideal fluid dynamic force; linear lift and Induced drag forces due to viscosity; lat-eral drag force of cross flow; longitudinal lift force of cross flow due to longitudinally asymmetric vessel shape; and frictional drag force of longitudinal flow.

This mathematical model Is pirticularly built up with a few specific factors of fluid-dynamics. It is also capable of estimating scalo effects in elemental forces. AccordngIy, some of the coefficients in this mathematical model of X,T,N fluid dynamic forces have some theoretical relationships to each other. This mathematical model could describe measured re-suits accurately. However, the relationship be-tween fluid dynamic coefficients does not hold In the analyzed results. This is especially true for some coefficients o! X force.

This fact is very undesirable whon at-temptlngto estimate all coefficients of X,Y,N forces from a few specific factors of (luid-dynamics.

The aims of this papere are to explain the causes of relationships which do not hold among fluid dynamic coefficients and to establish an estimating method of X.Y,N forces. For this purpose the measurements of X.T.N forces were conducted for various fishing boats during

their loading conditions as simulated in a

ThffiI

UMVERW

Laboratodum voor

Scheepehydromechaflica

ÑchIef

Mekelweg 2, 2628 CD Deift

MARSIM & ICSM 90Tokyo, Japan/June 4-7, 1990

TeL 015486873 . Fac 015 781838

Keucht Karasuno The Faculty of Fisheries, Hokkaido University Japan Aunisaburo Toneta The Faculty of Fisheries, Hokkaido University Japan Syunji Jyanuma The Faculty of Fisheries. Hokkaido University Japan

ABSTRACT

For simulating ship maneuvering cotton in wind, external forces should be describable by simple and accurate mathematical models. Such models should have physical meaning and be capable o! covering all ranges of drift angle or wind direction.

This paper separates fluid dynamic force from ideal fluid dynamic force (Hunk moment), lift and Induced drag forces due to viscosity, cross flow drag force, and frictional drag force when considering stall phenomena.

As the result, longitudinal force X, lateral force Y and yaw moment N of the fluid dynamic forces acting on the ship are described by the following mathematical model.

u.0 Iu'I 'u'fX',,,,

'2

towing tank or wind tunnel. Then the measured forces were separated into eleiental forces.

2. MATIIENATICAL MODELS OF ELEMENTAL. FLUiD DYNAMIC FORCES ACTING OH Snips. xoVInç IN AN OBLIqUE DIRECTION Fluid dynamic forces acting on a ship mov-ing slowly, steadily, and in an oblique direc-, lion are assumed to be composed of 'the six ele-.iental forces mentioned above. The rnathematical.

models o! these eleiental forces are described belov( see f lg.l and appendix ) .

Fig.i Elementary wodel of hydro- or. aero-dynamic:forces.

i) Ideal fluid dynamic force

Only Hunk moment Is generated 'from the oblique ship lotion in an ideal fluid. This mo-went is an entire aspect of yaw moment acting on a ship moving obliquely in an ideal flüid. As a result, liti or drag forces do not occur.

flunk Noient: ('m'-m',).u''v'.

'The following forces generated 10 viscoUs fluid 'are considered to modify the Ideal fluid .

dynamic force mentioned above. .

2) Linear lift force,

A lift force of a wing having an attack an-gle is generated by circulation dUe to viscOs-ity of fluid ,and Kutta-condition at .the.trail ing edge of the wing. The lift forte sheùld be added to the Ideal fluid force in order. to de-scribethls.actual,phenomena with a theoretical model. This lift force acts at right, angles to uniform flOw direction and its absolute is as

large as a conventional lift' force. However, its center of pressure Is located near the trailing edge and la different. from conven-tional center of pressure located near the leading edge. This lift torce is describùd with syìbol L'...and identifies the liflear'li!t

force. ' .

