**Suggested Citation:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." 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:**"Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions." National Research Council. 1990.

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

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Side-Wall Effects on Hydrodynamic Forces Acting on a Ship with Forward and Oscillatory Motions M. Kashiwagi and M. Ohkusu Kyushu University Fukuoka, Japan Abstract A rational slender-ship theory is presented for predicting the side-wall effects on the added-mass and damping coefficients of a ship, moving with forward velocity and performing heave and pitch motions in a waterway with vertical and parallel side walls. Satisfaction of the side-wall boundary condition in the far-field solution is acheived by the method of mirror images, with a closed-form expres- sion obtained of the resultant infinite se- ries. The inner expansion of the far-field solution dictates the asymptotic behavior of the near-field solution, thereby determining the near-field homogeneous component. This component accounts for the side-wall inter- ference in the inner region as well as the hydrodynamic interactions along the ship's length. Computed added-mass and damping coeffici- ents are presented for a half-immersed prolate spheroid. Validity of the proposed theory is confirmed by the comparison with the 3-D panel method for the special case of zero forward speed. Hydrodynamic forces acting on a ship model with forward and oscillatory motions are in most cases measured in a towing tank with limited width. When the forward speed and os- cillation frequency of a ship model are rela- tively small, reflection waves from the side walls of a towing tank affect the measured hydrodynamic forces: they must be different from what we expect for a ship model in open water. A diagram [1] is prepared for predicting whether the side-wall effect is expected or not, which gives the critical value of para- meter ~=Um/g (where U is the forward speed, the oscillation frequency, and g the gravi- tational acceleration) as a function of the ratio of tank width to ship length. This di- agram is, however, derived with heuristic ways and thus not entirely precise. There have been a few theoretical studies concerning the side-wall effects on hydro- 499 dynamic forces acting on a ship moving at a finite forward speed in waves. The pioneering work of Hanaoka [23 based on the thin-ship theory is a rational theory of the side-wall effects, but did not give reliable information on the effects. Hosoda ~3,4] and Takaki t5] are based on the strip-theory approach with a number of approximations in the mathematical evaluation of reflection waves from the side walls. Accuracy of their numerical results are therefore questionable despite their math- ematically complicated expressions. In the present paper, the slender-ship the- ory is applied to develop a rational method which is able to predict the effects of side- wall interference, particularly when a ship has a finite forward speed. The theory desc- ribed in this paper may be regarded as an tension of Newman's unified theory [6] to case of side-wall effects present. In the inner region close to the ship hull, since the side walls and the radiation condi- tion are absent, the inner solution can be identified with that in the unified theory for the open-sea problem. Namely it can be const- ructed by the superposition of the particular solution given by the strip theory plus a homogeneous solution giving three-dimensional effects. The latter component plays an impor- tant role in accounting not only for longitu- dinal flow interactions along the ship hull but also for the side-wall interference in the inner region. In the outer region far from the ship, the ship may be seen as a segment on the longitu- dinal axis, but the side walls are present. Thus the solution is represented by a line distribution of 3-D wave sources with unknown strength along the ship's length. The veloci- ty potential of 3-D wave source satisfying the side-wall boundary condition is derived by considering an infinite number of image singu- larities and a closed-form expression of the resultant infinite series. The source strength in the outer solution and the coefficient of a homogeneous component in the inner solution are to be determined from the asymptotic matching procedure. The implementation of the asymptotic matching bet- ween the inner and outer solutions leads to an ex- the

integral equation for the strength of 3-D wave sources, the solution of which then settles the coefficient of the near-field homogeneous solution; thereby completing the velocity po- tential necessary for the calculations of added-mass and damping coefficients. Solving the integral equation obtained re- quires the numerical evaluation of Cauchy's principal-value integrals involved in the ker- nel function representing the side-wall ef- fects. Numerical implementation of these in- tegrals is performed by firstly subtracting the singular behavior from the integrand, sec- ondly integrating analytically the subtracted singular part, and finally integrating numeri- cally the resultant non-singular part by means of an appropriate numerical technique. Clen- shaw & Curtis quadrature is employed in this paper with a tolerance of absolute error less than 10-5. Computational results are presented of the heave and pitch added-mass and damping coef- ficients, for a half-immersed prolate spheroid of length-beam