Hydrodynamic interactions between two lifting bodies

ARCHIEF

Hydrodynamic Interactions between Two Lifting Bodies

Katsuro KIJIMA, Member

ì 58 - Z FJ 54 8 Reprinted from TRANSACTIONS OF

THE WEST-JAPAN SOCIETY OF

NAVAL ARCHITECTS No. 58 AUGUST 1979

Lab. y. Scheepsbouwkunde

Technische Hogeschool

* Faculty of -Engineering. Kyushu University

(lU545

187

Hydrodynamic Interactions between Two Lifting Bodies

Katsuro KIJIMA*, Member

Summary

Ship-to-ship interactions that the forces and moments will act on each body

moving in close proximitr will be of impdrtance in ship maúoeuvrability,

articu-lrly in traffic contról or in hip colÌisioñ in résti4cted! atèrwys such as

harbour.--dr canal. -,

In this paper, the forces and moments acting on two identical thin wing bo-. dies in arbitrary forward direction and identical forward direction with respect

to each other are computed as a two dimensionaiproblem. . This case includes a

unsteady problem for determining the vorticity on the each body, but is here

solved as a series of steady flow problem, i.e. as step by step in time.

According to the computed results, for identical bodie moving with the same

forward speed and forward direction, the both forces ánd moments acting on

each body are greatly affected by the stagger representing the difference of

lon-gitudinal sitùation, and have an effect on separation distance between ships at the range of lower value than the drde fship's length';

-For bodies moving in arbitrary forward direction and identical forward speed,

the above interactions are affected muh more than that to the identical forward speed and direction because of th added effects due to -the wake vorticity, In general, it seems hydrodynamic interactions between ships will be greatly affected

in closer proximity range than the order of ship's length.

1. Introduction

The problem of hydrodynamic interactions between ships moving in äloe proximity has been

of interest in shit manoeuvrability. Particularly, in the restricted waterways such as- canali harbour, bay etc., where crowded with shipping, these are also -of importance in the field of

ship's handling or traffic control. The traffic control in these restricted waterways will be

seri-ous problem in recent years. We can see some problems occurred when many ships are in meeting or passing with respect to each other, or that relating to the ship's handling. And in

the past, we have also known many instances of the collisiòn in ships movihg in close proximity.

However, these problems are not so new one in recent years because some studies have

already dofle in this field. There have been â few theoretical analyses or. experimental wons

of these problems in thé past.

'ortheteadystate bank sñction problem, or the equivalent

refuelling problem, slender body results have been developéd by Newman [13 for axisymmetric bodies, and two dimensional approach have -also been extended by Tuck & Newman [2] for this problem.

From a theoretical póint of view, the problem of ship interactions can be readily divided into

the steady and unsteady one. The works on those in the past have been done almost in steady problem, i. e. ships are traveling side by side with respect to each other Recently, Yorimasa [3] has computed interactions between ships moving in different velòcity and arbitrary direction, as a quasi-steady problem. The problem on interactions of slender ships in shallow water has

been reported by Yeung [4], and those between a stational ship and a moving ship has been

performed by King [5]. .

In this paper, the forces and moments acting on two thin wing bodies moving in arbitrary

direction with respect to each other as. a two dimensional problem are computed by applying the method of Tuck & Newman and King.

2. Derivation of Integral Equations

The problem to be here considered is that of two dimensional thin wing bodies moving in arbitrary forward direction and in steady velocity with respect to each other We know that

the lifting effects are modeled by vortices and thickness effects by soúrces, by using a

conventi-onal lifting surface theory. When these lifting bodies move in close proximity with identical forward direction and vélocity, the i'orte distributids on the surface of bodies are constant during their thotios. Therefôre, in this case we can treat as a steady problem. However, this

problem must be considered as the i:insteady problem when these bo&es move in arbitrary for-ward direction because the vorticty along the body surface depeñd oñ timè. In a unsteady case,

the vortex shed in the wake will be produced by each

body.