When a.shipmoves laterally. tais linear lift force Is assumed to become zero, and 1f s'tali effects :occuring at 'large 'attack angles are not'considered rigorously, then linear lift force L' areassu.edto be the following: On the other hand, the center of pressure of this force Q '. is 'assumed -to be as follows:

Q. .,

, .

underthis'condition the f011ow assumption

hold: ..

.,..

'

-' a is a costant valUe, ,, ''. -1 a <:O.' ' .'

a is approximately equal to -1.

This i's because the centerof pressure may move

from near' the ,trailingi edge 'to' near midship ac-cording tothe increase ofattack angle. flovev-er. this shift is assuied to be insigflificant.. 3) Induced drag- force '

:

j

. Owing to. the theory of small aspect ratio wings, the 'induced 'drag 'force due t'o fre vor-ttces and its down wash velocity induced, is assumed to be the following:

D',C'pi.U'5'y'm

where,

C'ei+C'i,.

' '

-The centerof pressure of this induced drag force is described 'below;

-

'Q'oi+a

s ,

under this condition the follow assumption

hold: ' '

a u is a constant value,

-la,l,

a s depends on the separating position of free vortices'owing to the ship shape. This la a results froi the generation of free vortices which are created out of the bound vortices that make the lift force. Folliowing this1 the increase of drag force due to stall 'effects should be described.

4) Cross flow drag force

Cross flow, drag' pr.inciple is the idei of lateral force acting on a slender body, in this case a ship This force results from1the eddy caking resistance of two dimensional flow as lt occurs around the vessel.. in studies on ship maneuvering motion, the cross flow drag force'coeffi,cient Co was occasionally diséussed in relation to functions based on motion vari-.ables and longitudinal position. In this, paper

Co is assumed to be

C0C1.,e

lain El for ships with small drift angles. Accordingly normal force Yc and yaw moment Mc. due to cross drag force, are as follows:

Y'c--IColv'l.v'.d''dx'

-5

Cooev'''d''dX'e-Co,e'm'o Co v'i 'v''x''d' 'dx'

CDoQV'.3'x'd'di,''CDgo'i', 'y'' 5) Longitudinal frictional resistance

- The frictiOnal resistance is very

impor-tant for the lonitudinaI force X, even in the case of obliqUe ship motion,. Especially in the

case of ahead or astern ship motion at fi 0'

or 180', most of the X, force is due to fric-tional resistance. Convenfric-tionally the eddy niking resistance force is included within frictional resistance. This frictional resiste ence IF is due to the force tangential to

ship's surface. Therefore by using longitudinal speed u' and the coefficients of conventiänai ship resistance,

Ir

is assumed to be described by the following equation.

-C'F' lu'l .j'

However. in the case, of oblique motion. t1i-s may not be correct. This is possible becaùse the distribution of tangential velocity nar the ship's surface during oblique motion is ditferent from that which occurs during

straight motion. , , .

6) Lift 'force due to ship's fore-aft asymíetry When fishing boats with longitudinally a-symmetric foras move laterall',y'at fi-90'., urge longitudinal 'forces, nòrìal tO flow direction,

may act on the ship. This may occur at the same time when the cross flow drag force acts on the flow direction. This longitudinal torce resulting from the ship's asymmetry, is assumed

to be parted from linear lift force. It is de-scribed by the following equation:

X' .s e-CLD, V'

This equation is based on the assumption that linear lift force Is zero at ß 9O.

The six elemental forces mentioned above may not accurately include the stall effects which dramatically decrease the linear lift force and significantly increase the induced drag force.

3. MATHEMATICAL MODELS OF X.Y.N AND

INTRODUCTION OF TIlE STALL EFFECT TO LIFT. DRAG AND X-FORCE

Summing the six elestal forces mentioned above, the following mathematical models of I, T,N are obtained.

X'

iDi

IU'i u'4I'.,,v'°

iN' where X .,,nCtt _{X. --0O3} X' IDI X'

DDC

LRS

y ',_c' .

T' -C',, N .,--+ a a N'D a-Co ge - e' TABLE l-1

Coefficients of bydrodynamic forces obtained from towing tank test for the M-1 model ship, the Kotobuki-maru and the Ilokushin-maru.