ratio 8.0 advancing at a Froude number 0.1 in the waterway of width twice the spheroid's length. The appearance of side-wall effects is closely related to the wave pattern generated by an oscillating and translating ship and to its reflection from side walls of the water- way. Starting from the ring wave at ~=0, the wave pattern changes to the complicated one dominated by the diverging-wave component, as the parameter ~ increases across the critical value 1/4. Corresponding to this complicated variation of the wave pattern, the added-mass and damping coefficients including the side- wall effects show complex variations. In order to check the validity of the theory, numerical results for the special case of zero forward speed are compared with inde- pendent "exact" calculations based on the 3-D panel method, for example, by Kashiwagi [73. The results of the present theory agree excel- lently with the 3-D panel-method predictions. The present paper is restricted to the radiation problem of heave and pitch oscilla- tions, but the diffraction problem may be analyzed in a similar manner with the know- ledge of Sclavounos' diffraction theory t11] for the case of open sea, which is left for future work. 2. Formulation of problem As shown in Fig.l, we consider a ship in a waterway with vertical and parallel side walls. Let L, B. and d denote the length, breadth, and draft of a ship respectively, and BT the breadth of a waterway. The ship is as- sumed to move at constant forward velocity U and to oscillate sinusoidally with circular frequency ~ in heave and pitch; the depth of waterway is assumed deep enough, with no shal- low-water effect in the water-wave phenomena. A coordinate system used is shown in Fig.1. The x-axis is coincident with the centerline of the waterway and positive in the direction of the ship's forward velocity, the y-axis is Side Wall BT 1' L 'I B: ~ X ............................................................. Side Wall ~ AX z Fig.1 Coordinate system and notations horizontal, and the z-axis is vertical and positive downward, with the origin placed at midships and on the undisturbed free surface. Assuming the flow to be inviscid with irrota- tional motion, the flow field can be described in terms of the velocity potential ¢(x,y,z,t) satisfying the Laplace equation of the form ~ L ~ ~xx + tyy + fizz = 0 (1) in the fluid domain Z>O, IYI <BT/2. In order to justify the linearization of the problem, the amplitudes of ship's oscillatory motions are assumed small. Then the velocity potential can be linearly decomposed as = Ut-x+X~x,Y,z)] + Ref 3£ 5i(~)4jtj(X,y,z)e ~ (2) where x(x,y,z) is the steady perturbation po- tential due to the forward motion of the ship, and the second line represents the unsteady part of the velocity potential. Mode index j is used, with j=3 for heave and j=5 for pitch, and (j denotes the amplitude of the j-th mode of motion. Re means the real part of the expression to be taken and the time dependence el~t will be hereafter understood and deleted in the analysis. In this paper, we restrict the ship hull to be slender with small beam and draf~ compared to its length, and thus we are allowed on the free surface to neglect the contributions of steady perturbation potential. With this taken into account, the boundary conditions to be satisf fed by the normalized unsteady velocity potential ~j can be summarized as 500

[F] ~jz + K~j + i2l~jx - K tjxx [R] - ip( Ktj + il~jX ) = 0 (3) where K=w2/g, I=Um/g, Ko=g/U2 (4) [B] fjz ~ O as z ~ ~ (5) [A] fjy = 0 on y = +BT/2 (6) [H] Jan = nj + ~~ mj on ship hull (7) Here ~ in (3) is Rayleigh's artificial visco- sity coefficient to ensure the appropriate radiation condition [R] being satisfied. The subscript n in (7) denotes normal differentia- tion, with the unit normal vector defined positive when pointing into the fluid domain (see Fig.l), and nj is the components of the normal vector parallel to the xj-axis with extended definition of ns=znl-xn3. mj is the so-called m-terms representing the forward- speed effect due to the oscillatory motions in the steady flow, which has been originally derived in Timman and Newman [12] and can be explicitly written as (ml, m2, m3) = -(n V)VX (m4, m5, m6) = -(n V)(rxVX) (8) In order to obtain a solution of the above three-dimensional boundary-value problem, we exploit a slender-ship theory. In this theory, the flow field to be analyzed will be divided into the inner and outer regions, and in each region the governing equation and boundary conditions may be simplified, making it pos- sible to obtain the inner and outer solutions respectively with relative ease. However, both of these solutions include indeterminate coef- ficients, since nothing has been prescribed about the respective asymptotic behavior far away in the inner problem and close to the ship in the outer problem. These indetermi- nate coefficients can be settled by requiring the two solutions to be compatible in an over- lap region between the inner and outer fields. 