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The Influence of a Slowly Oscillating Movement on the Velocity Potential C. van der Stoep, A.J. Hermans (Delft University of Technology, The Netherlands) Abstract The central problem problem in this paper will be the deter- mination and calculation of the influence of a slowly oscillat- ing movement on the velocity potential. This is for instance important when you want to calculate the dynamic swell-up and the added resistance of a sailing ship. Two boundary integral equations for the velocity potential are derived to solve the steady and the insteady problem. It will be shown that the potential can be represented as a area source sin- gularity distribution over the ship hull and a source line distribution on the ship waterline. The equations that have been obtained in this way can be solved using an iterative scheme. First and second order solutions of the equations have been obtained. Nomenclature. B U g p 71(x) 710 ~1 L H x z n Fn G F.S. c' a Tn Jn7 Kn, Yn Beam at midship Wetted surface area at rest Ship speed Ship block coefficient Gravitational acceleration Pressure Wave elevation along hull steady wave elevation unsteady wave elevation Oscillator amplitude Length at water line Draft at midship coordinate system moving with velocity U in forward direction vertically upward unit vector normal to ~ in outward direction Froude number(U/~) Green function Free Surface steady wave potential unsteady wave potential total potential motion vector frequency of motion dipole strength source strength Chebychev polynomial of order n Bessel functions 119 1 Introduction The estimation of a ship speed and power was usually based on still water performance. Assuming a potential flow the stationary problem of calculating the ship wave restistance is described by the Laplace equation and the conditions on the ship hull and the free surface. Numerous computer pro- grams have been written to tackle this problem. In order to be able to predict ship performance in seaway it is also desirable to be able to calculate the instationary problem of a sailing ship. For example a ship sailing in waves or an oscillating vessel. In preliminary design studies the use of a fast computer algorithm could help to assist the tradi- tional model testing methods. The central problem in this paper will be the calculation of the added resistance and the dynamic swell-up of a ship when it is slowly oscillating. Dy- namic swell-up is the effect of water being pushed up around a bow higher than can be accounted for by considering heav- ing, pitching and incident wave alone (see also figure 1.1). The swell-up is given by the quotient of (amplitude of rel- ative motion) and za(oscillator amplitude). ~1 . / ,ho it' ~ Id / ) ~ Ha Figure 1.1: The dynamic 'swell-up'. Some work in this area has been done by Blok ([1]) and Tasaki ([2]). Tasaki was the first to introduce the term swell- up and on the basis of experiments obtained some sort of emperical formula for the calculation of the dynamic swell- up, when the wave frequency, ship speed and ship block coefficient CB are between some give limits. The problem will be divided into several subproblems: The treatment of the free surface condition. The treatment of the body boundary condition.

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The numerical treatment of the associated Green func- tion. The numerical solution of the resultant integral equa- tion. The calculation of the added resistance. Every subproblem will be dealt with in this paper. The research is carried out in cooperation with MARIN. 