**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

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The Effect of the Steady Perturbation Potential on the Motions of a Ship Sailing in Random Seas R. H. M. Huijsmans Mantime Research Institute Netherlands Wageningen, The Netherlands A. J. Hermans Technical University of Delft Delft, The Netherlands Abstract In this paper results will be presented of model tests and calculations of the wave drift force on a 200 kDWT tanker and half immersed sphere. The the- ory of small forward speed motion computations is ex- tended in order to allow larger horizontal distances in the Green's functions by deriving a proper asymptotic expansion of the low speed Green's function. Also an alternative formulation for the wave drift forces has been derived, based on the momentum balance as e.g. derived by Newman or Marno. This alternative for- mulation is derived for the small forward speed case. 1. . Introduction Recently we derived a formulation for the descrip- tion of the motions of a floating body with a small velocity. The reason for such a formulation is related to the wave drift damping phenomena. Large moored tankers offshore exhibit low frequency resonance be- haviour. These resonant forces are associated with the slowly varying wave drift forces. These forces can be computed with the help of linear diffraction the- ory and taking into account the second order effects of the pressure and the wave height. An extensive study among many other studies was published by Pinkster (14. Also second order wave excitation may be taken into account in an approximative way, see Benshop et al. t24. In an early paper Remery and Hermans t3] indicated that for an accurate descrip- tion of the low frequency motions not only the drift forces are important but also the accurate prediction of the damping coefficient near resonance. In a later study by Wichers t4] he showed that this damping co- e~cient was quadratic with respect to the wave height' thus leading to the concept of wave drift damping co- e~cient, which have been shown by Wichers et al. L5] to be related with the forward speed dependency of the wave drift forces. The effect of the varying wave drift forces with speed has been described by a num- ber of authors nowadays, beginning from Hermans and Huijsmans t64 to the more recent publication by Nossen et al. [7], Sclavonous [8] and Hu and Eatock Taylor A. In the paper of Hermans and Huijsmans the speed was restricted to be low, due to non-uniform character of the asymptotic expansion scheme. In the paper of Sclavonous the problem was solved by deriving explicit Green's functions for the wave drift damping, with a proper account of the disturbance of the steady poten- tial Us In order to solve the forward speed problerr~ Zhao and Faltinsen t10] showed that tl~e treatment of the speed dependent boundary conditions (depending on the steady potential ¢~) have to be handled care- fully. As soon as one tries to use the expansion scheme in t111 for the unsteady potential with respect to for- ward speed at a small but fixed forward speed one is confronted with the non-u~iformities in the asymp- totic expansion. In short ogle fields for a point source that the second order results behave like (~2 where ~ _ up is the small parameter and R is the distance g to the point source. In former studies only the speed effect due to a uniform flow has been attempted. However the in- fluence of the steady perturbation potential resulting from the stationary fluid flow around the ship, on the ship motion problem is not well understood. For the case the current is head on or the ship's course is at zero drift angle, then the influence can be ne- glected. In case of a ship moving at a certain drift an- gle it then appears to be of considerable influence, see Huijsmans et al. (124. In our study we have incorpo- rated the steady foreyard speed perturbation potential into the ship motion problem. The forward speed ship motion problem is solved using an efficient algorithm 375

for the Green's functions involved. The solution of the boundary integral equations comes from a very effec- tive iterative solver. The calculation of the wave drift forces is done in two alternative ways, one is the inte- gration of the pressures over the mean wetted surface of the body and secondly using the Maruo expression for the wave drift forces corrected for the small forward speed parameter. The nature of the non-uniformity in the asymptotic expansions will be studied in this pa- per and extended to uniform expansions. We keep in mind that one of our goals is to arrive at a formulation that makes use of the zero speed source potential. In this paper we present a uniform approximation valid for small values of the small parameter but also for fixed finite values of OR. 2. Mathematical Formulation We first derive the equations for the potential func- tion 4~,t), such that the fluid velocity n(z,t) is de- fined as u~x,t) = grad ¢(x,t). The total potential function will be split up in a steady and a non-steady part in a well-known way: ¢~2:, t) = Up + Add; U) + ¢~:e, i; U) (1) In this formulation U is the incoming unperturbed ve- locity field, obtained by considering a coordinate sys- tem fixed to the ship moving under a drift angle ct. In our approach this angle need not be small. The time dependent part of the potential consists of a incoming wave at frequency ~ an a diffracted and/or radiated wave contribution. To compute the wave drift forces all these components will be taken into account. 7 MY At the free surface we have the dynamic and kinematic boundary condition 9( + At + EVE · Vie = constant ~ at z = ~ (3) ~z-~-¢yc-G = 0 ' We assume that the waves are high compared to the Kelvin wave pattern, but that they are both small in nature, hence the free surface boundary condition can be expanded at z = 0. Elimination of ~ leads to the following non-linear condition: Bt2~ + 9oZ¢ + ot(V~ V¢) +V~ V [ ¢2 ¢| = 0 at z = 0 (4) To compute the wave resistance at low speed the free surface elevation must be treated more carefully, be- cause the wave height is of asymptotically smaller or- der. This problem has been studied extensively by Eggers [13], Baba [14], Hermans t154 and Brandsma [164. The velocity field is well described by the double body potential with a small wave pattern.Therefore we take the double body potential into account and we neglect the stationary wave pattern. For the wave potential Aid, t; U) the free surface condition now be- comes: ¢~t + 9¢z + Oust + +2V¢. V¢t + (U: + 2U¢= + ¢ +2(u+~¢y¢2.y+¢y¢egy +~3U¢~ + ¢,.~= + ~y~ry)¢r + +~2U¢~v + my + (y~yy)~ + {(2) {be} at z = 0 = ~ (5) The boundary conditions on the hull can be written in a similar way for all radiating and diffracted modes. We therefore treat the following general form, keeping in mind that the actual form has to be used in the computations. Generally we have the condition: Fig. 1. Axis of coordinate system (Vet. n) = V(~)e ibex ~ S (6) The equations for the total potential ¢' can be writ- ten as: /\f = 0 in the fluid domain De (2) where S is the mean wetted area of the ship hull. The non-linear operator {(2) on ~ win be neglected as well. The first term in equation (5 ~ contains linear terms in U. 376

