Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 A Flow Model for a Displacement-Type Fast Ship with Shallow Draft in Regular Waves M. Kashiwagi (RIAM, Kyushu University, Japan) ABSTRACT A linearized 2-D boundary-value problem is studied for the flow around a displacement-type shallow-draft ship advancing and oscillating in waves. In this problem, a homogeneous solution of the wave elevation exists satisfying zero vertical velocity on the free surface, which is introduced as a homogeneous component in the expression of the wave elevation. With this equation used as an in- tegral equation for the pressure distribution associ- ated with the disturbance of a ship, we can satisfy the Kutta condition requiring a smooth flow at the stern in addition to the kinematic condition of wa- ter surface being equal to the vertical position of a ship. An accurate numerical solution method is also presented, using-Chebyshev polynomials for the un- known pressure and employing a Galerkin scheme. Excellent performance of this method is confirmed by checking numerically Hanaoka's reciprocity theo- rem and the energy conservation principle. Through comparison of the results between satisfying and not satisfying the Kutta condition, it is confirmed that computed results based on the present flow model are reasonable. INTRODUCTION Hydrodynamic problems of a shallow-draft ship oscillating in waves with forward speed are of im- portance not only as actual engineering problems for developing high-speed ships but also as a mathe- matical boundary-value problem concerning the ex- istence of a unique solution. In the unsteady problem for a displacement- type ship, there is a lingering question associated with the so-called line-integral term at the intersec- tion of the body and free surfaces, which has made it difficult to get a reliable solution. When a ship is of shallow draft and thus can be represented by the pressure distribution on the water surface, no line- integral problem arises, thereby making the mathe- matical treatment relatively easy. However, there exists another problem peculiar to a planing boat (Bessho, 1977~; not only the nor- mal velocity but also the elevation of water surface must be specified on the bottom of a planing boat, and in addition the Kutta condition of smooth flow at the stern must be satisfied as in the airfoil theory. Theoretical investigations on this boundary-value problem have been made by Bossho (1992) in two dimensions. Bessho has pointed out that a solution is unique if the elevation of water surface is speci- fied as the body boundary condition; thus there is no room to impose the Kutta condition, unless the wetted length changes as in a gliding plank (Bessho & Komatsu, 19844. Therefore, for a displacement- type ship which is nearly vertical at the bow, it is impossible to satisfy both the kinematic body- boundary condition and Kutta condition. In order to surmount this difficulty, Bessho pro- posed a flow model (Bessho, 1992), in which a source singularity is added at the bow intersection, corre- sponding physically to a flow dammed in front of the bow and then streaming out along the bottom of a ship. However, computed results (Kashiwagi & Bessho, 1993) based on Bessho's flow model ap- peared eccentric in the magnitude of obtained phys- ical quantities. Therefore, we need to reconsider the flow model and propose a method enabling us to impose both of the kinematic condition of water surface being equal to the vertical position of ship's bottom and the Kutta condition of smooth flow at the stern. A new flow model proposed in this paper is physically similar to Bessho's (1992) or another flow model proposed by Bessho & Suzuki (1986), but dif- ferent in mathematical expressions. Since the ho- mogeneous free-surface condition must be satisfied, no additional terms can be allowed in the expres- sions of the velocity potential and the vertical ve- locity on the free surface. However, a homogeneous wave elevation satisfying zero vertical velocity may

