By Hajime Maruo, Kogakushi, Member Abstract
In this paper, a general formula for the forces acting on a body moving under the surface of water is obtained as an application of the extended Lagallys formula. The problem of the wave resistance experienced by a body moving under the regular sea waves is also discussed.
1.
Introduction
The formula given in the first report is quite incomplete, because, first, we can apply this to the pure translatory motion but it is inodequate for the motion including rotation. The other reason is that it is applicable
only when the body is represented by a special distribution of singularit-
Fig. i
ies since the expression involves integrals over the surface of the body. Recently W. Cummins extended Lagallys formula to the general unsteady motion. i J t Applying this result, we can find more general expression for the forces acting on a body moving under the surface of water with a quite arbitrary manner.2.
Extended Lagally's formula
Consider an arbitrary potential flow outside a moving body whose surface S is realized by some distribution of singularities imagined within S. Assume a control surface S' within S which en-closes the singularities, and consider the region V1 between the surfaces S and S'. (Fig. 1)
When the unit normal to the surface S or S drawn inward to V1 is denoted by n, the force acting on the surface S is expressd by
f pndS=f[-1 ----(q.q)gzndS.
r(1)
where q=p, the velocity of the fluid.
From the last term in the brackets we obtain thestatical buoyancy in the direction of z taken vertically upwards. F,,=pgV.
(2)
The surface integral of the first term can be transformed by means of Gauss' theorem. If we assume the surface S or S' moves or deforms with the normal velocity V, we can write.
n dS=
-f.r (-
)dT_f,-
dS_df
qdr+f
vmqdS_f --ndS.
dt s, et
Again making use of Gauss' theorem
dc
jqdr=----j
dr
pndS,
dt .' y1
dt 's+s'
7 11 *
Assistant Professor of The NationalUniversity of Yokohama. t Number in the square brackets means the number of the reference.
v
Scheepsbouwkund
Technische Ho3scIooI
Deift
The Forces on a Body moving under the
Surface of Water (second
report)
-Ic
-
ç,1.1f'fr-Ot1?
Zc:)k
Vt.
f1l
28 t.ç100J.
If the position vector of a point On the surface is denoted by rl, we can express
n=a
ri/&n,and making use of Green's reciprocal theorem, we have
dr
--IqndS-
dt
-j
ç---,. -dSdtJ+'
Ond
j
ridS=__d
j
(nq)ri dS.dt s+s' di
ss,
Since on the surface of the body
v= nq, we can write
v,3q ds=f(nq)qds.
If the control surface moves with the velocity y, we have
V=nV on S'
and the last term of (3) is transformed into d
n ds= n ds-f (vv)p. n ds_j5,P da
dr
r ra,
=
----j pnds+i (vq.nds-j p-ds.
dt /s'
.1,5, - s, diIf the control surface has a velocity of translation V1 and the angular velocity of rotation Wi about a point r0=r-ri, e have
v=V+oixr and an/6t=oixn,
(10)and we can readily prove
Jr(vn)q (vq)n+ p
s'l
otJ
- - ds= O - (11) Collecting the above results and making use of the relation (2 Jri
I
[----(q.q)n-nq)qJds=
o.s+s'
2 we obtain finally L.(5)
(9)
F
= -f [(nq) q-
---q .q) n] da- e-j-f [(n)ri +cpn] ds-p- f v,r1ds
(13)When the boundary form cf the body is given, the last term of (13) is a known quantity and the force is expressed completely by the integrals over the control surface.
If the singularities generating the boundary surface S are discrete, the control surface is taken as infinitely small spheres having their centres at the position of each singularity and the formula becomes a quite simple form. Consider a body moving with the velocity of trans-lation V5 and the angular velocity o about a point ro, and the boundary surface being gener-ated by a distribution of sources of strength and doublets of the vector moment ¡, which are situated at points denoted by the vector ro+r fixed to the body. Now we write the velocity potential in the form
(14)
in which pj is the velocity potential due to singularities within the body. Then the force is
ex-pressed by the formula
(12)
do)
F= 4
pf[{+(r)}q_(Vo+oxri)+ri d+4]dr_{roX
dt dt dt 4. dV0 io)(rø')+r5(ow)--
di f"
(15) where q=-VPe at rj, (16)r0 is the position vector of the centre of gravity of the body relative to the centre of
rota-tion, and V is the volume of the body.
3.