J

(1)

TABLE i-2

Coefficients of aerodynamic forces obtained from wind tunnel test for the image model of the Ilokusej-maru above it's waterline.

( Ship ahcad motion )

7 0.446 -0.195 0fl5 -1.227 -1.463 1235 -0.274 -0.024 0.01Z 0.819 0.780 1.268 1.192 0.636 0.307

0l152

The analyzed coefficients of measured forces X,T and N by eq.(l) can describe very well the experimental results. However, the theoretical relationships between the

coeffi-cients X'. and X' with Y'DD and Y' were notevident in the expermental resultsllj This is possibly a result of the experimental

coefficient values being significantly less

than the theoretical values (see tablai).

These differences should now be described by

AC'L or AC',,, that is X'..,C'L-AC't.

X' --(C',,-AC',,), T'.,D-C'L, T' R

-C',,.

This means that it is desirable to make a hypothesis that there are large forces which do not effect .Y',N' forces but adding to X' force. In order to pursuit a cause of this, it

is desirable to see the components X,T from the view point of lilt and drag force.

Eq.(l) is rewritten as follows: 'u''-v'-C'ç

Iu'I-u'-C'105'v''

-C'ta-v'-(AC'1.u'.y'2 - A C' 1 u' -

y' 5)

(2) -2 'oi'C'o, 'u'2-v''-C,,g'm' -v's

-+a iC',,.u'2.v3c,,g.;'.ijs

The term

force modifying X torce in eq,(l).

As a result, the followt'ng components are ob-tained.

lui. u2 .C' -

li'V'

T',-D'-C', u'5 -

y'2

+ C'oca s' e

y'

J

(

Ship_ahead motion >

fl-i iiiodel

Kotobukfllolçushjn

Condition

1-2 2-2 3-2 X' X' , D 0.229 0.218 0.163 -0.273 -0.371 -0.290 -0.046 -0.035 -0.044 -0.060 -0.038 -0.027 -0.017 0.130 -0.022 -0.019 0.172 -0.169 -0.032 -0.092

r00

-0.588

-Ó3f4

-0.596 -0.433 -O.43F Y' -1.283 -1.213 -1.308 -0.492 -1.164 Y'yvv 1.163 1.152 1.244 -1,034 0.901 -0.034 -0.008 0.024 -0.127 -0.OgÖ H' -0.576 -0.620 -0.575 -0.316 -0.481 0.068 0.060 0.049 -0.004 0.018 CD.I0 0.966 0.930 0.951 1.066 O.833 4C'1 0.359 0.355 0.433 0.449 0.261 AC'pu 1.009 .0.842 1.018 0.621 0.995_ C'01/C',. 2.183 2.115 2.194 1.137 2.688 4C'. /C'L 0.611. 0.619 0.726 1.038 0.603 AC'i'/C'çi 0;787 0.694 0.778 1.263 0.855 N . Cm -m

+N u)

Conditlon 1 2 3 4 5 6

X'.

0.100 0.238 0.321 0.411 0.402 0.458 X' -0.122 -0.242 -0.204 -0.275 -0.240 -0.106 X',. Q254 -0.243 -0.314 -0.314 -0.217 X',

Q

-0.111 -0.108 -0.136 -1300 -0.065 -1.247 -1.227 Y',,, -1.258 -1.274 1207 Y' -1.529 -1.513 -1.604 -1.552 -1.572 -1.410 Y'D -1.338 -1382 -1.402 -1.425 -1.256 -1.220 N'D -0.294 -0.300 -0.256 -0.252 -0.178 -0270 N' -0.018 -0.082 -0.016 -0.059 -0.001 -0.051 0.005 0.006 0.001 -0.005 0.030 -0.023 C,go 0.014 0.019 0.900 0.870 0.819 0.701 1.068 1.036 0.077 0.889 0.845 0.760 ¿C',, l.40L.J.27l 1.188 1.311 1.237 1.277 1.194 1432 1.261 1.214 1.140 C'DI/C'I. 1.215 4C'L /C't 0.849 0.813 0.753 0.684 0.678 0.62.7 4C'o,/C'oi 0.920 0.840 0.817 0.823 0.847 0.891 } (3)