t k4 k3 k. , ~ k2 In,_ ~ X ~=21 , TV Figure 1 Coordinate system and schematic representation of generated waves on the free surface be introduced in the expression of the wave eleva- tion, which is used as an integral equation for the unknown pressure distribution. The coefficient of this homogeneous wave elevation is determined by imposing the Kutta condition at the stern. The integral equation is solved by representing the unknown pressure distribution with Chebyshev polynomials and employing a Galerkin method to enhance the numerical accuracy. Excellent numeri- cal accuracy is confirmed through the check of theo- retical relations derived from Hanaoka's reciprocity theorem (Hanaoka, 1959) and from the energy con- servation principle associated with the damping co- efficients in heave and pitch modes. Computed re- sults are shown for the pressure, wave profile, hydro- dynamic forces, and wave-induced motions, which seems reasonable both in magnitude and variation tendency, implying the validity of the present flow model. THEORETICAL FORMULATION With the coordinate system shown in Fig. 1, we consider a 2-D shallow-draft ship which is advancing at constant velocity V while oscillating with circu- lar frequency of encounter we. In the analysis to fol- low, all physical quantities are nondimensionalized in terms of the velocity V, the half length of a ship = L/2, the gravitational acceleration g, and the fluid density p. Consequently the nondimensional parameters K = V2 ~ ~ = V ~ ~ = K = e (1) will be used, where ~ is known as the reduced fre- quency and ~ is Hanaoka's parameter. The fluid is assumed to be inviscid with ir- rotational motion, introducing the velocity po- tential, and the boundary conditions are lin- earized. All first-order quantities are assumed to be time-harmonic, with time dependence written by exp~i~et). Then the relations to be satisfied on y = 0 among the velocity potential ¢(x,y), the wa- ter surface elevation Next, and the pressure pox) are expressed as (it—~ )~)(x'O) +Kr1(x) =—pax), (2) ,3y¢~(X,O)= (it- i3~)71(X). (3) Here the pressure pox) on the free surface is associ- ated with the disturbance of a ship. Eliminating Next from (2) and (3) gives ( 0~ ) BY ( 0~ ) ~ on y= 0 (4) A solution satisfying (4) and the radiation con- dition at infinity is given by ~1 onyx ye = J ply, F(x—A, y) did, (5) -1 where `3 \ F(x, y) = iw—~, J Gas, y), (6) 1 t°° elkly—it Gfx,y) = Hi ~mO) Ok+—ip)2—K~k~ ~ ~ Gfx,y) is called the Green function, physically the velocity potential due to the source singularity of unit strength, which satisfies (it—~ ) F(x, O_ ~ + KGy~x' O_ ~ =—5;`xy <8' in the limit of y ~ 0_, where (6) has been used and 6(x) denotes Dirac's delta function. From (2), (3), (5) and (8) the following relations may be obtained: ~ 0(x,O) = | p(~)Fy~x—(,O_)d(, (9) ~1 r/(x) = ~ p(~)Gy~x—(,O_)d(. (10) -1

