E L S E V I E R P U : 8 0 1 4 1 - 1 1 8 7 ( 9 6 ) 0 0 0 1 8 -1
Printed in Great Britain. A i l rights reserved 0141-1187/96/$15.00
Hydrodynamic loads on a slender
cylinder moving unsteadily in a
m^m^eimniFcDrm l o w ield.
A. R. Galper, T. Miloh & M. Spector
Faculty of Engineenng, Tel-Aviv University, Tel-Aviv, Israel 69978 (Revised 4 January 1996)
In order to evaluate tlie wave loads on large ocean structures we consider the hydrodynamic force and moment acting on a slender rigid cylinder moving unsteadily in a non-uniform ambient potential flow field of a perfect fluid. The motion consists of both translation and rotation. The leading oder loading terms of the corresponding theory of perturbation are found and the influence of the cylinder's ends is accounted for. Copyright © 1996 Elsevier Science Limited
1 INTRODUCTION
In this paper we present a concise analysis of the hydrody-namic loads exerted on an thin slender cylinder moving unsteadily in a time-dependent spatially non-uniform ambi-ent potambi-ential flow field of a perfect fluid. The motivation for this study is the current interest in the 'ringing' phenomena experienced by large ocean structures (gravity based or ten-sion leg platforms) in extreme waves, studied by Rainey/ Foulhoux & Bernitsas,^ Jeffreys & Rainey,^ Newman/ Faltinsen/ Faltinsen et al.^ and Malencia & Molin.^ A typi-cal value for the radius of a vertitypi-cal circular column of such a platform is of the order of 10 meters and the wave length of the incident wave is 100-200 meters. Thus, the wave number and the pile radius are of the same order and, for this reason, the commonly used Morison formula has to be modified to account for higher-order diffraction terms.
The inviscid inertia (added-mass) first-order term in the Morison relationship is associated with the weakly non-uni-form or the long-wave flow assumption. It is valid for a flxed rigid 3-D body of size much smaller than the characteristic wave-length of the ambient stream. One of the earlier attempts to incorporate curvature (non-uniformity) effects of the streamlines of the imposed flow-field about a station-ary rigid body (leading to the so-called 'buoyancy-force'), is due to G. I . Taylor.^ This analysis has been recently extended by Galper & Miloh^ for the case of both rigid and deformable 3-D shapes moving unsteadily with six degrees of freedom in a weaicly non-uniform flow field. The classical Kirchhoff-Lagrange equations of motion, which determine the forces and moments acting on the
moving body, have been generalized to include the effect of weak fiow non-uniformity. In a subsequent study (Galper & Miloh^°), tlie wealc assumptions has been released and a general formulation for the same problem, within the frame-work of Hamiltonian dynamics, has been provided.
It is interesting to note that the 3-D weakly non-uniform flow theory can also be applied to a long cylinder, provided the scale of the non-uniformity along the axis is large com-pared with the cross-section radius. Among previous attempts to provide a consistent second-order diffraction correction for the Morison formula in the case of vertical circular cylinders, we mention Lighthill,^^ Sarpkaya & Isaacson,^^ Madsen,^^ Manners,^'* Manners & Rainey,^^ Kim & Chen^'' and Rainey.^' All these papers lack a theo-retical investigation of the limiting-process where 2-D cylindrical surfaces shrink towards an effectively 1-D hydrodynamical line. The present paper can be considered as a generalization of the above-mentioned works by pre-senting a concise derivation of the sectional hydrodynamic loads (forces and moments) acting on a long vertical cylin-der of arbitrary cross-section placed in an arbitrary non-uniform flow field. In addition, the cylinder is allowed to move and to rotate in an arbitrary manner. The correspond-ing theory of perturbation is rigorously constructed here for the first time. The important role of the ends of a cylinder is also presented.
The outline of the paper is as follows; in Section 2 we consider the hydrodynamic loads acting on a slender cylin-der of arbitrary cross-section moving in a non-uniform ambient flow field. General expressions for the total force and moment acting on a thin fixed cylinder of arbitrary
cross-section in an arbitrary stream are first presented in Section 3. Tiie physical force distribution (i.e., force per unit length) acting on a cylinder is first derived in Section 4 (eqn (36)). The case of a weakly non-uniform flow field is next treated as a special case in Section 5 and some well known results for the total force and the physical force dis-tribution acting on a fixed cylinder are rederived. The expression for the hydrodynamic loads are then extended in Section 6 (eqns (56), (57) and (61)) to include the effect of a moving and rotating cylinder. In Section 7, we treat the influence of the ends of a cylinder whereas the point loading on the ends is presented further in Section 8.