In this paper. *e shall here consider about only two

lifting bodies. The (X1, Yi) and (x2, Yz coordinates on

body i änd 2 respectively aré shown in Figure 1 The

velocity of body 1 and 2 are denoted by.Uj and U. and

it has the difference in -anglé 'O' to theif forwar'd

direc-tion. Fig; i Coordinate systems on bodies:

The body surfaces are here given by .

y, =ff(x,),

L,/2

X L,/2

(j

=

1,2) (1)

where j represents body's notation, with a +

sign for the upper surface and a - sign for

the lower surface, and L, for' the chord length of body'j. And here two bodies are .assuthed to

be thin, i. e. f, are smalÏ compared with L,.

The geometrical relatÍons betweén two bodiès Iiay be written asfoliws,

X0

=

x1-- U1t

=

(x2+U21)cosöy2 sinô+f'(t)

Y0

=

y1=(x2±Úz t)sinO+yz cosO(r) f(t)

=

(U1U2 coso5r±e(0).

(t) =U2tsinO+(0) (2)

5(t) and ij(t) represent stagger and separation distance between two bodies, and the initial loca-tion is denoted by 5(0) and 17(0). By applying a convenloca-tional wing theory, the disturbance velo-city potential due to the two lifting bodies can be written

q(x, y, t) =_L ,f[mi(x. t)logi/(X,_x)2_jy +r,(x, t)tan' Y

2 x i-i

x,x

Ç3)

where m,(x,t) is the source strength änd r(X,t) the vo'rtex strength on body j. The boundary coñditión to be satisfied on the surface of body j is

hm'

=

d f1(X1) (4)

and

= Ui-/_J?(xi)

and from equation (3) and (5) for body 2,

rL1/2

±--mi(xi, t)+i-- I m1(x, t) (x2 sinO+ii)cosO(xi cosûEx)sinû dx

2 ir l_-L1/2 (x2 cosôf _x)i+(xz sinO+)2

L2/2 Li/i

_-__{2 r J}

xx

t)dx+----_{2 ir J}I rj(x, t) (x2 cosox)cosô+(x2 sinô+i7)sinO_{(x2 coso_e_x)2+(xi 5iflO+i)2} dx

=

_dx2

Li/i

II

_r m1(x, r) (Xi sinê+7)cos8(x2 cosoex)sinedx

-Li/i (x2 coso_e_x)i+(xi sinO+V)2

Li/i

+-_Íuh2 ri(x, r) dx+-i_{I TiX, t)}(Xi cosûEx)cos8+(x sinô+rùsinê_dx

r)

xx

r)

(x2 coso_f_x)i+(xi SinO+)2

= Ui_-_If(x2,r)+f(xi,t)]

_ax2

In the right hand side of equation (10) and (11), the differential terms represent the camber of each body. Then, if these bodies are assumed to be symmetric with respect to its longitudinal

axis, this differential term will be zero. In this paper, two bodies are modeled by symmetric

form.

On the other hand, the essential point for treatment of unsteady wing will be a wake vor-tex, i. e. vortex shedding from the trailing edge, that is related to rate of change of circulation

around bodies. And we know well the Kelvin's theorem and Kutta-Joukowski condition for this problem. Now Sears [6] has introduced on this as follows: - The condition usually is presented Hydrodynamic Interactions between Two Lifting Bodies 189

Therefore, on substitution of equation (3) into (4) we obtain for body 1,

±--m1(x1, t)+---

I

m2(x, t)C(x1+e)coso sinOx)sine ((x1+e)sino+ coso)cosOdx ir J L/ ((x1+e)cosoi7 sinox)2+((xi+e)sino+ cos8)

L2/2

i L/2 L2/2

+ _{2 r}

_f

ri(x)dx+_

_x1x

_{2 r-_j'} r2(- )((xl+e)coso_7isino_x)cose+[(xl+e)sino+coso)sinodx (xj+OcosOij sinO_x)2+((xi+f)sinO+ cosO)

Subtracting the + and - parts of equation (6) and (7) implies that

m(x1,t)= (8)

m2(x2, t) = Ui-[f(x2, t),ft(x1, t)].

