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Hydrodynamic loads on a slender cylinder moving unsteadily in a 3-D non-uniform flow field
Applied Ocean Research 18 (1996) 29-36 Copyright Q 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved PII:
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SOl41-1187(96)00018-1
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Hydrodynamic loads on a slender cylinder moving unsteadily in a 3-D non-uniform flow field A. R. Galper, T. Miloh & M. Spector Faculty of Engineering, Tel-Aviv Universiry
Tel-Aviv, Israel 69978
(Revised 4 January 1996)
In order to evaluate the 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 theorv of uerturbation are found and the influence of the cvlinder’s ends is accounted ’ for. Copy&n 0 1996 Elsevier Science Limited
1 INTRODUCTION
moving body, have been generalized to include the effect of weak flow non-uniformity. In a subsequent study (Galper
In this paper we present a concise analysis of the hydrodynamic loads exerted on an thin slender cylinder moving unsteadily in a time-dependent spatially non-uniform ambient potential 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 tension leg platforms) in extreme waves, studied by Rainey,’ Foulhoux & Bernitsas,2 Jeffreys & Rainey,3 Newman4 Faltinsen,5 Faltinsen et a1.6 and Malencia & Molin.7 A typical value for the radius of a vertical 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-uniform or the long-wave flow assumption. It is valid for a fixed 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 stationary rigid body (leading to the so-called ‘buoyancy-force’), is due to G. I. Taylor.8 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 weakly non-uniform flow field. The classical Kirchhoff-Lagrange equations of motion, which determine the forces and moments acting on the
& Miloh”), the weak assumptions has been released and a general formulation for the same problem, within the framework 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 compared 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,12 Madsen,r3 Manners,14 Manners & Rainey,” Rim & Chen16 and Rainey.17 All these papers lack a theoretical 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 presenting a concise derivation of the sectional hydrodynamic loads (forces and moments) acting on a long vertical cylinder of arbitrary cross-section placed in an arbitrary nonuniform flow field. In addition, the cylinder is allowed to move and to rotate in an arbitrary manner. The corresponding 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 cylinder 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 29
A. R. Galper, T. Miloh, M. Spector
30
cross-section in an arbitrary stream are first presented in Section 3. The 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 distribution 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 I S x [- HJII U T+ U T_ , where T * 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+(x, t). The corresponding 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 & Mi1oh.l’ 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 velocity potential up,induced by the presence of the moving body, can be uniquely decomposed into
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
and
where sv is a 3-D integral over the volume of the cylinder. In the above we have introduced the substantional derivative symbol, & D $ + V*V, denoting liquid acceleration. The harmonic vectors $J and $ which appear in eqns (6) and (7) are the common Kirchhoff potentials satisfying the following boundary conditions gi~/~,
and g=x~
nl,,
(8)
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
(9)
ca=+++o,
(1) where 40 represents the additional disturbance potential satisfying a proper decay condition at infinity, i.e., ,j’m 40(x) = 0.
(2)
Here, x is the position vector in the above mentioned cylindrical coordinate system. It is important to note that +. is harmonic outside L whereas I$ 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) = 4x6(x - y)
,$s”~G(x,Y) -0,
one can express r$oin an integral form as, $0(x) = -
A,@r, i, k) = 4d(r - 6, ) + k2Z?(r,f, k),
(10) where r = (x,y) and f = (IG,j) are the corresponding 2-D position vectors. The boundary conditions for the Fourier transformed Green function Z‘(r,i, k) are
(3)
with the corresponding boundary conditions on both L and infinity
-$3x, ~11~=0,
where ISI is a characteristic length-scale of the crosssection, say its perimeter or the largest distance between two contour points. Next, fmd the Green function G(x,y) to the leading order in the small parameter E. Applying the Fourier transformation in the direction (z-4 to (3) one obtains
G(x, Y)(V-D)(Y)~(Y). (5) sL Here, n denotes a unit normal vector to L directed outward into the fluid.
(11) and ,fi~ G(r, f, k) - 0.
(12)
We are mainly interested here in the value of b(r,i,k) evaluated on the curve S, i.e., in the near field, where, in accordance with the constructed singular theory of perturbation, one can neglect the boundary condition eqn (12). The Green function @r, i, k) in eqn (10) actually depends on klSl and to leading order one can neglect the p-term in
Hydrodynamic loads on a slender cylinder
eqn (lo), which leads to @r, i, k) = Gz(r, f).
(13)
Thus, the leading-order
approximation for the Green function, obtained by inverse Fourier transformation, is given by G(x, y) = 6(z-i)G2(r,
ti) + 0(e210ge),
(14)
where G,(r, f) is the appropriate 2-D Green function for the cross-section S and it can be shown that the next term in the theory of perturbation for the Green function is 0(e210ge). 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 ISI with its centre in the middle of T * (see Section 6). After the substitution of eqn (14) into eqn (5), one can split $0 into 40 = @ + 48’ + r$@) 0 ’
(1%
where +@‘(r,z) = -
ss
G2(r, f)V(ti, z)~~(i)dS(i),
(16)
and &‘(r,z)=jT
f
G(r,i,z,i=
*H)V#,z-
*H)d2T,(i), (17)
where G(r, f, z, i = 2 H) is the exact 3-D Green function of cylinder applied near its ends and we understand ST,( * )d2T, as jr_ ( * )d2T_ + &-+( * )d’T+ . In splitting eqn (15), c#$’ is an asymptotic expression for +o based on the asymptotic eqn (14), 4,” incorporates the bases effect (resulting from integration over the two bases T * ) and ~$2 accounts for the correction needed for the Green function near the ends. Note that $@)is a finite (in 3-D) function and J/@)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
Fi =
I-H ’
fi.(z)dz + F!T) I + i+’I
7
(18)
with FsT’ I given by I and F!E’
F!T’ I az
(19) and
31
where Eii I ViVj is the rate-of-strain tensor of the ambient non-uniform flow. Finally, one obtains Pi(Z)
a ff s
s
V&An,&)G2(r, ~)E,,(~,z)n,(i)dS(r)dS(~)
z)n&)Gk, f)niWWWi) -If %r, s s
+ $$r,z)drfl,y-1,2, f
i-1,2,3.