2 GENERAL EXPRESSION FOR THE FORCE In this section we present a direct method for calculating the hydrodynamic loads acting on a rigid slender cylinder of arbitrary cross-section S (the surface of which is L = 5 x
[- H,H\ U r+ U r_, where are the upper and the lower
bases of the cylinder and 2H is its total length). The cylinder is moving unsteadily in an arbitrary direction and rotating about arbitrary axes. It is placed in a non-uniform ambient unsteady potential flow field V(x, t) = V<i)(x, t). The corre-sponding expressions for the force and moment are given in a moving (body-fixed) coordinate system. The derivation is based on the general methodology recently developed by Galper & Miloh. ^° The axis of the cylinder coincides with the z-axis of a corresponding cylindrical coordinate system with an origin coinciding with the centre of mass of the cylinder.
The total velochy potential if, induced by the presence of the moving body, can be uniquely decomposed into
(p = 4) + <j)Q, (1)
where <^o represents the additional disturbance potential satisfying a proper decay condition at infinity, i.e.,
lim <^o(x) = 0. (2) Here, x is the position vector in the above mentioned
cylindrical coordinate system. It is important to note that 00 is harmonic outside L whereas <t> is harmonic inside L.
Using the outer Green function G(x,y) for a cylinder (depending only on the body's geometry), which represents the solution of the following Poisson equation;
V2G(x,y) = 47rÓ(x-y) (3)
with the corresponding boundary conditions on both L and infinity
^ G ( x , y ) l i = 0, limG(x,y) = 0, (4)
an I x l ^ c o one can express 4>a in an integral form as.
<^o(x)^ G(x,y)(V'n)(y)dL(y). (5)
Here, n denotes a unit normal vector to L directed out-ward into the fluid.
The hydrodynamic force F and moment M acting on a fixed body immersed in a potential stream V = V<^ can then be written (see Galper & Miloh^") as
4>o F(<#.) = dn av dL + dt) D V , (6) and M(0) = a(x A V) dn . ^ n . ^ d L . dt) (7) where is a 3-D integral over the volume of the cylinder. In the above we have introduced the substantional deriva-tive symbol, ^ = | + V-V, denoting liquid acceleration. The harmonic vectors <^ and \l/ which appear in eqns (6) and (7) are the common Kirchhoff potentials satisfying the following boundary conditions
- = XAnli, (8) — = nli, and
dn dn
and a proper decay condition at infinity.
3 SLENDER CYLINDER
Let us consider further the case when the length-scale of the non-uniformity of V(z) in the z-direction is much larger than the body's size. There exists in this case a small parameter d^V dz2 151 dV dz (9)
where ISl is a characteristic length-scale of the cross-section, say its perimeter or the largest distance between two contour points. Next, find the Green function G(x,y) to the leading order in the small parameter e. Applying the Fourier transformation in the direction (z-z) to (3), one obtains
A2G(r, r, ii) = 47rö(r - f , ) + Ic^Gir, r, Ic), (10)
where r = (x,y) and r = ix,y) are the corresponding 2-D position vectors. The boundary conditions for the Fourier transformed Green function G(r, f, Ic) are
öG(r,f,A:), dn = 0, and lim G{r,r,lc) = 0. l r l - » " » (11) (12) We are mainly interested here in the value of G(r, f, Ic) evaluated on the curve S, i.e., in the near field, where, in accordance with the constructed singular theory of pertur-bation, one can neglect the boundary condition eqn (12). The Green function G(r, f, k) in eqn (10) actually depends on k\S\ and to leading order one can neglect the Ar^-term in
eqn (10), which leads to
G(r,r,A:) = G2(r,r). (13)
Thus, the leading-order approximation for the Green function, obtained by inverse Fourier transformation, is given by
G(x,y) = ö(z-f)G2(r,f) + 0(e'loge), (14)
where G2(r,r) is the appropriate 2-D Green function for the cross-section 5 and it can be shown that the next term in the theory of perturbation for the Green function is O(e^loge). The approximation eqn (14) is valid when x and y are far from the ends of the cylinder and within our leading-order theory of perturbation we imply that eqn (14) can be applied everywhere on the surface of the cylinder except on a semi-sphere of radius 151 with its centre in the middle of T j . (see Section 6).