(9)

From the above equations the source strength are entirely determined. On the other hand, on adding the + and - parts, we can obtain the following equations,

I

_r)

I mi(x, t)((x1+e)cosO sinox)sine((xi+e)sine+vì cosO)cos8dx

-L i/i

C(x1+e)cos sinox)i+((xi+Osino+ cosO)i

Lili L2/2

r1(x, t) dx+I_{I ri(x, t)}C(xi+Ocoso sin8x)cosO+((xi+e)sjnO+i cosO)sine_dx

r

xix

rJ

((x+e)cos8 sin8_x)i+((xi+f)sjnO+ cose)i

as the statement that the pressures above and below the trailing edge must be equal.: In the absence of the wake of shed vortices, this means that the vortex strength at the trailing edge

must vanish, but it is easily verified that, in the pressure of the wake sheèt, it. means instead that the vortex strength of airfoil and wake must be equal at the trailing edge.: as the

con-ditión into the wake for a moving body.

In this paper, the condition at tle trailiri edgè of each bod is applyed by. Sears method

that the vorticity at the trailing edge be equal to the vorticity in the wakeimmediately ajacent.

that is

t) = rwJ(Xr.E.) (12)

where

represents the position ofx axis at the trailing edge.

This is equivalent to

Kutta- Joukowski condition, that the pressure above and below at the traiÏing edge must be

equal. By using this condition as required, we may be obtained the unique solution. And by

letting.-x , letting.-x , letting.-x = , x2 ,x/(L/2),

f', ' = Ç, /(L/2),

L1=L2L,

if these parameters are used for non-dimènsi6n. wè havé the follówing equations from (10), (11) and (12) 1 11 _Ti(x',t) dx'+ f

(/ t)

((x+f1)çosO_1sino_x1)cosO+C(xç+f')sinO+'cos6)Sin8dX,

xix'

j Tz _{((xÇ+')cosO_'sinO_x')2+((x+f')sin8+cosO)2} = M((xÇ,t)±W((xI, t), (13) where and

M'(x' t -

1 _{m (X'} t) [(.+e)cose_'sino_x')sine_((xç+e')sino+'cos0)cos8dX i 1. ) - _j _ Z W((x,

t) = - :

r:) dx'

1

f'

/ ) dx' i Ç' 1(x',t) _dx'+1 _{'(X' r)} (X,' cosOf'x')cos8+(x,' sin8+')sinO dx'

j, ±,'--x.

r J ' (x,' cosof.'x')'+(x,' sinO+7')2 = M,'(x,' O+W,'

(4 t)

, , , - (14 where

M'( /

_{2 x2} r) 1 1 (x' (x cosOf'x')cos6±(X sinO±,')sinO d /

- ---

m1 ₎

(x cos6f'x')'+(x sin6+7)')2

W,'(x r) Ç Twz(X',t) dx' . 7J

X'X'

-I (X''t)

cosOf'x')cosO±(x sine+')sinè'dx,

j

TI

_'

_{(x cosOf'x')'+(x sinO+')2}

We obtaiñed here the integral equation tó be solved in the above equations.

3. Numerical Calculation .

'sòlu-tions are non unique until the vortex shedding

oditioñ'i

aplièd. Th& .N*nian's or King's

method are here used lii the numerical procedure for solving this integral equation.

Integrating the equation (13) and (14) with respect to x,, we have ri(x', t)log(xÇx')dx' irJ - i, -1 .