Equation (9) is an integral equation for the un- known pressure distribution, with the normal ve- locity 0<b/0y specified on the bottom of a flat ship. Since the kernel function Fy~x-(, 0_) has the singu- larity of 1/~r~x—(), a homogeneous solution exists, as well known in the airfoil theory, which makes it possible to satisfy the Kutta condition requiring a smooth flow at the stern. In this case, however, the resulting surface elevation at X ~ 1 may not be equal to the vertical position of ship's bottom, which must also be satisfied in the problem of a flat ship. On the other hand, (10) can also be regarded as an integral equation for the pressure distribu- tion if the vertical position of ship's bottom, ~(x), is specified on y = 0. However, as noted by Bessho (1977), this integral equation gives a unique solu- tion because the kernel function Gy (x—if, 0_ ~ has a logarithmic singularity; hence the Kutta condition at the stern can not be imposed. As is clear from (3), the wave elevation re- lated to the homogeneous solution of (9) satisfying 0~/0y= 0 is given by 9(X) = eiWX . (11) Therefore, (11) may be added to (10) as a possible homogeneous wave elevation, with which the Kutta condition can be satisfied at the stern. It is clear from (2) that this homogeneous wave elevation is equivalent to the hydrostatic pressure given by pox)=—K71(x)=_Keiwx (12) It seems necessary to consider a physical flow model, justifying inclusion of (11) multiplied by an unknown coefficient in the integral equation (10~. For a displacement-type ship, the flow may be dammed at the bow and then stream down along the ship's bottom. This periodical phenomenon may be represented by the pressure distribution of (12) from the bow to downstream infinity, which is equivalent to considering the homogeneous surface elevation of (11) in the same region, as noted in relation to (12~. With this hypothesis, we consider the following integral equation: ~1 71(X) = / p(~) Gyps—if, 0_) do—FeiW=U(1—x) -1 (13) Here ~ is the unknown coefficient of the homoge- neous component to be determined by the Kutta condition and U(1—x) denotes the unit step func- tion, equal to 1 only in the region of x < 1. (E is associated with the surface elevation at the bow, which is in fact given by A = Rein.) KERNEL AND KOCHIN FUNCTIONS The kernel function in (13) can be computed with the exponential-integral function as follows: Gy(X,y) = - 27r [k k ~(—1) knSn~x,y) k3—kit ~ ~ 1) kn Sn (X, y)], (14) where Snip y) = e-E—k (,) Tori U(~(kn(] ~ U(1 - 2~) ~ for n = 1,2 (15) Sniff, y) = e-k=; f En ~—keg,)—27ri Ut—x) ~ for n = 3,4 (16) with ~ = bye + ix, ~ its complex conjugate, and the complex sign in (15) taken according as n = 1 and 2. Ed (z) is the exponential-integral function, and the symbol ~ in (15) means taking the imaginary part. In the above, four different wave numbers have been introduced, which are defined as kl}=K[1-2~]' (17) k4} 2 [1+27~]. (18) Considering the behavior of Ed (z) in the limit of r = >/~ ~ 0, it is easy to show that Gyps' y) has the singularity of—(1/7r~lnr. On the other hand, at a large distance from the origin, E~(z) decays rapidly and only the sinusoidal exponential terms remain. Thus substituting the asymptotic form of Gyps' y) into (13) gives the asymptotic form of the water-surface elevation as follows: 9(x) ~ k ~ kink H(k2) e-ik2X U(1 - fir) ik, ash >+x, (19) r1(X) ~ k ilk H(`k~ ~ e-ikix Uf 1 - 4~) —k 3k H(k3) eik3x + Eke H(<k4) eik4X —Rein as x ~—x, (20)

where H(kn), n = 1 ~ 4, is referred to as the Kochin function, defined as rl H(kn)= J p(x) eiik7iX do, (21) -1 where the plus sign in (21) is to be taken for n = 1, 2 and conversely the minus sign is for rz = 3, 4. According to (19) and (20), there exist four dif- ferent wave systems on the free surface in addition to the homogeneous component expressed by the last term in (20); these characteristics are schemat- ically shown in Fig. 1. BASIC SOLUTIONS A shallow-draft ship is considered, heaving and pitching in a regular incident wave, but for simplic- ity only the case of head wave will be described here. The basic solutions