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Numerical Evaluation of a Ship's Steady Wave Spectrum F. Noblesse David Taylor Research Center, Bethesda, USA W. M. Lin Science Applications International, Annapolis, USA R. Mellish David Taylor Research Center, Bethesda, USA ABSTRACT The study presents a modified mathemati- cal expression for the wave-spectrum function in the Fourier representation of the wave pattern of a ship advancing at constant speed in calm water. This new expression is obtained from the well-known usual expression via several applications of Stokes' theorem resulting in the combining of the integrals along the waterline and over the hull surface of the ship. The modified expression for the wave-spectrum function is considerably better suited than the usual expression for accurate numerical evaluation because the significant numerical cancellations which occur between the waterline and hull integrals in the usual expression are automatically and exactly accounted for in the modified expression, as is demonstrated mathematically and confirmed numerically. INTRODUCTION Near-field potential-flow calculations about ships advancing at constant speeds in calm water are routinely required for evalua- ting their hydrodynamic characteristics, both in calm water and in waves, and for determin- ing the required propulsion and control devices. Calculations of far-field ship wave patterns are also important in connection with wave-resistance predictions and remote sensing of ship wakes. In particular, the latter practical application requires the capability to determine the short divergent waves in the wave spectrum having wavelengths between 5 cm and 40 cm associated with Bragg scattering of the electromagnetic waves in typical SAR systems used in remote sensing of ship wakes. No meaningful predictions of such short waves can be obtained on the basis of currently available numerical methods. More generally, numerical predictions of the steady wave pattern at large and moderate distances behind a ship are notoriously dif- ficult and unreliable, as was recently made clear at the Workshop on Kelvin Wake Computations [1]. Ship wave-resistance cal- culatio-ns are also known to be unreliable. 145 Several alternative numerical methods have been developed for evaluating near-field flow about a ship, that is, flow at the hull surface and in its vicinity. These include finite-difference methods, e.g. Coleman [2] and Miyata and Nishimura [3], and the more widely used boundary-integral-equation methods, also known as panel methods. The latter methods can be divided into two main groups, according to the Green function that is used. These two groups of methods are the Rankine-source method and the Neumann-Kelvin method, which are based on the simple Rankine (free-space) fundamental solution and the more complex Green function satisfying the linearized free-surface boundary condition, respectively. The Rankine-source method was initiated by Gadd [4], Dawson [5] and Daube [6], and has since been adopted by many authors. The Neumann-Kelvin approach has a long history. A survey of recent numerical predictions obtained by a number of authors on the basis of the Neumann-~elvin method may be found in Baar [7]. This study and that by Andrew, Baar and Price [8] also contain extensive comparisons of the authors' own Neumann- Kelvin numerical predictions with experimen- tal data. An approximate solution, defined explicitly in terms of the value of the Froude number and the hull shape, to the Neumann-Y~elvin problem was proposed in Noblesse [9]. This slender-ship approximation was recently used by Scragg et al. [10] and compared to both Neumann-Kelvin predictions and experimental data in [7] and [8] and to experimental data in [1] and [11]. The aforementioned alternative numerical methods for predicting flow in the vicinity of a ship are not all directly suitable for predicting the far-field wave pattern of a ship. More precisely, the finite-difference method and the Rankine-source panel method require truncating the flow domain at some relatively-small distance away from the ship and therefore can only be used for near-field f low calculations. (However, these near-field f low predict ions can be used as input to the far-field Neumann-Kelvin flow representation considered in this study. ) On the other hand,

the Neumann-Kelvin theoretical framework is equally suitable for near-field and far-field flow predictions. Indeed, the far-field Neumann-Kelvin flow representation is a simplified particular case of the correspond- ing near-field representation. The problem considered in this study is that of evaluating the steady wave spectrum and the wave pattern of a ship at moderate and large distances behind it in terms of the near-field flow on the hull surface. The near-field flow thus is assumed known for the purpose of the present study, which is con- cerned with the prediction of the steady wave spectrum and the wave potential behind a ship stern within the Neumann-Kelvin theoretical framework as was just noted. This theory expresses the wave potential in terms of a Fourier representation, as is well known and is specifically indicated by Eq. (3) in this study. The wave-spectrum (or wave-amplitude) function in this Fourier representation is defined by the sum of an integral along the mean waterline and an integral over the mean wetted-hull surface. This expression for the wave-spectrum func- tion is ill suited for accurate numerical evaluation because the waterline integral and the hull