+1f

Tz(x', t)log (xÇ+f)sin8+cosO)u] dx' = M1(xÇ , t)+W1(x , t)+ci(t), where M1(x1 ₎ i 1'

- ---j

m2(

_)j

ccxç+e')cose-='sinox')2+((x(+e')sino+'coso)2 W1(xÇ ,t)

_{f_:''1" t)iog(xx')dx'}

_.IJ

r2(x', t)log "2dx', and

rz(x', t)log(x x')d'+_J Ti(X', t)log [(x SiflO+,7)2] i/idx

=M2(x2,t)+W2(x2,r)+c2(t),'

.,

where

M2(x, ,t) =

if

rn1(x',t)Í (xcosOe')cosO+(xsinO+')sin8 dx'd /

ir j

(x cosôe'x')2+ (x sin8+i/)2

X

W2(x2 , t) =

-1f

r2(x', t)log(xx')dx'

_f_:(x/.

t)log [x cosee'x')2±(x sinO±i)u1 "2dx'

In the above equations. ci(t) and c2(t) are arbitrary coñstant of integration, which are deter-mined by the vortex shedding condition For the numerical procedure the problem of unsteady potential flow is solved as a series of steady-flow problems. So. the solution proceeds step by step in time and at each step a small segment of vorticity is shed from each body

Now, we shall divide the body segments and wake segments into interval in which the vorti-city can be represented by step functions So we divide the body segments into intervals

x,1<x<x, with vorticity T,i(t) where

x =

_cos(_.);

i = 0,1,2, And as the control points we take

I

i\

=

_cos(J');

i = 1,2, ...M.

After some arithmetic operations for the equation (15) and (16), we -may have the fôllowing

equation s.

1 r- _1XXi.ft i r .jXXi.k

ri.s(t)l Y11(iÇ1,X',t) L

+--r2(t) [Y12ÇÇ1,x',t)

I -.

ir ft-i L

- irk-1 - L

= M1(ÎÇ, , t)+W1(, , t)+c1(t), on body 1, (17) ZXj XX2ft

x',r)]

-

rl.k(t) I k I r2.(r)[Y22(xi ' X X2 k 1 J r X k-1 M2Çt, , t) + W2ÇE, , r)+c2(t), on body 2, where

Y11 = (5Ç,x')logI5,x'I x'

Y12 = [(Î,+ e')cosO'sino---x')log [(tÇ,+ e')coso'sinex')

+((xÇ,+e')sino+'coso] 1/2

+((,+5')sinO+'cos6)tan-

I (,+e')s8+,71cos8 l,

L

(,+')cose'sinox'J

Y21 = - ( (2,cosO x')log [(Î,cosO - e'x')2+ (zsin8+)h] 1/2

I

.,sinO+'

+(2,sine+!)tan_lL

_,cosoe'x'

Y22 = (,x')logIÎ,x'

Ix'

we have finaly the following algebraic equations.

-

i.,,(t)Ai.,,+Er2.k(t)AI.k+U = M1(., r)+W1(5Ç, , t)+c(t) Irk-I

E

rl.k(t)AI+M.k+ET2.k(t)At+M.kf u = M2(î, , t) W2(, , t)+C2(t) Irk-i by letting r A,,, = [Yii(czx1.r)j XXi.k_1 r 1XX2. = LY12(î,, x', r)JXX2, k-1 r 1XX1.k =LY21(2j,x,t)] 11.k_I XXI.k

=

[Y22(i,x', t)]

This equation can be written by the following matrix, A(t) = [A,,(t)] as A(t)r(t) = g(t)+c1 e1"+c2e121 (20) where Tiu (t) Tzi (t) T2M (t) g(t) = M1(21, )+ W1(11, t) M1(î1, t)+W1(.i5.t) M2Ç21, t)+W2Ç21, t) M2(x2, t)+ W2(î2, t) and

Hydrodynamic Interactions between Two Lifting Bodies 193

N3(t)

=

--Jx'

:1J r,(C, t)dC, dx'

4. Computed Results

In the first, the computation as a steady problem that two thin wing bodies traveling in the

identical forward velocity and forward direction, i. e. the both stagger and separation distance between two bodies are constant, was here carried out to compare with the experimental

re-sults in two dimensional thin wing model. The body form used in the numerical procedure is given by

f(x') =(1_X'2), 1x1.