2 Mathematical formulation 2.1 Introduction We consider an object moving horizontally with forward speed U in an infinitely extended fluid. Viewed from an inertial frame~x,y,z) attached to the ship, there is an inci- dent stream of velocity U in the positive direction. The formulation will be done in a Cartesian coordinate system moving with the object. With the x coordinate in forward direction and the z coordinate vertically upward. The free surface elevation ~ will be given by: Z = 71(X, y) (2.1) The total velocity potential ~tota' can be split in a steady and an unsteady part: total(X' y, Z. to) = US + (X, y, Z) + O(X, y, Z. t) (2.2) steady unsteady We are especially interested in the influence of the steady part ' on the unsteady part . Subtotal has to satisfy the following three conditions: Subtotal = 0 outside the object, z < Mix, y) (2.3) At the free surface y~x,y) we have the dynamic and the kinematic boundary conditions: ~ ~3., ~ `~?2 + ~ 2 + t3~2' + gz = 0 at z = l at ~ 8c By At) = 0 at z = and a boundary condition on the ship hull: ~ = Vn, atE (2.5) The main difficulty in this problem is to find a solution of the Laplace equation using boundary conditions at a free sur- face which is still unknown. Various attempts to solve this problem have been done by Brandsma ([3~+ [4]), Baba [5], Sakamoto [6i, Hermans [7] and many others. Most of them are using some sort of expansion in a small parameter usu- ally the small ship velocity or some sort of ship slenderness parameter. To derive an approximating solution of the problem we con- tinue by linearizing the free surface condition (2.4) (as will be done in the next section) and the body boundary condi- tion (2.5~. 2.2 Linearizing the free surface condition The free surface condition consists of two parts: The dynamic boundary condition: .9' + EVE V4} +gz = 2U2 at z =~(x,y) (2.6) and the kinematic boundary condition: ~ -~9~1 _ ~ ~971 _ ~ ~971 = 0 at z = ~7(x,y) (2~7) One way of dealing with this problem could be neglecting the higher order terms in 4~. Then equation (2.7) leads to: B~ = i971 (2.8) and the linearized free surface height will be given by: ~ 041, (2.9) combining equations (2.8) and (2.9) leads to the well known free surface condition: ~f2 + 90 = 0 at z = 0 (2.10) In this paper we will obtain an asymptotic solution for the free surface condition. Let ~ << 1 and expand the free surface elevation around a 'known' solution ?1 = ho: 71 = ho + 6773 + ~. (2.11) Denote the total velocity potential by ~tota', and split this total potential in a steady and an unsteady part as follows: .. ~'Ota`(x' y, Z' t) = UP ~ (x, y, z) + c(x, y, z, t) (2.12) All the terms in the free surface condition (2.6) and (2.7) have to be expanded. This leads to the following free sur- face condition: For the first order problem (the steady state solution): ~ OZ + 2} V(uX + 4)) V[v~uX + O) V(uX + (/)~] = 0 | atz=yO | (2.13) and for the second order problem (the unsteady state solu tio~ ): - - - _ - - g(z - Lyon - ~yHoy - Bust + vex + ) vat+ 2V(Ux+o).Vot+V(Ux+o) ~V[V(Ux+o) EVE]+ ~ Ott = 0 at z = yO 120 (2.14,

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2.3 The body boundary condition The body boundary condition for the stationary potential reals as: or equivalently: .9 = Vn at the body ~(2.15) n.V(Ux+o) = 0 atE (2.16) And now the calculation for the instationary part (see also for instance Timman t8] et al.~. The displacement of the ship hull is given by: x - x' = ~ or~x,t) (~.17) where x denotes the coordinate system moving with velocity U. and x' denotes the coordinate system fixed to the ship. The body boundary condition states that: at the momentary position of the hull, the normal velocity of the fluid is equal to the normal velocity of the hull. Expansion of ~ round the rest position with a small parameter ca yields: Momentary = Total + ~ (t Subtotal (2.18) so for the body boundary condition this leads to: n (total + ~ A` ~total) = ~ n it,` (2.19) combining equation (2.16) and (2.19) leads to: ~ = n ~ ~ At, - V(a ~ V(Ux + ~) (2.20) now using the assumption of or~x,t) and (x,t) to be oscil- and latory: onyx, t) = o~x' .-i~ >(x,t) = ,~x).