Our Ansatz is that in order to obtain the first or- der approximation with respect to U the higher order terms in U may be neglected in the free surface con- dition. In the next section we show that in general this is true, but first we discuss the construction of the regular part of the perturbation problem, with the complete linear free surface condition. We assume ¢(x, t; U) to be oscillatory. f~x,t; U) = `~(x; u~e-iw' The free surface condition is then written as: _~~72~_ 2i~U¢~ + 2+ gig = D(U; ¢) {~) at z= 0 (8) where D(U;~) is a linear differential operator acting on ~ as defined in equation (5~. We apply Green's theorem to a problem in Di in- side S and to the problem in De outside S. where S is the ship's hull. The potential function inside S obeys condition (8) with D = 0, while the Green's function fulfills the homogeneous adjoins free surface condition: -~2G + 2i~UG`+ +U2G`` + gG: = 0 at ~ = 0 (9) This Green's function has the form Gf x, g j U) = - - +,¢(x, A; U) (10) where r = |xAl and r' = |xA'|, where A' is the image of g with respect to the free surface. Combining the formulation inside and outside the ship we obtain a description of the potential function defined outside S by means of a source and a vortex distribution of the following form: 47r¢(x) = //s^/(g)~3nGtX'<)dS~ 9 /T! L-rtg)Gfx,g~dr'+ //S (g)G~X'()ds: + 9 /T! ~ [~(g)0gG(2,(~+ { t)~(g) + ~T^IT(g)} Gtx, g)] do/ + + 9 /T~ ~ ~na~g)Gtx' (ids + 9 //s Gin, ()D {(a) dSt (11) with c', = cos(O2, t), AT COs(OX,_) and ~x,, COS(OX,7~), where n is the normal and t is the tan- gent to the waterline and T = t x ~ the binormal. It is clear that the choice of high = 0 for the integral along the waterline will give no contribution up to or- der U. The source distribution we obtain in this way is not a proper distribution, because it expresses the function ¢, in a source distribution along the free sur- face with a strength proportional to derivatives of the same function ¢. However this formulation is linear in U and moreover the integrand tends rapidly to zero for increasing distances R. So finally we arrive at the formulation: 2xa (A //s a (g) ~ G (x, () dSf +U / ``na (a) an G (2,g) do + g / /FS ;3~Z G (x'() D {<I)) d cot = 47rV ran), xE S and 4,ri (`x`) = (12) //S~(g)G(X'()dse+ 9 //r s G (x, () D {g7) dS: x ~ Do t13y where D {if} = 2Vi ~ V¢. We now consider small values of U. keeping in mind that there are two dimensionless parameters that play a role in the limit. We consider ~ _ flu << 1 and u= AL :3, 1. It turns out that the source strength and potential function can be expanded as follows: or (x) = a,) (x) + TCr1 (x) + ~(x; U) (14) ¢~:~) = To (X) + rf1 (X) + ¢(X-; U) (15) where ~ and ~ are (~(T2) as T ~ O. while the expansion of the Green's function is less trivial. 3. The Green's Function In this section we present an asymptotic expan- sion of the Green's function. The Green's function fol- lows from the source function presented in Wehausen and Laitone [17]. In case of T ~ 1/4 the function ~ (x, 6; U) is written as follows: 377