+C'c'IU'I.U+C'L.u'.v'2+D'f _{J AND ITS RESULTS} vhere

XLC'L.U.V3-C'DI.U'3.V'I

C',. 1u1

T'L_CL.U'1.V+CDI.U3.V

₍₄₎ + C'D D 5' u u ' V' LCL.U.v'GCD,,-..I'S.U'.V3

CF

Iu'I

U''V'C'LAS'V''LJJ

vhere L' C' L

U' , V' 2+C0.

.'s. u' .

C'p.Iu'I.u'.v'2C.v'4

a'U''V''

C','

Iu'I.u'$_CLDS.U2.v2 ' (5)

C'Ds.U'.V'

C't.u''.y' X.CrL.U. 'V'2C'ip .U.3.VrT

_C'L.U'.V'2+CiI.U3.V2

lu 'I u'

y' ,C' LIS' U' 'Y'3

Cboi.u'm y' a.

_{C' i, 'u '}

'y '3

D' ,-C i u y 200

.s L'i y

C'F'

IU'l 'U'2y'C't#i''.y'

C'e. .U'' y'

C' ,'U'2

d

The physical mechanism of the codifying X'

force - C'

U'V-AC'e.'.y'2

_should

be generated by stall phenomena. This is

be-cause the decrease of linear lift torce L's, and the increase of induced drag force D'51

ow-ing to stall phenomena, should result in a ion-gitndinaliy directed force, Eq.(5) and (6)

de-scribe how a decrease In linear lift force L'3,

and an increase in the induced drag torce

_D'e,

will strengthen the X torce and cancel _{the T}

torce. This does not effect N' moment _provided the longitudinal centers of pressure are in the same position.

Furthermore in order to examine the hypoth-esis of stall effect, the experimental _results are separated to elements and compared vith synthesjed results. This vas done at the stages of the X,Y components of lift or drag, and vice versa by using analyzed fluid _dynamic Coefficients. Fig. 2 shovs that

Ls,, D,,

or L51+D1, _{are generated starting at the range}

near ß 3Q, Its- maximum Is near ß 6O and vanishes Completely at ß9O. In addition, _the

components of De, or Lii occupy lesser parts of lift or drag force but _{a greater degree of} X torce.

This phenomena should be similar to.stall phenomena, because _{DIT and L31, due to stall} effect, cake a larger component of X' torce than due to the frictional force Xp _vhlch could not cake large changes by the distribu-tion of velocity u'. As a result, the stall phenomena can explain the elements of X force having the term

- ( C i. u

y' .

C' u' y

'3)

4. A METHOD OF DIVIDING ELEMENTS FROH MEASURED FLUID DYNAMIC FORCES

(6)

The physicalmathematical iode] eq.(2) which incorporates the physical characteris-tics of fluid dynamic forces, should _{be checked} to verify Its applicability to vessel fluiddy-namic forces, For this purpose _{some ship scale} models vere tested In a towing tank or a wind tunnel. The principal particulars and _test con-ditions of these ships are _{presented in table 2}

and fig,3.

-The results of these experiments _are ob-tained in fig.2 and nondimensionalization _of fluid dynamic forces are as follows:

X'-X/(+ p S'U), Y'-Y/(+ p 'S'U), N''N(+p 'S' L,,

.IJ2)

where U : ship speed or wind speed

S : representative projected

profile area (-L,,'d.)

d. : _{mean draft or mean of from board}

plus bulwark height A,: projected profile area

longitudinal position of center of A,,

4,1 METHOD OF ANALYSIS

The fluid dynamic coefficients _{of C'(, C5,} etc. are analyzed from X.Y,N forces _measured

from ranges between ßO'-90' or 90' 180' by

using regression analysis method of eq,(2). However the accuracy of this method is poor when the drift angle is small. For this reason this papar adopted an essential _{point analysis} method, This approach provided _accurate coeffi-cients with good fitting results from _the ex-perisental data, especially within the impor-tant ranges of drift angle in _{longitudinal and} lateral motion,