-iwt (2.21) then the following is valid: 4' = -n (i`.Ja + V(or ~ V(Ux + f)~) (2.22) with the function G written as: Figure 3.1: The mathematical domain. G = (- - + - - ~(x, (; U)) (3.2) In the same way as Brard t9] we consider the following inte- gral formulations: IE = /! (me () _ G Me ~ d5 EFe' + Ee + r I] = I! (Oi {) _ G {) i) dS EFi + Ei (3.3) (3.4) this finally leads to the following body boundary condition: I { 41rSt)e 27 C De (3 5) I = { 4~Vi x C D' (3.6) and free surface condition (2.14): 20 - 2ic~Uo~ + US + 9z = ((U;~){~} at z = 710 (2.23) where [(U;~) denotes a linear differential operator acting on a. 3 The leading equations 3.1 Calculation of the singularity distributions Introduction of a Green's function G and application of Green's theorem to the domain as can be seen in figure 3.1. The function G has to fulfill the following condition: -HUG + 2i~UG~ + U2G<~: + gG: at ~ =0 (3.1) At EFi and Life the following equations (2.23) and (3.1) are valid: ~0 = _{~(~' + ~2> + 2icoU~ _ u2 ~ 'it} (3.7) ;,G ~ {HUG - 2i~U ~ - U {~2 } (3.8, Combining equations (3.7) and (3.8) leads for equation (3.3) to: dG 4~ e (me .~ - G ~' ~ = 121 -OeG - -Use- g g 0* U 4)e~ G _ -{(o)G

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- g )eG - -U ~ eG + + _ `9 Ve G (3 9) regrouping the expressions at the right hand side of equa- tion (3.9) leads to the following formulation: 2i~ut3~(~)eG) + U 0( ~c G (e 8= ~ _ - C`oe'G 9 dx 9 dx g (3.10) and almost the same for equation (3.43. The ~F, part of equation (3.3) leads to: |/ (e ~'n - G ~) ~dS = - / -U 4>e G dy ~Fe' J 16e J y C -Cf and for the ~Fi part: g |'-at c,~ - aT ~T + ~na(~)Gfx,()dy Cf - - J| 13~)Gf x~ ()dSt = 47r<~(x) (3.15) F.S. with: CYt = COS(OX, t) OT = COS(Ox, T) Ctn = COS(OX, n) (3.16) with a choice of y(~)-O the following expression will be obtained: 4,ro~x) =- /. [cJ(~)Gtx,()dst- + - /c~nat()Gtx,()d /* u2 d~e dG ~ .1 -. 9 C _ / ~ G - e~ ~dy 1 1 1 ^ c~J c, g {J ~0~ + g J~ {(o)G(z,()dS~ (3.~7' - - || {(ve)G dS (3.11) , using the body boundary condition (2.22) we may obtain a description of the potential funtion ~ by means of a source distribution of the following form: ||~i~ - G~i~dS = |-~ ,,~iG - >i ,, ~dy EFi CF | 2i~U Gd (3.12) adding equations (3.11) and (3.12) results in (with ~8i = ~ _ oN. _ ~ _ ~ J. Ie + Ii = IJ~ ~(~)e _ ~'iGdS +-U /`Oe-oi~Gdy ~ Cf U2 /~li _ 0le'Gdy + U /',e-'i) ;3~ dy Cf C' -|| {~)e ~Gd S - - || l:~'i ~Gd S rFe ~Fi + | j`<;6i-e) ~3 dS The source and vortex distribution are defined as: = >e-`/)i Bii 0e a = ~ _ _ equation (3.13) will now transform into: | 7~) d G(z, f ) d St - | | a(~)Gf x, t!~)dS~ g | ~y(~)Gtx, (,d71 + _ | ?~) aG(~ . f) (3.14) 122 - 41rn (i~or + V(~. V(Ux + ~) = - 2lrafx) | | ~) 6 ( 'f ) dS~: + -| ~ c7~) 6 f( ' f) d - ~ || ~C(~) f ( ' f ) d St F.S. Remark: We consider the following potential ~: (3.18) ~ = ||-dS (3.19) the following is valid (Kellogg [10]~: if the density a of the distribution on ~ is continuous at x, the normal derivative of the potential ~ approaches limits as X approaches x along the normal to ~ at x from either side. (3.13) These limits are: ( ~'n )+ = -2~a~x) + || a(~) ,9/ dSt (3.20) ( ~n )- = +2~a~x) + || a(~) ~/ dSt (3.21) So when using the R sources the resulting equations looks like: + 21ra(x' - ||a ~ '<;)dSf + ... = Vn (3.22, and when the -R sources are used (as in Brard [9]) the resulting equation transforms into:

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-2,ra(x) - ||a ~ '()dSf t = Via (3.23) 3.2 Calculation of source strength and potential function We will use an expansion of a, ~ and G in the small param- eter w: a(x) = aO(x) + wai(x) + (x) = fO(X) + woi(x) ~ G(x,() = Go(x,() + h7Gi(x,<) + (3.24) substitution of these expansions in equations (3.17) and (3.18) leads for the first order problem to: - 4lrn ~ V(cx V(Ux + )) = - 21raO(x) aO(~)0 ~( '()dSt + ~ Now) t~( ~ ()dr ~Cf - ~ / / ~Co(~)~ ~( ' () dSt (3.25) F.S. and Jo is given by: - 41r~o(X) = -| | ao(~)Go(x, ()dS~ + -| ('naO(~)GO(X' ()d?1 ~ of +~ || t~o(O)Go(X, ()dSf (3.26) F.S. and for order h': -4~i (x)-~ J a~(~) aft (x,() dS~ + U2 J ol(g) ~G~o(x,() dr1 = -47rn in + ~ [aO(~3(='() dS~ _ rug Jo c~aO(~) (inn + + 9 ~ r (em ( ~ ,0 ~ + , O (< ,~ ~ ~ (A <) + ~0 (SO ~ ) of, (A <) dS,: F.S. and ~ is given by: 3.3 The incorporation of the body boundary condition The right hand side of equation (2.22) contains a vector ork or (but) = { rk(t) (ik X Xj