(21)
Here, and in the sequel, Greek symbols (CY,0, r) take on the values 1,2 and Roman symbols (i&k) the values 1,2,3. In order to estimate the time dependent term in eqns (19), (20) and (22), we use the relation
5-
fs
n%
at
(22)
.
We will show further in Section 4 that FjT’ also can be written in a distributed form, as
F!T’= 1
where the density function g(z) is of O(E).It is demonstrated in Section 6 that, up to O(E),one can neglect the force term F?’ I * 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., V,V(r, z) = - 2 - O(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 FjT’ which incorporates the socalled ‘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 perturbation (see Section 6). The effective force distribution g(z) given in eqn (21) does not actually represent the real physicaZ 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).
A. R. Galper, T. Miloh, M. Spector
32
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 ‘effective’ distributions can be expressed as
Substituting eqn (24) into eqn (27) and using eqns (26), (28) and (29), one obtains
Mf%)- (4 * f), -
A
d(N-z
f),
dz
4x(4 = (4 A f),(z) - 7
(31) 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 acting on the bases as full z-derivatives results from the fact that the function $a (included in the bases integrands), which is well defined on the bases, has singularities within the cylinder. Thus, it is generally impossible to express the bases-integration as integrands over the contours bounding the bases. Because of these difficulties and in order to determine the physical force distribution free 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 Q(z), which is decomposed as in eqn (18) into
JH
f,(z) = (i,
A
M&),(z) + d(iZ A~)“(z).
ti(z)dz + M(r) + M@)7
(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) +
dM”,dd(z) dz
A
i, + 0(c2),
(33)
dMadd(z) where I? is given by eqn (21) and Q! is found from dz eqn (7) as Midd(z) =
H
M-
where i, I fi is the unit vector in the z-direction. Equation (31) can be also written as
&’ a(r inv)*dS + / (r s
A
EV),ds
(25) + O(E),(Y= 1,2.
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”) 9 namely dN,(z) &l,(z)=M,(z) + dz. Using eqns (6), (7), (18) and (25) one can rewrite the effective moment distribution acting on the cylinder in the following form h,(z) = (x * p),(z) + M?(z),
(27)
where fi(z) (consistent with the approximation eqn (14)) is given by eqn (21). Similarly, by virtue of expressions eqns (6) and (7) (w h’rch incorporate integration over the ends), one can show that N,(z) = (x A f(z)), + l,(z),
(28)
&t!!&= fS
at
(29)
which simply follows from p(x
A n),dzdS,
F, = (30)
fS
r*n% at *
(35)
Finally, the required physical force distribution F,(z) for a slender cylinder is obtained as
where I(z) as well as M&(z) are some bounded functions, 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 cylinder by the relation (M),(z) - (x A F(z)),
(34)
It is important to note here that our eqn (33) coincides in the limit of a weakly non-uniform flow field with equation (15) of Rainey” derived for the limiting case of a very thin (hydrodynamical line) cylinder. Thus, the term M?(z) A i, 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 derivative terms contribute only to the z-component of M& and for this reason do not appear explicitly in eqn (34). In deriving eqn (34) we used a similar identity to eqn (22), namely
+ IS
a=1,2i=1,2,3,
rJ&V&)
+ O(e’loge),
(36)
where it can be shown that the next order term in the Eexpansion of the force is of O(&oge). Note that the only restriction imposed on eqn (36) is the weak flow non-uniformity in the z-direction.
Hydrodynamic loads on a slender cylinder
Manners & Rainey l5 for the effective force distribution (without accounting for bases and ends effects)
5 WEAKLY NON-UNIFORM ELOW 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 in the transverse (qy)-plane is much larger compared with the characteristic length scale S of the body’s cross-section (the so called ‘weakly-non-uniform’ field approximation). ln this case, one can introduce an additional small parameter IIv~IIIsI
*
33
f(z) - (si+??r) D$+ (a& -k!z)V+
O(S2),
where 1 is unitary matrix and I@ is the 3-by-3 added-mass tensor fi = diag(rit, 0). One can also write the effective force distribution eqn (44) in a component form as follows ~I(z)=(s+mI)D~+E12(mI
-
m2F2
- W31V3
(37)
=-im-
where I I(.) II 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 integration in eqn (21), one obtains for the first term in eqn (21)
(44)
+ 0(S2),
(45) @2(z)= (s + m2) Dg + E12(m2- mI)VI - m2E32V, +0(S2), (46) where x1, x2 are two unit vectors along which the 2-D added mass tensor fi is diagonalized, i.e., fi -diag(ml, mz). In order to determine the physical force distribution (force per unit length), we use eqn (36) and note that
V,(r,z)ng(r)G2(r,~)E,(i,z) ss JJ r&S~iV,dSBO(62 J ns(r)G*(r,~)n,t~)~(r) ss JJ 4f)a(r&), J an ssna(r)G2(r,i)n,(~)dS(r)d JJ Vo(r = 0, z)E,(i
= 0, z)
=E,(z)m,f3V&),
(38)
where the transverse added-mass tensor per unit length (neglecting end effects) is given by m,fl= -
172
s
which can be derived by placing .@in front of the above integral. Correspondingly, following Galper & Miloh,’ one obtains for the lirst integral in the RHS of eqn (36) -(rfrV),V3+Q48~V,V,)+o(62). B
s
(47)
(39)
After inserting the following Taylor expansion of V about the centroid of the cross-section V&z) - V@, z) +&JO, z)rT + 0(S2),
(40)
in the second integrand in the RHS of eqn (21), one obtains
The last term in eqn (47) can be neglected in eqn (21) being a term of O(e8). Thus, by gathering eqns (45), (46) and (21), we finally obtain the sought expression for physical force distribution acting on a cylinder with an arbitrary cross-action (see equation (1) of Rainey” and the discussion in Rim & Chen16), namely DVt ~t(z)=(s+mr)~+&(mr
-m2)%+@33Vr
ss~r,z)na(r)G2(r,i)n,(i)dS( -JJ =map dt rynrs(r)G2(r,~)n,(i)~(r) JJ +O(max(e’, S2)),
av,
-Es~QM+
(41)
where we introduce the third-order tensor Q=s~
Qc&r= s s
which is equal to zero for a symmetric cross-section (Galper & Miloh’). The second term in the RHS of eqn (41) is of O(e) in comparison with the first term and thus can be neglected. Finally, the last integral in eqn (21) is equal to
JD+r,4~-sD+0,z), s
DV2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
(42)
(48)
0(max(e2, S2)).