¢(X'<;U) - T' lo do| dkF(8,k)+ ~ 9 | do | dkF (8, k) ~ 7r/2 L2 where: F(f' k' kexp(k(z + ~ +~(~() cos8]) ·cos ik(y'7)sin8] (17) The contours {~ and (2 are given as follows: kl k2 I ~ O k3 k4 O <~_L2 find: Fig. 2. Contours of integration ( -) The contours are chosen such that the radiation conditions are satisfied. The radiated waves are out- going and the Kelvin pattern is behind the ship. For small values of T the poles of equation (17 ~ behave as: 4;:,~ ~ ~ + 0(T) as ~ ~ O (18) /,A/ ~ ~ ~ C) (1) as T ~ O (19) A careful analysis of the asymptotic behaviour of Ax, (; U) for small values of U leads to a regular part and an irregular part: ¢(X,<; U) = Jo (X, () + ¢] (~ () ...~t, (~,() + ~ ¢' (x,6) + where (20) ¢t, (x, () - 2k ~ k exp Katz + () J .`kR'~dk (21) ¢'1 (z,<) 4ik A, ~ k2 exp Katz + i) J ~kR)dk (22) where R2 - (~ -- (12 + (y -- 9)'> and b' - arc tan and ¢0 (x,<) = (16) 4u f exp iL,tz + () sect 8] sin Ax() sec 84 ~ · cos tidy71) sin ~ sec2 8] sec2 Ode (23) The expression in equation (23) gives the interaction of the translating part of the Green's function with the oscillatory part. In Hermans and Huijsmans t6] it is shown that due to the highly oscillatory nature the influence of equation (23) may be neglected in our first order correction for small values of a. The non-uniformity character of equation (20) for large values of R becomes clear, if we analyse ¢~ (x, A) a little bit further. The contour of integration L2 is chosen well underneath the singularity k`' = w2/g and performs a partial integration of equation (22~. The end points give zero contribution. Hence we 4ik,,cos8/ Ok k ~dk [k2ek(Z+~)J~(kR)] dk (24) We are mainly concerned with small values of T(Z+~) because the pressure is calculated at the ship's hull and we assume the horizontal length scales large compared to the vertical length scale. To get more insight in the structure of the source function we deform the contour L in the complex plane. ko , ~ - -- k iL Fig. 3. Contours in complex plane 378

We use the relation Jn(z) = ,~ {H(l)(z) + H,(2)(z)}. For large values of z the Hankel functions behave like Half ~ jei(z-:n~-~/~) d H(2) ~ I/ 2 e-i(Z-:n=-'r/~) (25) We find as an approximation of equation (20) ~6 ~ ¢0 + r~6~ = 27ri {kOek° (a+) Ha) + +2ricos8'd,k t~k2ek(Z+~) H(~)(kR)] k k ~ + 2 too (eik(Z+~) e-ik(z+~) ~ + - / kit, + ~ Ko(kR)dk+ Trio Ok +zko kLo J 4r cos 8' t°° k2 | eik(Z+~) eik(z+~) fir ·Kl(kR)dk Jo 1~ (k + iko)2 (kko)2 J (26) This expression is then studied for large values of R. The two integrals can be expanded with the help of the following integral representation of the function Kn (z) Kn(z) = / e-ZC"Sit~coshntdt fly) Jo We first apply the method of steepest descend with respect to the t integral and perform a partial inte- gration with respect to k. It then turns out that both integrals behave like (~(R-3/2), hence they lead to uni- form expansions with respect to T. The integrals are (~(1) as T ~ O. ~RE[O.~) - The terms that follow from the residues give rise to the expected non-uniform behaviour.The second term in equation (26 ) may be written in the form: T¢] (X, () |R = 2,ri * 2iT COS 8' [koH(~) (koR) + +i,z + ¢,)k`2,H(~)(k,-'R! + Rk~2H!(~)(k,., + R)] ek°(Z+~) (28) The second term gives rise to non-uniform behaviour of large values of z + A, however we restrict ourselves to finite values of z + A. Our main concern is the last term. We compare this term with the first one in equation (26) ?,6~ie9 ~ 2~ik~-,ek° (my) H('t) ~ kO R) {1 + 2iTkoR COS ~ ~ + 0(T ), ~[0.~) (29) and for large values of R we have: ~ ~ 2~i~koek°(Z+~)ei(koR - =/~) {1 + 2rikoRcos8'3 + (~(R-3/2) + (A r2) (30) The origin of the non-uniformity is now clear. It is the well-known phase shift of the wave numbers of the PLK method. The residue of the exact source func- tions leads to the exact phase shift, in our case we have approximated: exp(2ikoT(xA)) by 1 + 2ikoT(xA) (31) This term originates from the a, t derivatives in the free surface boundary condition. Before treating methods to obtain uniform expansions we must keep in mind the way we like to use the Green's function. This leads to the insight that we need two different approaches. One for the computation of the far field wave and one for the computation of the integral equation. In the far field the exact value of the wave number has to be taken into account, while in the latter case a first order correction of the wave number is sufficient to arrive at solutions valid up to second order. 4. Expansion of Source Strength . . . In this section an approximate solution of equation (12) will be derived. Inserting (14) and (15) into (20) one obtains for like powers of T the following set of equations: 27raO (a)i/s JO (it) ,9~ Go (id, A) dS~ = 4~Vo(~), x ~ S (32) and 2~i (I) - //s ~i (I) '~ Go (it, it) dSf .//s ° (I) ^~ (ala) dS: + 41rV~(x)+ 2 i~ ~Go (x,<) V¢~(~) V¢o (it) dSt. (33) here Go (id 6) =r + r'160 (X, () iS the zero speed pulsating wave source and V(X) = Vo(X) + TVI (X) + 0(T ~ (34) 379