k - 49596 (3.29) here C>k(t) is the deflection in translational motion for k = 1,2,3 and for k = 4,5,6, Skit) represents rotation angles about the Xk_3 -axis. So for example for k=l(surge) this looks like: 4) = -icons-nl==-n2O=y-naves (3.30) An when ~ << 1 then the expansion (3.24) will start with leading order ~ (a = Day + h'2~2 + ) and then equa- tion (3.25) will transform into: - 21ra~ - ,/ JP(a] ~ + ~ on ) + ~=o =-no (3.31) when o = (~(1) then the expansion will start with leading order zero and equation (3.25) will look like: - 27raO - ,/ / aO i, o + = -n! O== - n2=y - n3O=z (3.32) it should also be noted that for the calculation of the steady wave potential ~ the same matrix equation (3.25) as for the unsteady potential ~ can be used! But of course the right hand side for in now looks like: ~ = -no U (3.33) (fin 3.4 Solution of the integral equation. (3.27) _ 47ri1(X)= (ao(~)~l(x~ A) + al(~)GO(x, ())dS~ +-| ~n(aO(~)~l(X, () + al(~)GO(x' ())dr' 9 C' -- || (Jo + L:l)Go(X,() + {o(o)~l(x,<)dS~ F.S. (3.28) The solutions obtained by the singularity distributions are two coupled integral equations. Equations (3.25) and (3.26) will be solved using an iterative scheme. In this scheme use has been made of the numerical evaluation of the wave re- sistance Green function as done by Newman (~11~+ t12~). The numerical solution of equation (3.25) and (3.26) is car- ried out using a finite element method. The wetted body ~ is divided into N triangular panels and integration is done using a piecewise constant variation of the source strength a(~). | | = ~( | | a(~) ~ ~ d So (3 .34) In this way a set of N linear equations for the N source strengths aj is obtained. The Green function contains a 1/A-term. The integration of these 1/A-terms is carried out by a subroutine developed at MARIN t13] using analytical expressions of those integrals in order to avoid large errors from numerical integration for points close to the panel. 123

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4 The Green's function 4.1 Behavior of the Green's function. In Wehausen and Laitone t14] the Green's funtion of an os- cillating source is given: G(x,(; U) = -- + - - ~(x,(; U) (4.1) with the function ~ given by: 2 7r ~(x,(;U) = 9 JlF(8,k)d~dk O L1 7r + 9 ||F(8,k)d~dk (4.2) 2 7r L2 where F(8,k) is given by: F(8, k) = k e ~ cos(k(y-A) sin 8) the paths Lo and L2 are given in figure 4.1. with Kit - k4 the poles of F(8,k): 1 - j1 -4T cos /; ~2r cos ~ -W 6' ~ (O' 72r) (4.4) ~ = 1 + AL 4rcos8~ ~ ~ (0, 72r) (4.5) Why = 1 - ;1 4rcos8~ ~ ~ (72r~1r) (4.6) ~ = _ `9 ~ 6' ~ (12r,~) (4. with ~ = ugly, so T ~ 1 this leads to: ~j: + ~(~2) (4.8) /;: it- Ucosp + + (~( ) ( ) when for instance U ~ O then k2 and k4 will go to oo and the paths Lo and L2 will coincide with there poles located at Kit = k3 = w2/9. see also figure 4.2. and when ~ ~ O the poles Kit and k2 will move to the origin and the paths Lo and L2 will look like as can be seen in figure 4.3. However this is not correct, when ~ ~ O a factor k can be removed from the function F(8,k) and looks like this: K1 K2 O ~ K4 Figure 4.1: The contours L1 and L2. K1 ~ K3 _ L ~ I \ ~ , L1 ~ L2 O Figure 4.2: The location of the poles when U ~ 0. K1 1 it.. o K2 K3 Ka I ~ J > L2 o Figure 4.3: The location of the poles when ~ ~ 0. F (8 k) ek(Z+~+i(~-~)Cs8)cos(k(y-77)sin8) (410 with only one pole, located at k2 = k4 = U2 C9os2 8 124

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4.2 Calculation of the Green function. Newman (~114+ t124) has written two papers in the Journal of Ship research about the evaluation of the wave resistance Green function: one of of the calculation of the double inte- gral and one for the calculation of the single integral on the centerplane. The Green function is written in the following form (see Wehausen and Laitone t14~: 1 1 Tn(X) = costn arccosx) -1 < x < +1 (4.14) so the function 95x; is approximated by: N f(x) = ~ cmTm~x) (4.15) m=0 The coefficients cm can be found according to: N 1 = ~ 4~ s(k~cos8)cos(ky sin y~dkd ~Cm = N-~ f(~)Tm(~n) (4.16) 0 0 1 7r - 4 | e-Z sec2 ~ sings sec 8) costs sec2 ~ sin 8) sec2 add o (4.11) The quantities are defined as can be seen in figure 4.4. > X .1 Ir . S( rage Ye 1 - \G? urce ~f ield Figure 4.4: Location of field-, source- and image point. with R = ~x2 + y2 + z2 As has been done by Newman this integral can be split in a Double and a Single integral as follows: 2 + 2 ~ e-kz+iklxl see +ky cam ~ Double = -i ~ / CoS ~ 2 . dEd; or J J k - COS ~ + tE _~x O (4.12) + 2 Single = 4iH~-x) ~ sec2 Be-zsec28+ixsec8+ilylsec28sin~db 2 (4.13) 4.2.1 Double integral The double integral as given by equation (4.12) will be ap- proximated by Chebychev polynomials as done by Newman. In order to approximate a function f of one variable x in the normalized range i-1,+1] the Chebychev polynomial of order n is defined by: with c0 = 1,em _ 2, the double prime indicates that the first and the last terms in this summation are multiplied by 2. The coordinates on are given by: (see Fox and Parker [15~) an = cost N ~(4.17) For the 3D-case the situation is completely equivalent. In equation (4.12) logarithmic singularities are present when R = 0. These singularities must be subtracted and approx- imated first in order to improve convergence of the approx- imation. The final approximation is given by: 16 16 8 D ~ S + ~CijkTiff(R)]Tj(-1 +-8)T2k(7ra) i=o j=o k=0 (4.18) The function f(R) is defined so as to transform the interval (O,oo) into (-1,+1) see figure 4.5. and ~ are defined as: x = Using z + iy = R cos heir = pew (4.19) Ti, Tj and T2k are Chebychev polynomials. S is the loga- rithmic part of the double integral. The Chebychev coeffi- cients Cijk are calculated and tabulated by Newman. Also the differentiated Green function has to be evaluated, which contains terms like dGfx,(~/dox (see equation (3.25) and f (A) 25 /1~1 /1~ / ~ / 4 / 10/ ~ Figure 4.5: The transformation function f(R).

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(3.27~. Each part of the expansion (4.18) has to be differed- where Ye (x) denotes the Bessel function of the second kind, tiated and evaluated analytically. The following terms have ~ equals - ~ and F(~) Dawson's integral: to be evaluated: ~, = A ~ (4.20) with the transformation matrix A given by: OR OR By BY OR LIZ LIZ (4.21) and also for the singular part S. For example the next fig- ure 4.6 is obtained. The singular character is well shown here. 4.2.2 Single integral The single integral as given by equation (4.13) is evaluated by Newman at the centerplane, i.e. a special case where the source and field point are in the same longitudinal plane. This is especially important when analyzing thin ships. The Figure 4.6: Function plot of the Double integral. case y 7t 0 will be dealt with later. For instance with the use of Fade approximations. The centerplane integral looks like this: 2 ~ Sax, Y. Z)lY=0 = -8H(-x) l sec2 ee-Z see ~ sinks sec 8)db o (4.22) In each of the different x-z regions (see figure 4.7), the inte gral will be approximated differently. region A: (small z) an expansion involving differentiated Bessel functions of the second kind: S -F( )- d2 4 23 n on! dz277 t2 i~ )+ ~: ~ ~ 126 o 1 2 3 4 5 6 O 1 2 3 4 5 6 7 8 9 x ~,> (C)' ' ' ' ' - (B) "~` (B) ~ Figure 4.7: Domains for the approximation of the single integral. :r F(x) = e-2 |et2dt 0 ~ x < oo (4.24) o region B: an expansion in Neumann series, products of Bessel functions of the first kind and modified Bessel func- tions of the second kind: S = 2e 2 ~(-1) J2n+1~) [K-n(2Z)+Kn+l(2Z)] (4.25) where Jew) denotes the Bessel function of the first kind and Kn~z) the modified Bessel function of the second kind. Region C: large distances form the origin, steepest-descent expansion. The final expansion looks like this: S ~ -2 ieh()+it am, (2n + 1 ~Bn(` _ in+ 2 (p + i`~'nr~n + 2 ~ (4.26) where only the coefficient Bn has to be evaluated. For all the expansions in the different domains (A)-(C) the differentiated Green's function has to be evaluated. An ex- ample for the Green function can be seen in figure 4.8 . The wave character is well observed here.