necessary in this problem may be obtained by considering the following body boundary conditions: a) Heave mode (j = 3): b) Pitch mode (j = 5): 03 (X) = 1, ¢3y (X, O_ ) = it (22) 95 (X) = X, ¢5y (X, O_ ) = iWX—1 (23) c) Diffraction mode (j = 7): 07(X) = - 710(X) = _ eikox l 24) ¢)7y (x, 0_ ) = -into eik°X, `~; = ale + ko J ( where A, we and ko in (24) denote the (nondimen- sionalized) surface elevation, circular frequency, and wave number respectively of the incident wave. In head wave, ko = k4. Each basic solution must be obtained so that the Kutta condition is satisfied at the stern (x = - 1); this condition can be imposed in the form xl~,m1 4[p; (X) ~) = 0 (25) To satisfy these condition, (13) may be decom- posed as follows: where j = 3, 5, 7. pj*(x) for each mode can be de- termined by specifying 71i (x) according to (22)-(24), and pH(X) associated with the homogeneous surface elevation can be determined from (28) irrespective of the mode. The coefficient of the homogeneous component, Ill in (26), may be determined by sub- stituting (26) into (25). FORCES AND MOTIONS Radiation problem It is clear from (2) that integrating the pressure over the ship's bottom gives not only hydrodynamic but also hydrostatic forces. Therefore the results are expressed as L3 _ ~ p3(x)dx = w2(A33—iB33) - 2K -1 ~1 L5 _ J Is (x) do = ~2 (A3s—i B3s) -1 ~1 M3 _ J pa (x)x do = w2 (As3—i B53) -1 Ms——J p5(x)x do = ~2 (Ass—Bs5) - 3 K (29) Here Aij and Bij denote the added-mass and damping coefficients in the i-th direction due to the j-th mode of motion; these are defined as Aij—iBij = e2ij _ i to ij , where 63 = 1 and 65 = e. We note that the hydrostatic restoring moment included in Me is obtained with the center of gravity assumed to be equal to that of buoyancy. Diffraction problem As is clear from (2), the diffraction pressure on the free surface, p7(X), includes the contribution from the incident wave. Thus the resulting force is equal to the hydrodynamic wave-exciting force, which can be written in the form L7 _ ~ p7 (x) do = KE -1 J (31) Pj(~)=Pj(~)+FjP (I)' (26) ]/~7 _ J. p7(~;)~5c=KE5 t1 -l 7Ij(x) = ~ pj(~)Gy(x—(,0_) dig (27) ~1 eden = J pH (a) Gy (x - I, 0_ ) d: (28) -1 where the nondimensional value is defined as E' Ei (32)

Heave and pitch motions The complex amplitude of heave (X3) and pitch (X5) may be determined by solving the coupled mo- tion equations of heave and pitch, given by (m'~2 + L3) 3 - L5—= L7 —Ma_ _ (I'm + M5)—= M7 J ~1 J5 - m,n— -1 Tm(X) d .~ is Tn (~) J 1 x/~=y~x—() d(, (38) 77m = | ,iF~ it(x) do. (39) Eq. (37) can be analytically evaluated, and the result can be written as where m' = m/pt2 = 0.2 and I' = I/pi4 = 0.25m' ~r are used, with m and I being ship's mass and mo- Lm~n = Am 2 6;m,n ment of inertia, respectively. NUMERICAL SOLUTION METHOD We must solve (27) and (28~. As noted before, the kernel function Gy~x,y) includes a logarithmic singularity—(1/~r) in r. In addition, since (27) and (28) are of the same form, we write these in the form ——J p(~)ln~x—(~d al 1 ~ + I path Gy (x—() df = it(x) ~ (34) J -1 ~ where Gy denotes the regular part of the kernel function Gy, and the right-hand side, R(x), is to be Ajax) or eider. Let the pressure distribution be expressed in terms of the first-kind Chebyshev function Tn(X)' or equivalently the Fourier series similar to that used in the airfoil theory: N T ( ) N ~ where x = cost and Pn (n = 0, 1, , N) are the unknown coefficients to be determined. Substituting (35) into (34) and employing a Galerkin method with weight functions of Tm(X)/~ (m = 0, 1, , N)' we have a linear system of simultaneous equations: N At, Pn { (m~n+5m,n } = firm n=0 where