After the substitution of eqn (14) into eqn (5), one can split <j)Q into where G2(r,r)V(r,z).n(r)d5(r), (15) (16) and G ( r , f , z , f = ±H)V^{r,z= ±H)êUr), (17) where G(r, r,z,z= ±H) is the exact 3-D Green function of cylinder applied near its ends and we understand Jr^ ( * )d2r as \j_ ( * )A^T_ + )d^r,. In splitting eqn (15), (f)'^^ is an asymptotic expression for based on the asymptotic eqn (14), (ji'p incorporates the bases effect (resulting from integration over the two bases T ±) and 4>f^ accounts for the correction needed for the Green function near the ends. Note that ^jP^ is a finite (in 3-D) function and i / ' ' ^ has a compact support near the ends. Correspondingly, using eqn (16) in eqn (6) and eqn (17), the total force acting on the cylinder can be written as
•H -H
(18) with and Pf^ given by
at J • / and f [ E , ^ n j p . n J - ^ ] é S , h \ dt J (19) (20)
where £y = V,V,- is the rate-of-strain tensor of the ambient non-uniform flow. Finally, one obtains
F,.(z)=- Vp{r, z)«^(r)G2(r, r)Ei^{r, z)n,(r)d5(r)d5(f)
, s. S at - / ( r , z)«^(r)G2(r, f )n,(f)d5(r)d5(r)
Dt (r,z)d^i3,7=l,2, f = l,2,3. (21)
Here, and m the sequel, Greek symbols (a, /3, 7 ) take on the values 1,2 and Roman symbols (ij,lc) the values 1,2,3. In order to estimate the time dependent term in eqns (19), (20) and (22), we use the relation
s dt ' n ^ d 5 . s dt $ - ^ d 5 = dt dn s dn dt (22) We will show further in Section 4 that F,
written in a distributed form, as
f dg(z) (T) H dz -dz. also can be (23) where the density function g(z) is of 0(e). It is demonstrated m Section 6 that, up to 0(e), one can neglect the force term
It is worth noting that the real pressure force exerted on the z-cross section S (within the thin cylinder approximation eqn (14)) should be the same as the one acting on a 2-D shape S embedded in a 2-D flow field V(r,z) with a nonzero 2-divergence i.e., VrV(r,z)= - ^ = 0(e). Consequently, the total force experienced by the cylinder is further given by the integral of the 2-D forces acting on the cross-sections placed in such a non-solenoidal 2-D flow field.
4 PHYSICAL FORCE DISTRIBUTION; BASES EFFECT
We consider now the force pf^ which mcorporates the so-called 'bases effect'. Other effects to be accounted for are connected with the fact that the approximation eqn (14) for the Green function is not valid near the ends of the cylinder. This is further denote as 'ends effects' and, as demonstrated can be neglected within the realm of our theory of perturba-tion (see Secperturba-tion 6).
The effective force distribution F(z) given in eqn (21) does not actuaUy represent the real pfiysical force (i.e., a force per unit length acting on the cross-section F(z)). The reason for this is that eqn (6) does not really correspond to a pressure integration over the surface of the cylinder, since the integral over any part of the cylindrical surface is not necessarily equal to the force (moment) acting on this part. Yet another reason for the difference between the two is connected with the presence of the T-terms in eqn (6).
Thus, there appear to be some fictitious forces acting on the cylinder bases which must vanish based on physical ground. The difference between the so-called 'physical' and 'effec-tive' distributions can be expressed as
df(z) F(z) = F(z) +
dz (24)
where the function f(z) is uniquely determined in the sequel. The full z-derivative term in eqn (24) contributes (after a z-integration) to the expression for the total force throughout the effective (nonphysical) force eqn (19) acting on the ends of the cylinder.