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Figure 4.8: The Single integral at y _ 0. 5 Pade Approximations. 5.1 Introduction The basis for the Pade approximation technique is the formal Taylor series expansion. From this basis a Pade approxima- tion can be found. It is also possible for a Taylor expansion to be divergent and the Pade expansion to be convergent and also vice versa. For instance the Taylor expansion of the exponential function: oo rat 1 1 en = at, a! = 1 + x + 2x2 + 6x3 + rC = oo (5.1) and for instance the Euler function: oO E(x) = 1 1 + tdt = 1-x + 2x2 - 6x3 + ~ ~ rc = 0(5.2) o The idea of Pade expansion is to approximate the function by a rational funtion of the following form: Definition: (see Baker t16~+ [17~) We denote the L,M Pade approximant of I(X) by: [LL/M] = Q (( )) (5 3) where P~,(x) is a polynomial of degree at most L and QM(X) is a polynomial of degree at most M. The formal power series of f(X) reads as: When we require: oO f (X) = ~ aixi (5 4) i=o f(X) - [L/M] = O(XL+M+1) (5 5) Then the coefficients of PL and QM can be found according wich will be denoted by: to the following scheme (Baker): 127 + aOql + at + aOq2 al + al-! at + aoq' = al+ + at + a'-m+~qm = al+m + ai+m-~ at + alum = an - O n M for instance the t1/1] approximant for en reads as: = Po = Pi = P2 o (5.6) end 2+ ~(5.7) inevitable a pole occurs at x = 2. So the Pade approximants seems worse here. The Euler function evaluated at x = 1 is wildly oscillating ( when using Taylor expansion). The answer however for E(1) is known: E(1) ~ 0.5963 . The Taylor expansion never reaches this value. The Pade approximants however do. The t2/2] approximant reads as: 2/2(x) ~ + 52: + 2~2 (5.8) So E(1) ~ 0.6154, only five Taylor terms have been used to get this accuracy. The Taylor expansion leads to the result: Taylors(1) = 20. (5.9) The t6/6] (x) approximant is even better: E(1) ~ 0.5968 So Pade approximant could lead to valuable results when Taylor expansion fails (see also van Ge~nert t18~). In the next paragraph use will be made of this when evaluating the single integral at y ~ 0. 5.2 Pade approximations in the single integral The single integral to be approximated is given by equa- tion (4.13) repeated here: +2 Single _ 4iH(-x) | seC2 6ie-zsec28+i~sec8+ilylsec28sin~d6, 2 ~ the real part will be used: (5.10) 1 7r -8H(-x) l sec2 be-, see ~ sin(x sec 8) cos(y sec2 ~ sin 8)db o (5.11) -8H(-x) f (x, Y. z) (5.12)

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n no Q ~ ~ ~A -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 n ~n-R 0 nls o -0.005 -0.01 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0.02 0.015 0.01 0 - non ~ . ~ ~ - 1 -0.8 -0.6 -0.4 -0.2 0 025 ~ 0.02 Q 0.015 0.01 0.005 o -0.005 -0.01 -0.015 _ . ~ . ~v . ~ . ~u u . c u . ~U . ~U . ~1 0.025 0.02 Q 0.015 0.01 0.005 o -0.005 -0.01 -n nip; t ; -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 no ~o no Q -0.01 -0.02 1 , . . . . . . . -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 6.3: Hull side wave profiles of WIGLEY. 128

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changing of variable to s = see ~ leads to: oo 2 se-ZS Ids (5~13) 1 ~ Taylor expansion of f~x,y,z) leads to: f(x,y,z) = f(x,O,z)ly=0 +Y (~'yY ~ IY=0 + ~ y2 02f(~, Y'Z) 1 + . (5.14) applying this to (5.13) leads to: oo 92 f (x, y, z) = | ,~ sin(sx)ds so 1 r _ 1 y2 J S3 >~e-zs sin(sx)ds + O(y4) (5 15) 1 This is the formal Taylor series expansion. Numerical in- vestigation reveals the fact that this series is not converging very well. Perhaps Pade approximation could lead to valu- able results. Numerical experiments give raise to the follow- ing table (see table 1): (x, y, I) (-1,.1,-1) (.1,.5,.1) (~1,.9,.1) (~1,.9,.9) (~5,.1,.9) (~5,.9,.5) f (x, y, z) .30124 .0053 .0029 .00252 .2132 .1608 ~- . Taylor .5179 2.7 102 1.1 105 .0044 .2132 l 2.11 Pade[2/2] .3019 .0073 .006 .00254 .2132 .168 . Pade[O/4] .3031 -.0018 -.00012 .00255 .2132 .177 Table 1: comparison between different approximation tech- niques. So in some cases Pade approximation leads to better results. 6 Computational results. 6.1 Wave profiles of Wigley Hull. Some examples for a parabolic Wigley Hull (Shearer t194) will be calculated. The Wigley Hull is a mathematically defined form (see figure 6.1~: y = By _ `227~2~1 _ ~ Z >~2~ with -2 -48;00 -32.00 -16.00 0.00 16.00 32.00 Figure 6.1: one side of \Arigley Hull. For the steady wave potential , matrix equation (3.25) to- gether with the right hand side (3.33) have been used. The solution of this matrix equation is the source strength a(~) . Now using equation (3.26) the potential ~ can be calculated. A first order approximation for the wave height ~(x) can be found using the following: nix ~ = --o=(x, y,0) (6.2) The potential ~ and the source strength ~ are known (have been calculated). The wave height or equivalently = can be calculated using