olo=21n2, cam=—(m>1) | (40) where dm,n denotes Kronecker's delta, equal to 1 for m = rip and zero otherwise. Next, the double integral in (38) must be eval- uated with an efficient and accurate method. The method employed here is as follows: 1) Values of Gym) at M points of X < 2 were computed in advance and saved, 2) on each segment Gym) was approximated by a linear variation, 3) then the in- tegrals with respect to ~ given by Bn An = | Van df l =~ Trig) Add J were analytically evaluated, and 4) finally the in- tegrals with respect to x in (38) were numerically evaluated using Clenshaw-Curtis quadrature with variable transformation of x = cos d. In the meanwhile, the integral of (39) can be given analytically as follows: a) Heave mode: b) Pitch mode: c) Diffraction mode: for m=0,1,. ,N, (36) d) Homogeneous mode: 1 ii T (x) or _l ~ x / \~ln~x—Bide, (37) 1Zm = ~5m,O (41a) Arm= 2(im7~ (41b) Him =—~—i Am ~ Jm (ko ~ (41c) Elm = (i~m~Jm(~,) (41d) where Junk) is the Bessel function of the first kind. Eq. (36) can be solved by a conventional Gauss elimination method. Once the solution of each

mode is obtained in this manner, the Kutta con- dition of (25) can be imposed in the form Hi = ~ + Fj ~ = 0, (42) where N _—lymph. (43) n=0 From (42), the unknown coefficient Fj can be determined. Here ~ and AH correspond to the val- ues of (43) to be computed with the results of (27) and (28), respectively. With these results, it is straightforward to com- pute hydrodynamic forces, the Kochin function and wave-induced motions. RECIPROCITY THEOREM Applying Hanaoka's reciprocity theorem (Hanaoka, 1959), we can derive several relations to be satisfied among basic solutions defined in (22~-~24~. These relations can be used to check the accuracy of numerical computations. Bessho (1992) showed some relations for flat-ship problems. However, since the present flow model is different from that of Bessho and we are concerned with differences between solutions satisfying and not satisfying the Kutta condition, we extend Bessho's results to the present case. Skipping the details of the derivation, the first and second theorems proposed by Hanaoka may be written for the present flow model as follows: The first theorem is I where where al | Pi ~X' vjy ¢-X' dX +—hi ~j -1 2 = | pj (x) (iy ~—x) dx + 2 dj se, (44) <y } x~1 [Pt my]- The second theorem is rl J pi Aft ~ x' do + rj Hi (w) -1 r = J pi x' do +rz Hj(~), (46) -1 I1 Hj(~) = pj(x~e-iwxdx (47) -1 Here suffix i or j denotes the mode index, which takes3, 5 and 7. For solutions satisfying the Kutta condition, ~ = 0 as shown by (25) and (43), thus the second term on each side of (44) vanishes. On the other hand, for solutions not satisfying the Kutta condi- tion, we can put ['j = 0 and thus the second term on each side of (46) vanishes. Considering all possible combinations of i and j for (44) and (46), we will have six different rela- tions. oat—x) in (44) and 7~-x) in (46) are given explicitly by (22~-~24), with which the six relations to be obtained can be summarized as follows: rl J pa (X) do + r3 H5 (~) -1 J-1 rl =—J p3(xjxdx + r5H3(w)' (48) -1 p3(X) do = - 2 i55~3—53a5} + id {r3H5~) - rats} ~ (49) J1 p7(x) do =—H3(ko) + F7H3~w ~ - r3 Hit), -1 (50) ho H3 (ko) =—i 2 {57 ~3—53 ~7} + A iE7 H3 (~ )—F3 H7 lo) ~ } , (51 ) I1 p7 (x) x d x = H5 ( ko ) + F5 H7 (W )—r7 H5 (~ ) ~ -1 koH5(ko) + iH3(ko) =—i 2 {57~5—55~7} (52) +r7 {Hit + iH3~} -Hit) {ire + ir3} . (53) With (29) and (31), we can envisage that (48) and (49) give the symmetry relations for the added- mass and damping coefficients and that (50) and (52) give the generalized Haskind relation for the wave-exciting force and moment. ENERGY-CONSERVATION RELATION As well known, the wave-making damping force in the potential flow can be computed either by the pressure integral or from the wave-amplitude func- tions from from the body by virtue of the energy- conservation principle. To derive this relation, let us consider the following equation in two different ways: al rl Wj = ~ pj ~j (x) do—~ pj (x) JO (x) do, (54) -1 -1 where the overbar means the complex conjugate.