2.1 The outer problem In the outer field far from the ship hull, the effects of three dimensionality and of side walls of the waterway must be accounted for. When the ship is seen from the outer field, it may be viewed as a segment on the x-axis, and thus the flow field is insensitive to the details of ship's hull geometry. There- fore the outer problem is defined by the 3-D Laplace equation [L], subject to the free- surface iF], radiation ~R], sea-bottom [B], and side-wall tW] boundary conditions, that is, (3)-(6~. Since these boundary conditions are homogeneous, the outer solution can be described in terms of the 3-D Green function with unknown wave sources distributed on the x-axis, in the form 501 ~j~x,y,z) = JLqj(~)G~X-t,y,Z)d; (9) Here G~x,y,z) denotes the Green function or the potential of a "translating-pulsating" source with unit strength, and q j(X) is the strength of the source to be determined. The Green function appropriate to the present problem can be derived by applying Hanaoka's approach t2], in which the method of mirror images is utilized. Considering the Fourier transform with respect to x of a unit source located at the origin and its mirror images with respect to both of the side walls of waterway, the Green function satisfying homogeneous boundary con- ditions (3~-~6) is given in the Fourier space as follows: G35(k;y z) = J. Gfx,y,z)eikKXdx _00 1 j~ ncostnKz)-vein~nKz) n .Z e-KIy-pBT|V~dn p=_= -~sgn~ 1+2kl) e ·£ e-iSgn~l+kl)~|y-pBT|/l_k2/v2 p=_= 1 _<z ~ -< | y-pBT| ik2 /V2 -1 ( 10) where v=(l+kl)2, <=Kv=K(l+kl)2 (11) and the upper and lower expressions in brack- ets are to be taken according as ~ k | <v and | k | >v, respectively. The values of k satisfying |k~=v give the branch points of the square-root function, which are written in the nondimensional form as kl 2 = -2/ ~ 1+21+VT;ZE) k3 4 = 2/(1-2l+Vi :~) 12) ( 13) Note that for l<l/4, k3 and k4 are real and positive, but complex-conjugate for l>l/4. At zero forward speed (I=O) k2 and k3 become -1 and +1 respectively, whereas k1 and k4 tend to respectively minus and plus infinity. In connection with the infinite series appearing in (10), we can obtain their closed- form expressions under the condition |Y|<BT/2 as follows: ~ e KiY-PBT~t = e-KIylt p=_= +cosh(QKy)~-l+coth( _ T)] (14) ~ e+iKIY-PBT~Q e+iKlylQ p=_a) +cos~ Y)[ KBT6(t' KBT) l_icot( 2 )](15)

The function appearing as the first term in brackets on the right side of (15) denotes the infinite series of Dirac's delta function, defined by ~ ' KBT) m- m6(Q KBT ~ (16) and therefore contributes only when Q=2nm/KBT. If we consider the limit of BT ~ ~ in (14) and (15), it is relatively easy to confirm that the second line on the right-hand side of each equation vanishes and only the first term remains. Thus the side-wall effects are repre- sented by the second line in (14) and (15~; this suggests that the Green function can be expressed in an addition form of the open-sea Green function Go~x,y,z) plus the side-wall- effect part GT(x,y,z). The final expression of the Green function can be given by consi- dering the inverse Fourier transform of the above expressions, with the following formula: Gfx,y,z) = Go~x,y,z) + GT(X,y~z) K lo {G*(k;y Z)+GT(k;y~z~}e~ikKXdk (17~ In the outer solution given by (9), the 3-D source strength q jinx) is unknown but will be determined by requiring the inner expansion of (9) to be compatible with the outer expansion of an appropriate inner solution. For this matching procedure, the inner expansion of the Green function must be sought. Following the method of matched asymptotic expansions, we seek the inner expansion with the following order of magnitude: Ky, Kz=O(~), k=0~1), ~<0~1) (18) Considering that the side-wall effects will be expected when the forward velocity and oscil- lation frequency of a ship are relatively small, the assumption on the order of para- meter ~ in (18) seems reasonable. It should be noted, however, that this assumption does not necessarily mean the applicability of the present theory is restricted to the range under the critical value given by ~=1/4. The transitional value of the parameter ~ where the theory becomes invalid may be determined from numerical computations and comparison of those with experiments. The analysis for the inner expansion of the open-sea Green function is identical to that in the unified theory devised by Newman [6], and hence with the present notations the desired result can be expressed in the form Go~k;Y,Z) = G2D(Y,Z) + ~ (l-Kz~fo~k) where + O(Kr~l-v),K2r2) (19) G2D(y,z) = Go(°;Y'Z) - ~ {(l-Kz)(logKr+y) + Kr~cosO+OsinO)}-i~l-Kz) (20) 502 fork) = logl 21 + Hi ~ ~ (cosh~~(lv~+nisgn: l +k~ )} L ~k2 / 2 { COS-\ ~ I V I )_~} (21) In (19) and (20), two-dimensional polar co- ordinates (r,6) are used with the relation (y,z) = (rsinG,rcosO), and y is Euler's cons- tant equal to 0.5772~. The expansion of the side-wall-effect part of the Green function can be obtained with comparatively simple reduction, in the form iE 1 * GT(k;y,z) = ~ (l-KZ)fT(k) where + O(Kr~l-v),K2r2) (22) fT(k) = J 2 2 (-l+coth; T ~ ' dn a,, -nisen(~+~1)( 2n ~ ~ ~ KBT) 1 + -i sgn ~ l+kl ~ cot ~<_k2 /V2 ~ } ~2/V2-l ~) _ (23) As in (10), the upper and lower expressions in brackets in (21) and (23) correspond to | k | <v and | k | >v, respectively. The final result of the inner expansion of the Green function will be obtained by substi- tuting (19) and (22) into (17) and performing the inverse Fourier transform. Regarding the open-sea Green function (19), the analysis for the inverse Fourier transform shown in Newman and Sclavounos [9] can be directly applied. Therefore we have Go~x,y,z) ~ 6(X)G2D(y~z)+ ~ (l-Kz~dx Fo(KX) where Fl(Kx)+F2(Kx) for x<0 Fo(Kx) = F2(KX) for x>0 (25) F1 (X) = ~-J =+lk je [~1 k2/ 2 -l]dk/k + Elk + El~i~k2X~) (26) 2( ~ 2 ~ JO +Jk je [~1 k2/ 2 -l] dk/k 2 Jk e take/ 2 1 -l]dk/k (27) Function El(z) in (26) denotes the exponential integral with complex argument. The expression (27) is applicable for ~<1/4, and should be understood for ~>1/4 with k3 = k4, where the wavenumbers kj ~ j=1~4) are given in (12) and (13~.

