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Interaction between Current, Waves and Marine Structures R. Zhao and 0. M. Faltinsen Norwegian Institute of Technology Trondheim, Norway Abstract A theoretical method to analyse current wave interaction on large-volume marine structures is presented. It is shown how to circumvent the problems associated with the mj-terms. Numerical results for mean wave forces on floating vertical cylinders are discussed. The results are based on using both direct pressure integration and conservation of momentum. Introduction Zhao & Faltinsen [9] have presented a two- dimensional theory that hydrodynamically ana- lyses the combined effect of waves and current on two-dimensional floating struc- tures. Zhao et al. [10] generalized the theory to three-dimensional flow. A hemisphere was analysed and satisfactory agreement between numerical and experimental prediction of linear and mean wave forces was documented. The theory is based on matching a local solu- tion to a far-field solution. In the far- field the waves "ride" on the undisturbed current velocity, while in the near-field the waves "ride" on the local steady flow. The theoretical solution for the velocity poten- tial is expressed as a series expansion in the wave amplitude (a and the current velo- city U. The problem is solved to first order in (a and first order in U. It is assumed that the wave slopes of different wave systems and the Froude number are asymp- totically small. In the free surface con- dition and the body boundary condition the interactions with the steady motion potential are taken care of. In addition a radiation condition is specified. In the numerical solution a boundary element method based on Green's second identity is incorporated. The far-field solution is represented by a sum of multipoles (including sources) with singu- larities inside the body. The multipoles satisfy the radiation condition and the far- 513 field free surface condition. For a general body several singularity points are used for the multipoles inside the body. The coef- ficients in the multipole expansion are determined by matching the local- and far- field solution. In the main text we will outline the theory in more detail. We will focus our attention on the m -terms in the body boundary con- dition. The mj-terms are due to interaction between the current and the oscillatory fluid motion. The same terms occur when ship motions at forward speed are analysed (Newman [8]. Numerical inaccuracies and unphysical effects of these terms can cause large errors in the numerical prediction, in particular for a body surface with sharp corners or local high curvature. It is shown numerically and analytically how problems with the mj- terms can be avoided. Numerical results for horizontal and vertical mean wave forces on floating vertical cylin- ders of finite length are presented. Both a direct pressure integration method and a method based on conservation of momentum are used. Convergence of the results are docu- mented as a function of number of panels approximating the body surface, the free sur- face and the control surface. It is demonstrated that special care has to be shown in modelling the cylinder surface around sharp corners. This is particularly true when the direct pressure integration method is used. Theory Consider a structure in uniform current and regular incident waves with small amplitude in deep water. The structure is restrained from drifting, but is free to oscillate har- monically in six degrees of freedom. We choose a right-handed coordinate system (x,y,z) fixed in space. z = 0 is in the mean free surface, positive z-axis is upwards and goes through the centre of gravity of the structure when the body is at rest (see Fig. 1). Surge (~1), sway (~2) and heave (~3) are

~3 - _= with respect to (a, ¢2 is proportional to (a. Far away from the body '~2 As - U(x case + y sing) ~1 Fig. 1 Coordinate system and sign convention for translatory and angular displacements. the translatory displacements of the body in the x-, y- and z-direction referred to the origin of the (x,y,z) coordinate system, when the body is at rest. Roll (~4), pitch (~5) and yaw (~6) denote the angular displacements about the x-,y- and z-axis, respectively. Consider the fluid to be incompressible and the fluid motion irrotational so that there exists a velocity potential ~ which satisfies Laplace equation. v2~= 0 (1) In reality the flow is always rotational in a boundary layer close to the body. In addition the flow may separate from the body and inva- lidate a potential flow description also in parts of the flow outside the boundary layer. This depends for instance on the shape of body, the Reynolds number, the Keulegan- Carpenter number (KC), a non-dimensionalized frequency of oscillation, the roughness ratio and the ratio between the current velocity U and the maximum horizontal oscillatory ambient fluid velocity UM in the current direction. Obviously the flow will always separate from a body with sharp corners in any type of ambient flow. This is also true for the flow around any blunt-sharped marine structures in current without waves. However, when the free stream velocity along the current changes direction with time, i.e. U/UM < 1, the flow around bodies with curved surfaces will not separate for small KC- numbers (Zhao et al. [10]). The potential flow solution will be written as a series expansion in the wave amplitude (a of incident waves. It is assumed that (a, the body motion and the steepness of the dif- ferent wave systems are asymptotically small. We write ~ = As + ¢1 + ¢2 (2) where As is independent of (a, ¢1 is linear Here a is the angle between the current direction and the x-axis. We assume that the current velocity U is small