The main difficulty in expressing the fictitious forces act-ing on the bases as full z-derivatives resuhs from the fact that the function i^o (included in the bases integrands), which is well defined on the bases, has singularities within the cylinder. Thus, it is generaUy impossible to express the bases-integration as integrands over the contours bounding the bases. Because of these difficulties and in order to deter-mine the physical force distribution ^ee from any fictitious bases effect, it is useful to consider the expression for the total moment eqn (7) acting on the cylinder. The effective moment distribution M(z), which is decomposed as in eqn (18) mto
M
jH
(25) is connected with the physical moment distribution M(z) (in a similar manner to eqn (24)) by a full z-derivative term of a function N(z) (a z-integration of which is equal to
M ' ^ ) , namely
M„(z) = M„(z) + dN„(z)
dz ' (26)
Using eqns (6), (7), (18) and (25) one can rewrite the effective moment distribution acting on the cylinder in the following form
M „ ( Z ) = ( Z A F ) „ ( Z ) + M : ' ' ' ' ( Z ) , (27)
where F(z) (consistent with the approximation eqn (14)) is given by eqn (21). Similarly, by virtue of expressions eqns (6) and (7) (which incorporate integration over the ends), one can show that
N„(z) = (zAf(z))„ + l„(z), (28) where l(z) as well as W ^ ) are some bounded
func-tions, depending on a quadratic combination of V(z) and V(z) but otherwise do not explicitly depend on z.
Note now that the physical moment distribution M(z) is connected to the physical force distribution F(z) for a cylin-der by the relation
(M),(z)=(zAF(z))„ (29) which simply follows from
cH ( M ) „ = - p{xAa)^dzdS, F „ = •H • pndzdS. -H is (30)
Substituting eqn (24) into eqn (27) and using eqns (26), (28) and (29), one obtains
T^'"i<i(-r\ a A f^ d ( N - z A f ) „ dl„(z)
M„ (z) = (i^ A t)„ (i^ A f)„(z) •
dz dz
(31) where = ||| is the unit vector in the z-direction. Equation (31) can be also written as
f J z ) = (i,AM'«'''),(z) + d(i. A l)„(z)
dz (32)
Recalling again the fact that l(z) in eqn (32) depends explicitly on z only through V and V, we deduce that the second z-derivative of l(z) is O(e^) and thus can be neglected in eqn (24) (consistent with our theory of perturbation). The physical force distribution is finally expressed in terms of the effective force distribution eqn (6), as
F„(z) = F„(z) + d W f j z )
dz Aiz + 0(e'), (33)
where F is given by eqn (21) and eqn (7) as (A) a(r A V)„ dMfiz) , M f (z) = •^0 dn ïdS + dz (r A ÊV)„ds is found from + 0(e), « = 1,2. (34)
It is important to note here that our eqn (33) coincides in the limit of a weakly non-uniform fiow field with equation (15) of Rainey^' derived for the limiting case of a very thin (hydrodynamical line) cylinder. Thus, the term M''J''{z) A in eqn (33) has the physical sense of a transverse pressure force acting on the pointed ends (two singular points for a hydrodynamical line) of the cylinder, in accordance with Rainey^^ (Section 3a). We also remark that the time deri vative terms contribute only to the z-component of M' and for this reason do not appear exphcitly in eqn (34). In deriving eqn (34) we used a similar identity to eqn (22), namely add av ^^n dS-S dt r A n—^diS. S dt (35) Finally, the required physical force distribution F„(z) for a slender cylinder is obtained as
d_ AA)dV. d^l - + nr. dn s dn \dS + dV, dt ^^E,,Vi]ds + r^E^iVids ]+0{ehoge), a = l , 2 i = l,2,3. (36)
where it can be shown that the next order term in the e-expansion of the force is of O(e^loge). Note that the only restriction imposed on eqn (36) is the weak flow non-uniformity in the z-direction.
5 WEAKLY NON UNIFORM FLOW FIELD
Consider next the case when, in addition to eqn (9), the characteristic length scale of the non-uniformity of the ambient flow field V m the transverse (^.yj-plane is much larger compared with the characteristic length scale S ofthe body's cross-section (the so cafled 'weakly-non-uniform' field approximation). In this case, one can introduce an additional small parameter
WÈW ' (37)
where ll(.)ll denotes the norm of the tensor. For the case when the transverse non-uniformity of V is of the same order as the axial one we obtain e = 6.