where The inverse Fourier transform of the side- wall-effect part (22) can be expressed as GT(x,y,z) ~ rr (l-KZ)dd FT(KX) (28) FT(X) = 2~ ,: e ikX dk/k ·: 2n 2 { -l+coth( 2T~)} ins +kZ + ) Z · Z m - sGn(i+kT) e-ikX 2 KBT p-1 m_O [kldv/dk-k/vl ~k=kpm 2 [ ~k2+:k4] IT · cot( 2 1//l-k2/v2)dk/k i ~ {k2 (key -ikX 1 + 2 L Jkl Jk3J ,/k2/V2-l . coth(2ik2/v2-1 )dk/k [ ;- Jk2 Jk4] vS~ dk/k ~ [ Jk2 Jk4] ikX 2/V2_l / (29) and ~ 0 = 2 ~ Em = 1 (miO) (30) The first term on the right-hand side of (29) represents the contribution of non-radiating local waves. The second term is obtained from the infinite series of delta function in (23) and physically the contribution of the out- going waves at infinity. In this second term k m denotes the values satisfying KBT~ =5Trm (m-O,1,2,··), which exist at most four in number and when m=0 coincide with kj ( j=1~4) given in (12) and (13). A part of the radiated waves are reflected on the side walls, and changed in phase by the factor rr/2 and represented as the third term in (29). In other words from the standpoint of hydrodynamic-force calculation, the third term is originally to be contributed to the damping force, as is the same as the second term, but by the phase shift of ~r/2 due to the side-wall effects, this term will contribute to the inertia force. The integrals in the third term must be treated as Cauchy's princi- pal-value integral at the points of k sat- isfying sin(KB7~/2)=0, namely at k=kpm defined in the second term. The fifth and sixth terms in (29) are independent of BT and thus to be cancelled out by some terms given in the open-sea expres- sions (25)-(27). However as discussed previ- ously in connection with the infinite series of (l4) and (15), these two terms play a role in cancelling out respectively the third and fourth terms of (29) in the limit of BT ~ Go. When ~>1/4, the expressions from third to sixth terms should be understood with k3=k4. Since we got the desired inner expansion of the Green function, by substituting it in (9) we have readily the inner expansion of the outer solution in the following form: ~j(x,y~z' ~ qj (X)G2D(Y'Z) - ~ ( 1-Kz )lLqj (A )dE,: [Fo{ K(x-: )} + FT{K(x-E,)} id: (31) 2.2 The inner problem Since the ship hull is assumed slender, changes of the flow in the x-direction are small in the region close to the ship hull, by comparison to changes in the transverse plane. Thus the flow in the inner region may be described by the 2-D Laplace equation subject to the free surface condition which is inde- pendent of forward velocity and applicable to the 2-D problem in the y-z plane; this can be mathematically justified by the coordinate stretching argument with the assumption of x=O(1), y=O(~), and z=O(~). In the inner prob- lem, the radiation condition and the side-wall boundary condition can not be specified, be- cause the side walls are absent in the inner region. With these taken into account, the boundary-value problem can be written as [L] tjyy + Jazz = 0 for z>0 (32) [F] 4jz + Kjj = 0 on z=0 (33) tH] IN = Nj + 7.U Mj on ship hull (34) Here we note that the subscript N in (34) de- notes the normal differentiation on the sec- tional contour in the 2-D transverse plane, and Nj and M denote the slender-body approxi- mation of tie 3-D quantities in (7) and (8), which are explicitly given as N5 = -xN3 ~ M3 = -N2Xzy~N3Xzz ~ (35) M5 = N3-XM3 J The inner boundary-value problem defined from (32) to (35) is the same as the conven- tional 2-D formulation except that the radia- tion condition is absent. Thus the inner solution can be identified with Newman's uni- fied-theory solution t6], composed of a par- ticular solution commonly used in the strip theory plus a homogeneous solution multiplied by a three-dimensional-effect coefficient. To be more specific, (X;y~z) =ijP(y z) + U9P( ) + Cj(x) ~jH(y,z) (36) 4J(Y,z) = ~J(Y'Z)- ~j(Y,Z) (37) where the overbar~denotes the complex conju- gate, and Liz and 98 are the particular solu- 503

Lions determined to satisfy the following boundary conditions on the body profile at station x: P ED 4jN = Nj ~jN = Mj (38) The coefficient of the homogeneous solution Cj (x ~ in (36) is indeterminate at this stage, but may be settled by matching the outer expansion of (36) with the inner expansion of the outer solution already given by (31~. Far from the ship hull in the inner region, (36) reduces to ~x;y,z) ~ [Oj(X)+1,U ~j(X) + Cj~x)~i~x)-~j(X)) iG2D(Y,Z) - e cos(Ky) 2iCj (x)oj (x) (39) Here ~j(X) and Dj(X) denote the 2-D effective source strengths; these can be given by sol- ving the 9~- and -problems respectively. G2D(y,z) is the 2-D Green function and iden- tical to the one shown in (20) or (31~. 2.3 Matching In the analysis described above, the un- knowns are the 3-D source strength q j (x ~ in the outer solution and the coefficient Cj~x) of a homogeneous component in the inner solu- tion. These will be determined by the match- ing of the inner and outer solutions. Compar- ing (31) with (39) and equating the factors of G2D, the following relation can be found: Ajax) = ~j(X)+'U ~j(X)+Cj~x){oj~x)-oj~x)l ~ Equating the remaining terms in (31) and (39) gives i2rrCj~x)Oj~x) = ~Lqi(~) · d: [For Kfx-t )} +FT[ Kfx-` )} id: (41 ) with the error of order O(K2r2 ). Eliminating Cj (x) from (40) and (41 ) we have an integral equation for the 3-D source strength q jinx) of the form q jinx) - 2~ t(>j~x)/oj~x)-l]J.Lqi(~) · d<; ~Fo{Kfx-E~}+FT{Kfx-E~ ids = ~j(X)+~U Ajax) (42) Once q jinx) is determined by solving (42 ~ with an appropriate numerical method, the coeffi- cient Cj~x) can be readily determined from (40) and thus the inner solution will be completed. In the case of no side-wall effects, i.e. BT ~ ~ the function F - Kfx-~} becomes zero as already mentioned, and the integral equa- tion (42) reduces to the corresponding one in Newman's unified slender-ship theory in the open-sea case [63. In the special case of zero 504 forward speed, Kinoshita and Saijo [8] derived an analogous equation to (42 ~ in the study on a multi-body-type floating breakwater, consis- ting of an infinite array of slender bodies. The inner solution (36) appears formally to be invariable regardless of whether the side walls are present or not. However, through the 3-D source strength qj~x), which includes the side-wall effects as a solution of (42) the coefficient of homogeneous solution Cj (x ~ accomodates not only the 3-D interaction ef- fects between transverse cross-sections but also the side-wall effects of the waterway. 