The potential functions in (14) now become: F(8, k) = To (x) = - 4,r ils so (I) Go (x, A) dSf (35) ¢)1 (a) = - 4 |/ crO (g) Al (a, () dSk 41r || ~1 (a) Go (it, A) do + 21rg ||FS Go (xi A) V¢(~) ¢0 (a) do (36) In principle the solution of the problem is now solving ¢0 (x) and ¢1 (:~) using the steady perturba- tion potential. The steady perturbation potential is determined using a boundary integral technique for the steady double body flow, which originally comes from a Hess and Smith type of algorithm. The steady double body flow is calculated separately and is then incorporated into the free surface integral. In reference t6] it is shown that the non-uniform term with respect to z' in the Green's function leads to contributions that are asymptotically small compared to the terms we have taken into account. 5. Uniform Asymptotic Expansions In principle we have to solve (32) and (33) where the source function suffers non-uniform behaviour. If the size of the ship is order one with respect to T it is sufficient to use (10) with (20). The question remains how to compute the far field. This will be dealt with in part 1 of this section. If the size of the vessel becomes large with respect to T. TR = o(1), we have to modify (20) in order to obtain proper approximations of the source strength from (32) and (33~. This problem is stated in part 2 of this section. 5.1 The Far Field In the case where art) + Tin iS known, the far field may be computed with the aid of equation (13~. Be- cause for R large we cannot use (10) with (16~. As explained before the contribution of the "Kelvin" resi- dues from kit and kit are neglected. Contributions of the wave residues are dominant but first we apply the method of stationary phase to the integral with respect to 8. The integrand F(8, k) for large values of R with the notation xR cos 8, y = R sin 8: 380 with k exp~k~z + (~) 2 gk(w + kU COS 8~2 {exp ~ik~g cos ~ + ,7 sin 8~] · exp [ikR cost8~] + exp ~it cos ~ ~ sin 8~: exp [ikR cost + 8~] } (37) We have to distinguish between the four quadrants at infinity. We choose O < ~ < ~r/2. AD other quadrants can be treated in a similar way, the results are the same. We obtain: ~(x, (; U) ~ 9 1~_7r2 e 4 [/ ~ exp {kid + ()it cos ~ + ~ sin 8) + ikR) 'dk gk(w + Uk cos 8)'2 ;- Ad; exp {kid + () + it cos ~ + ~ sin 8)ikR} d (2 gk(wUk cos 8~2 ~ (38) The first integral may be closed in the upper quarter plane whilst the second one may be closed in the lower quarter plane.The integrals along the imaginary axis are of C'( R--:~/2 ). We now finally obtain: Ace, (; U) ~ /; ~ ei(kl (OR A/ ]) ~ 27r! ~ / c . 1 . V 1rR | (1 _ 2U COS 8~ + Kit MU cos 8)) exp {k~)(z + ()ik~)~<cos~ + rising)} (39) - 9 - 2~U COS ~ - 97~1 - 4T COS 8) - 2U2 COS2 ~ (4o)

Hence, to obtain the potential in the far field we ?,6~, = use (13) with = Large Vessels - ~ ,¢(x, A; U) and (41) = too (id) + TO (I) (42) In the case of large vessels (41) is not a good ap- proximation for the source strength. We now have to take care of the non-uniformity as described in (30). A practical requirement is that we want to make use of the zero speed oscillatory Green's function and its derivatives. The function ~6, (it,<) can be computed for the major part using algorithms as e.g. developed by Newman t184 or Noblesse t194. One minor con- tribution has to be evaluted separately. Keeping in mind that the PLK method requires the omission of ~6 ~ To + Tip + (~(T2R) (48) the most severe secular term to obtain uniform expan- sions we may conclude that the procedure only needs to avoid approximations as (31~. The following proce- dure makes it possible to use the zero speed algorithms with a slight modification. A proof of the validity can be given rigorously with the same analytic manipu- lations as described in Section 3. For instance the following correction may be performed: 1602~koek° (z+~) ~ [e ( ° / ) ~1 exp~2ik('r(~ (~3] (46) and ~~ = ?,6'2,rk,,ek°(Z+~) . ~ [e ( ° / ) 12ik~(xAlp] (47) The correction of TO can easily be performed in the zero speed Green's function algorithm, while the cor- rection of ¢~ can be performed either analytically or numerically. It can be shown by inspection also that: Hence the region of validity is extended in a proper way. If one wants to higher order approximations the procedure has to be reconsidered. Corrections can be obtained along the same line. However the advantage of reduction to the zero speed algorithms is not avail- able anymore. One has to devise a fast algorithm for ~72- ~JRes ~ (~°rcs + ~i ~ P~ °(Z (~) (43) It can be shown by inspection that this multiplicative correction yields the correct uniform asymptotic ex- pansion up to (~2R) as ~ ~ 0 t204. The interval of validity is properly extended. It is also possible to apply the correction to Mores alone and to show that: ~rr.s = (¢orc:~(l 2ikOT(~ Air) A- T¢lr'!~) exp(2irkO(~(~) + (27~2R) (44) The correction is only needed for large values of KoR, therefore the correction may be performed at asymp- totic level. This leads to the following simplified re- sults for the total ¢. This result is rewritten in asymp- totic form where we made use of the explicit form of the residues (45) where 6. Wave Drift Forces In Hermans and Huijsmans t6] we described a way to compute the first order forces and the second order wave drift forces. The method we used there was based on a direct pressure integration of the first and second order pressures respectively. It has been shown before (e.g. see Pinkster t14) that this method works well and is even necessary in order to compute the slowly varying wave drift forces. At this moment we are mainly interested in the constant component of the wave drift force. In this section we recapitulate a method that leads to results that possibly are more accurate numerically, because when using the pressure integration technique one has to use derivatives of the potential function over the mean wetted surface. This is even more the case if one uses pressure integration in the case of ship motions with forward speed (see Huijsmans t21] ). Newman t224 and Marno [23] have derived an expression for the wave drift forces and moments. 381