By pulling the product of the rate-of-strain tensor times the ambient velocity out of the mtegration in eqn (21), one obtains for the first term in eqn (21)
Vp{Y, z)n^{r)G2{r, r)£„^(r, z)«^(r)d5(r)dS(r) = jS , s
y^(r = 0,z)£„^(f = 0,z)f [ «^(r)G2(r,r)«Jf)d5(r)dS(r)
=E^^iz)m^fiV^{z), (38)
where the transverse added-mass tensor per unit length (neglecting end effects) is given by
« 7 / 3 = - n^{r)G^{r,r)n^{r)éS{r)éS{Y). (39)
jS jS
After mserting the following Taylor expansion of V about the centroid of the cross-section
V^ir,z) = V^{0,z) +Ep^iO,z)r^ + 0(6^), (40) in the second integrand in the RHS of eqn (21), one obtains f f SVR — £ ( r , z)«^(r)G2(r, f )«„(r)dS(r)d5(r) , s Js or dVp dt ^m^p^-EfiyQcpy, (41)
where we introduce the third-order tensor Qa^y
f r^n^(r)G2(r,i^)«„(f)dS(r)dS(r), (42)
s Js
which is equal to zero for a symmetric cross-section (Galper & Miloh^). The second term in the RHS of eqn (41) is of 0(e) m comparison with the first term and thus can be neglected. Finally, the last mtegral in eqn (21) is equal to
(43) where s is the area of the cross-section S.
Combining eqns (38), (41) and (43) and then adding and subtracting the term mapE^yVy, we rederive equation (35) of
Manners & Rainey^^ for the effective force distribution (without accounting for bases and ends effects)
n y
F(z) = {si + m)~ + ( M -MÊ)\ + 0(0^), (44)
where / is unitary matrix and M is the 3-by-3 added-mass tensor M ^ diag{m,0). One can also write the effective force distribution eqn (44) in a component form as follows Fi(z) = {s + m,)^ + E^Mi -mjW^ -m.E^^V^ + 0(6^),
(45)
F2{z) = (5 + , « 2 ) ^ + En{m2 -m{)V^ -m2E^2V3 + 0(5^),
(46) where Xi, are two unh vectors along which the 2-D added mass tensor m is diagonalized, i.e., m = diag{mi,m2).
In order to determine the physical force distribution (force per unit length), we use eqn (36) and note that
r^E^iViés = 0{b\a = l,2
which can be derived by placing Ê in front of the above mtegral. Correspondingly, following Galper & Miloh,^ one obtains for the first integral in the RHS of eqn (36)
dn dXf,
(47) The last term in eqn (47) can be neglected in eqn (21) being a term of 0(eö). Thus, by gathering eqns (45), (46) and (21), we finally obtain the sought expression for
phy-sical force distribution acting on a cylinder with an
arbi-trary cross-action (see equation (1) of Rainey^'' and the discussion m Kim & Chen"), namely
Fi(z) = (s + mi)^+Ei2imi -m2)V2 + ^1^33^1 + 0(max(e^6^)), F2(z) = (s + m2)~+£12(^2 -mx)Vi+ /n2£33 ^2 + 0(max(e^ö2)). (48) (49) For a circular cylinder mi=m2 = s and eqns (48) and (49) reduce to
F,{z) = 2s'^ + sE,,V^, F2{z) = 2s^^sE^^V2, (50) Dt " ' '''' " Dt
which are the corrected inertia terms in the Morison for mula for circular slender stmctures.