(49)
For a circular cylinder ml = mz = s and eqns (48) and (49) reduce to F~(r)-zsD~+~E33V~,
F~(z)‘~F~~+sE~~V~,
(50)
which are the corrected inertia terms in the Morison formula for circular slender structures.
(43)
where s is the area of the cross-section S. Combining eqns (38), (41) and (43) and then adding and subtracting the term m,~EO,V,, we rederive equation (35) of
6 MOVING AND ROTATING CYLINDER For a cylinder moving with a velocity U(t), there arises an additional effective force distribution denoted here by @),
A. R. Galper, T. Miloh, M. Spector
34
due to the interaction between the moving cylinder and the ambient non uniform flow field (see Galper & Miloh’D)
given by $t’ = Uj s (E,@na+i - *JZflEfii)dS. (51) J For a circular cylinder Cp,= -r, and it follows from eqn (51) that
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 - (u) Fr =&(mz -mW2
-m&&F2
-(u)
=EIz(mI -m2)UI -mzEvU3.
(53)
(54)
which, substituted in eqn (32), renders fi(.d=wKV3+mlV1U3,
fiW-m2U2V3+m2V2U3.
(55)
Combining eqns (55) and (24) with eqn (36), we finally obtain the desired physical force distribution (see also equation (1) of Rainey17), Fl(r)-O+ml)~~+Elz(m,-m2)(V2-U2) +mlE,3(VI
- U1)+O(max(e210ge, S2)),
(56)
F2(z)=(s+m2)~~+El2(m,-ml)(V~-U1) + rn2E33W.2- U2) + 0(max(e210ge, S2)).
tic”‘(Z)
= fl A &v(Z)
(57)
For time-dependent U(t), the above have to be augmented by the traditional added-mass term
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 62.There exists now a new force component which is proportional to Q. The effective distribution of it is denoted by i?(*), and is given by the following integral (see Galper 8z Miloh”)
-A@
A v(Z))
+ o(s2),
(60)
or when expressed in a component form E?‘(z) = v2(Z)h
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 O(6) (see equation 3.18 in Galper & Miloh”) by A@(z) = (V A AU), -(@V(z)) A U), + O(6),
Note that all terms in eqn (59) should be expressed in the moving (with the body) coordinate system. The last two terms in the RI-IS of eqn (59) are of O(6), whereas the tirst two are of zero-order. For a cylinder with a symmetric cross-section moving in a weakZy non-uniform flow field, the last two terms in the RIB of eqn (59) are proportional to 0 and therefore vanish (Galper & Milohp. Thus, for a rotating cylinder in a weakly non-uniform flow field, eqn (59) renders the following effective force distribution
Fr3
=
- m2)Q3
- wQ2v3(Z),
V,(z)(m, -m2)n3 +m2QIV3(z).
(61)
Since the last terms in the RHS of eqn (61) are full zderivatives terms, they can be expressed after a z-integration as integrals over the ends. For a circular cylinder, eqn (61) reduces to E?‘(Z)
= - S&V3,
E?‘(Z)
= d-4 V3.
(62)
Proceeding along the same lines as for the force distribution in Section 4, one should notice that the additional moment acting on a rotating cylinder with a Jymmetric cross-section is of O(6) and therefore (see eqn (32)) the effective force distribution eqn (61) actually coincides with the physical force distribution up to O(&), i.e., F(“) 01 =F’(*) Lx + O(&).
(63)
Equation (60) is also in full agreement with Rainey ((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 with an arbitrary cross section, in view of the additional force distribution introduced by the last two terms in eqn (59). In the above, one should also add the common Coriolislike force distribution. FCC) a = (0
A hu)
(YY
(64)
which is included in the Kirchhoff equations (see, for example, Milne-Thompsonr8 chapter 18). Thus, the physical force distribution acting on a moving and rotating cylinder is given by eqns (56), (57), (61) and (64). As a concluding remark to this section it is worth mentioning that, as far as the total force on an endless cylinder is concerned, one can use (up to O(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 & Milohp.
7 ENDS EFFECT
(59)
Ends effects arise due to the correction for the leading-order Green function eqn (14) which is required near the ends of
Hydrodynamic loa& on a slender cylinder
the cylinder (i.e., in the zone of 0(&l) 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 40 remains finite in the limit of a hydrodynamical line for a 3-D cylinder (in contrast with the corresponding 2-D case). Thus, one can approximate a needle-like cylinder near its ends by the limiting case of a prolate spheroid Pa (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, ~)l(~,r)g~= z ~A~m’~(C(X))~(a~))cosm(JI(X) - #cv))Y
35
distribution near the ends (where the simple leading-order Green function eqn (14) does not hold any more) and should be effectively applied in the zone of order ISI below the bases. This point loading force is given, say for the upper basis z = H, by F$! =
a=l,2
i-1,2,3.