Aspects of this model are as follows; I) establish the coffjcient C'r('-X',u,u) _{in X'}

equation at the drift angle ß.Q' _{or 180'} establish the cofficjent

C's(.-X',,,,)

_{in X'}

equation at the drift angle ß9O

establish the cofficlent

in T' equation at the drift _{angle ß'90'} establisi, the cofficient C't(u-Y',,,,,,) _and

C'0,

('-y' ) _{in T' equation by means of} regression analysis method using the

coeffi-cient Y' _{from the range of ß-O' 30}

or

ß O' 90'

establish the cofficient (C'L-C'L)(x',.,) and

(C'i;-AC'oi)(.-X'

_{) in X' equation}

by means of regression analysis _{method using} the-coefficients C', and C'1,,3 from _{the range}

of ß'O' 90'

-establish the coffjcient N'u,.(am',m',,-+a, 'C'1) and N' _{(--- a iC'p,) in N}

equa-tion by means ofregresslon _{analysis method}

from the range of ß'O'. 30'

_{or ß'O' SO'}

4.2 ANALYSIS RESULTS

The coefficients obtained by the essential point analysis method described aboye are pre-sented in table I,

The arrangecent of elemental _{forces In X,} Y,H components are drawn in fig. 2. This fig,2 shows good agreements between _{synthesis results} and measured experimental _{results. The roles of} each of the elementa.l forces _{are shown very}

O.. 0' 'i

L'..

D'.

Pc

'O'K.i_ e' 00f 01001 F s L., ,Sd Ix f'; C,

Fig. 2-1 Hydrodynasic forces of the fish carrier boat Kotobuki-saru

Simulated bydrodynasic forces (D' and' L'). Longitudjnal(X') and lateral (T') forces are resolutod for each component factor.

Simulated bydrodynsiic forces(X,T' and N'). Drag(D') and 1111(1') forces are resoluted for

eack component factor.

linea.r.lift force induced drag force cross flow drag force

lift force due to longitudinally asymmetric ship's shape

fricional res is tance

stall effect of induced drag force

o e.0

L'5,

N' -0.12 W

,

,'

(a)

(b)

Fig. 2-2 Aerodynamic forces of the T.S. Hokuset-maru

Simulated aerodynamic forces (D' and L'). Longitudjnal(X') and lateral (Y') forces are

resoluted for each component factor.

Simulated aerodyna.jc forces(X',y' and N'). Drag(D') and lift(L') forces _{are resoluted for} each component factor.

stall effect ot 1lf t force N' derivative of added mass and linear lift

N' _{H' derivative of induced drag}

force

N'..,., _{N' derivative of cross flow}

drag force Heasured value and thin solid line of

The relationships between the coefficients

CO3,C,C'01,C

obtained, are plotted in fig.4. For the purpose of estimating these co-efficients, these figures show that both ratio

IjC'os/C'es and

C'/

C'L.

bave nearly con-stant values. However, the ratio C'p,/C'1 is auch larger than its theoretical value of 1/2 which occurs for saallaspect ratio wings. The obtained values of ratio CD /C vary with the vessel shape or draft conditions. For example:

TABLE 2

Principal dimensions of model ships.

U-I,..ed.I

.e,I,. I

...dIIl.. I

e.I.b.II e...

H

:...

Fig. 3 Conditions 0f model ships for towing tank test andwind tunnel test.

1.2 for a fish carrier boat or a fisheries training ship. 2.2 for a small coastal fishing boat, and 2.9 for a salmon drift gilinet boat.

--iCM1,.Q73C.,

ou.I Ahead L kolabalil £ Hobuihin Hohussi

5Z

C,

r

a- O,.,

---5c,72cM.