3. Added-mass and damping coefficients Since the inner solution has been deter- mined, we proceed to the calculation of hydro- dynamic pressure force and moment acting on a ship with forced heave and pitch motions. The linearized hydrodynamic pressure is given from Bernoulli's equation. Then the hydrodynamic force in the i-th direction due to the j-th mode of motion can be provided by integrating the pressure over the mean wetted surface of the ship hull, and can be expressed in terms of the added-mass (Aij) and damping (Bij) coefficients, in the form Pi = - 3 5~(iw)2Aij + (i~)Bij}`j ' (i=3,5) (43) ~..~, ~ , Aij+Bij/itl) = J Ltaij(X)+bij~x)/im~dx (44) a i j +b i j l it) = ~ P :CNi ~id' + ip ~ ~ ~ (Ni(j)j-~i~j Do - p (`~ ~ ~ Ni~jdQ - pCj~x)l (Ni-~ ~i)( $j~ $J)di ~ Here p is the fluid density, and in deriving the above, Tuck's theorem t13] has been used. aij and bid defined in (44) and (45) denote the 2-D added-mass and damping coefficients respectively, involving the 3-D interaction effects and the side-wall effects, and c to the integral sign in (45) denotes the submer- ged portion of the contour of the transverse section. In order to perform the calculations of (45), the term M3 defined TAX (35) and the related velocity potential 95 must be known, besides the velocity potential 93 commonly calculated in the strip theory. If 95 and 95 are obtained, the remaining velocity poten- tials for pitch (j=5) follow from (35) 95P=-x93P ~ (46) 95 = $3 -X93 Sclavounos [lO] studied in the open-sea case the relative importance of the contribu- tions frog the M3-term and related velocity potential 95, by comparing the numerical re- sults with experiments. His results reveal that the inclusion of the M3-term leads to a

substantial overprediction of the damping co- efficients. This overprediction may be at- tributed to the inaccuracy of the m-terms near the ship ends, which have been evaluated with slender-body approximation. Therefore the m- terms should be evaluated from the 3-D precise calculation for the steady perturbation poten- tial. Fortunately, according to his numerical study, a better agreement with experiments is provided by simply omitting the m-terms in the unified theory. Thus in the numerical calcu- lations with side-wall effects presented here too, it was decided to neglect the M3^term and consequently the velocity potential 95 in (45~. It should be noted that the last term in (45), multiplied by the coefficient Cj~x), plays an important role in accounting for the unified-theory corrections in the open-sea case and for the effects of side-wall inter- ference in the presence of waterway. Without this last term, the remaining expressions in (45) are identical to those in the strip theory. 4. Numerical calculation method An improtant task in the present theory is to solve the integral equation (42) for the 3-D source strength q (x). For this purpose, after dividing the swipes longitudinal axis into NX segments of equal length, the 2-D boundary-value problem for heave (j=3) in each divided transverse plane must be solved; which gives 03(X) necessary in calculating the right side of (42~. Since we neglect the contribu- tion of steady perturbation potential, 93(X) becomes zero, and the 2-D effective source strength for pitch (j=5) can be evaluated directly from o3(X), with the result of o5(X) =-xc3 (x) and oryx)= 03(x). The 3-D source strength q jinx) , which is to be determined, has been assumed to vary linearly in x between adjacent nodal points, with the value of fix) at station x=xk denoted by qk where ~ (x-xk 1 ~ / (Xk-Xk- 1 ~ Xk- 1 <X<Xk Ak~x) = ~ (x-xk+1 )/ (xk-xk+1 ~ xk<x<xk+1 l O elsewhere (48) qfx) = k£2Akfxjqk ~ ~ With this approximation, the integral on the left-hand side of (42) is analytically evalu- ated over each segment. The remaining task after this procedure is the numerical evalua- tion of the integrals associated with the two functions, Fo{Kfx-~) and F/Kfx-~}. In re- lation to the third term in Fig Kf x-; ~ } given by (29), Cauchy's principal-value integral must be numerically evaluated. In the numeri- cal implementation of this integral, singular contributions are subtracted from the integ- rand and analytically integrated. For this manipulation we have used the following integ- ral formula: ~ ~)/dk-k/v v dk J sink Tig7~) ~ = KB logltan K4BT ~1 (49) The resultant non-singular integral is evalu- ated using Clenshaw & Curtis quadrature with an absolute convergence requirement of 10-s applied. This quadrature is known to give an accurate result even for highly oscillating functions with less computing time. Other single integrals, appearing in (29) and in (27) for the open-sea kernel function, are evaluated similarly with Clenshaw & Curtis quadrature, with appropriate truncation cor- rections based on asymptotic expansions of the integrand for large values of valuable k. The double integral appearing as the first term in (29) is well-behaved, since its integ- rand rapidly reduces to zero as the radial distance in the n-k plane increases. Thus the evaluation of this integral is performed also by means of Clenshaw & Curtis quadrature for the double integral, with an absolute error less than 10-s required. After all these procedures, the integral equation (42) is transformed into a linear system of simultaneous equations for the dis- cretized values of qua) at NX-1 nodal points. (The source strengths at end points k=1 and k= NX+1 have been postulated to vanish and thus treated as known.) The simultaneous equations obtained are solved by the matrix inversion with Gauss' elimination method. Once the solution of the 3-D source streng- th qj~x) is obtained, it is straightforward to calculate from (40) the coefficient Cj~x) of the near-field homogeneous solution and from (45) the 2-D added-mass and damping coeffici- ents. The final results of the 3-D added-mass and damping coefficients can be obtained from (44), using the strip-wise integration along the x-axis. 5. Numerical results and discussion . . . 