A body moving under the surface of water
Now we assume a body moving under the surface of water, and the boundary condition of the rigid surface is satisfied by a surface distribution of sources of strength c' (t) and doublets of the vector moment p. (t) whose position is denoted by r'ro+ri. Then the velocity potent-ial is
ds-
1 ds1deI(+KE,)exp[,c(t)]d,c
2zJ )_, Jo +- I dsl(ir)}VdK
J (17)where ri=!rr'j,
E is a vector (icos9, isinO, 1), and(t) =rE+r'(tE,
(18)When we define the integral
H (K, O, t)=f {o'(t)+,E1(t)}exp [Kr'(t)E] ds, (19)
the velocity of the external motion becomes
q4,= 21 fdOf H (jc,O,t)exp(,crE)KÈdK
___-ftd'rftdef°°H(ic, O,'r)exp(KrE)sin{Vgh (t r) } VgK x'd,c. (20)
Then the first part of the force due to the external motion, viz, the effect of the free surface, beco mes
F1 = 4 1rafo.+ (pv)} q4s = - 2pf Thef'7H (K, e, t) ii(c, O, t),cE dK
+4f drf dOfJ(K, O, i) H(ic, O, r)sin{y'g(tr)}/'
gic xEdK. (21)Now we consider a pure translation. According to Green's formula, the density of the surface distribution of sources and doublets on the surface S is given by
= (1/4 or)(äp/an) (22) ¡j = - (1/4 irpn (23)
If the body is regarded as a very slender symmetric body with respect to the direction of mo-tion, the contribution of the doublet is comparatively small. Hence the usual approximation in such a case is to take only the Sources into account. Thus we have
(24) where V0 is the velocity of the body, on accunt of the boundary condition, or we have a uni-form space distribution of doublets of the vector moment (1/4 1v) V0 within the body. In this case we can write
s H(K, O, i) = (1/4 ir)(Vo(t)E)exp[,cro(t)EJ G(K,O) (25) where G exp(cri E)dr (26) Then we find
F1=
87r° +f'dûfG(K, O)
i2[(vo(t)E)]E'
gícxf[vc(T)Elexp[KEro(e) +Ero(r)]sin{/
(!r)}dr,
(27)where ro (t) jS the instantaneous position of a point fixed to the body whose z componenent is z0 The remainder of the force is given by the formula
F2 = _4irpf(iVo+rith4dt)ds±pVdVo/dt. (28)
and substituting (24) we find
F2=0. (29)
Thus the inertia force does not exist and the force is solely due to the effect of the free sur-face. This result is paradoxical. The reason lies in the fact that the inertia force is induced by the doublet distribution of (23) which has been omitted. In fact, substitution of (23) into the formula (15) shows
F: = pf(ap/an)nds. (30)
that is the inertia force. If we wish to express the fluid motion by means of the Source distri-bution only, we have to adopt a source strcngth-(/4 ir) (1+k) (Von) where k is the inertia coefficient of the body in the direction of notion. Then the resulting wave resistance is
aug-mented by the rate of (1+k)2. This result has been obtained more explicitly by M. Bessho.1 3) On the other hand, if we compare the effect of the free surface given by (27) with the result in the previous paper, some difference still exists.
This is also due to the omission of the doublet distribution. Now consider the doublet
(31)
which is attributed to the external motion. Since Pe is harmonic within the surface S, Green' s
theorem gives
r
q'- - --
f1\
repejds= ¡
-fl\Yi/
-'sOn'i
and the doublet distribution given by (31) is equivalent to the source distribution of the inten-sity.-(1/4 ir) Op/On on the surface S, or by virtue of Gauss' theorem, it is equivalent to the volume distribution of doublets of the t'ector niomerit-(1/4 2!) q, within the body. Again taking an approximation of (25) we find the force due to the effect of the free surface being
F =
IC V (VoE)}IG@c O)e2r.o,c2EdK- 8
_fdef'IG(c,9)JieoLVoE2Eicdicwh
The Forces on a Body moving under the Surface of Water (second report) 31
---
f
dOf ¡G(c,O) ic3dicf (Vo(r) E)exp [ickro(i)f- IcEro(T)J
42V3 J... J
X cosli/
(t)}dr,
(32)
vhere V, is the z component of V0. This result coincides with theprevious one.* 1f the motion ncludes rotation, G is also a function of time and the expression for the force becomes some-vhat complicated.
4.
A body moving under
wavesThe wave resistance of a ship in a seaway has been considered first by T.Hanaoka( 4 3.
11e
reated the effect of the on-corning waves in a rather approximate way. Now we can apply the ormula developed in the previous paragraph to this problem.