and solve the problem correctly to O(U). A consequence of neglecting terms of o(U2) is that we disre- gard the effects of the steady wave system generated by the current flow past the body. In practice this is expected to be a good approximation. On the free surface, As satisfies the rigid free surface condition, '.e. a¢S - = 0 on z = 0 (4) On the mean position of the body surface SB, As satisfies a zero-normal velocity con- dition, i.e. a¢S an = 0 on SB (5) In general a numerical method has to be used to find As In our case we used Hess & Smith's method [2]. This is based on distri- buting sources over SB. Due to linearity we can decompose the first- order velocity potential ¢1 into separate components due to the rigid-body motions ski the incident waves and a diffraction poten- tial theist. We can write ¢1 = woe t + Ageist + ~ ok (6) k=1 The incident wave potential can be written as At tOe = 9<a "i(~t - kx cost - ky sing) + kz ~0 (7) Here i is the imaginary unit, t the time variable, ~ the angle between the wave propa- gation direction and the x-axis, ~ the cir- cular frequency of oscillation and k the wave number. ~ and k are connected through the relations = ~0 + kU cos(~-a) (8) k = ~0/9 (9) It is understood that the real part is to be taken in expressions involving east. 514

By using the dynamic and kinematic free sur- face conditions, it follows that Ski k = 1,6 satisfy correctly to O(U) a2¢S a2¢S k sieves V.k ~ id ~ 2 ~ - ] atk + gaz = 0, on z = 0 all) Also (~0~67)e;~t satisfies equation (10). Far away from the body equation (10) becomes 2 ask ask ok ~ 2ieU (cos a a + sin a ay ) O. on z = 0 (11) Equation (10) expresses the fact that the waves interact with the local steady flow As around the body. Equation ( 11 ) resembles the classical free-surface condition with forward speed, which for a = 0 can be written as 2 ark 2 amok ax ask gaz = 0 , on z = 0 (12) The difference between equation (11) and (12) is that terms of o(U2) are neglected. Zhao & Faltinsen [9] found for two-dimensional flow that it is appropriate to use equation (11) when T = WU/9 ~ ~ O. 15. In the three-dimen- sional flow case we expect a similar limita- tion. An example on calculations of the real part of the velocity potential due to a har- monically oscillating source satisfying either free surface condition (11) or (12) is shown in Fig. 2. The curves in the upper half plane correspond to calculations with equation (11). The T-value is 0.1, s ~/9 = 0.6773 where zs is the z- coordinate of the source point. The calcula- tions in Fig. 2 are for points on the mean free surface. We note a small phase dif- ference between the wave systems when con- dition (11) or (12) is used. This is more evident on the upstream side of the source. The amplitudes are in good agreement. The consequence of neglecting terms of o(U2) in the free surface condition is that fewer wave systems occur far away from the body. However, the consequence of this is of no practical importance for small T-values below 0.15. A further simplification of the source potential G is sometimes used. One writes G = G. ~ T aG | + U Fig. 2 Calculated values of the real part of the Green's function on the free surface. In the upper half part of figure equation (11) is used. In the lower half part the classical free surface con- dition (12) is used (T ' 0.1, me; ~ 0.6773) ( The ca l cu l a t i ons i n the l over ha l f p l ane has been prov i deaf by J. Hoff). This simplification can lead to large errors at some distance from the source point. This is illustrated in Fig. 3. The conditions are the same as in Fig. 2. The curves in the upper half plane correspond to that equation (13) has been used, while the curves in the lower half plane corresponds to that the free surface conditions (11) has been used. Fig. 3 Calculated values of the real part of the Green's function on the free surface. In ache upper half part of the figure equation (13) has been used. In the lower half part free surface condition (11} has been use ~ T ~ 0.1, ~j - O.6773 ~ . 515

The body boundary conditions can be written 4~.k(Xo'Yo,Zo) as ark - an lank + ok ' k = 1,6 ~ an , k - (14) Equation (14) applies on the mean wetted body surface Sg. The nk- and ink-components are defined by - n = (n1'n2,n3) ran = (n4,n5,n6) (15) _ ~ m = - n ~ VW = (m1,m2'm3) ~ _ ~ -n ~V(rxW) = (m4,m5,m6) ~ ~ ~ ~ _ where W = V¢s and r = xi + yj + zk. Positive normal direction is into the fluid domain. The ink-terms in equation (14) arise because the steady motion potential does not satisfy the body boundary condition on the instan- taneous body surface correct to °((a) The derivation is based on a Taylor expansion, which means the formulation is breaks down at sharp corners. This will lead to dif- ficulties which will be further discussed later in the text. It is also necessary to specify a radiation condition. When the free surface condition (11) is used, it means that the waves are propagating away from the body. A solution to the boundary value problems for ok can be found by applying Green's second identity to the functions ok and 1/R, in a fluid domain enclosed by the boundary S defined by Sg U SF U Sc U SO (see Fig. 4). Then we obtain the following expression ~ DOMAIN ' HI SC 'HI! DOMAIN I 550 Fig. 4 Oefinit10n of surfaces used in the integration of equation (16). 