5.1 Results of forward-speed problem Numerical computations are performed for a prolate spheroid of length-beam ratio L/B=8, moving at the Froude number Fn=O.1 and oscil- lating sinusoidally with circular frequency in the waterway of width 16 times the ship's breadth, i.e. BT/B=16. These ratios of ship and waterway dimensions are selected to cor- respond to a typical experiment situation, where a ship model of L=2 m and B=O. 25 m is towed at the velocity U=0.443 m/s in a towing tank of 4 m in width. The heave added-mass (A33) and damping (B33) coefficients are shown respectively in Fig.2 and Fig.3, in the nondimensional form against the nondimensional wavenumber KL. With a similar style of representation, the added- moment of inertia (Ass) and damping (Bss) coefficients in pitch are presented in Fig.4 and Fig.5, respectively. The nondimensional 505

40E 2.0 1 n ^3s~V _ in 0~ See _- _ - - S ~ ~ ~ i~ ELI ~ (-e=le} G.0 Slender Ship Theory 4.0 n , . ~ ~ O KL 15 Fig.2 Heave sided mass of a prolate spheroid (~/B=8) at Fn=O.1 in waterway of B~/B=16 ^~V~ 0.10 0.05 -0.05 _ Elate Sphe~ld ~8 = e 81 Fn =0d In Open See ~ ~ 0 4 --------- Slender Shlp Theory ( ~ ~ ) Slender Shlp Theory O . , , , , I 5 0.3 0.2[ Fig.4 Pitch added moment of inertia of a prolate spheroid (~/B=8) at F~=O.1 in waterway of B~/B=16 506 F Ba3/~V/~7[ Fig.3 Heave damping coefficient of a prolate spheroid (~/B=8) at Fn=O.1 in waterway of B~/B=16 s~V~ . ~ ~ `, Fig.5 Fitch damping coefficient of a prolate spheroid (L/B=8) at Fn=O 1 in waterway of B~/B=16

form of these coefficients are displayed in the ordinate of each figure. In all of these figures, thick solid lines indicate the numerical results in the presence of side-wall effects, computed by the slender- ship theory described in this paper. In order to show the magnitude of side-wall effects, the values in open sea are shown by short- dashed lines, which were obtained with the side-wall-effect part of the kernel function Fly Kfx-; )} set to be zero in the integral equation (42), and therefore must be identical to the unified-theory solutions t94. Also shown in the open-sea case are the strip-the- ory predictions, which are indicated by dash- dotted lines. Comparing the predictions of the strip theory with those of the unified slender ship theory, we can understand that the ef- fects of three dimensionality are prominent only in the low frequencies. Since the forward velocity is present, with the incensing wavenumber KL, the parameter =Um/g-FniKL increases and takes the critical value ~=l/4 at KL=6. 25. The position of this critical wavenumber is shown by the vertical thin solid line with a downward arrow. In the frequencies less than ~=1/4, the effects of side walls are considerable not only in heave but also in pitch modes. In particular, Ass and Bss change drastically in the frequency range slightly less than the critical frequen- cy ~=l/4, and Ass takes a negative value. It should be noted, however, that the damping coefficients B33 and Bss predicted by the present theory are definitely positive, al- though they vary greatly in magnitude and become nearly equal to zero at some frequen- cies. This non-negative damping force seems quite reasonable, judging from the considera- tion on the energy flux radiating in the longitudinal direction of the waterway. In some published results by a heuristic method [5], negative damping-force coefficients are predicted in the low frequencies; this is not the case. It is known in the case of zero forward velocity that the wavenumbers corresponding to the tank-resonant mode in the transverse di- rection can be given by KBT=2nm (m=1 , 2, · · ~ and thus in the present case by KL-nm ; at which the ratio of wavelength to tank width is equal to the inverse of an integer. When the forward velocity is present, the wavelength of the wave component radiating in the transverse direction is diminished in comparison to the wavelength at U=O, due to the effects of forward speed. With this knowledge, we can observe particularly in the range of ~<1/4 that the tank-resonant frequency is shifted to the lower frequency than the zero-speed tank- resonant frequency given by KL=nm. The wave pattern generated by a ship with forward and oscillatory motions in open sea is known to change drastically, dependent on the value of ~ [143. In particular for ~ close to but larger than 1/4, the angle of the sector in which no radiating waves exist increases rapidly from zero to more than 90 degrees. This leads to the conjecture that, in the range of ~1/4, there exist the short waves which originate from the cusp part of the wave pattern and propagate in the transverse direc- tion of the waterway. These waves reflect on the side walls and may exert a complicated influence on hydrodynamic forces on a ship. In the numerical results of the added-mass and damping coefficients shown from Fig.2 to Fig.5, we can observe fast variations in the short range of the wavenumber approximately between KL=7.3 and 7.8. The parameter ~ cor- responding to these wavenumbers takes the values ranging from 0. 27 to 0. 28. Therefore the fast variations in the added-mass and damping coefficients might be attributed phys- ically to a contribution of short waves origi- nating from the cusp part of the wave pattern. As the motion frequency increases across the range where the fast variations occure, the effects of side walls gradually decrease, and the added-mass and damping coefficients reduce to the corresponding values in open sea shown by short-dashed lines around the nondimensional wavenumber KL=14. O. In this range, i.e. between KL=8. 