6 MOVING AND ROTATING CYLINDER
For a cylinder moving with a velocity V(t), there arises an additional effective force distribution denoted here by f^"\
due to the interaction between the moving cylinder and the
ambient non uniform flow field (see Galper & Miloh^°) given by
(51) For a circular cylinder = -r„ and h follows from eqn (51) that
V^n^éS, (52)
for an arbitrary flow field. Under the assumption of a weakly non-uniform flow field, one can pull the rate of strain tensor outside the integral in eqn (51), which gives
=En(m2-m,)U2-m,E,,U„F'i'^
= £ i 2 ( w i - m 2 ) U i -m2E23U3. (53)
In order to determine the physical force distribution, one should note again that the corresponding addition M^"\z) to the moment acting on the moving cylinder, is given up to 0(0) (see equation 3.18 in Galper & Miloh^°) by
M(«)(Z) = ( V A /«U)„ - {{m\{z)) A U)„ + 0(0), (54)
which, substituted in eqn (32), renders
Mz) = t/iV3 + m,ViU3, f2{z) = /«zC/aV3 + mj^af^s-(55) Combining eqns (55) and (24) with eqn (36), we finally obtain the desired physical force distribution (see also equation (1) of Rainey^^), DV^ F^{z) = {s + m,)^+Eu{m, - W2)(^^2 - U2) + miE23{V, - f / i ) + 0(max(e2loge, Ö')), (56) F2{z) = {s + m2)^ + Euim2 - m,){V, - U,) + «2^33(^2 - U2) + 0(max(e2loge, 6^)). (57)
For time-dependent U{t), the above have to be augmen-ted by the traditional added-mass term
.dU
F^"' ^ -M
dt' (58)
which is the added-mass part of acceleration in the Kirchhoff equation for a cylinder moving in a quiescent fluid.
To generalize the present formulation, we include here also the effect of rotation of the cylinder with an arbitrary angular velocity 0. There exists now a new force component which is proportional to fi. The effective distribution of it is denoted by F'"^, and is given by the following integral (see Galper & Miloh^°)
( « ( 3 y ; 3 ( * A n ) „ - # „ « ^ ( V A O ) ^
+ # „ « , 3 % ( r A O); +£„p«^^^0^)d5. (59) ^ f i z ) :
Note that all terms in eqn (59) should be expressed in the moving (with the body) coordinate system.
The last two terms in the RHS of eqn (59) are of 0(5), whereas the first two are of zero-order. For a cylinder with a
symmetric cross-section moving in a wealily non-uniform flow fleld, the last two terms in the RHS of eqn (59) are proportional to Q and therefore vanish (Galper & Miloh^). Thus, for a rotating cylinder in a weakly non-uniform flow field, eqn (59) renders the following effective force distri-bution
F*"'(Z) = A MV(z) - M ( f i A V(z)) + 0(6^),
or when expressed in a component form
F f \ z ) = V2{z){m^ -m2)n, - m,^2V^{z),
7(n)
Vi{z){mi -m2)n^+m2Ü^V^{z).
(60)
(61) Since the last terms in the RHS of eqn (61) are full z-derivatives terms, they can be expressed after a z-integra-tion as integrals over the ends. For a circular cylinder, eqn (61) reduces to
Ff\z)=-sÜ2V3, Ff\z)=sü,Vs. (62) Proceeding along the same lines as for the force distribu-tion in Secdistribu-tion 4, one should notice that the addidistribu-tional
moment acting on a rotating cylinder with a symmetric
cross-section is of 0(5) and therefore (see eqn (32)) the effective force distribution eqn (61) actually coincides with the physical force distribution up to 0(óe), i.e..
F r + 0(öe). (63)
Equation (60) is also in fufl agreement with Ramey ((1995) equation 1) for the physical force distribution acting on a rotating cylinder with a symmetric cross-section. However, it differs from the corresponding expression for a cylinder whh an arbitrary cross section, in view of the addhional force dis-tribution introduced by the last two terms in eqn (59).
In the above, one should also add the common Coriolis-like force distribution.
FL'^>=(nA/?w)„, (64)
which is included in the Kirchhoff equations (see, for example, Milne-Thompson^^ chapter 18). Thus, the physi-cal force distribution acting on a moving and rotating cylin-der is given by eqns (56), (57), (61) and (64).
As a concluding remark to this section it is worth men-tioning that, as far as the total force on an endless cylinder is concerned, one can use (up to 0(e)) the theory of weakly non-uniform (in all directions) flow fields for the expression of the effective force distribution, in spite of the fact that the cylinder is an infinitely long body (Galper & Miloh^).