516 1l (ok an R ~ ank R) ds (x,y,z) (16) where R = ~ x-xO) + (y-yO) ~ (z-zO) , Sc is a vertical cylindrical control surface, SF is the mean free surface between Sc and Sg, SO is a bottom surface inside Sc. We separate the total fluid domain into two parts. Part I is the fluid domain inside the boundary S while part II is outside S. In the outer domain (far-field) the free surface condition (11) is assumed valid, while in the inner domain (near-field) the complete free surface condition expressed by equation (10) is used. We assume that the velocity potential ok in the outer domain can be represented by a sum of multipoles (including sources) with singu- larities inside the body (see Fig. 4). A m + A m am ax am By m m aG(x;Xm) a G(x;x A+ at A 4m az Am 2 m ax m a G(X;xm) a G(X; AT + A em 2 A7m ax By By m m m a G(x;xm) a G(x;x + A8m ax az ~ Age By a m m m m (17) where Xm = (xm,ym,zm) is the coordinate of a singularity point. The Green's function G and its multipoles satisfy the far-field free surface condition (11) and the radiation con- dition. In the numerical solution SB, SF, Sc and So are divided into plane quadrilateral ele- ments. The velocity potential is assumed constant over each element. At SB the term ask/an is replaced by the body boundary con- dition (14). At SF the term B.k/8n is replaced by the free surface condition (10) which includes the velocity potential and its first order derivatives along the free sur- face as unknowns. The first order derivatives of ok are numerically approximated by a Taylor expansion which are only function of ok on the free surface. The approximation is

correct to 0(~), where ~ is a characteristic length of the elements. At SO and SO the term 3tk/an is replaced by equation (17). By letting the point (xO,yO'zO) in equation (16) approach the mid-points of each element on the boundary surface S. we obtain a Fredholm integral equation of the second kind. This results in N number of equat,ions. The total number of unknowns is N+NII, where NII is the number of terms used in the multi- pole expansion (17). The NII additional equations are obtained by matching the inner and outer solutions at the control surfaces SO and SO. This is done by a least square condition. It means we require the aEr aE ( ( nary)) _r _ where Er = 2 ~ [Re(.kI _ ~k)]2 + ~Im(.II ~I)]2~ 0 (18) Further ok is the inner domain solution, And and ok are defined by equation (17) and No is the number of elements on ScUSO which is going to match the outer solution. This leads to the following condition N ~ {-i~k(xj) G I(xj;xM:) + i ~ 2 2 AnmG (xj;xm)] GNI(Xj;xMI)} = 0 where GO is defined by writing equation (17) as k 21 2 And G (x;xm) . . and GNI(xj;xMI) is the conjugate of G (xj;xMI). In the following equations the sign means the time average. Equation (19)is satisfied for NI = 1,L and MI = 1,K. This means a total of NII = LxK number of equations. Number of multipole terms have to be much smaller than number of control surface elements. As an example with the results presenting in Fig. 10, NII=10 and NC=56 . surface condition ( 11 ) and the radiation con- dition. The radiation condition is taken care of by introducing a Rayleigh viscosity p. The Green's function Gent can be written as G(x,y,z;xO,yO,zO) = (x - xO ) + (Y - Yo ) + (Z - Zo ) ] (20) - [(x-xO) + (y-yO) + (Z+zo)2] 36 A(z+zO+ircos(U-~+a) ) + ~ f du r dA Ae1 2 ~ -a O A~g(~ +2~UAcosu-2ip(~+UAcosu)) where x = Jocose and y = rsin8. Expression (20) can be simplified similarly as Grekas [1] did by using the residue theorem and introducing exponentional integrals. The derivatives of the Green's function (Multipoles) were obtained by numerically evaluating the analytic expressions for the derivatives. Having found the velocity potential by the method described in the previous text, the added mass, damping and first order excita- tion forces can be obtained by integrating the fluid pressure over the mean wetted body surface correctly to °((a) and O(U). When we have solved the equations of the first order motions, we can find the mean wave forces and moments correctly to o((a2) and O(U) either by directly integrating the pressure or by using the equations for conservation of momentum. We will show how this can be done if the equation for conservation of momentum is used. We start with expressions given by Newman [7]. The rate of change of momentum M(t) in the fluid volume Q inside S = SBUSFUScUSO (see Fig. 5) is dt M = - P iit(p + gz) n + V (Vn-Un)] ds (21) Here V is the fluid velocity vector, Un is the normal velocity of surface S. n is the normal vector to S (positive direction out of the fluid) and Vn = V.n. The total fluid pressure is given by awl P = -PsZ - P at ~ PV¢S veil - Q EVES ~2 _ p 8~2 (22) - pV. ·V. + 0(~3) + o(U2) The Green's function (source function) that s 2 a we need should satisfy the far-field free 517

at' ~ ~cc ' $, n '' \~ So \ disc Fig. 5 Def initions used of surfaces in the calculation of mean wave f orces. By time averaging equation (21), assuming that SO is a horizontal plane at great depth and using boundary condition on S it follows that the mean wave drift forces can be writ- ten F = -p || ( ~ n + V V ) ds m S P m m n c + 1 O , m=1,2 - | | pgzn3 ds, m=3 SB+SF The integration is over the instantaneous surface S. We have to be careful when ana- lysing the problem and keep all contributions which are correctly to second order in the incident wave amplitude and first order in the current velocity. For the first term of the first integral in equation (23), we should first integrate up to the mean free surface, in which only the terms -up ~V¢1~2 - pV¢s.V¢2 in equation (22) have contribu- tions. In the integration from the mean free surface to the instantaneous free surface A, the first three terms of equation (22) have contributions. In the second term of the first integral, we can write Vm = 8(.s+~1+~2)/8Xm and Vn = a(¢S+~1~2)/8n. In the integration up to the mean free surface, the terms a~s/an.~2/axm' a~l/an~a~l/axm and a¢2/an~a~s/axm have contributions. In the integral from the mean free surface to the instantaneous free surface A, the terms which have contributions are B¢S/an.