0 and 14 .0, the di- verging-wave component may be dominant in the side-wall effects on hydrodynamic forces on a ship. We have a conventional diagram [1] which can be used to judge whether the side-wall effects are expected or not, by means of the parameter ~ and the ratio of tank width to ship length BT/L. In the present calculations the ratio of BT/L is 2.0 and thus the critical angle of a sector, Fc' is given as Oc = tan~: (BT/L) = 63.4 deg., where the critical angle Fc is determined geometrically such that the wave emitted from the ship bow will strike the afterbody of the ship by the reflection from tank walls. This critical sector angle, on the other hand, is estimated from the calculations of the wave pattern generated by an oscillating and trans- lating source [14], and is depicted in the diagram as a function of ~. Using this diag- ram with the critical angle Fc as the input, we get ~=0.365 as the predicted critical fre- quency. In the frequency range lower than this point the side-wall effects will be expected. For Fn=O.l, the value ~= 0.365 gives the crit- ical wavenumber of KL= 13.3. Looking at the computed values shown in Figs. 2-5, this crit- ical wavenumber turns out to be a good approx- imation. 5.2 Accuracy check and validation The items to be checked for the accuracy of the present calculations are the 2-D solution in the transverse plane, numerical evaluation of the kernel function (25~-~27) and (29), and the solution of the integral equation (42~. Since the 2-D boundary-value problem is well posed, no discussion is needed. Haraguchi & Ohmatsu's method [15] is utilized in the present work, which easily get rid of irregu- lar frequencies and give an accurate solution. 507

Numerical integrations in (25~-~27) and (29) are, as described earlier, performed using Clenshaw & Curtis quadrature, with an abso- lute convergence requirement of 10-s applied. Therefore the remaining thing to be checked is the accuracy of the solution of integral equa- tion (42~. Fig.6 presents the added-mass and damping coefficients when the number of divisions in the x-direction NX was changed from 10 to 70, under the same computational conditions as in Figs. 2-5. Computed results are plotted with the values of NX=70 set to 1.0. The upper results in Fig.6 are for KL=5.0 (~=0.224<1/4), and the lower ones are for KL=10.0 (~=0.316 >1/4~. For ~=0.224, all computed coefficients ap- pear to converge as the number of division increases, although the coefficients associ- ated with the pitch mode dictate a finer dis- cretization relative to that necessary for the heave-mode calculations. (Here we note that the relative error in Bss might be noticeable but its absolute error is not so large, be- cause the value itself is small at KL=5.0 as seen in Fig.5.) Prolate Spheroid of L/B=8 in waterway of BT/B=16 1 If _ ~ r 1.0 0.95 1.05 1.0 - 0.95 B55 ,' "` $,, " : B33 "` \ , ~ ~ - ~ ~ / . . . . 10 30 K L = 5.0 Fn = 0.1 ~ = 0.224 Values of NX=70 are set to 1 .0 l 50 NX 7 0 "~x B55 B33 .-------+ ~ _ A55 1_ . , , , 10 30 K L = 1 0.0 En =0.1 ~ =0.316 Values of NX=70 are set to 1 .0 A 70 Fig.6 Number of division in the x-direction (NX) vs. added-mass and damping coef- ficients of a prolate spheroid, at KL= 5.0 (upper) and KL=10. 0 (lower). Results are plotted with the values at NX=70 set to 1.0. 508 For ~=0.316, the results at NX=60 are slightly different from those at NX=50 or NX=70 with relative error of approximately 1.0 %. This suggests that the solution of the integral equation (42) tends to be unstable as the value of ~ increases beyond 1/4. Sclavou- nos t10] has found this kind of instability occurs in the unified theory for large values of I, and proposed an alternative scheme, using the Chebyshev-polynomial expansion for the unknown source strength. However in the range of ~ calculated here the instability seems not so serious, and Fig.6 reveals that 40 segments along the ship's length are suffi- cient to give a solution with relative error less than 2.0 %. On this basis, all of the computations shown in this paper have been carried out with NX=40. In the special case of zero forward veloci- ty, 3-D "exact" calculations based on the integral-equation method may be available, with the Green function modified to satisfy the side-wall boundary condition. A 3-D cal- culation method of this kind has been develop- ed by Kashiwagi ~7], in which almost perfect agreement is shown between the calculated and experimental values for a hemisphere and a ship model with fore-and-aft symmetry. If we compare the zero-speed results computed by the present theory with the corresponding ones by the 3-D integral-equation method, the valida- tion for less complicated case can be accomp- lished. Fig.7 and Fig.8 present respectively the heave added-mass (A33) and damping (B33) coefficients, for a prolate spheroid of L/B=8 floating at zero forward speed in the waterway of BT/B=16. Similarly Fig.9 and Fig.10 show the pitch added-moment of inertia (Ass) and damping (Bss) coefficients under the same conditions. In these four figures, the same scale of the ordinates and the same line sym- bols are used as those in the corresponding figure for the forward-speed case shown from Fig.2 to Fig.5. Also included in Figs. 7-10 are the results of 3-D integral-equation (panel) method, which are presented by plus symbols for the case of open sea and by open circles for the case of side-wall effects present. A certain amount of inaccuracy should be expected in the results of 3-D panel meth- od, too. However numerical accuracy is be- lieved to be fairly good, because the hull surface of spheroid and the normal vector on it can be mathematically given and some ana- lytical manipulations are thus used to improve t'ne numerical accuracy. We see that very good agreement exists between the results of the slender-ship theory and of the 3-D panel method, showing the validity of the present theory. We proposed a new rational theory for predicting the hydrodynamic forces on a ship, moving at constant forward speed and oscilla- ting in heave and pitch in a restricted water-