1.

-Go4,o.eIc,t,.o.oe

ON-I Ahead aulalobehI a HoI.e,hS,,

:','

2. s 0.5

«

0.5

(c)

o 0.5 1.0 IS Ct'siW.

(a)

o 0.5 1.0 cIm

(b)

Fig.4 a) Relation between coefficient of induced drag(C'01/.o) and stall coefficient ( C os la.).

Relation between coefficient of lift(C'1/i.)

and stall coetficient(C'1/..),

Relation between coefficient of 1ift(C'/i.) and induced drag(C'gs/..).

5. CONCLUSIONS

In this paper fluid dynamic force which acted upon ships moving obliquely vere sepa-rated into ideal fluid dynamic force, linear lift force, induced drag force, cross flow drag force, lift force due to cross flow on longitudinally aymmefric form, frictional drag force, and stall effect on both the linear lift and Induced drag force. Furthermore, the

mathe-matical models constructed for the elemental forces mentioned above, all have fluid dynamic terms with clear physical leaning. As e result, they are able to accurately describe the exper-imental results.

In estimating fluid dynamic coefficients. It is useui to recognize that during ahead motion the following was observed.

Cli'ooO,72

x(C'L/i'uI)

cross flow on longitudinally aymmefric form, frictional drag force, and stall effect on both the linear lift and Induced drag force. Furthermore, the

mathe-matical models constructed for the elemental forces mentioned above, all have fluid dynamic terms with clear physical leaning. As e result, they are able to accurately describe the exper-imental results.

In estimating fluid dynamic coefficients. It is useui to recognize that during ahead motion the following was observed.

For future investigations occurring in this fieldof study, it vili be interesung to research the relationship between C'o, and C', the'center of pressure due to induced drag force, and the development of this model from oblique lotion to circular motion of ships ( see appendix ).

R EF FE RENC ES

Karasuno, K. Toneta. K. Jyanuma. S., " A

nov atbematical iode! of hydrodynamjc force and moment acting on a hull in ma-neuvering motion at slow speed and oblique direction - , Journal of the Kansai society

of naval architects. Japan. No.209. 1988.

pp. 111-122.

Karasuno, K. Tonata, K,. - Wind forces acting on a model of a fishing boat and new mathematical lodai for its analysis" Journal of the Kansai society of naval ar-chitects. Japan. NO.212. 1989. pp.133-143. 1153 Comittee in the society of naval

archi-tectsof Japan. Reports of Comjttee for investigation of a mathematical model of ship steering motionin shallow vater and/or at slow speed I -.1V " , Bulletin of the

society of naval architects of Japan. No.717. 718.719,721. 1g89, pp.

Oltaan. P. Sbarma. S.D., _{Simulation of} Combined Engine and Rudder flaneijvers Using an Improved Hade! of Hull-Propeller-Rudder Interactions, the 15th OHR Symposium on

Na-val Hydrodynamics. l984,pp.l-24 ".

Karasuno, K. Jyanuma. S. Taneta. K.. Experi.ental studies on hydrodynamjc force and moment for maneuvering motions a! fishing vessels (4) - on the mathematical model of ship hydrodynamic force and moment while drifting - , Bulletin of the faculty

of fisheries. Hokkaido University, Vol.39. No.1, 1988, pp.45-52,

APPENDIX

VORTICES HODEL.OF SHIPS IN OBLIQUE NOTION AND TIlE EXTENSION OF

TIlE HATHEHATICAL HODEL TO CIRCULAR NOTION OF SHIPS

In the fluid dynaaic forces vhich act on the ships moving in oblique direction, the linear lift force mentioned above is assumed to be located near the trailing edge of a ship. This lift force is related to the sway velocity y' at the edge which is connected with the Kutta -condition. On the other hand, the induced drag

force is assumed to be located at the generat-ing point of the Irme vortices which separate from the ship hull, and is related with the flow direction of the free vortices and the intensity of the bound vortices. Therefore.

the direction and intensity of the induced drag forces depend on the sway velocity y' at the position where the free vortices separate from the ship hull. Under these assumptions the

au-thais adopted the vortices model shown in fig.