~1/3xm and a¢S/axm. a~l/an. By using the body boundary conditions and Stokes theorem (Ogilvie & Tuck [5]) it can be shown that the following terms a. a¢2 a¢2 a. | | ( ~pv~s ~ V¢2 em + P an aX + P an ax ) m m cO (24) l + f ax an ( ~ 1 atl) do cc m will partly cancel. ScO means the control surface up to the mean free surface and Cc is the water line between the mean free surface and the control surface (see Fig. 5). For m = 1,2 expression (24) is zero. For m = 3 it is equal to p8~5/8n ~ ¢2dl. However, this will Cc be cancelled by a similar term in the last integral in equation (23). When we integrate over SB in equation (23) we follow a similar procedure as outlined by Ogilvie to] for zero current velocity. The final expressions for mean wave drift forces are m 29 f ( at + v~sev~l ) nmdQ c 2 r I lV¢1 I no ds (23) cO a¢1 a¢1 a¢1 a. P J J axm an ds ~ P f as ~ an do cO c O , m = 1,2 | ~ Jr P9znm ds ~ (25) P llt~(atl)+v~s aZ(V¢l)~+\ I V4)ll2]nmds SO to - P f (at + v~s~v~l)(n + a x r) e n do C r 111 = 3 where SFo is the mean free surface. CB is defined in Fig. 5. In the integral over CB r n is the normal vector to C8 in the horizontal plane with positive direction out of the body volume. Further ~ = (A ~2 r 03 ) and a = (~4' q5rq6). The derivations is based on the body surface is wallsided at the intersection bet- ween the mean free surface and the body sur- 518

face. We note that the second order potential does not contribute in equation (25). The mean wave forces can also be obtained by using direct pressure integration. By following a similar analysis as outlined by Ogilvie [6] for zero current velocity we find that . . Fm = - P I I {\ I V¢1 I + SB o . (r1 + a X r).V(at} + V. ·V¢1)+V. ·V¢2]n . _ . _ . . at + (a x n)m (at ~ V¢s.V¢1)} ds (26) Fig. 6a Fixed Circe flow. 2 f r1(rl3 + y,74 - xr'5) ] n dl U - || pgznmds, m = 1,2,3 where See is the mean wetted body surface. When the body is fixed it is possible to show that the second order potential does not I contribute. where r = I/(x-xl)2+(y-y )2 and Sg is the body surface. We use plane panels with constant singularity density over each panel. One possible source of large inaccuracies in the procedure outlined above as well in other procedures is the presence of the mj-terms in the body boundary conditions (see equation (14)). This will be further discussed in the following section. Discussion of the mj-terms We will illustrate the difficulties with the mj-terms by giving same simple examples with two-dimensional bodies in infinite fluid. We will start with studying the detail of the behaviour of the first and second order deri- vatives of the velocity potential at the body boundary. This will be done by a similar panel method as we used in the three- dimensional flow case. We write the velocity potential as 2,r¢S ( X1 ~ Y1 ) I (an log r ~ Us an log r) ds(x,y) (27) 519 U=1 (< W'(h lar cylinder in infinite fluid and in steady incident by FIR Ma ' I' x We -: ~2 Fig. 6b Definition of parameters in the analysis of local two-dl~ensional f low around a sharp corner with an laterlor angle 8. We will choose a simple case with uniform current past a two-dimensional circular cylinder (¢s = Ux+¢sB with radius 1 and U = 1 in an infinite fluid domain (see Fig. 6a). The potential due to the body ~sB is cosO/r and the normal derivative B¢sg/8n is -cosO/r2 at the body boundary. We can then divide the boundary into line elements and for each element assume ~sB and 3¢SB/3n are constant with values which are equal to the correct values at the mid-point of the ele- ment. The potential and its derivatives out- side the body boundary can be obtained by eq. (27) and derivatives of eq. (27). Fig. 7 shows the result of ~sB' B¢SB/3n, r~18¢sB/aO, a ~SB/an2 as a function of the distance along the normal vector to the body boundary at the mid-point of the element. The results are for ~ = 45° (see Fig. 5). The effect of different number of elements NB is investi- gated. The horizontal axis is the ratio bet- ween the distance Al from the boundary and the length As of the elements. The results show that we get convergence and correct results Of tsb and B¢SB/8r at the boundary. However, for r~l8¢SB/aO and 32¢SB/Br2 we can- not obtain correct results at the boundary. The reason is that we are not integrating

8 ~ - 8 _~ A LIZ ^ ~ ~ _ _~ i. ++ . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~^ ~ ~ ~ ~ ~ ~ 3 ~ Hi ~ ~ ~ B ~ ~ ~ ~ ~ ;' ~ 0 AL 'n ~' 0.250 O.500 0.750 Moo N8 =*° + I_ + ~ ~ ~ + + + ~ +- 1 - 1 + + U) I C~_ ~ · · · ~ *__,& ~~h ~ O. ~ ~ . ~ `_, ~ ~ ~ ~ ~ . ~ at. . ~ ~ ~ ~ ~ ~ o 8_ o .r 8 ID ;- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ E E ~ ~ ~ ~ ~ ~ ~ D I I I '0 =0 0.250 0.500 o.?So to AS Fig. 7 Calculated values of velocity potential and its derivatives for the case presented in fig. 6. (The steady incident flow is exclu- ded). Analytical solution. Numerical values: Us: 6, 8.S/Br: A, r~1 B`pS/ar: x, a2.S/3r2 : +. (AL and as are def ined in Fig. 6) NO . number of pane l s . with correct curvature and with correct variation of tsB and a¢SB/8r over the ele- ments. From Fig.7 we can see that r~18¢SB/aB' 82~5~/8r2 are satisfactorily estimated at a distance of O(As) along the normal vector of the element. That means we may use an extra- polation method to calculate the velocity along the body and the second order derivati- ves of the velocity potential. After some tests with a circular cylinder and a sphere and an ellipsoid in the three-dimensional case we found that the velocity along the body, the second order derivatives and the mj-terms are in good agreement with the ana- lytical solutions. The other way to integrate the term mjlogr(or 1/R) over the body boun- dary is to apply the formula given by Ogilvie & Tuck (1969). , ~ JJ m. log r(orl/R)ds = ~-44 SO J - || V¢sVlog r(or 1/R)nj ds (28) This formula is valid for a body without sharp corners, wall-sided at the free surface and when