(70)
As shown in Section 5 for a weakly non-uniform flow field eqn (70) simply reduces to F$), = V,(fiV),I,_,+
(71)
In order to find the corresponding physical loading in the z-direction, we use eqns (18) and (21) which, for i = 3, leads to
(65)
where
(ss
(66) e(p), E(t) 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 argument. We consider a tripley-orthogonal spheroidal coordinate system, (p,[,$) which is related to the Cartesian system by x1 = Hpt ; x2 + ix3 = H(t2 - l)“‘( 1 - p2)1’2e”,
(67)
where the two foci of the spheroid are located at ( f H,O,O) and the needle-like spheroid is given by ‘$=1+&U<
1.
(68)
In order for the surface the cylinder to coincide with that of the prolate spheroid we chose P--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
s s
1
V,(r,z)n~(r)G2(r, i)V,(f,z)n,(f)dS(r)cLS(i)
s.[
zss Vp(r,z= H)na(r)G2(r,
FA@1’ = $V*i@V)i,.
The transverse point loading on the ends F$“, which incorporates the bases effect, results from the actual pressure
(73)
By converting the last term in the RHS of eqn (21) into a surface integral and using the Gauss theorem, one obtains IT+(iv’+
$)d2T+,
(74)
from where we find the additional z-pointed end loading 7 FA@2’ pd2T+. JT+
(75)
For a weakly non-uniform flow field eqn (75) gives F$.. = - spi.?’
(76)
Finally, by combining eqns (71)-(76) we rederive Rainey’s (1995) equation (3) as a limiting case for an infinitely thin fixed cylinder F@’= v3Av
8 POINT LOADS AT THE ENDS
(72)
For a weakly (in the transverse plane) non-uniform flow field, there exists an axial point force Fjp,) acting on the ends which is given by
$2 = -i,
Similar calculations for a cylinder with a Green function given by eqn (14) lead to a term of O(e). Nevertheless, since the end support is of O(u), one finds that the additional force representing end effect is one of the order u x utf2 = u312 and thus the ends effect can be neglected in the present considerations.
f)V,(i, z = H)n,(i)&(r)&(i).
+ ( +fiv-sp)izizsH.
(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).
A. R. Galper, T. Miloh, M. Spector
36
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 3D body placed in an arbitrary non-uniform stream V(x, t) + VC#J of a perfect fluid. The total force can then be written as an integral of a force distribution including the socalled 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 flow non-uniformity in the axial direction is the small parameter used in the theory of perturbation constructed in Section 3. Using physical arguments, related to determining the moment exerted on the cylinder, a unique expression for the force distribution (correct to second-order) is found (see eqn (36)). This expression 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 arbitrary cross-section we obtain eqns (48) and (49), in agreement with Rainey.17 Nevertheless, it should be noted that the present rigorous analysis is more general than Rainey’s since it treats, from the start, a finite cylinder in contrast to using a 1-D hydrodynamic line model and arguments of a Mm&-moment type. The expressions for the force distribution are then extended for a moving and rotating 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 nonsymmetric cross-section, due to the non-vanishing of thirdorder tensor 0 appearing in eqn (42). Finally, it is remarked that the general framework presented here can be also extended to non-slender (‘fat’) cylindrical ocean structures.
REFERENCES 1. Rainey, R.C.T., A new equation for wave loads on offshore structures. J. Fluid Me&., 204 (1989) 295-324.
2. FouIhoux, L. & Bemistas, M.M., Forccs and moments on a smaII body moving in a 3-D unsteady flow (with applications to slender structures). J. Offshore Me&. Arctic Engng, 115 (1993) 91-104. 3. Jeffreys, E. R. & Ramey, R. C. T., Slender body models of TLP and GBS ringing. In Proc. 7th Intl Conf on #he behaviour of offshore sirucrures, MIT, Cambridge, MA, 1994. 4. Newman, J. N., Nonlinear scattering of long waves by a vertical cylinder. In Proc. Symp. on Waves and nonlinearprocesses in hydrodynamics, Oslo, Norway, 1994. 5. Fakinsen, 0. M., Ringing loads on gravity based structures. In Proc. 10th Int. workshop on water waves and eating bodies, Oxford, UK, 1995. 6. Fakinsen, O.M., Newman, J.N. & Vinje, T., Nonlinear wave loads on a slender vertical cylinder. J. F&d Mech., 289 (1995) 179-199. Male&a, S. & Molin, B., Third order wave diffraction by a vertical cylinder. J. Fluid Mech., 302 (1995) 203-229. Taylor, G.I., The forces on a body placed in a curved or converging stream of fluid. Proc. Roy. Sot. London A, 120 (1928) 260-283. GaIper, A. & Miloh, T., Generalized Kirchhoff equations for a rigid body moving in a weakly non-uniform flow field. Proc. Roy. Sot. London A, 446 (1994) 169-193. 10. Galper, A. & MiIoh, T., Dynamical equations for the motion of a deformable body in an arbitrary potential non-uniform flow field. J. Fluid Me&., 295 (1995) 91-120. 11. LighthiII, 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, CA, 1980. 13. Madsen, OS., 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. Sot. London A, 438 (1992) 331-339. 15. Manners, W. & Rainey, R.C.T., Hydrodynamic forces on a fixed submerged cylinders. Proc. Roy. Sot. London A, 436 (1992) 13. 16. Kim, M.H. & Chen, W., Slender-body approximation for slowlyvarying wave loads in multi-directional waves. Appl. Ocean Res., 16 (1994) 141-163. 17. Rainey, R.C.T., Slender-body expression for the wave load on offshore structures. Proc. Roy. Sot. London A, 450 (1995) 391-416. 18. Milne-Thomson, L., Theoretical Hydrodynamics, Macmillan, London, 1968. 19. Morse, P. M. & Feshbach, H., Merhodr of theoretical physics, Vo12, McGraw-Hill, Maidenhead, UK, 1953.