The mean forces and moments may be expressed ,¢,( t)= as t22]: F 1' - || (p cos ~ + pVI~(Vl' cos ~Vo sin 8)] Rd~dz (49) F - v || [psin ~ + PVK(VK sin ~ + Ve cos 8)] Rd~dz (50) sop US + e{kl (~)Z+i(kl (~)(~:'s~+9sirl~)W[)} +F(~)eiS`~1) ~/ie(k1 (fl)Z+i(kl (a)At)) here (a iS the amplitude of the incoming wave and F(~)eis(~) results from the asymptotic expansion of the far field potentials in (53) with M: = 4(j) (x; U) = fly VRVcR2dEdz (51 ) fls ~ (I) G (I, A) dSe where p is the first order hydrodynamic pressure, V is the fluid velocity with radial and tangential com- ponents V'2,Vc and SO is a large cylindrical control surface with radius R in the ship-fixed coordinate sys- tem. Faltinsen and Michelsen t24] derive from these formulas expressions in terms of the source densities of the first order potentials. From: 6 {or (if) = ~(7) (if) + ~ (I) (if) HI (52) j=l where CYj = c~je-iWt j=1~1~6 are the six modes of mo- tion and superscript 7 refers to the diffraction com- ponent of the source strength. In our case we follow the same reasoning to obtain similar results for the slow forward speed case. Our velocity potential has the form: flex, t) = Us + ¢(xi U) + Act, U)e is 9 /IFS V; V¢(i)G (X () dS'` (55) where G (x, A) is approximated by (30~. Due to the fact that the function Vow) decays rapidly as |~| ~ °° we assume that R = Axe is large enough to take the asymptotic expansion of G (x,() in the last term as well. The function V¢(i) may be replaced by V¢~(-,,). This leads to - |~ e-31ri/4 27r 1 _ 2U COS 8~ + kl(~)U COS 8) · [us ~ (a) exp {k1 (~(ibl (~< cos ~ + ,7 sin 8~} dS~ 2i~' || V¢' V<~'oT)exp{ibl(~) 9 FS (< cos ~ + ~ sin 8~} do] (56) = Ux+~;U)+ 6 ~ where: ~ ¢(0) (X; U) + ¢(7) Em; U) + ~ ¢(i) (X; U) JO ~ e-iwt ¢(T) (I) = ¢~() ) (I) + ~ If) (I) j j=1 (53) where the potentials ¢(j) (x; U), j=1,7 have the form (13) and are the potentials due to the motions and diffraction effects. We assume that they are all deter- mined by means of a source distribution where cr(j) (x) = (I) ( ~ ) + Ta(i) (~ ). In the far field R ~ 1 we neglect the influence of the stationary potential ¢(x; U) in (53) hence we approximate (53) by: 382 (57) This result is of (CAT ~ as T ~ O VRe[O,=)- The upper integration boundary in (49), (50) and (51) is the free surface ~ -- -Re [(iw¢>(x, y, O)U¢~.(x, y, O))e-i=~] (58) It follows from the pressure term that we may write: - 1: d Pg (, ~ _ P| (I VI 2U2) dz (59)

We find the following expression for Fit.: - Fx = P ~ {kooky + 2TIm(~)) R cos add + 4 / ~ t ( R2 ~°~8~R¢R + Liz f z) R cos 8+ Fit = F( " + F( 2) + R (¢R¢8 + BROW) R sin 8] dEdz and for Fy F.', = 4 / {kudos + 2TIm(~)) R sin Ode 4 Jo ~~ ~(R2 ~ jR¢R + ~zfz) Rsin8+ R (¢R¢8 + BROW) R cos 85 dude (61 ) and for M:: Mz = +4 1 | R (¢R¢8 + BRIE) dude (62) The integration with respect to z needs some ex- tra attention due to the fact that the exponential be- haviour of the incident wave and the radiated or dif- fracted wave is different, due to the dependence of the wave number on ~ and 8, respectively see (54). The asteriks in equation (60) denote the complex conjugate and A= go exp(k1 (,l3)z + ik~ (~)(~ cos ~ + y sink)) +F(~)eis(~)4 exp {kl (~)z + iEl (WRY (63) We formally write (63) as sum of two components: ~ = HI + SIB (64) For the wave number kit (a) we can write neglecting U terms in the wave number depending on b: k~(~) = k~(1 + 2~ cos 8) (65) The first term is related to the incoming wave field while the second one describes the diffraction and ra- diation ellects. A closer look at (63) and (60) shows that the con- tribution to Fc consists of two parts. The first one, F(.~), originates from those parts of the cross products that behave like R-1/2 while the second one, F(2), orig- inates from those square terms of the second part of (63) they behave like Ret. We formally write: (66) (60) We now can write: F(1) _ plan Ri/2 /; F(~)(cos ~ + cos if) ~ · cos {(k1 (it) cos(8,B)kl (~))R - S(~)) do + +rP 2(n R1/2 | F(~) [(cos ~ + cos JIB) cos +- (cos2 ~sin2 8))sin ~ sin p~ · cos {(k~(/3) cos(8,0)k~(~))R - S(~)3 do +(2) (67) We now apply the method of stationary phase for large values of R. To do so we have to determine the sta- tionary points of the integrands. We determine the point where: ~~; {k~(~) cos(8it) -- k~(~)) -- O (68) We find two points, 8~ = ~ + 2r sink and 82 =- 3 + ~ + 2r sin 3 (69) The second point gives a zero contribution, where we remark that the last term in (67) is cancelled by the T- dependent contribution of the last term. It is obvious that the other terms give zero contribution due to the factor or in 82. We use the asymptotic result: r2= J l(~) cos {(Al (I) cos(8p) -- kit (~))RS(~)) do tRko ] ( 1 - 2T COS 3)1/2 {1(3 + 2T cos 3) · COS ( S (d + 2T cos ~ ) + ~ /4 } ( 70 ) As mentioned before the second contribution is zero. A critical reader will notice that formally we have to deal with a non-uniformity due to the fact that we have a first order 'uniform' solution. The phase correction is of first order. Hence in our asymptotic result of (70) we neglect the second order phase correction. The only 383