As satisfies the rigid free surface condition. From a numerical point of view this formula is more simple to calculate because it only includes first order deriva- tives of the steady potential. It is expected to give more accurate numerical results than by direct integration of the mj-term. For a body with sharp corners the mj-terms are singular. The consequence is that eqs. (27) and (28) are not integrable. For example in the case of uniform current past a two dimensional section with a sharp corner the complex potential W(z) in the vicinity of the edge can be approximated as (see. Fig. fib). W(z) = ClZl/A + C (29) where A = 2 - 8/~ and C1 and C are constants. For a rectangular section the first order and second order derivatives of the potential are o(|z |-1/3) and o(|z ~~4/3). In the vicinity of the corner it is possible to show m2 = C2 R-4/3 for x = 0 and m2 = 0 for y = 0, where C2 is a real constant (see Fig. fib). This means egs. (27) and (28) are not integrable. Actually, this is true for all corners with internal angles less than a. However, if we solve the wave-current-body interaction problem in the time domain by using for instance Green's second identity and satisfy the body boundary condition on the exact body boundary, the expressions are integrable. The reasons why the integrals are not integrable when the body boundary con- dition is satisfied on the mean position of the body boundary is that the formulation of the body boundary condition is wrong. The mj_ -terms have been derived by a Taylor expression. This is not valid at a corner. We will show how we can avoid the difficulty with the mj-terms. We divide then the velo- city potential Ok into two parts k k ok where (30) elk atkb an k ' an ilk (31) The following solutions of Ok satisfy the body boundary conditions and Laplace equation a 8¢S a 8¢S a a¢S .1 = ~ ax ' t2 = ~ By ' .3 = ~ az 520

a 8¢s ads t4 = ~ Y az + Z By a. a. .5 = xazs - z aS a ads a¢S .6 = ~ x - ~ Y ax (32) By using Green's second identity we obtain tahe following expression for Ok = ~k-.k and ok (see Fig. 8). ( k ok) |x=x. Al (ok ~ Ok) an R) - an R]ds(x,y,z) 4~ Ma 1 C = k x=x1 ( t [ok Ban R ~ an R] ds(x,y,z) (33) (34) where x1=(xl~yl~zl) is inside S~=SBUSFUSCySo' C=1 when x1 is inside S2=(SF-SF )USCUSoUS and C = 0 when x1 is outside S2. INF The integration surface is closed. S is part of the free surface and does not need to coincide with SF. By subtracting equation ( 34 ) frOm equations (33) we find that ( k k)| X X IJ ~ ( k ok) an R an (.k ~ Ok) R] ds I +l +S (ok an R an R) ds ~ I J ( k as 1 _ k l) ds SF+SINF n R an R (35) where C=1 when x1 is outside S2 and C = 0 x1 is inside S2. The last integralais a known quantity. The unknowns are Ok-. on the body and ok on SF, Sc and So. By writing the integral over SB like it is shown in equation (35) the integrand of the integral over SB is integrable. The solution procedure to find SF \7 ~ SINF I tsO I Fig. 8 Def 4nition of surfaces used in the integration of equation (33) and (34). the unknowns can be done similarly as in the previous section. What we have done now is to analytically isolate the difficulties with the mj-terms. This procedure is also valid when ship motions at forward speed is eva- luated. The same procedure can also be used to solve the second order potential problem where a similar difficulty occur. We also have to be careful when we find the added mass and damping coefficient. We will illustrate this by a simple example. Fig. 9 presents results which shows the effect of bilge radius r on sway added mass for a two- dimensional body in infinite fluid. The para- meters of the body is given in the figure. From Fig. 9 we can see how the added mass is dependent on the bilge radius in the cases with and without current. The added mass with current is going to infinity when the bilge radius r ~ O. This is an unphysical result. The reason to this behaviour can be found by studying the dynamic pressure part used in finding added mass and damping. Correct to °((a) and o(U2) we can write A22 p{B/2)2 1nD 7S An T ~ B=H Current _: _ x u L 1 Err; MOB ~ y 1 . . , - g id, ~ Do -- 0 00 0 25 0.50 0 75 1 00 r/( Bl2 J - _~ 'I ~ ~~ _ Fin. 9 Illustrati.^n .^.~ .^~1AII1O.4~ X X X U =0 167 Bw -~G~ U=0000 ficients for a body in a current can lead to unphysical results. The calculations are done for infinite fluid. (A22 ~ two- d i pens i one 1 added mass 4. n sway ) . 521

P = ~ P[ Bt Ilk + V¢s V.k~k ~~g~z~ (36) 2 (a V (V.s) )k] where a= (rat ~ zrl5 - yr16)i (37) (~2 ~ Z~4 + xq6)~ + (03 + yq4 x05) The index k in the last term in equation (36) ~ means that we only consider displacement in -- mode k. What we have done in the calculations pre- sented in Fig. 9 is to use the two first terms in equation (36). In this way we have included singular terms of o(U2), which are cancelled by the last term in equation (36). Actually we can write equation (36) as P = ~ P ( at Ilk+ V¢s · V.k Ilk) (38) This means that V. ·V.(a)pk cancels the last term in equation (56). If we use equation (38) we will find that the results for added mass at U ~ O is the same as for U = 0. However, this is not generally true when a free surface is present. What is true then is that the singular corner behaviour when the radius of curvature goes to zero is can- celled. Since our theory is currect to O(U) we can also write P = ~ P ( at Ilk ~ V¢s.