The solution of equation (20) is obtained by using a standard matrix inversion method, and may be assumed

(t) = r (0) (t)+cir1 (t)+CzT2> (t) (21)

where

=

A'(t)g(r), and rw(t)

=

A-'(t)e"> (i= 1,2,).

We have determined a numerical approximation to the general solution to the pair of singular

integral equation, with e1 , e2 as arbitrary constant.

If we follow the procedure used by Tuck [7] such as aplied also by King, we can determined the arbitrary constants C1 and c2 by fitting the following function at the trailing edge,

=

?/XT.E.-X D

+E+0(1/xr.E.x)

Therefore, if the above function is used as the Kutta condition, the condition to determine the

constants and the vorticity at the trailing edge can be obtained as follows, D°' +c1D'>+c1D2'

=

O

D°> +c1D +c2D21 =O

and

T,,i(Xr. E. t) = E° +c1E' +c2E2> r,,2(XT.E., r)

=

E°> +c1E' +c2E2

If the vorticity r(t) is obtained, the sway force (Y) and yaw moment (N) on body

j (j =

1,2) are given by

Y1(r)=Pf -i-f r(C, r)dC, dx'

(22)

(25)

The computed results are shown in Figure 2 and 3 for the lateral force and moment acting on

= 1 i o o M terms

=

M terms

o-o O

1'

1

lj

M terms M terms ) (23) } (24)

body i respectively. The positive value iñ stagger 'f'" represents that the body 1 is traveling in forward position to the body 2. The positive value of Y' means a suction force, negative one a repulsion. And also positive in N' means a bow inward moment,, negative a bow outward

mo-ment. The results show the peak value in lateral force at C'=O, and in yaw moment at C'=±

0.75.

In this Figure 2 and 3, computed results are compared with the experimental ones obtained by Oltmann[8J for elliptic sectioned- bodiès with beam/length =0. 125, and it seems to give good

qualitative agreement with the experiments. However, it seems the peak point on the suction

force occurs experimentally at a lower value of the stagger e' than that on the computed results. Although the experimental results on yaw moment in Figure 3 show the bow outward .thoment

at f'=O, the- computed one is zero, i. e. the suction force is acting on mid-station of body 1. For.this difference, the distribution of the vortex along the chord length is symmetry with res-pect to the mid chord in the present computation, and it seems on of the reasons of the effect

of viscosity in the actual flow around the bôdy.

- Then, examplesof the-compiited résult depending-onthe difference of the body's length are

also shown in Figure 4 and 5 for the forces and moments acting on body 1. These resulis show

larger ihteÏadtioñs' to compare with the results of the identical body.' It is to be noted that-the

bow inward moments act on body -1 aV C'=Ö.

'

Figure 6 to 9 show the computed results in 0 =30' and 60' as the arbitrary forward direction

with respect to each body. The numerical method in the previous section was done by taking twenty mesh points on each body, and the time step were such that a change in the stagger about one twentieth of the length of the body occured in each time step However in fact this numerrical method must be tested for convergence by varying the number of mesh point on each body. But, the mesh point presented in this paper is twenty, and this is the same with the case of computation for the identical forward direction, based on the convergence wasfound to be satisfied by twenty mesh points.

Thus, Figure 6 and 7 show the computed result for the lateral force to that the différence

(0) of forward 'direction is 30' and 60' degree respectively, and Figure 8 and 9 for yaw moment.

These results are given by the value of it at the instant of just '=0.8 or 1.Ö, 2.0 during the

travel with different forward directions.- Comparing with the ,res,ultsfor. constant-stagger, the,

both lateral forces and yaw moments have large value. As the reason for this, we can - here consider each body are moving in more closer proximity than that of the constant stagger, and

also body i is greatly affected by the wake vortex due -to the body 2. In the yaw moment, we see the quite different character comparing with that of constant stagger.