way to do the analysis properly is to use the exact value of the wave number in (40) instead of our asymptotic approximation. Summarizing we now have: Fat) ~ A ilk ~ cos(S(~*) + x/4) ·F(~*) {cos if* + cos id+ -2r cos(2 p)) where A =PA and A* = ~ ~ 2T sin it. (71) A similar analysis can be followed for the sway force and leads to the following result: Fat) ~ A ilk ~ cos(S(~*) + ~/4) ·F(~*) {sin it* + sin 3+ 2T sin(2 if)) (72) The second part of the wave drift force may now be written as: F(2) = 4ko/ F2~{cos~2r cos(2 8~) do (73) For the sway force this leads to: F(2) = 4koJr F2~{sin~2T sin(2 8~) do (74) 7. Numerical Results For the validation of the mathematical model pre- sented in the previous sections computations have been performed on a fully loaded 200 kDWT tanker and a half immersed sphere. The description of the tanker can be found in Huijsmans et al. t124. The numerical results of the wave drift forces on the tanker have been validated against results of model test experiments. The cal- culations for the 200 kDWT tanker are applicable to the situation of a tanker in both head-on current and waves as well as the case of the waves and current co- parallel under 135 degrees, as displayed in the next figure. w c: ~ tow Fig. 4. Definition of waves and current The calculated steady double body flow using the Hess and Smith algorithm apply to the situation of a tanker under a drift angle of 135 degrees (in principle the solution can be made applicable to any drift angle Oc) The boundary conditions on the mean wetted hull for the radiation problem have been applied with the steady perturbation potential set to zero, which then results in the well-known expression for the m-terms of the radiation boundary condition. A more proper way of handling the radiation boundary condition as published by Zhao et al. t10] was not attempted here due to the difficulty of the splitting the boundary con- ditions into linear forward speed dependent boundary conditions. For the results of computation for the wave drift force on the sphere a comparison was made with results of computations as was obtained by Nossen et al. A. The panel discretization of the tanker amounted to 238, 720 and 1610 panels in order to check the nu- merical accuracy of the pressure integration technique. The result of the computation are depicted in Fig. 15. The number of panels for the sphere amounted to 744. To solve the integral equations for the source strength a(j) (x) and cr(i) (x) large algebraic equations have to be solved. In the present solver use is made of a newly devel- oped iteration scheme. Details of this new solver for Boundary Integral Equations will be published in the near future. To solve a large system of 1610 unknowns for the radiated and diffracted modes only required 4 to 6 iteration steps to gain an accuracy of 10-4 in the Euclidian norm. The timing was approximately 28 CPU seconds per wave frequency for 1610 panels on an ETA lOP mini supercomputer from ETA systems. The analysis of the results of the wave drift forces us- ing the pressure distribution integration, as shown in 384

Figs. 9 to 12 leads to the observation that the numer- ical accuracy of the pressure distribution integration greatly depends on the panel discretization. Detailed results are presented in Table 1 for the frequencies of 0.5 and 0.7 rad/s. Table 1 Fir for 180 deg. waves and current 238 720 0.5 15.5 17.3 0.7 11.2 15.3 1610 l 14.9 15.1 7.1 Half Immersed Sphere to 7. The results of computation for the sphere are pre- sented for the mean wave drift force. The mean drift force on the sphere has been calculated in two ways, i.e. one describing the correct influence of the steady perturbation potential and the other is using the slen- derness approximation in which the influence of the steady perturbation potential has been neglected. The region of integration over the free surface was from ~ = R to r = 3R, where R is the radius of the sphere. The number of free surface panels amounted to 810 panels. These results are presented in the following Figs. 5 Cat 40 1 .C 20 X Lie Drift force on sphere V = 0.0 m s1 4' _ ~ ~rcssur~f Hi\ 0.00 0.25 V 0.50 0.75 tOO Frequency in rod s1 Fig. 5. Zero speed wave drift forces L 8.oo ~ . . . 0.25 ~ 0.50 0.75 1.00 Frequency in rod s1 60 E 40 o · _ X 20 Drift force on sphere Pressure integration ~ v _ o.o ms1 1 it, L J 8.1 )O Q25 0.! SO 0.75 1.00 Frequency in rod s1 Fig. 6. Forward speed wave drift forces 60 Drift force on sphere Influence stationary potential . . . 1F~ __ _ 11 ~ stat.po eq 0.0 D E} stat.pot ne 0.0 Fig. 7. Influence of steady potential Drift force on sphere V = 1.0 m s1 nor 1 1 1 E 40 · _ X 20 1_ r-1~ mpuls Nossen ~1 lo_ 8. 0 0.25 0.50 0.75 1.00 Frequency in rod s1 in, ~ _ Fig. 8. Forward speed results Nossen 385