(V.k)~u O ok ( ) This discussion illustrates that false effects can be created due to the singular corner behaviour if we are not careful in analysing the results. Numerical results for mean wave loads Calculations of mean wave loads require in general higher accuracy than computations of linear wave loads. We will therefore con- centrate our numerical studies on mean wave loads. Both a direct pressure integration method and the equations for conservation of momentum have been used. When the current effect is incorporated, the procedure is correct to O(U). Calculations have been per- formed both for horizontal and vertical mean wave loads. The first case we will discuss is incident regular waves on a fixed vertical cylinder that is penetrating the free surface. The draught of the cylinder is 0.25 R where R is 522 ~ = 0.000 ~ based on \/9 ~ momentum and OFF = 0 0479 J energy relations -D-> OFF =~°°°° 1 direct pressure integration JO- ~ = 0 0479 _ 1 ¢~ W. ~ ~ 1~/,~ =~ 0 3~ 0~ 0~ 11~ 9 Fig. 10 Numerical results of horizontal driftforces F2 for U/~i ~ 0.000 U/~ ~ 0.0479 with direct pressure integration method and a method based on conservation of momentum' and energy. The body is a fixed vertical cylinder with draugth-radius ratio 0.25. Element distribution: NN1 ~ 16. NN2 ~ 12. NN3 ~ 14, NN4 - 4 NN5 ~ 8 R ~ 1.0, R1 ~ 3.0, H = 1.2A (see Fig. 11). Element lengths on the body are nearly constant. ~1 Hi= Side view ~ {opv~ew Fig. 11 Def inition of number of elements and dimensions of control sur- faces used in the numerical solution of flow around a vertical cylinder of finite length. the cylinder radius. The cylinder bottom is impermeable. The current direction coincides with the wave propagation direction. An example on results for horzontal mean wave loads are presented in Fig. 10 as a function of ~02R/g where o0 is the circular frequency of oscillation of the waves without current present. Both for zero and non-zero current speed we note an important difference in the calculations based on direct pressure integration and the results based on the equations for conservations of momentum in

the fluid. The panel distribution used in the calculations can be illustrated by means of Fig. 11. By referring to the nomenclature in the figure, NN1 = 16, NN2 = 12, NN3 = 14, NN4 = 4, NN5 = 8, R = 1.0, R1 = 3.0 and H = 1.2 A (A = incident wave-length). The panel dimen- sions on the body were of nearly constant equal length. The reason to the differences in the results is that the direct pressure integration method is sensitive to the distribution of the element in the vicinity of the corner at the bottom of the cylinder. This can be illustrated by Fig. 12 where the calculations are presented as a function of R/AL when e02R/g = 0.8. AL means the length of the element nearest to the corner on the vertical side (see Fig. 11 ) . NN1, NN2 and NN3 were the same as used in the calculations presented in Fig. 10 while NN4 varied from 4 to 12 and NN5 from 8-12. This means that the total number of elements were quite similar in the calculations presented in Fig. 10. It is the size of the elements that differes significantly. The height of the elements on the vertical side were selected so that Ln'1/Ln is a constant, there n = 1 means the element closest to the corner, n = 2 the ele- ment next closest and so forth. The constant ratio was always below 1.5. On the horizontal bottom the length of the element in the radial direction was selected in a similar manner, starting with an element closest to the corner. The length of the element on the bottom closest to the corner was the same order of magnitude as the height of an ele- ment on the vertical side closest to the corner. However, the most important parameter in the calculations by the direct pressure integration method was the distribution of the elements on the vertical side close to the corner. The reason was associated with the contribution from the velocity square ~2 2 P9:o {2R) CD 1 ~ _ o term in Bernoulli's equation, which is singu- lar, but integrable at the corner. In Fig. 13 are shown numerical results for vertical drift forces on a vertical cylinder that is free to oscillate in surge and heave and restrained from oscillating in pitch. The incident wave propagation direction is in the positive x-direction. The draught h of the cylinder is equal to the cylinder radius. The Fit 9`a (2Rl 4. see ~ 3°°°1 1.500 - aom -1.soo - ~ (~ ) . Direct pressure integration Based en momentum and energy relations 1Q 000 Q300 0600 O900 1200 Fig. 13 Numerical results of vertical mean wave force F3 with direct pressure integration method and a method based on conservation of momentum and energy. The body is a vertical cyl Under that is free to oscillate in surge and heave and restrained from oscillating in roll. The draught-radius ratio is 1.0. Element distribution: NN1 ~ 16, NN2 ~ 10, NN3 ~ 14, NN4 . 8, NNS - 8, R ~ 1.0, R1 ~ 2.5, H ~ 1.2A (see Fig. 11). Element length on the body is nearly constant. Zero current velocity. F3 2 PI {2Rl u = o.ooo ~ based on ) momentum and ~ = Q.0479 J energy relations At-> OFF =°°°°°1 direct pressure ~ intenmtion 2.000 >0- ~~ = 0.047g J loon Fig. 12 Numerical results of horizontal drift forces F2 with direct pressure integration method and a method based on conservation of momentum and energy. Data presented as a function of R/6L (AL defined in the figure 11) NN1 - 16, NN2 - 12, NN3 ~ id, NNd 4-12, NN5 ~ 8-12, R ~ 1.0, R1 ~ 3.0, H - 1.2A (see Fig. 11). High density of elements close to the cylinder corner. The body is the same as used in Fig. 11. 523 -1.000 -2 000 U N N - FIR =0.70 based on Aid ~1 momentum ~R=0.58 J and energy O 0 _ relQtlons ~ FOR - 070 l direct ¢0-> ~ R = 0.58 J integration -~ - i ~ I 1 10.000 200.000 400.000 60Q000 800.000 ~ L Fig. 14 Numerical results of vertical mean wave forces F3 with direct pressure integration method and a method based on conservation of momentum and energy Data presented as a function of R/6L (AL defined in the figure 11). NN1 - 16, NN2 ~ 10, NN3 ~ 1d, NN4 - 8-12, NN5 . 8 - 12, R ~ 1.0, R1 ~ 2.5, H ~ 1.2A (see Fig. 13). High den- sity of elements close to the cyl inder corner. The body 15 the same as used in Fig. 13. Zero current velocity.