ELSEVIER
SOl41-1187(96)00018-1
0141-1187/96/$15.00
Hydrodynamic loads on a slender cylinder moving unsteadily in a 3-D non-uniform flow field A. R. Galper, T. Miloh & M. Spector Faculty of Engineering, Tel-Aviv Universiry
Tel-Aviv, Israel 69978
(Revised 4 January 1996)
In order to evaluate the 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 theorv of uerturbation are found and the influence of the cvlinder’s ends is accounted ’ for. Copy&n 0 1996 Elsevier Science Limited
1 INTRODUCTION
moving body, have been generalized to include the effect of weak flow non-uniformity. In a subsequent study (Galper
In this paper we present a concise analysis of the hydrodynamic loads exerted on an thin slender cylinder moving unsteadily in a time-dependent spatially non-uniform ambient potential 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 tension leg platforms) in extreme waves, studied by Rainey,’ Foulhoux & Bernitsas,2 Jeffreys & Rainey,3 Newman4 Faltinsen,5 Faltinsen et a1.6 and Malencia & Molin.7 A typical value for the radius of a vertical 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-uniform or the long-wave flow assumption. It is valid for a fixed 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 stationary rigid body (leading to the so-called ‘buoyancy-force’), is due to G. I. Taylor.8 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 weakly non-uniform flow field. The classical Kirchhoff-Lagrange equations of motion, which determine the forces and moments acting on the
& Miloh”), the weak assumptions has been released and a general formulation for the same problem, within the framework 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 compared 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,12 Madsen,r3 Manners,14 Manners & Rainey,” Rim & Chen16 and Rainey.17 All these papers lack a theoretical 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 presenting a concise derivation of the sectional hydrodynamic loads (forces and moments) acting on a long vertical cylinder of arbitrary cross-section placed in an arbitrary nonuniform flow field. In addition, the cylinder is allowed to move and to rotate in an arbitrary manner. The corresponding 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 cylinder 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 29
A. R. Galper, T. Miloh, M. Spector
30
cross-section in an arbitrary stream are first presented in Section 3. The 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 distribution 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 I S x [- HJII U T+ U T_ , where T * 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+(x, t). The corresponding 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 & Mi1oh.l’ 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 velocity potential up,induced by the presence of the moving body, can be uniquely decomposed into
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
and
where sv is a 3-D integral over the volume of the cylinder. In the above we have introduced the substantional derivative symbol, & D $ + V*V, denoting liquid acceleration. The harmonic vectors $J and $ which appear in eqns (6) and (7) are the common Kirchhoff potentials satisfying the following boundary conditions gi~/~,
and g=x~
nl,,
(8)
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
(9)
ca=+++o,
(1) where 40 represents the additional disturbance potential satisfying a proper decay condition at infinity, i.e., ,j’m 40(x) = 0.
(2)
Here, x is the position vector in the above mentioned cylindrical coordinate system. It is important to note that +. is harmonic outside L whereas I$ 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) = 4x6(x - y)
,$s”~G(x,Y) -0,
one can express r$oin an integral form as, $0(x) = -
A,@r, i, k) = 4d(r - 6, ) + k2Z?(r,f, k),
(10) where r = (x,y) and f = (IG,j) are the corresponding 2-D position vectors. The boundary conditions for the Fourier transformed Green function Z‘(r,i, k) are
(3)
with the corresponding boundary conditions on both L and infinity
-$3x, ~11~=0,
where ISI is a characteristic length-scale of the crosssection, say its perimeter or the largest distance between two contour points. Next, fmd the Green function G(x,y) to the leading order in the small parameter E. Applying the Fourier transformation in the direction (z-4 to (3) one obtains
G(x, Y)(V-D)(Y)~(Y). (5) sL Here, n denotes a unit normal vector to L directed outward into the fluid.
(11) and ,fi~ G(r, f, k) - 0.
(12)
We are mainly interested here in the value of b(r,i,k) evaluated on the curve S, i.e., in the near field, where, in accordance with the constructed singular theory of perturbation, one can neglect the boundary condition eqn (12). The Green function @r, i, k) in eqn (10) actually depends on klSl and to leading order one can neglect the p-term in
Hydrodynamic loads on a slender cylinder
eqn (lo), which leads to @r, i, k) = Gz(r, f).
(13)
Thus, the leading-order
approximation for the Green function, obtained by inverse Fourier transformation, is given by G(x, y) = 6(z-i)G2(r,
ti) + 0(e210ge),
(14)
where G,(r, f) is the appropriate 2-D Green function for the cross-section S and it can be shown that the next term in the theory of perturbation for the Green function is 0(e210ge). 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 ISI with its centre in the middle of T * (see Section 6). After the substitution of eqn (14) into eqn (5), one can split $0 into 40 = @ + 48’ + r$@) 0 ’
(1%
where +@‘(r,z) = -
ss
G2(r, f)V(ti, z)~~(i)dS(i),
(16)
and &‘(r,z)=jT
f
G(r,i,z,i=
*H)V#,z-
*H)d2T,(i), (17)
where G(r, f, z, i = 2 H) is the exact 3-D Green function of cylinder applied near its ends and we understand ST,( * )d2T, as jr_ ( * )d2T_ + &-+( * )d’T+ . In splitting eqn (15), c#$’ is an asymptotic expression for +o based on the asymptotic eqn (14), 4,” incorporates the bases effect (resulting from integration over the two bases T * ) and ~$2 accounts for the correction needed for the Green function near the ends. Note that $@)is a finite (in 3-D) function and J/@)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
Fi =
I-H ’
fi.(z)dz + F!T) I + i+’I
7
(18)
with FsT’ I given by I and F!E’
F!T’ I az
(19) and
31
where Eii I ViVj is the rate-of-strain tensor of the ambient non-uniform flow. Finally, one obtains Pi(Z)
a ff s
s
V&An,&)G2(r, ~)E,,(~,z)n,(i)dS(r)dS(~)
z)n&)Gk, f)niWWWi) -If %r, s s
+ $$r,z)drfl,y-1,2, f
i-1,2,3.