7 E N D S E F F E C T
Ends effects arise due to the correction for the leading-order Green function eqn (14) which is required near the ends of
the cyhnder (i.e., in the zone of 0(l5l) below the bases). It is shown below that, contrary to the bases effect, one can neglect the end effect for a thin cylinder. In order to prove this assertion it is first noted that 0o remains finite in the limit of a hydrodynamical line for a 3-D cylinder (in con-trast with the corresponding 2-D case). Thus, one can approximate a needle-like cylinder near its ends by the hm-iting case of a prolate spheroid PQ (for which there is no bases effect) and then use the exact Green function for a needle-like prolate spheroid of length 2H. Both surfaces coincide in the limit as the slenderness ratio approaches zero and one can thus expect that the difference between the two solutions near the ends will tend to zero.
The Green function for a prolate spheroid is given (following Morse & Feshbach^^) as
G(x,y)l(,,y)^„= 2 lA^n^KMx)r'M))cosm{^P{x)
where
^''''^.^^'''^\n.m)\ieo-l)mL)
K'ilJ'),Q",'(^) denote the Legendre polynomials of order
n and type m of the first and second kind, respectively, and
the upper 'dot' denotes differential with respect to the argu-ment. We consider a tripley-orthogonal spheroidal coordi-nate system, (/*,?,!/') which is related to the Cartesian system by
X, =//^i? ;x2 + ix, =H{e - if'^il - fiY'e't (67)
where the two foci of the spheroid are located at
( ± H,0,0) and the needle-like spheroid is given by
^ = l + u,u<l. (68)
In order for the surface the cylmder to coincide with that of the prolate spheroid we chose /i~e.
Direct calculation of the loading per unit length near the needs of such a prolate spheroid (based on eqn (6)) shows that it is proportional to
IT in-m)\ Q:'iU
(65)
(66)
lim , (69)
Simüar calculations for a cylinder with a Green function given by eqn (14) lead to a term of 0(e).
Nevertheless, since the end support is of 0(M), one finds that the additional force representing end effect is one of the order uxu^^^^n^^^ and thus the ends effect can be neglected in the present considerations.
8 POINT LOADS AT THE ENDS
The transverse point loading on the ends F ^ \ which incor-porates the bases effect, results from the actual pressure
distribution near the ends (where the simple leading-order Green function eqn (14) does not hold any more) and should be effectively applied m the zone of order l5l below the bases. This point loading force is given, say for the upper basis z = iï, by
^ < P ^ J U / ^ , S . \ r ^ E , ^ V , é s
\js an Js
a = l,2 ! = 1,2,3. (70)
As shown in Section 5 for a wealdy non-uniform flow field eqn (70) simply reduces to
F ^ U F 3 ( M V ) „ U ^ . (71)
In order to find the corresponding physical loading in the z-direction, we use eqns (18) and (21) which, for / = 3, leads to F d_ dz 2 in V^(r, z)n^(r)G2(r, r)Vy{r,z)«^(f)d5(r)d5(r)) dz = M [ V0ir,z=H)npir)G2{r,r)Vyir,z=H)ny{r)dS{r)dSif). (72) For a weakly (in the transverse plane) non-uniform flow field, there exists an axial point force F ^ j acting on the ends which is given by
1,
(73) By converting the last term m the RHS of eqn (21) into a surface integral and using the Gauss theorem, one obtains
-H
DV-Dt •(r,z)dsdz dt J (74)
from where we find the additional z-pointed end loading
p(p)
pd^T,. (75)
For a wealdy non-uniform flow field eqn (75) gives
FÏl=-^Ph- (76)
Finally, by combining eqns (71)-(76), we rederive Rain-ey's (1995) equation (3) as a limhing case for an mfinitely thin fixed cylinder
F(P' = y , M V + (77)
Note that there is no physical reason for the transverse point loading F ^ ' and this loading should be accounted for only through the additional force distribution (see Section 4).