approximately the same formulas as Kel- log (~3.20) and (3.21~: B' = 2CJ(X)n] + A| ~ dSt = 2af In + | | a(~) ,~ dSt B~ = 2a~x~n3 + ||a(~) ~ dS~ (6.3) The differentiated Green functions dG/0xi are known be- cause they have also been used in order to calculate matrix equation (3.25~. So when setting up the computer program a data set with the dG/8xi-must be preserved. Now the wave heights can be calculated easily. For a series of Froude numbers (0.20 - 0.45) the steady wave potential A of a Wigley Hull have been calculated as can be seen in figure 6.3. The results have been compared with the measured and calculated values of Kitazawa and Kaji- tani t20~. In our calculations a grid size of 24 x 8 has been used, so discrepancies could be also due to this relatively large grid size. Especially near the bow this could lead to substantial errors. See for instance figure 6.2. These calcu- lations have been performed on a 48 x 8 grid. The values near the bow agree more with the measured values. Starting from the bow the wave height first increases a little and then decreases rapidly as can be seen from the measurements and can also be observed in figure 6.2. 129

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0 . 02 0.015 0.01 0.005 O -0 .005 -O . 0 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0.06 n 0 04 0.02 o -0 .02 -0 .04 -n n n n 0.02 -0 .04 O . u 0.04 n 0.02 o -0 .02 -0 .04 -O . VIE -1 . ~ ~ -1 -0.8 -0.6 -0 .4 -0.2 0.04 -n no 4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 . ~ _A 6 2 o -2 4 ~ -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 . v 0 ~ ~ -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 6.6: Wave heights for the unsteady motion (En = 0.31). 130

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A ~ ~ ~ Figure 6.2: Wave profile on a 48 x 8 grid (Fn = 0.20~. It can also be observed that only near the ship bow the wave height changes substantially when decreasing the grid size. So local grid refinement near the ship bow will lead to more accurate results. For a specific value for the Froude number (Fn = 0.348), the calculation can be compared with values calculated by Tsutsumi t21] and Dawson t22] as can be seen in figure 6.4. In all our calculations the wetted body ~ is divided into triangular panels. This has been done such that to obtain a symmetrical grid with respect to the y-0 plane. In order to test the grid-independence of the algorithm, the symmetry with respect to this y-0 plane has been removed. Results of this can be observed in figure 6.5. The algorithm seem to be relatively grid-independent. For the unsteady motion 0 the water heigths for the six dif- ferent types of motion have been calculated using the same matrix equation (3.25) as for the unsteady motion o but now with different right hand sides (for instance equation (3.30~. see figure 6.6. For this figure the following right hand sides for the unsteady ~ wave motion ~ have been used: surge:-And TOil: w~zn2-yn3) sway:-con2 pitch:-Un3 heave:-~n3 yaw: Un2 (6.4) Of course as mentioned is the paragraph dealing with the incorporation of the body boundary condition, these right hand sides are not complete. For instance when ~ = (~1) the right hand side for surge looks like: _ -no ~=-name-n3\2Z (6.5) But already some interesting features of the unsteady ware character or the dynamic swell-up factor can be observed here. In these figure there is a sway-yaw and a pitch-heaste correspondence. This can also be seen from the right hand side of equation (6.4~. But a different order for the magni- tude: sway O(~) and yaw (ECUS is evident. 6.2 Added Resistance. Once all the charactertic quantities are known, the pressure can be determined from Bernoulli's equations. (added) Re- sistance can be calculated directly by integration of first and second order pressure, or by means of conservation of mo- mentum, derived in a similar way as done by Huijsmans t23~. - Figure 6.4: Wigley Hull - Wave Profiles for Fn = 0.348 0.02 0.015 0.01 0.005 O -0 .005 -0.01 Figure 6.5: Grid independence of the algorithm (Fn=0.2663. 7 Concluciing remarks. We presented an asymptotic method for the calculation to the influence of a slowly oscillating movement on the velocity potential. Especially the steady~unsteady wave interaction is important. The free surface condition for the unsteady wave component was derived and the body boundary con- dition. The potential function, source strength and Green function are expanded in the small parameter w. A first order approximation for the unsteady wave have been ob- tained using calculation that only involves the evolution of the steady wave Green function. This Green function must also be used when calculating the steady state character- istics. Future plans involves the calculation of (added) re- sistance in the computer program as well. Also at the Ship Hydromechanics Laboratory at the Delft University of Tech- nology some model tests will be performed. So the calcu- lated results can be compared with these test. Acknowledgments The author wishes to thanl: R.H.M. Huijsmans and H.C. Raven of M GRIN for their valuable comments. 131