When a body is moving in the direction of x with a uniform
speed of advance V0 under waves vhose direction of propagation makes an anglo a with the axis of x, it makes small oscillations. rhen the velocity potential is written in the form,
(33) vhere is a velocity potential of the incident waves,
q is that due to singularities generating
he surface of the body and is the elfect of the free surface. Then ve have = (gh/c)exp(kz ikxcosa ikysina± ici),
(34)
=fro-_,.r]ri1ds
(35) y he re k = 27V/A = f we pot H(ic,O,1) f[a.+ic(,E)1cxp[c(r(1)E)Jds =exp(iicVo1cosd){Ho(,c,O)+e'Hj(K, 6)}bstitute in (17) and perform the integration with respect to r, we find
i rr
= dO) {Ho(ic, O-i- e"'tHi(,c, O)}exp
[it-(riE)] dic
¿7V - O
_-ic-0-f7rdO
P. V. f(ic cos0dxo) 1H0(ic,O)exp [«riE)] dic
( j.-2 ç2
+ iK0 J
-J
+J Ho(,cosec"O,O)exp[icosecW(riE)]secddO
-
2
2)-
-ed8 P. vf
¿13+&'i/V0)KoKHz(,c, O)exp tK(rlE)] dic
+iicoeiníf
-f -f +f
1r--_-_Hj(aj,O)expfaj(rjE)]seco Odd
t.
t
t2
j2) aa2
ere
There are errors of aigris in the expressions of the previous report.
(36)
(37)
J
______L_' __- - --
--.-wj=c,.,kVocosx, ri
rVnt,
tco=g/V02 KO-2CUSC(wi/ V0) ±/2i4ocos G (wi/5
a,J 2 cos8
= COS(VoKo/4Wi)
Then the mean value of the force during one period is given by
F=F0+F,
whe re
'0the mean value of 4
2fdOf{Ho
(K,O)J2+_ Hi(,c,_4ptcof'dO P. V.f (KCOS Gko)-'IHo(K,0)!cdK
( -*2 I2
*'
+47ripxoJ
-,-J
-*2+1 J.IIIo(kosec20e)2Esec4edO*2
_2pKofde P. V. f {(Kcos G+U,i/Vo)---KOk}'IHl (K, 0)!2KdK
( ,*Z ,-i
p*2 r" a2--+2rip,c J-x
-J
-2
-J
+j
*2) a1a2
- Hj(aj,G)!2EsecOdO_2iPKo{f+1:
'= the mean value of 4
=2 ivp( gh/oo)Hi(k, a)kE(c). (42)
E (a) is the value of E when 0=a, and only the real part is to be taken. The x component of F0 gives the usual wave resistance which is the same thing as Hanaoka's findings. On the other hand, the x component of gives the resistance increase due to waves indicated by
Havelock. C 5] When the translatory oscillation is expressed by the vector h exp iwjt and the rotational oscillation by the vector O exp isit, the velocity on the surface S becomes
v=Va+ioie"1'(h+0xri), (43)
where V0 means the uniform velocity in x direction. If we take the approximate source distri-bution due to the motion of the body
(44)
on the surface S (the slender body assumption), we get H1(k, a) = - (1/4 i) (kV0E(a) +ic,i} Vif exp [kriE(a)] n ds
+of exp [krjE(a)] (ri xn)ds} (45)
On the other hand, the force evaluated by the undisturbed wave pressure (Froude Kriloff hy-pothesis) may be written as
Ld the moment is expressed by
ipghe
itf
exp [kry '(a)J (ri X n)ds_ eotmw.(47)
ien the mean force becomes
= ieal part of
E(a) (h+9n),
(48)
ncc
kV0E(a) + 101 io.
lis is equivalent toHavelock's result.
5. Con elusion e author has obtained a general formula for the
force acting on the body moving under the nace of water, taking the
surface disturbance of water into consideration,
ist applications of this formula are permitted if a suitable distribution of singularities is
found
as to satisfy the boundary condition on the surface of the body. e problem of the wave resistance of a ship in
a sea way is an application of practical impor-ncc and will be discussed in future.
This work is a part of the researches about the propulsive
performance of a ship in a sea iy subsidized by the Ministry of Education.
References
L ) W. Cummins, The forces and moments
acting un a body moving in an arbitrary
potential stream. T. M. B. Rep. No.780 (1033)
) L. M. MimeThomson,
Theoreticai hydrodynamics, 3 rd ed. (1955) p.84 J M. Bessho, On tile wave resistance theory of
a submerged body. J. Z.K. vol.99 (1956)
) T. Hanaoka, The motion of a ship among waves and the theory of
wave resistance. J. Z. K. vol.98 (1956)
J T. H. Havelock, Notes on the theory oy heaving
and pitching. T. I. N.A. vol.87 (1945) J H. Marue, The forces on a body moving under the surface of
water J. Z. K. vol.97 (1953)
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---,-The Forces on a Body moving under the Surface of Water (second report)