(21)
Here, and in the sequel, Greek symbols (CY,0, r) take on the values 1,2 and Roman symbols (i&k) the values 1,2,3. In order to estimate the time dependent term in eqns (19), (20) and (22), we use the relation
5-
fs
n%
at
(22)
.
We will show further in Section 4 that FjT’ also can be written in a distributed form, as
F!T’= 1
where the density function g(z) is of O(E).It is demonstrated in Section 6 that, up to O(E),one can neglect the force term F?’ I * 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., V,V(r, z) = - 2 - O(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 FjT’ which incorporates the socalled ‘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 perturbation (see Section 6). The effective force distribution g(z) given in eqn (21) does not actually represent the real physicaZ 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).
A. R. Galper, T. Miloh, M. Spector
32
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 ‘effective’ distributions can be expressed as
Substituting eqn (24) into eqn (27) and using eqns (26), (28) and (29), one obtains
Mf%)- (4 * f), -
A
d(N-z
f),
dz
4x(4 = (4 A f),(z) - 7
(31) 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 acting on the bases as full z-derivatives results from the fact that the function $a (included in the bases integrands), which is well defined on the bases, has singularities within the cylinder. Thus, it is generally impossible to express the bases-integration as integrands over the contours bounding the bases. Because of these difficulties and in order to determine the physical force distribution free 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 Q(z), which is decomposed as in eqn (18) into
JH
f,(z) = (i,
A
M&),(z) + d(iZ A~)“(z).
ti(z)dz + M(r) + M@)7
(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) +
dM”,dd(z) dz
A
i, + 0(c2),
(33)
dMadd(z) where I? is given by eqn (21) and Q! is found from dz eqn (7) as Midd(z) =
H
M-
where i, I fi is the unit vector in the z-direction. Equation (31) can be also written as
&’ a(r inv)*dS + / (r s
A
EV),ds
(25) + O(E),(Y= 1,2.
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”) 9 namely dN,(z) &l,(z)=M,(z) + dz. Using eqns (6), (7), (18) and (25) one can rewrite the effective moment distribution acting on the cylinder in the following form h,(z) = (x * p),(z) + M?(z),
(27)
where fi(z) (consistent with the approximation eqn (14)) is given by eqn (21). Similarly, by virtue of expressions eqns (6) and (7) (w h’rch incorporate integration over the ends), one can show that N,(z) = (x A f(z)), + l,(z),
(28)
&t!!&= fS
at
(29)
which simply follows from p(x
A n),dzdS,
F, = (30)
fS
r*n% at *
(35)
Finally, the required physical force distribution F,(z) for a slender cylinder is obtained as
where I(z) as well as M&(z) are some bounded functions, 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 cylinder by the relation (M),(z) - (x A F(z)),
(34)
It is important to note here that our eqn (33) coincides in the limit of a weakly non-uniform flow field with equation (15) of Rainey” derived for the limiting case of a very thin (hydrodynamical line) cylinder. Thus, the term M?(z) A i, 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 derivative terms contribute only to the z-component of M& and for this reason do not appear explicitly in eqn (34). In deriving eqn (34) we used a similar identity to eqn (22), namely
+ IS
a=1,2i=1,2,3,
rJ&V&)
+ O(e’loge),
(36)
where it can be shown that the next order term in the Eexpansion of the force is of O(&oge). Note that the only restriction imposed on eqn (36) is the weak flow non-uniformity in the z-direction.
Hydrodynamic loads on a slender cylinder
Manners & Rainey l5 for the effective force distribution (without accounting for bases and ends effects)
5 WEAKLY NON-UNIFORM ELOW 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 in the transverse (qy)-plane is much larger compared with the characteristic length scale S of the body’s cross-section (the so called ‘weakly-non-uniform’ field approximation). ln this case, one can introduce an additional small parameter IIv~IIIsI
*
33
f(z) - (si+??r) D$+ (a& -k!z)V+
O(S2),
where 1 is unitary matrix and I@ is the 3-by-3 added-mass tensor fi = diag(rit, 0). One can also write the effective force distribution eqn (44) in a component form as follows ~I(z)=(s+mI)D~+E12(mI
-
m2F2
- W31V3
(37)
=-im-
where I I(.) II 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 integration in eqn (21), one obtains for the first term in eqn (21)
(44)
+ 0(S2),
(45) @2(z)= (s + m2) Dg + E12(m2- mI)VI - m2E32V, +0(S2), (46) where x1, x2 are two unit vectors along which the 2-D added mass tensor fi is diagonalized, i.e., fi -diag(ml, mz). In order to determine the physical force distribution (force per unit length), we use eqn (36) and note that
V,(r,z)ng(r)G2(r,~)E,(i,z) ss JJ r&S~iV,dSBO(62 J ns(r)G*(r,~)n,t~)~(r) ss JJ 4f)a(r&), J an ssna(r)G2(r,i)n,(~)dS(r)d JJ Vo(r = 0, z)E,(i
= 0, z)
=E,(z)m,f3V&),
(38)
where the transverse added-mass tensor per unit length (neglecting end effects) is given by m,fl= -
172
s
which can be derived by placing .@in front of the above integral. Correspondingly, following Galper & Miloh,’ one obtains for the lirst integral in the RHS of eqn (36) -(rfrV),V3+Q48~V,V,)+o(62). B
s
(47)
(39)
After inserting the following Taylor expansion of V about the centroid of the cross-section V&z) - V@, z) +&JO, z)rT + 0(S2),
(40)
in the second integrand in the RHS of eqn (21), one obtains
The last term in eqn (47) can be neglected in eqn (21) being a term of O(e8). Thus, by gathering eqns (45), (46) and (21), we finally obtain the sought expression for physical force distribution acting on a cylinder with an arbitrary cross-action (see equation (1) of Rainey” and the discussion in Rim & Chen16), namely DVt ~t(z)=(s+mr)~+&(mr
-m2)%+@33Vr
ss~r,z)na(r)G2(r,i)n,(i)dS( -JJ =map dt rynrs(r)G2(r,~)n,(i)~(r) JJ +O(max(e’, S2)),
av,
-Es~QM+
(41)
where we introduce the third-order tensor Q=s~
Qc&r= s s
which is equal to zero for a symmetric cross-section (Galper & Miloh’). The second term in the RHS of eqn (41) is of O(e) in comparison with the first term and thus can be neglected. Finally, the last integral in eqn (21) is equal to
JD+r,4~-sD+0,z), s
DV2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
(42)
(48)
0(max(e2, S2)).