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References t1] Blok, J.J. and Huisman, J.,'Relative motions and swell- up for a frigate bow.' The Royal Institution of Naval Architects (1983~. t2] Tasaki, R.,'On shipment of water in head waves', 10th ITTC. London, (1963~. Brandsma, F.J., Hermans, A.J.,'A quasi linear free sur- face condition in slow ship theory,' Shiffstechnik 32 Heft 2 pp. 25-41. [4] Brandsma, F.J., Low froze number expansions for the wave pattern and the wave resistance of general ship forms. Phd. Thesis. Delft (1987~. t5] Baba, E., 'Wave resistance of ships in low speed.' Mit- subishi Technical Bulletin 109 (1976~. t6] Sakamoto, T. and Baba, E.,'Minimization of resistance of slowly moving full hull forms in short waves.' (1985~. t7] Hermans, A.J. and Huijsmans, R.H.M., 'The effect of moderate speed on the motion of floating bodies.', Schiffstechnik, 34 (1987~. t8] Timman, R., Hermans, A.J. and Hsiao, G.C., Water Waves and Ship Hydrodynamics. Delft University Press (1985). [9] Brard, R., 'The reprensentation of a given ship form by singularity distributions when the boundary condi- tion on the free surface is linearized.' Journal of Ship Research, 16 (1972) 79-92. t10] Kellogg, O.D., Foundations of Potential Theory, Springer Verlag, Berlin (1929~. t11] Newman, J.N., 'Evaluation of the wave-resistance Green function: Part 1 - The double integral.' Jour- nal of Ship Research, 31~1987) 79-90. t12] Newman, J.N., 'Evaluation of the wave-resistance Green function: Part 2 - The single integral on the cen- terplane.' Journal of Ship Research, 31 (1987)145-150. t13] Dercksen, A., 'Pa,~el integration of 1/A-function for nearbylocations.' IvIARINreportno.50831-1-RF, Wa- geningen (1988) t14] Wehausen, J.V. and La~tone, E.V.,'Surface waves', Hand~ok of Physics ( 1960~. [15] Fox, L. and Parker, I.B., Chebychev polynomials in nu- merical analysis. O~ford University Press U.K. (1970) [16] Baker, G.A. jr.,Essentials of Pade appro~cimants, Aca- demic Press, New York (1975~. t17] Baker, G.A. jr. and Graves-Morris, P., Pade approxi- mants, part I 1~ II, Encycl. of Mathematics (1981~. t18] van Gemert, P.H., A linearized surface condition in low speed hydrodynamics. Delft University of Technology (1988) t19] Shearer, J.R. and Cross, J.J., 'The experimental de- termination of the components of ship resistance for a mathematical model', Transactions of the Royal Insti- tute of Naval Architects, vol.107 London, pp 459-473 (1965~. t20] Kitazawa, T. and Kajitani, H., 'Computations of wave resistance by the low speed theory imposing accurate hull surface condition', Proc. Workshop on Ship Wave- Resistance Comp., Bethesda, Marylalld (1979) 288-305. t21] Tsutsumi, T., 'Calculation of the wave resistance of ships by the numerical solution of Neumann-Kelvin problem',P~c. Workshop on Ship Wave-Resistance Comp., Bethesda, Maryland (1979) 162-201. t22] Dawson, C.W., 'Calculations with the XYZ free surface program for five ship models', Proc. Workshop on Ship Wave-Reststance Comp., Bethesda, Maryland (1979) 232-255. t23] Huijsmans, R.H.M. and Hermans, A.J., 'The effect of the steady pertubation potential on the motions of a ship sailing in random seas', 5th Int. Conf. on Numer- ical Ship Hydrodynamics, Hiroshima (1989~. t24] Jensen, G., Mi, Z.X. and Soding, H., ' Rankine source methods for numerical solutions of the steady wave re- sistance problem,' Proc. 16th Symp. on Naval Hydrody- namics., Berkeley (1986~. DISCUSSION . Nicholas Newman Massachusetts Institute of Technology, USA It is very interesting to see this expansion about the w=0 solution complementing the studies at Delft where n = 0 is the basis. But here the applications are less obvious. Added resistance is mentioned by the authors, but it is generally negligible for small w. Perhaps seakeeping problems in following or quartering seas are more appropriate applications? AUTHORS' REPLY We would like to thank Professor Newman for this discussion; it will give us the opportunity to clarify this. The problem is not only the calculation of the added resistance, which can indeed be called small for small w (but will be calculated just as well), but also the determination of the dynamic swell-up. This swell-up is shown to be an important factor-for instance, when looking at the phenomenon of shipping of W green waterW. About your last remark, seakeeping problems in, for instance, following seas, is also an interesting thing to investigate. But when the incoming waves catch up with the ship, this ship is actually ~ridingW on a wave. A highly unstable situation occurs which cannot be considered by us thus far. 132