Firstly, substituting (22) and (23) for ~j(x) in (54), it follows from (29) that Wj= - 2iw2B~j. (55) Secondly, utilizing (13) for lo (x) and transform- ing the result with (14) and (22), we can show that Wj = h k ~ ki ~Hj (ki ) ~2 + k2 ~Hj (k2) ~2 + k3—ki ~ k3 THE (k3)~ —k4 ~Hj (k4)~ ~ +Ej Hj (I)—Fj Hj (w) . (56) Equating (55) to (56), the following relation for the damping coefficient in the j-th mode can be obtained: B:i=2 2tk k iki~Hj(ki)~2+k2~Hj(k2)~2) —b k ~ k3 ~Hj (k3) ~2 _ k4 ~Hj (k4) ~2 ~ —2 A{ Pi Hj (I) } ~ (57) 20 O cg CO ~ -10 10 -20 - 20 . 10 ~ O CO -10 -20 with Kuffacond at Eh=0.5, A/.r--2.0 -1.0 4.5 0.0 x/ (L/2) 0.5 1.0 with Kuffacond. at Eh=0.5, 1/L=2.0 .... .. ~ ~ 1 .... .... , ,,, = ;~= ~'~~~~-'" ~ PI tch ~ 1~ Witch (T--g) I .... 1 1 .... -1.0 -0.5 0.0 x/ (L/2) with Kuffa coed. at Fn=O. 5 20 . . . . . . . . 1 . . . ~ 10 . b0 ~_ ~ , . ,jC -10 _ DO diffraction (}~1J _ . ~ Diffraction (T~gJ -20 _ .... 1 .. . ~ 1 .... . -1.0 -0.5 0.0 0.5 x/ (L/2) 0.5 1.0 1.0 Fig. 2: Pressure distribution at Frr = 0.5 and )/L = 2.0 (with Kutta condition) Table 1: Accuracy of numerical results, errors in Hanaoka's reciprocity theorems and the energy con- servation principle (Frz = 0.5, A/L = 2.0 in head wave; w = 3.343, ~ = 1.672) Eq. No. (48) (49) (50) (51) (52) (53) (57) j =3 j =5 with Kutta coed. Error (%) 0.1918E-04 0.1643E-02 0.3692E-04 0.4653E-03 0.7245E-04 0.1491E-02 0.2117E-O1 0.2382E-01 RESULTS AND DISCUSSION w/o Kutta coed. Error (~o) 0.4755E-05 0.3412E+OO 0.8947E-05 0.3424E+OO 0.2063E-04 0.3423E+OO 0.1334E+O1 0.1822E+O1 An example of numerical errors in six equations shown by (48~-~53) and the energy-conservation re- ~ without Kuttacond. at Eh=0.5, 4/~-2.0 L ~ II= - I ~ fl ev (jag) I - 0.0 0.5 1.0 x/ (L/2) -0.5 without Kuffacond at En=0.5 4/~2.0 100, , .... , ' . | ~ patch (jug) | 0.5 1.0 100 without Kuffacond. at Eh=0.5, 1/L=2.0 O ~ -50 ~ ~ Di~frac"~ (my) ~ . ~ L Diction (Tag) | -100 .... 1 .... ~ .... ~ .... -1.0 -0.5 0.0 0.5 x/ (L/2) 50 ~ O -50 -1.0 -0.5 0.0 x/ (L/2) Fig. 3: Pressure distribution at Frz = 0.5 and A/L = 2.0 (without Kutta condition) ,.o

4 2 g O -2 6 4 2 ~ O 5 -2 ~ ' with Kuffacond. at Fn=0.5, 2/~2.0 ~ .. ~ :~_~ -3 -2 -1 0 1 2 3 x/ (L/2) at Fn=O. S. 4/0—2. 0 0 1 2 3 with Kuffa coed. ' 1 ' 1 .- t~,f ix~;~ ~ l -5 ~ -3 -2 -1 x/ (L/2) ~ with Kuffa coed. , ~ ~ 1- 1 ~ 1 L' ~ W. , amp---? ,xX^~ ~ . ,,, , -5 ~ -3 -' 6 4 2 o -2 with Kuffa coed. .,, 2 I I0 ~ -5 ~ -3 ^~ ph=n ~ 2/L=2. 0 ~ Diffraction | 1 1 . 1 . ~ ~ _ -1 0 1 2 3 x/ (L/2) at Fn=O. 5, 2/~—2. 0 ~ ' . ~ 1 ' 1 ~ ~ 1 mix 1 = _ . 1 incident breve + | I .— Diffraction brave | j , . 1 . ~ . 1 . i, _ -1 0 1 2 3 x/ (L/2) Fig. 4: Wave profile on the free surface (with Kutta condition): Solid and dotted lines are by the exact expression, symbols to and x) are by the asymptotic expression. ration for Bjj shown by (57) is presented in Ta- ble 1 for both cases of satisfying and not satisfying the Kutta condition, for En (= V//;' = 0.5 and A/L = 2.0 in head wave. We can see that the accuracy with Kutta con- dition is virtually perfect. For the case of not sat- isfying the Kutta condition, the accuracy is a lit- tle worse but still within allowable tolerance. The accuracy for other parameters of En and A/L was confirmed to be of the same order as Table 1, im- plying that the present boundary-value problem is without Kuffa coed. En 50 without Kuffa coed. 25 ~ O -25 -50 50 25 ~ O -25 on eln 25 ~ O -25 -50 - -5 at Fn=O. S. 1/I^2. 