(49)
For a circular cylinder ml = mz = s and eqns (48) and (49) reduce to F~(r)-zsD~+~E33V~,
F~(z)‘~F~~+sE~~V~,
(50)
which are the corrected inertia terms in the Morison formula for circular slender structures.
(43)
where s is the area of the cross-section S. Combining eqns (38), (41) and (43) and then adding and subtracting the term m,~EO,V,, we rederive equation (35) of
6 MOVING AND ROTATING CYLINDER For a cylinder moving with a velocity U(t), there arises an additional effective force distribution denoted here by @),
A. R. Galper, T. Miloh, M. Spector
34
due to the interaction between the moving cylinder and the ambient non uniform flow field (see Galper & Miloh’D)
given by $t’ = Uj s (E,@na+i - *JZflEfii)dS. (51) J For a circular cylinder Cp,= -r, and it follows from eqn (51) that
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 - (u) Fr =&(mz -mW2
-m&&F2
-(u)
=EIz(mI -m2)UI -mzEvU3.
(53)
(54)
which, substituted in eqn (32), renders fi(.d=wKV3+mlV1U3,
fiW-m2U2V3+m2V2U3.
(55)
Combining eqns (55) and (24) with eqn (36), we finally obtain the desired physical force distribution (see also equation (1) of Rainey17), Fl(r)-O+ml)~~+Elz(m,-m2)(V2-U2) +mlE,3(VI
- U1)+O(max(e210ge, S2)),
(56)
F2(z)=(s+m2)~~+El2(m,-ml)(V~-U1) + rn2E33W.2- U2) + 0(max(e210ge, S2)).
tic”‘(Z)
= fl A &v(Z)
(57)
For time-dependent U(t), the above have to be augmented by the traditional added-mass term
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 62.There exists now a new force component which is proportional to Q. The effective distribution of it is denoted by i?(*), and is given by the following integral (see Galper 8z Miloh”)
-A@
A v(Z))
+ o(s2),
(60)
or when expressed in a component form E?‘(z) = v2(Z)h
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 O(6) (see equation 3.18 in Galper & Miloh”) by A@(z) = (V A AU), -(@V(z)) A U), + O(6),
Note that all terms in eqn (59) should be expressed in the moving (with the body) coordinate system. The last two terms in the RI-IS of eqn (59) are of O(6), whereas the tirst two are of zero-order. For a cylinder with a symmetric cross-section moving in a weakZy non-uniform flow field, the last two terms in the RIB of eqn (59) are proportional to 0 and therefore vanish (Galper & Milohp. Thus, for a rotating cylinder in a weakly non-uniform flow field, eqn (59) renders the following effective force distribution
Fr3
=
- m2)Q3
- wQ2v3(Z),
V,(z)(m, -m2)n3 +m2QIV3(z).
(61)
Since the last terms in the RHS of eqn (61) are full zderivatives terms, they can be expressed after a z-integration as integrals over the ends. For a circular cylinder, eqn (61) reduces to E?‘(Z)
= - S&V3,
E?‘(Z)
= d-4 V3.
(62)
Proceeding along the same lines as for the force distribution in Section 4, one should notice that the additional moment acting on a rotating cylinder with a Jymmetric cross-section is of O(6) and therefore (see eqn (32)) the effective force distribution eqn (61) actually coincides with the physical force distribution up to O(&), i.e., F(“) 01 =F’(*) Lx + O(&).
(63)
Equation (60) is also in full agreement with Rainey ((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 with an arbitrary cross section, in view of the additional force distribution introduced by the last two terms in eqn (59). In the above, one should also add the common Coriolislike force distribution. FCC) a = (0
A hu)
(YY
(64)
which is included in the Kirchhoff equations (see, for example, Milne-Thompsonr8 chapter 18). Thus, the physical force distribution acting on a moving and rotating cylinder is given by eqns (56), (57), (61) and (64). As a concluding remark to this section it is worth mentioning that, as far as the total force on an endless cylinder is concerned, one can use (up to O(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 & Milohp.
7 ENDS EFFECT
(59)
Ends effects arise due to the correction for the leading-order Green function eqn (14) which is required near the ends of
Hydrodynamic loa& on a slender cylinder
the cylinder (i.e., in the zone of 0(&l) 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 40 remains finite in the limit of a hydrodynamical line for a 3-D cylinder (in contrast with the corresponding 2-D case). Thus, one can approximate a needle-like cylinder near its ends by the limiting case of a prolate spheroid Pa (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, ~)l(~,r)g~= z ~A~m’~(C(X))~(a~))cosm(JI(X) - #cv))Y
35
distribution near the ends (where the simple leading-order Green function eqn (14) does not hold any more) and should be effectively applied in the zone of order ISI below the bases. This point loading force is given, say for the upper basis z = H, by F$! =
a=l,2
i-1,2,3.