current velocity is zero. The panel dimen- sions of the body were of nearly equal length. By referring to the nomenclature in Fig. 11, NN1 = 16, NN2 = 10, NN3 = 14, NN4 = 8, NN5 = 8, R = 1.0, R1 = 2.5, H = 1.2 A. The large differences between the two different methods occur in the vicinity of heave reso- nance. The reason to the differences is again that the direct pressure integration method is sensitive to distribution of the elements in the vicinity of the corner between the bottom and the vertical side of the cylinder. This can be illustrated by Fig. 14 where the calculations are presented as a function of R/AL when e02R/g = 0.58 and 0.7. AL means in this case the length in radial direction of the element closest to the corner on the horizontal side (see Fig. 11 ) . The distribu- tion of elements were selected similarity as in the previous example. Total number of ele- ments are nearly the same for all calcula- tions presented. When the direct pressure integration method is used to calculate the vertical mean wave force, around heave reso- nance, the contribution from the velocity square term in equation (26) is large and of opposite sign to the other contributions in the integral over SBo. The absolute values of these terms are nearly equal to the velo- city square term. This means a high accuracy is needed in the integration over SBo. In Figs. 15 and 16 are presented numerical results for horisontal drift forces on a ver- tical cylinder with draught-radius ratio 3.0. The method based on conservation of momentum was used. The effect of using higher density of elements close to corner between the bot- tom and the vertical side of the cylinder was investigated. There was a maximum of 1% dif- ference in drift forces. The influence of number of singular points inside the body (see equation (17)) was investigated. Also the effect of number of multipoles was studied (see equation (17)). In the calcula- tions presented in Figs. 14 and 15 number of singular points is two and number of multipo- les for each singular point is 10. If only one singular point was used there was a maxi- mum of 1% diffrence in drift forces. If number of multipoles was increased to 16 there was a maximum of 0.2% differences in driftforces. The effect of increasing R1 (see Fig. 11) from 3.0 to 5.0 was studied. The difference in results were less than 2%. CONCLUSIONS A theoretical method to analyse current-wave- body interactions is presented. It is demon- strated that the mj-terms arising in the body boundary conditions can cause large errors in the numerical result. A theoretical way to provide stable numerical solutions is pre- sented. A method to calculate mean wave forces based on conservation of fluid momentum is pre- sented. It is demonstrated that a direct pressure integration method can lead to large errors in prediction of mean wave forces when the body has sharp corners. -0-0- ~ = o.oooo _O_O_ u = 0 0319 ~ = Q0638 7_~ ~ =-0.0319 Vim =-ao63s O O F: 2 P9~ (2RI ._ ._ o A_ o A_ n Aid 8t · ~ R on 000 3.~ 0.6m o.soo t.200 9 Fig. 15 Numerical results for horizontal wave drift forces on a fixed ver- tical cylinder with draugth-radius ratio 3.0. Current velocity U is in the same direction as the wave propagation direction. Ele- ment distribution: NN1 ~ 16, NN2 ~ 16, NN3 ~ 14. NN4 ~ 12. NNS . 5, R1 ~ 1.0, R1 ~ 4.0, H ~ 1.2A (see Fig. 11). Element length on the body is nearly constant. - > ~ = 0.0000 _O_O_ u = 0 0319 O ~ U = ao63a _ ~ 3952 211 `°~ ~~~ ~~ 0.0319 c, . _ o A_ o To ._ o ~ a- ,,~ =-ao63s ma car ~ ~0 /~/'~W §.A ~ W~ ,1 ~02 R Fig. 16 Numerical for horizontal wave drift forces on a vertical cylinder with draught-radius ratio 3.0. The cylinder is free to oscillate in surge only. Element distribution is the same as used in Fig. Is. 524

References 1. Grekas, A. 1981, Contribution a ['etude Theorique et Experimentale des Efforts du Second Ordre et du Comportement Dyna- mic d'une Structure Marine Sollicitee par une Haul Reguliere et un Courant, These de Docteur Ingenieur (Ecole Nationale Superieure de Mechanique). 2. Hess, J.L., Smith, A.M.O. 1962, Calcula- tion of Non-lifting Potential Flow about Arbitrary Three-dimensional Bodies, Report No. E.S. 40622, Douglas Aircraft Division, Long Beach, California. 3. Mavrakos, S.A. 1988, The Vertical Drift Force and Pitch Moment on Axisymmetric Bodies in Regular Waves, Applied Ocean Research, Vol. 10, No. 4. 4. Molin, B., 1983, On Second-Order Motion and Vertical Drift Forces for Three- dimensional Bodies in Regular Waves, Proc. Int. Workshop on Ship and Platform Motion, Berkeley, pp. 344-357. Ogilvie, T.F., Tuck, E.O., 1969, A Rational Strip Theory of Ship Motion: Part I, Report No. 013. The Department of Naval Architecture and Main Engineering, The University of Michigan, College of Engineering. 6. Ogilvie, T., 1983, Second-Order Hydrody- namic Effects on Ocean Platforms, Proc. Int. Workshop on Ship and Platform Motions. Berkeley, pp . 205-265 . Newman, J.N., 1967, The Orift Force and Moment on Ships in Waves, Journal of Ship Research, Vol. 11. 8. Newman, J.N., 1978, The Theory of Ship Motions, Advances in Applied Mechanics, Vol. 18. 9. Zhao, R., Faltinsen, O.M., 1988, Interac- tion Between Waves and Current on a Two- dimensional Body in the Free Surface, Applied Ocean Research. Vol. 10, No. 2. 10. Zhao, R., Faltinsen, O.M., Krokstad, J.R., Aanesland, V., Wave-Current Interaction Effects on Large-Volume Structures", BOSS '88, Trondheim. 525