5. Conclusion

In this paper, hydrodynamic interaction between ships moving in close, proximity was com-puted by means of two dimensional thin body. By. this results, in general it seems that the hydrodynamic interactions' betWeén ships ôccuré iü th rangé of stagger from

=-2.O'to +2.0.

For the sepàratioñ' distârie the interactioñs re ffèctéd bj region of i' 1.0 in the same

for-.ràrd direction ön èch' bäay.

Acknowledgments

-The author express his sincere thanks to Professor S. moue, Kyushu University,. for his

encouragement, and also to Professor J. N. Newman, Massachusetts Institute of Technology, for his kind -guidance. Thanks are -exprsséd to- Professor-- R. W. -Yeung, MassaChusetts Institute, of

Hydrodynamic Interactions between Two Lifting Bodies 195

Technology, for his discussions, and to Mr. P. Jones who made his effort for the computer pro-gram. to Mr. Y. Nakiri who helped me in the drawing.

References

J. N. Newman, "Lateral Motion of a Slender Body between Two Parallel Walls", J. of Fluid Mechanics, 1969 vol. 29.

E. O. Tuck & J. N. Newman, "Hydrodynamic Interactions between Ships", Tenth

Sym-posium on Naval Hydrodynamics, 1975.

H. Yorimasa, "Ship Manoeuvring in the Waterways Crowded with Shipping", Master Thesis of Kyushu University, 1978.

R. W. Yeung, "On the Interaction of Slender Ships in Shallow Water", J. of Fluid Mecha-nics, vol. 85, 1978.

G. W. King, "Unsteady Hydrodynamic Interactions between Ships". J. of Ship Research. vol. 21. 1977.

W. R. Sears, "Unsteady Motion of Airfoils with Boundary Layer Separation". AIAA Journal. vol. 14, 1976.

E. O. Tuck. "The Effect of Stream Wise Variations in Amplitude on the Thrust Generating Performance of a Flapping Thin Wing". Symposium on Swimming and Flying in Nature, Cali-fornia. 1974.

P. Oltmann, "Experimentelle Untersuchung der Hydrodynamishen Wechselwinkung Schif-fsahnlicher Korper". Schiff und Hafen. vol. 22. 1970.

Nomenclature L, : Chord length of body j

Stagger

Lateral separation distance between body's longitudinal centerlines

O : Difference in the forward direction between two bodies

U, Velocity of body j

r, : Vortex strength on the surface on body j

Twj Vortex strength in the wake

Y : Lateral force acting on body

N : Yaw moment acting on body

Y' : Non dimension value for

(' = Y/(-pLU2))

N' : Non dimension value for N(N' = N/(.pL2U2))

-0.4

Fig. 2 Lateral force acting on body i

at the various separation

dis-tance (a')

-1.2

Fig. 4 Lateral force acting on body I

(L1) moving side by side with body 2 (L2=2L1) 9:0

- r

04 r .0.6 r 1.0

,2.0

oOItmans Exp. (2'.0625)

-

E' 80 LfL, .2.0 r Q

2.0 -

4.0 ex30 ' .08 r .1.0 - 0.6

Fig. 6 Lateral force acting on body I

moving in different forward di-rection (O = 3O) from body 2

-0. 1 2

Fig. 3 Yaw moment acting on body i at the various separation

dis-tance (ij')

20 '40

Fig. 5 Yaw moment acting on body i (L1) moving side by side with body 2 (L2=2L1)

Fig. 7 Lateral force acting on body i moving in different forward di-rection (O = 60)from body 2

9x0. .0.4 r .0.6

-7x2.0

aoltrnan's Exp. (r0-625)

Hydrodynamic Interactions between Two Lifting Bodies

N'

83O'

- r

T '1-0 r2-O

Fig. 8 Yaw moment actingon body i

moving in different forward di-rection (O = 30') from body 2

Fig. 9 Yaw moment acting on body i moving in different forward di-rection (O = 60') from body 2