(70)
As shown in Section 5 for a weakly non-uniform flow field eqn (70) simply reduces to F$), = V,(fiV),I,_,+
(71)
In order to find the corresponding physical loading in the z-direction, we use eqns (18) and (21) which, for i = 3, leads to
(65)
where
(ss
(66) e(p), E(t) denote the Legendre polynomials of order n and type m of the first and second kind, respectively, and the upper ‘dot’ denotes differential with respect to the argument. We consider a tripley-orthogonal spheroidal coordinate system, (p,[,$) which is related to the Cartesian system by x1 = Hpt ; x2 + ix3 = H(t2 - l)“‘( 1 - p2)1’2e”,
(67)
where the two foci of the spheroid are located at ( f H,O,O) and the needle-like spheroid is given by ‘$=1+&U<
1.
(68)
In order for the surface the cylinder to coincide with that of the prolate spheroid we chose P--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
s s
1
V,(r,z)n~(r)G2(r, i)V,(f,z)n,(f)dS(r)cLS(i)
s.[
zss Vp(r,z= H)na(r)G2(r,
FA@1’ = $V*i@V)i,.
The transverse point loading on the ends F$“, which incorporates the bases effect, results from the actual pressure
(73)
By converting the last term in the RHS of eqn (21) into a surface integral and using the Gauss theorem, one obtains IT+(iv’+
$)d2T+,
(74)
from where we find the additional z-pointed end loading 7 FA@2’ pd2T+. JT+
(75)
For a weakly non-uniform flow field eqn (75) gives F$.. = - spi.?’
(76)
Finally, by combining eqns (71)-(76) we rederive Rainey’s (1995) equation (3) as a limiting case for an infinitely thin fixed cylinder F@’= v3Av
8 POINT LOADS AT THE ENDS
(72)
For a weakly (in the transverse plane) non-uniform flow field, there exists an axial point force Fjp,) acting on the ends which is given by
$2 = -i,
Similar calculations for a cylinder with a Green function given by eqn (14) lead to a term of O(e). Nevertheless, since the end support is of O(u), one finds that the additional force representing end effect is one of the order u x utf2 = u312 and thus the ends effect can be neglected in the present considerations.
f)V,(i, z = H)n,(i)&(r)&(i).
+ ( +fiv-sp)izizsH.
(77)
Note that there is no physical reason for the transverse point loading F$‘) and this loading should be accounted for only through the additional force distribution (see Section 4).
A. R. Galper, T. Miloh, M. Spector
36
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 3D body placed in an arbitrary non-uniform stream V(x, t) + VC#J of a perfect fluid. The total force can then be written as an integral of a force distribution including the socalled 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 flow non-uniformity in the axial direction is the small parameter used in the theory of perturbation constructed in Section 3. Using physical arguments, related to determining the moment exerted on the cylinder, a unique expression for the force distribution (correct to second-order) is found (see eqn (36)). This expression 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 arbitrary cross-section we obtain eqns (48) and (49), in agreement with Rainey.17 Nevertheless, it should be noted that the present rigorous analysis is more general than Rainey’s since it treats, from the start, a finite cylinder in contrast to using a 1-D hydrodynamic line model and arguments of a Mm&-moment type. The expressions for the force distribution are then extended for a moving and rotating 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 nonsymmetric cross-section, due to the non-vanishing of thirdorder tensor 0 appearing in eqn (42). Finally, it is remarked that the general framework presented here can be also extended to non-slender (‘fat’) cylindrical ocean structures.
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2. FouIhoux, L. & Bemistas, M.M., Forccs and moments on a smaII body moving in a 3-D unsteady flow (with applications to slender structures). J. Offshore Me&. Arctic Engng, 115 (1993) 91-104. 3. Jeffreys, E. R. & Ramey, R. C. T., Slender body models of TLP and GBS ringing. In Proc. 7th Intl Conf on #he behaviour of offshore sirucrures, MIT, Cambridge, MA, 1994. 4. Newman, J. N., Nonlinear scattering of long waves by a vertical cylinder. In Proc. Symp. on Waves and nonlinearprocesses in hydrodynamics, Oslo, Norway, 1994. 5. Fakinsen, 0. M., Ringing loads on gravity based structures. In Proc. 10th Int. workshop on water waves and eating bodies, Oxford, UK, 1995. 6. Fakinsen, O.M., Newman, J.N. & Vinje, T., Nonlinear wave loads on a slender vertical cylinder. J. F&d Mech., 289 (1995) 179-199. Male&a, S. & Molin, B., Third order wave diffraction by a vertical cylinder. J. Fluid Mech., 302 (1995) 203-229. Taylor, G.I., The forces on a body placed in a curved or converging stream of fluid. Proc. Roy. Sot. London A, 120 (1928) 260-283. GaIper, A. & Miloh, T., Generalized Kirchhoff equations for a rigid body moving in a weakly non-uniform flow field. Proc. Roy. Sot. London A, 446 (1994) 169-193. 10. Galper, A. & MiIoh, T., Dynamical equations for the motion of a deformable body in an arbitrary potential non-uniform flow field. J. Fluid Me&., 295 (1995) 91-120. 11. LighthiII, 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, CA, 1980. 13. Madsen, OS., 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. Sot. London A, 438 (1992) 331-339. 15. Manners, W. & Rainey, R.C.T., Hydrodynamic forces on a fixed submerged cylinders. Proc. Roy. Sot. London A, 436 (1992) 13. 16. Kim, M.H. & Chen, W., Slender-body approximation for slowlyvarying wave loads in multi-directional waves. Appl. Ocean Res., 16 (1994) 141-163. 17. Rainey, R.C.T., Slender-body expression for the wave load on offshore structures. Proc. Roy. Sot. London A, 450 (1995) 391-416. 18. Milne-Thomson, L., Theoretical Hydrodynamics, Macmillan, London, 1968. 19. Morse, P. M. & Feshbach, H., Merhodr of theoretical physics, Vo12, McGraw-Hill, Maidenhead, UK, 1953.