In Fig. 8 the results of computation are presented for En = 0.032 against other results as obtained by Nossen et al. t74. The observed difference between the results of Nossen and ours is due to the fact that the results of Nossen apply to a fixed cylinder in waves whereas our results apply to the free floating case. In the range where diffraction dominates, the results tend to be the same. The zero speed wave drift forces on the sphere are calculated in two ways; i.e. one using a pressure dis- tribution integration technique and the other one uses the momentum flux analysis (Maruo [23~. Drift force in 135 degrees V = 2.0 m s1 30 a) ~ 20 C) modeltest pressure 1 It 8. 0 o. 25 0 iO 0. 75 t00 Frequency in rod s1 Fig. 9. Mean surge drift force Drift force In 135 degrees V = 2.0 m s1 100. ~ ' ' ~ ' ~ modeltest - pressure 50 ~1 ' '5a~b' 1 1 Too 0.2 0.50 0. 75 t00 Frequency in rod s1 Fig. 10. Mean sway drift force For the forward speed case this was also done, how- ever now the equations (71) and (73) have to be used for the momentum flux analysis. One can see from Figs. 5 to 7 that the calculations using the momentum balance are not so much influ- enced by the panel discretization. This can be made plausable by observing that the expression for the wave drift forces from the momentum balance only simple source distribution integration over the mean wetted hull is required. It is our opinion that due to the in- tegration of higher order derivatives in the pressure distribution for the forward speed case the results are more sensitive for the way the integration is performed. Drift force in 135 degrees v = 2.0 m s1 30 l 20 o c L,= 10 m 1 - 1 ~ N : _~__ 8. lo o. o. 0 o. '5 t' 10 Frequency in rod s1 Fig. 11. Mean surge drift force Drift force in 135 degrees v = 20 m ,:1 ~Q5 1 1 1 1 . 100 _ _ ;~e moddtest presto Cal E 75 1 o ~ 1 __ 3.00 025 OSO- Q75 t00 Frequency in rod s1 Fig. 12. Mean sway drift force 386

7.2 Tanker At MARIN a number of tests have been conducted. The test comprise in short regular wave test on a sta- tionary moored 200 kDWT tanker in 82.5 metres of water depth with waves and current parallel to each other. The results of model tests and computations for head waves and 2 knots current have been exten- sively described by Wichers [25] and Huijsmans [21~. The results of model tests for the tanker with the waves and current coming from 135 degrees are presented in Figs. 9 and 12 for the mean surge and sway drift force respectively. 30 r 20 o · _ x lo Drift force In 135 degrees V = 2.0 m s1 ~ stat~ot eq 0.0 Cl stat .pot ne 0.0 r _~_- 8.1 )O 0.' 2 o' sO 0. 5 tt ~0 Frequency in rod s1 Fig. 13. Influence of steady potential Drift force In 135 degrees V = 2 0 m ~1 _! E a so c 11 Ho 1 1 STAT.POT ED 0. ~ Ci STATFOT NE 0. to_ _ / ~ 8. so o. 25~ 0.! SO 0. r5 t O Frequency in rod s1 Fig. 14. Influence of steady potential The resulting drift forces are calculated in two ways; i.e. one using the pressure integration technique and one using the momentum flux analysis. The results have been presented in Figs. 11 and 12 for the surge and sway wave drift force respectively. The influence of the stationary potential on the wave drift forces has been studied.The results are displayed in Figs. 13 and 14. From these results one is tempted to conclude that the influence of the stationary potential on the drift forces in this case is restricted to the sway drift force results, however, previous observations also in- dicate that the use of a pressure integration for the calculation of the wave drift forces with forward speed are somewhat doubtful, due to the large dependency on the panel discretization of the body. Drift force in 135 degrees Influence of discretization V = 2.0 m s1 30 1 20 c · _ x 10 o _ x 1610 panels - __ C) 720 panels A 1B 238 panels ~ ~ 0.00 0.2~ 0.50 0.75 1.00 Frequency in rod s1 Fig. 15. Influence of panel discretization References [1] Pinkster, J.A., Low frequency second order wave exciting forces on Boating structures, PhD thesis, University of Delft ( 1980). [2] Benschop, A., Hermans, A.J. and Huijsmans, R.H.M., "Second order diffraction forces on a ship in irregular waves", Journal of Applied Ocean Re- search, Vol. 9 (1987). [3] Remery, G.F.M. and Hermans, A.J., "Slow drift oscillations of a moored object in random seas", Proceedings of the Offshore Technology Confer- ence, Houston, Paper 1500 (1971). 387