Al. As a result, the fluid dynamic forces _model was obtained as shvn in fig.A2.

"t

--I

.-.- ¡

-

- ., nome shoe e

+

and Ring vortices

i:.

_{-s_} J Stall vortices Fig. Al Scheme of the vortices model

II 'Flow Ideal Fluid vortices

Dl

-Ç>

7Flow

flunk moment

-.__-<

Induced Drag Ost L51 Stall Effect

Fig. A2 Schema of fluid dynamic forces due to the vortices model

In the extension of4this mathematical model for the fluid dy:namic forces _{to the}

circu-lar ship motion, the six elemental _{forces may} be described as-(olows, This _{occurs provided} both the linear lift force and the induced _drag force in circular motion with drift angles, is generated in the same manner as the oblique motion C See fig.A3 ).

Fig. A3 Scheae of linear lift and induced drag in circular cotion

asy..etry

N',,,, _C,ee'3i'5

N',,, -0001.3.', C'Las.(y'2,+r'2)

N'.,

_Ceg..i'4

These mathecatical lodels should be

where _{C'LASC'LaI.-C'LAÇ}

applied to experjienta and _{investigaje in the} Suijng these forces and adding the stall near future.

effects, the lathecatical odels are obtained as follows

-(C',-A C',)'u' ' (v'+sgn(a _{u) + r')'} C'P1u'l.u'...C'Les.(y'2++r'z)

(C'La,.+C'L,,,).v' 'r'

C', _{u''. (v'.z gn (a j) . -4 r')}

3w',.v' 'r' 2+.', r'

3)

N'(l'.,i',).u' .y'

- + a 'C'e.. u''. (y' + r')

-+a ,.C',u.u'3.(y,5g(« i)' + r')' Ceos(?'a.v'303i',.v'2.r'

These equations are rewritten in the following conventional fores

+X',,r,u'.r'a,X' u'''v''

1'

_{U''.v''r'sX',,,,,,,u'S.r's}

l,.I j.lU', u'4X'.,v''

Y' -- e' . u ' ' r' Y' ., u' y + y'j.j., u' ' r' sT' u'''y''.y' T'.,,,v'+y',,,,v'w.r' -sN' u''v''.N' H' j.,,.,, u'

2.

' r' N' ., , ,, u . r'' +R',,.v'1+N.,,,v2.r +N' ,, , w' r''sN' .,, r'' where (c'i.-A C'1) '(C'L-

C'1). 1/4

X' _{(C'oIAC'ei).sgn(ai).} X' _{-(C'1-C', i) 1/4}

X' u.s.,

Ideal fluid dyna.jc forces

X'.. .C',,, I'

: l',v'r'

X'., _{(C' C'1,,p)} Y' -w','u'r' X'., N' : (.'..',).u'.y'

Linear lift (orce

Y'..., C'L

L - C' L u (y + r')

Y .,. C L I / 2

2 '.- + a where a .,-1 Y _C'DI

Induced drag force

Y' _{C'ei.egn(« s)'3/2}

D'I_C'Dau2.(ys5gfl(«,).)2

Y' --C'05.3/4

2 'u--a i where -1

«s

1 _Y'..,,,

-C'oi'sgn(a s)'1/8 Cross flow drag forces

Y',,,.

'=Ceei'e

Y'.1CDeeIsjnß.p.(y',x'.r').iy' _Y',,,

_=-C,3',

x'r'i d'.dx' T',,, -Con'3i', Y',,, - Cee.i', -N'..,, _-C'L.a,,.1/l N'.,., C'L'a.,.1/4-N' N' _{-C'es.sgn(a;).«5.3/4} Longitudinal frictional resistance

R'

_Cos.ai.3/8

N' Lift force duo to ship's fore-aft

N',,. -C,e,''1

X,. X.. X e.,,