DISCUSSION by R. Huijsmans I first like to congratulate the authors on their very concise treatment of the low forward speed problem. I have a few questions. 1) Can the authors elaborate on how to obtain the low frequency drift forces from their "far field" expansion of the drift force? 2) The authors identify the well known problem in using double derivatives of the stationary potential on the body. They used a kind of extrapolation procedure to avoid the problem. Have they now used the potential As on B. described by a c2 function by using some linear variation? (results of Fig.7 of their paper for /&L/AS - 0) 3) The authors experienced some problems even for the zero speed case in determining the wave drift forces based on the pressure integration, because of the presence of sharp corners. Have they some experience on how "sharp" these corners must be in order to get into troubles. 4) The authors did not mention actually solved the systems of equations (directly?) and what the computational burden of their method is with respect to the number of panels. How much more expensive is the treatment of the non-zero speed case with respect to the zero speed case? 5) And the final question is regarding the use of their method without the low forward speed restriction. Can the authors give some idea how to their method can be applied for high speeds. Author's Reply We thank Huijsmans for his comments. The replies are as follows: 1) Our "far" field expression is based on conservation of momentum and energy which can not be applied to calculate low frequency drift forces. When we study low frequency drift forces one should also include second order potential. A discussion about this problem is given by Faltinsen and Zhao[A1]. 2) We have not applied high order panel method to predict the second order derivatives of stationary potential on the body. We think for a body with sharp corners one will also get numerical problems even we use high order panel method. In our another formulation (see eq.(35)) we can avoid to calculate the second order derivatives on the body. 3) When one calculates wave drift force based on pressure integration, it is difficult to predict the contribution from u2-term, because the velocity will be infinite at sharp corners. We did not investigate how "sharp" these corners must be to get into serious problem. But from our experience we think the most important thing is due to cancellation effect of the contributions from different terms. This will depend on the whole body configuration the incident wave system as well as the local sharpness of the corner. In some cases only a few percent error in predicting the contribution from u2-term will give 100% error in wave drift forces. 4) The usual direct equation solver was used to solve the systems of equation. The CPU time is almost the same for the case with or without current velocity. 5) Our method may apply to high speed problems. In that case one should obviously use another free surface condition and Greens function. [Al] Faltinsen, O. and Zhao, R.: Slow-Drift Motion of a Moored Two-Dimensional Body in Irregular Waves, J. of Ship Research, Vol.33, No.2, June 1989, pp.93-106. DISCUSSION by A. Hermans I only shall make some remarks about Figs.2 and 3 and the text just before those figures. It looks like the authors are not aware of a large amount of literatures about the typical nonuniform behavior of that show up in "both" figures. The way they think that (13) has been applied leads to nonuniformities at the distance of order I R. while the application of (11) leads to a nonuniform behavior at the distance of order 1 2R. Already in 1882 Lindstedt noticed and remedied this kind of nonuniform behavior in the computation of the trajectories of planets. In 1892 Poincare in his book on "Mecanique Celeste" proved that the remedy that was given is correct. In 1949 Lighthill extended their theory to problems in fluid dynamics. The approach suggested in (13) is uniformised quite easily because the exact phase relation can be used, while the approach according to (11) always will have a phase error in the far 526

field. The uniform version of (13) and application of (11) both lead to a correct approximation of the amplitude. Author's Reply We would like to thank professor Hermans for his comments. We think one has to have in mind what one should calculate when we compare the two different approaches. If one should calculate the wave drift damping coefficient that is proportional to the slow drift velocity, the two approaches should be equal. However if one want to study wave current interaction and in particular the wave elevations around the structure, the two different approaches are different. The approach that we are using, is then a better approximation. DISCUSSION by H.J. Choi On this occasion, I would like to ask a 527 question which I have had for a long time. It is a well-known fact that incident waves deform significantly in amplitude and propagation angle, depending on the magnitude and incidence angle of current. As a result, a considerable amount of radiation stresses is to be built up in water, which might contribute to the second-order forces on marine structures, too. My question is if we could discard the effect. If it is not the case, how can you incorporate it into your method? Author's Reply The effect of radiation stresses is included when ye calculate mean wave forces. A discussion of this is given by Longuet- Higgins[Al]. [Al] Longuet-Higgins, M.S.: The Mean Force Exerted by Waves on Floating or Submerged Bodies with Applications to Sand Bars and Power Machines, Proc. R. Soc. Lond. A.352, pp.463-480.