9 SUMMARY AND CONCLUSIONS
The general methodology of the present analysis is based on using eqns (6) and (7) for computing the hydrodynamical force and moment, respectively, acting on a general fixed 3¬ D body placed in an arbitrary non-uniform stream V(x,f) + V0 of a perfect fluid. The total force can then be v/ritten as an integral of a force distribution including the so-called basis and end effects eqn (18). The ambient flow field is assumed to be weakly non-uniform in the axial direction. The ratio between the effective radius of the cylinder and the length scale of the fiow non-uniformity in the axial direction is the smaU parameter used in the theory of per-turbation constmcted in Section 3. Using physical argu-ments, related to determining the moment exerted on the cylinder, a unique expression for the force distribution (cor-rect to second-order) is found (see eqn (36)). This expres-sion may be further simplified if the flow non-uniformity is assumed to be weak in all three directions (see eqn (37)). Thus, for a stationary cylinder of arbhrary cross-section we obtain eqns (48) and (49), in agreement with Rainey." Nevertheless, it should be noted that the present rigorous analysis is more general than Rainey's since h treats, from the start, a finite cylinder in contrast to using a 1-D hydro-dynamic hne model and arguments of a Munlc-moment type. The expressions for the force distribution are then extended for a moving and rotatmg cylinder (see eqns (56), (57), (61) and (64)) Again, they confirm in principle the corresponding results obtained recently by Rainey^^ for a symmetric cross-section, however, they differ for a non-symmetric cross-section, due to the non-vanishing of third-order tensor Q appearing m eqn (42). Finally, it is remarked that the general framework presented here can be also extended to non-slender ('fat') cylindrical ocean stmctures.
REFERENCES
1. Rainey, R.C.T., A new equation for wave loacis on offsliore structures. J. Fluid Mech., 204 (1989) 295-324.
2. Foullioux, L . & Bernistas, M.M., Forces and moments on a small body moving in a 3-D unsteady flow (with applications to slender structures). / . Offshore Mech. Arctic Engng, 115 (1993) 91-104.
3. Jeffreys, E . R. & Rainey, R. C. T., Slender body models of T L P and GBS ringing. In Proc. 7th IntI Conf on the behaviour of offshore structures, MIT, Cambridge, MA, 1994.
4. Newman, J. N., Nonlinear scattering of long waves by a vertical cylinder. In Proc. Symp. on Waves and nonlinear processes in liydro-dynamics, Oslo, Nor\vay, 1994.
5. Faltinsen, O. M., Ringing loads on gravity based structures. In Proc. 10th Int workshop on water waves and floating bodies, Oxford, U K , 1995.
6. Faltinsen, O.M., Newman, J.N. & Vinje, T , Nonlinear wave loads on a slender vertical cylinder. J. Fluid Mech., 289 (1995) 179-199.
7. Malenica, S. & Molin, B., Third order wave diffraction by a vertical cylinder. J. Fluid Mech., 302 (1995) 203-229.
8. Taylor, G.I., The forces on a body placed in a curved or converging stream of fluid. Proc. Roy. Soc. London A, 120 (1928) 260-283. 9. Galper, A. & Miloh, T., Generalized Kirchhoff equations for a rigid
body moving in a weakly non-uniform flow field. Proc. Roy. Soc. London A, 446 (1994) 169-193.
10. Galper, A. & Miloh, T., Dynamical equations for the motion of a deformable body in an arbitrary potential non-uniform flow field. J. Fluid Mech., 295 (1995) 91-120.
11. Lighthill, M.J., Fundamentals concerning wave loading on offshore structures. J. Fluid Mech., 173 (1986) 667-681.
12. Sarpkaya, T. & Isaacson, M., Mechanics of wave forces on offshore structures. Van Nostrand Reinhold, Scarborough, C A , 1980. 13. Madsen, O.S., Hydrodynamic force on circular cylinders. Appl.
Ocean Res., 8 (1986) 151-155.
14. Manners, W., Hydrodynamic force on a moving circular cylinder submerged in a general fluid flow. Proc. R. Soc. London A, 438 (1992) 331-339.
15. Manners, W. & Rainey, R.C.T., Hydrodynamic forces on a flxed submerged cylinders. Proc. Roy. Soc. London A, 436 (1992) 13. 16. Kim, M.H. & Chen, W., Slender-body approximation for
slowly-varying wave loads in multi-direcdonal waves. Appl Ocean Res., 16 (1994) 141-163.
17. Rainey, R.C.T., Slender-body expression for the wave load on off-shore structures. Proc. Roy. Soc. London A, 450 (1995) 391-416. 18. Mibie-Thomson, L . , Tlieoretical Hydrodynamics, Macmillan,
London, 1968.
19. Morse, P. M. & Feshbach, H., Methods of theoretical physics. Vol 2, McGraw-HiU, Maidenhead, U K , 1953.