1 o - -1 s Prolate Spheroid L/B=8 at U=0 in Open Sea - St rip Theory ---- Slender Ship Theory + 3-D Panel Method with Side-Wall Effect ( BT/B=16 ) i? - Slender Ship Theory 'TV I ~~- ~ 1 0 3-D Panel Method 7_ . C~ BI ~ / KL Fig.7 Heave added mass of a prolate spheroid, the same as Fig. 2 except for Fn=O.O. Comparison with the 3-D panel-method predictions. 0.05 6 )0 2 ~ J All 15 Ass/PVL Prolate Spheroid L/B=8 at U=0 B55/pVL 0.1 5 -. in Open Sea | - Strip Theory ~ ~ Slender Ship Theory 0.10 j + 3-D Panel Method \ with Side-Wall Effect (BT/B=16) i. Slender Ship Theory 0 3-D Panel Method 0 2 ~'~ I;__ O _ 0.3 10 KL 15 Fig.9 Pitch added moment of inertia of a prolate spheroid, the same as Fig. 4 except for Fn=O. O. Comparison with the 3-D panel-method predictions. 10 KL Fig.8 Heave damping coefficient of a prolate spheroid, the same as Fig. 3 except for Fn=O. O. Comparison with the 3-D panel- method predictions. Hi 1 1 10 KL 15 Fig.10 Pitch damping coefficient of a prolate spheroid, the same as Fig. 5 except for Fn=O. O. Comparison with the 3-D panel-method predictions. 509

way. Only the slenderness of the ship hull is assumed, and thus the proposed theory is valid for all frequencies and forward velocities of practical interest where the side-wall effects are prominent. Furthermore the theory is cor- rect even for the case of a narrow waterway, because the side-wall effects are taken into account not only on outgoing waves but also on evanescent local waves. Validity of the proposed theory is confirm- ed for the special case of zero forward velo- city by comparison with the numerical results of 3-D panel method. In the case of non-zero forward velocity, however, the present theory may be the first one which is able to give precise predictions of the side-wall effects on hydrodynamic forces. Therefore with this theory, we are ready to make quantitative discussions on the effects of tank-wall inter- ference included in the results of experiments for a ship model. Computations were performed for the heave and pitch added-mass and damping coefficients of a prolate spheroid of length-beam ratio 8.0, moving at the Froude number 0.1 in the waterway of width twice the spheroid's length. The computed hydrodynamic forces show complex variations as the frequency increases. It is noted that these variations correspond to the appearance of complicated wave pattern, which starts from the pattern dominated by the ring wave and changes to the markedly different one dominated by the diverging wave, as the param- eter l=mU/g increases across the critical val- ue 1/4. References 1. Vossers, G. and Swaan, W.A., "Some sea- keeping tests with a victory model", I. S. P. Vol.7, No.69, pp.189-206 (1960~. 2. Hanaoka, T., "On the side-wall effects on the ship motions among waves in a canal", J. Soc. Nav. Arch. Japan No.102, pp.l-5 (1958~. 3. Hosoda, R., "Side wall effects of towing tank on the results of experiments in waves (1~", J. Soc. Nav. Arch. Japan No.139, pp.23-30 (1976~. 4. Hosoda, R., "Effect of side-wall inter- ference of towing tank on the results of experiments in waves (2~", J. Soc. Nav. Arch. Japan No.143, pp.52-60 (1978~. 5. Takaki, M., "Effects of breadth and depth of restricted waters on longitudinal mo- tions in waves", J. Soc. Nav. Arch. Japan No.143, pp.173-184 (1979~. 6. Newman, J.N., "The theory of ship motions" Adv. Appl. Mech. Vol.18, pp.221-283 (1978~. 7. Kashiwagi, M., "3-D integral-equation me- thod for calculating the effects of tank- wall interference on hydrodynamic forces acting on a ship", to be published in J. Kansai Soc. Nav. Arch. Japan No.212 (1989~. S. Kinoshita, T. and Saijo, K., "On the mul- ti-body-type floating breakwater", J. Soc. Nav. Arch. Japan No.149, pp.54-64 (1981~. 12 . 510 9. Newman, J.N. and Sclavounos, P.D., "The unified theory of ship motions", Proc. 13th Symp. on Nav. Hydrodyn. Vol.4, pp.1-22 (1980~. 10. Sclavounos, body theory: 15th Symp. (1984~. 11. Sclavounos, P.D., "The diffraction of free surface waves by a slender ship", J. S. R. Vol.28, No.1, pp.29-47 (1984~. 12. Timman, R. and Newman, J.N., "The coupled damping coefficients of symmetric ships", J. S. R. Vol.5, No.4, pp.34-55 (1962~. . Ogilvie, T.F. and Tuck, E.O., "A rational strip theory for ship motions", Dept. Nav. Arch. Mar. Eng., Univ. Michigan, Rep.No.13, pp.1-92 (1969~. 14. Hanaoka, T., "On the velocity potential in Michell's system and the configuration of the wave ridges due to a moving ship", J. Zosen Kiokai, No.93, pp.l-10 (1953~. 15. Haraguchi, T. and Ohmatsu, S., "On an im- proved solution of the oscillation problem on non-wall-sided floating bodies and a new method for eliminating the irregular frequ- encies", Trans. West-Japan Soc. Nav. Arch., No.66, pp.9-23 (1983~. P.D., "The unified slender- ship motions in waves", Proc. On Nav. Hydrodyn., pp.177-192

DISCUSSION by C.M. Lee I think that it is the first paper which presented a 3-D theory for an oscillating ship, including the side-wall effects. The authors should be congratulated for their excellent work. Similar to the case of twin-hull ships, this paper shows negative added mass at certain frequencies. This kind of phenomenon does not occur in the open-sea case, and, therefore, induces puzzlement to those who cannot accept the motion of "negative added mass". My advice to those people has been that one should not get too excited by just observing unusual hydrodynamic coefficients alone but should wait until the computed results of ship motion in waves are shown. My prediction is that although the hydrodynamic coefficients may look quite different from those of the open-sea case, the motion results may not show significant differences, particularly for the tank width being twice the ship length as chosen in the sample calculations in this paper. I would like to encourage the authors to compute the ship motion to check if my prediction is correct. Author's Reply Thank you for your comment. We are now applying the proposed theory to the diffraction problem with side-wall effects. If the calculations of wave-exciting force and moment are completed, the ship motion in waves can be readily computed from them, using the added-mass and damping coefficients predicted by the present theory. Therefore, I think that the computed results of ship motions in waves can be shown in the foreseeable near future. 511