0 . ~ ~ ' 1 ' ~ i__ I ~ ~ -tin ~ , I ~ i 1 1 : I | Heave ~ ., . 1 ., . -1 0 1 2 3 x/ (L/2) at Fn=O. 5, A/~—2. 0 , ~ 1 1 1 1 1 1 = L_0 -- I hi I ~ I lPitchl i ! '. I .'i 1 2 ^ -5 ~ -3 -2 -1 0 x/ (L/2) without Kuffacond. at Fn=0.5, A/~-2.0 ~ ., ., . ~ ., 1 ~ 1 1 — 1 1 ~ ~ I ~ I Di ffraction ~ 1 . 1 ., . 1 .': -3 without Kuffa coed. -2 -1 0 1 2 3 x/ (L/2) at Fn=O. 5, 2/ -—2. 0 ' 1 ' ' ' 1 ' 1 1 1 1 ~ A- -t------r~ ~----~ !~ r| I ! ~ i | | Incident drive + Li | | Diffrection - diva | i 1 .'1 . ~ . i .'1 -1 0 1 2 3 x/ (L/2) Fig. 5: Wave profile on the free surface (without Kutta condition): Solid and dotted lines are by the exact expression, symbols (o and x ) are by the asymptotic expression. successfully solved. Computed pressure distributions for Fn = 0.5 and A/L = 2.0 are shown in Fig. 2 for the case of satisfying the Kutta condition and in Fig. 3 for the case of not satisfying. As expected a big difference can be seen between Fig. 2 and Fig. 3 near the stern (~ = -1), and the pressure distribution with Kutta condition looks reasonable. Figures 4 and 5 show the wave profiles for the same parameters as that in Figs. 2 and 3. The wave amplitude is normalized in terms of the maximum of

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Gap in Wave Amplitude at Bow at Eh=O. 5 ~ . , . . ~ - —— fioeve | r | /X3 - Pi I:` oh ~ /X! (L/2J ~ ~ 1 l " ~ | Diffraction | r | /a _ ~ —~ _ -~;'~; ,_. ......... , ,.,., , ,,.,.,.,.,., , ,,.,.,,, ,, ,.,,, ,,.,.,.,.,.,,,. 1 Motion Free I r I /a , I I 0 1 2/L Fig. 6: Amplitude of homogeneous component of the wave elevation, corresponding to the gap in wave amplitude at bow, at Fn= 0.5. 10.0 an an 2.0 on -2.0 — 1 n ns on .5 -1.0 with Kuffa coed. at Fn=O. 5 - 1 ~ 1 ~ ~ ! ~ . T T 1 A33/p (L/2) -2~/a' - --- B33/p (L/2J a>. : . 1 _ , Ll 0 1 2 Reduced frequency: with Kuffa coed. ^~ Fn=0 5 ___ . i it,: __ 1 an an l l l _ ~~ _ . 1 . _ 3 4 0 a. (L/2) /V . 10 . o -10 -20 B55/p (L/2) -2/3a ----- B55/p (~/2) a). ! 1 1 1 1 _ 1 2 3 4 Reduced frequency: w.(L/2)/V Fig. 7: Added-mass and damping diagonal coefficients in heave and pitch (with Kutta condition), at Fn = 0.5. each mode in the radiation problem and in terms of the incident-wave amplitude in the diffraction prob- lem. In the case of not satisfying the Kutta condi- tion (Fig. 5), the amplitude is of order of 25, which seems quite unrealistic. In contrast, the results with Kutta condition (Fig. 4) seems reasonable, judging from the order of wave amplitude. The open circles and crosses in Figs. 4 and 5 are the results computed by asymptotic expressions shown as (19) and (20~. Slight difference from the exact results based on (13) can be seen only in close proximity to the stern and bow, which means that the evanescent wave term is relatively very small. As shown by (13), the flow model in the present paper gives necessarily a gap in the wave ampli- tude at the bow, corresponding mathematically to the amplitude of a homogeneous component of the wave elevation, Ace. The nondimensional value of this amplitude is shown in Fig. 6 for various modes at Fn = 0.5. For the realistic case of a ship freely -10 _ without Kuffa coed. at Fn=O. 5 -em 1 _ 03/p(~/2,2-2X/~2 1 ~ , ~: ,~ 1 2 3 4 Reduced £=equency: a'. (L/2) /V without Kuffa coed. at Fn=O. 5 1 1 1 , ~ : B55/p (L/2J -2K/3Q) B55/p (L/2) Jane ~ 1 ~ 1 1 1 0 1 2 3 4 Reduced frequency: ~ (L/2J /V Fig. 8: Added-mass and damping diagonal coefficients in heave and pitch (without Kutta condition), at Fn = 0.5.