[4] Wichers, J.E.W., `'On the low frequency surge motions of vessels moored in high seas", Off- shore Technology Conference, Houston, Paper 4437 (1982~. t54 Wichers, J.E.W. and Huijsmans, R.H.M., "On the low frequency hydrodynamic damping force acting on offshore moored vessels", Offshore Technology Conference, Houston, Paper 4813 (1984~. t64 Hermans, A.J. and Huijsmans, R.H.M., "The ef- fect of moderate speed on the motions of floating bodies", Schiffstechnik (March 1987~. t74 Nossen, J., Palm, E. and Grue, J., "On the solu- tion of the radiation and diffraction problems for a floating body with a small forward speed", Inter- national Workshop on Water Waves and Floating Bodies, Oslo (1989~. [8] Sclavonous, P., "The slow drift wave damping of floating bodies", International Workshop on Water Waves and Floating Bodies, Oslo (1989~. t9] Hu, C.S. and Eatock Taylor, R., "A small forward speed pertubation method for wave-body prob- lems", International Workshop on Water Waves and Floating Bodies, Oslo (1989~. t20] t104 Zhao, R. and Faltinsen, O., "A discussion of the m-terms in the wave-body interaction prob- lem", International Workshop on Water Waves and Floating Bodies, Oslo (1989~. t11] Huijsmans, R.H.M. and Hermans, A.J., "A fast algorithm for the calculation of 3-D ship motions at moderate forward speed", Proceedings of 4th Numerical Ship Hydrodynamic Conference, Wash- ington (1985~. [12] Huijsmans, R.H.M. and Wichers, J.E.W., "Con- siderations on wave drift damping of a moored tanker for zero and non-zero drift angles", Pro- ceedings of the PRADS Symposium, Trondheim (1987~. Eggers, K., "Non-Kelvin dispersive waves around non slender ships", Schiffstechnik, Vol. 8, (1981~. t144 Baba, E., Wave resistance of ships in low speed, Technical Report 109, Mitsubishi Technical Bul- letin ( 1976). t15] Hermans, A.J., The wave pattern of a ship sailing at low speed, Technical Report 84A, University of Delaware (1980~. t164 Brandsma, F.J. and Hermans, A.J., "A quasi- linear free surface condition in slow ship theory", Schiffstechnik (April 1985~. t17] Wehausen, J. and Laitone, E., Surface Waves, Volume 9 of Handbuch der Physik, Springer Ver- lag (1960~. t184 Newman, J.N., "The evaluation of free-surface Green's functions", Proceedings of the 4th Inter- national Conference on Numerical Ship Hydrody- namics, Washington ( 1985~. t19; Telste, J. and Noblesse, F., "Numerical evalua- tion of Green's functions of water wave diffrac- tion and radiation", Journal of Ship Research, Vol. 1111 (1987). Hearn, G., Priv. Communication ( 1989 ). [21] Huijsmans, R.H.M., "Wave drift forces in cur- rent", 1 6th Conference on Naval Hydrodynamics, Berkeley~ 1986~. t22] Newman, J.N., "The drift force and moment on a ship in waves", Journal of Ship Research, Vol. 10 (1967~. t23] Marno, H., "The drift of a body floating on waves", Journal of Ship Research, Vol. 4 (1960~. t24] Faltinsen, O. and Michelsen, F., "The motions of large structures in waves at zero Froude number", Symposium on the Dynamics of Marine Vehicles and Structures in Waves, London (1974~. t25] Wichers, J.E.W., "Progress in computer simula- tions of SPM moored vessels", Offshore Technol- ogy Conference, Houston, Paper 5175 (1986~. 388

DISCUSSION by R. Zhao ( 1 ) Discuss about direct pressure integration method (2) Discuss about wave drift damping Author's Reply The direct pressure integration method is based on the integration of local hydrodynamic quantities like water velocities and gradients of pressures over the mean wetted hull. This procedure i s demons bated in our papers presented at the ONR Conference in 1986, [Al ] and at the UTAM symposium, [A2 ]. One of the main problems in using the direct pressure integration method i s based on the accuracy of the underlying hydrodynamic data. In the zero speed case there is sufficient evidence that this procedure, using 3-D diffraction theory results, will lead to meaningful results. In the non-zero speed case however this evidence is lacking and to the author's opinion the accuracy of the f irst order results based on 3-D diffraction results with forward speed must be even more accurate than the zero speed results due to the fact that in the evaluation of the pressure distribution, gradients of the water velocity distribution are required. In pursuing the direct pressure integration one has to resort to e.g. higher order panel methods to obtain the required accuracy. The question regarding the wave drift damping has not been addressed in this paper, however information can easily be obtained using the results of our paper. At the moment we only calculated the drift force at two speeds, which is in principle enough to obtain wave drift damping data. However, if one wants to obtain reliable results for the wave drift damping coefficient several more speeds (also negative speeds) have to be calculated. This will be done in the near future. [A1 ] R. H. M. Hui jsmans, "Wave Drift Forces in Current", 15th ONR Symposium, Berkeley, 1986. [A2] R. H. M. Hui jsmans, "Slowly Varying Wave Drift Forces in Current", IUTAM Symposium on non-linear water waves, Tokyo, 1987. 389