with Kuffacond. at Fn=0.5, head rave without Kuffacond. at Fn=0.5, head wave 40 ~ _ lE3l/pUa(L/2) | | I ~~ 100 l ~ — ga.o~ ~KSl/2 And 2 °~Isal/2ga~ i' ~~ :=T- ~2 20$ 0 1 2 3 4 0 1 2 3 4 4/L 2/L 1.0 0.8 0.0 - withKuffacond. at Eh=0.5, head wave 1 ~ 1 1 ~ _ Heave ( I X3 I /a) _ In Pitch ( I X5 1 /kOa) _ i0.6 . _ __ 0.4 __ _ _ 0.2 ~ /, /, al' . .~ _ _ I L 0 1 2 1/L _ _ ~ 3 4 Fig. 9: Upper: Wave exciting forces in heave and pitch, Lower: Motion amplitudes of heave and pitch (with Kutta condition), at En = 0.5 in head wave oscillating in waves (denoted as motion free), the gap in amplitude at the bow decreases as the wave- length increases. We note that the value of Is in (13) corresponds physically to the amount of fluid periodically dammed at the bow and then streaming down along the bottom of a ship. The added-mass and damping coefficients (only for diagonal coefficients) versus the reduced fre- quency (w = we{/V) are shown in Fig. 7 for the case of satisfying the Kutta condition and in Fig. 8 for the case of not satisfying. All results change dras- tically around ~ = 0.5, which is equal to ~ = 1/4 because ~ is given by we(/V = 2wFrz2. Since no experimental results are available, it is difficult to make a definitive judgement. However, at least we can say that the results satisfying the Kutta condi- tion seem much better than the results without the Kutta condition. 1.4 1.2 1.0 0.8 ~ 0.6 .~ no no 0.0 . without Kuffacond. at Fn=O.S, head cave ~T , 1 ! ~ it' --it 1~ -~---1 ~ 1 ~ , 1 1 . ,, I ~' I | Heave (lX3l/a) ;- | I ~ -- Pitch ( I X5 1 /kOa) n I I 1 ' I I r I 0 1 2 3 4 2/L Fig. 10: Upper: Wave exciting forces in heave and pitch, Lower: Motion amplitudes of heave and pitch (without Kutta condition), at Fn = 0.5 in head wave Computed results of the wave-exciting forces and wave-induced motions are shown in Fig. 9 for the case with Kutta condition and in Fig. 10 for the case without Kutta condition. We note that the results without Kutta condition are unrealistic judging from other past computations for a surface- piercing body and related experiments. On the other hand, the results for the case of satisfying the Kutta condition seem to be reasonable in magnitude and variation tendency, which supports the validity of the proposed flow model. CONCLUDING REMARKS In the hydrodynamic problem of a displacement type ship with shallow draft, a solution must satisfy the kinematic condition of water surface being equal to ship's vertical position and the Kutta condition of

flowing out smoothly at the stern. However there has been no convincing theory accomplishing this requirement, and in fact it is known that a solution satisfying only the kinematic condition gives very large pressure and wave elevation near the stern. To overcome this problem, a new flow model was proposed in this paper, which introduces a homoge- neous wave elevation satisfying loo/by = 0 into the integral equation for the pressure distribution, spec- ifying the surface elevation as the boundary con- dition. The coefficient of this homogeneous com- ponent can be determined by imposing the Kutta condition at the stern. Numerical results were confirmed to be very ac- curate through checking several relations derived from Hanaoka's reciprocity theorem and the energy- conservation principle. It was also confirmed that computed results satisfying the Kutta condition were reasonable judging from the magnitude and variation tendency of the pressure, wave profile, hy- drodynamic forces, and wave-induced motions. Fi- nally we should note that conventional results with- out Kutta condition, which were also shown in this paper, were much different from the results by the present flow model and thus not acceptable even as an approximate solution. REFERENCES Bessho, M., "On Hydrodynamic Forces Acting on an Oscillating Gliding Plank," Journal of Kansai Soc. of Nav. Arch., Vol. 165, (1977), pp. 49-57. Bessho, M. and Suzuki, K., "Porpoising Instability of Two-Dimensional Oscillating Planning Plate," Mem. of National Defence Academy, Vol. 25, No. 1, (1986), pp. 25-43. Bessho, M., "Hydrodynamic Forces Acting on an Oscillating 2-D Shallow Draft Ship Advancing with a Constant Speed," Mem. of National Defence Academy, Vol. 30, No. 1, (1992), pp. 41-49. Bessho, M. and Komatsu, M., "Two-Dimensional Unsteady Planing Surface," Journal of Ship Re- search, Vol. 28, No. 1, (1984), pp. 18-28. Hanaoka, T., "On the Reverse Flow Theorem Con- cerning Wave Theory," Proceedings of 9th Japan National Congress for A pplied Mechanics, ( 1959~. Kashiwagi, M. and Bessho, M., "A New Flow Model for a 2-D Shallow-Draft Ship Advancing at High Speed in Waves," Proceedings of the 2nd Japan- Korea Joint Workshop on Ship and Marine Hydro- dynamics, (1993), pp. 45-52.