integral largely cancel out, as is shown further on in this study. Errors which inevitably occur in the numerical evaluation of the waterline and hull integrals cause imperfect numerical cancellations between these integrals and corresponding large errors in their sum. This fundamental diffi- culty was recognized in [9] and in [12], where attempts to remedy it were presented. However, these ad hoc approximate numerical remedies, based upon combining the waterline integral with the contribution to the hull integral stemming from the upper part of the hull surface are not satisfactory. A conceptually simpler and numerically more effective remedy is presented in this study, in which a modified mathematical expression for the wave-spectrum function is obtained via several applications of Stokes' theorem resulting in the combining of the waterline integral and the bull integral. This new expression for the wave-spectrum function is considerably better suited than the usual expression for accurate numerical evaluation because the significant numerical cancellations which occur between the water- line and hull integrals in the usual expres- sion are automatically and exactly accounted for, via a mathematical transformation, in the modified expression obtained in this study. The fundamental advantage of the new expression over the usual one may readily be appreciated from Fig. 4. Another interesting feature of the modi- fied expression for the wave-spectrum func- tio-n is that it only requires the tangential velocity at the hull, not the potential, whereas the usual expression requires the values of both the velocity potential and its gradient at the hull. The modified expres- sion thus defines the wave-spectrum function in terms of the speed and the size of the ship, the shape of the mean wetted-hull sur- face and the tangential velocity at the mean bull surface. This expression is suitable for use in conjunction with any near-field flow-calculation method, including boundary- integral-equation methods based on source distributions and other numerical methods in which the velocity vector is determined directly on the mean hull surface rather than derived from the potential. It provides a practical and reliable method for coupling a far-field Neumann-Kelvin flow representation with any near-field flow-calculation method, including methods based on the use of Rankine sources or finite differences. NEUMANN - ELVIN REPRESENTATION FOR THE WAVE POTENTIAL As already noted, this study considers steady potential flow about a ship advancing at constant speed in calm water of infinite depth and lateral extent. Nondimensional coordinates and flow variables are defined in terms of the length L and the speed U of the ship and the water density p. The undisturbed sea surface is chosen as the plane z = 0, with the z-axis pointing upwards, and the x- axis is taken in the ship centerplane (port- and starboard-symmetry is assumed) and point- ing toward the bow, as is depicted in Fig. 1. The Froude number and its inverse are denoted by F and v, respectively, and are given by F = U/(gL) = 1/v , (l) where g is the acceleration of gravity. Within the so-called Neumann-<elvin theo- retical framework, the velocity potential ¢(~) at any point ~ = (I, n , ~ < 0) strictly outside the ship hull surface is defined in terms of an integral representation [9, 13] involving integrals along the mean waterline and over the mean wetted-hull surface of the ship. The integrands of these integrals involve the values of the potential ~ and of its gradient at the mean hull surface as well as the Green function and its gradient. The Green function may be expressed as the sum of three terms [14] corresponding to a Rankine source/sink pair, a nonoscillatory near-field flow disturbance and tile wave pattern behind the singular point in the Careen function. Likewise, the potential ¢(~) may then be expressed as the sum of three terms, as follows: |(~) = lS(~) + ANT) + We) ~ (2) where IS ~ IN and LOW correspond to the singu- lar source/sink term, the nonosciliatory near-field term and the wave term, respec- tively, in the expression for the Green function. This study is restricted to the numerical evaluation of the wave potential |~ , which is the most complex of the three components in Eq. (2) and is dominant at some distance behind the ship. 146

More precisely, the problem considered this study is that of evaluating the wave potential () at any point (5, n , ~ < 0) behind the ship stern. It is shown in [9] and [13] that the wave potential may be defined in terms of the following Fourier integral representation: W(~) = (2/~) |o exp(v2`p2) cos(v2npt) Im exp(iv (p) K(t) dt , (3) where p is defined in terms of the Fourier variable t by the relation p = (1 + t2) 1/2 (4) and K(t) represents the wave-spectrum func- tion defined further on in this study. The wave potential () in Eq. (3) is expressed in terms of a familiar Fourier superposition of elementary plane waves propagating at angles ~ from the ship track given by tang = t . (5) The amplitudes of these elementary plane-wave components are essentially given by tile func- tion K(t), which may thus be referred to as the far-field wave-amplitude function or as the free-wave spectrum function. The wave- spectrum function K(t) contains essential information directly relevant to a ship's wave resistance and wave pattern. In partic- ular, the wave resistance, R say, experienced by the ship is defined in terms of the wave- spectrum function by means of the well-known Havelock formula R/(pU L ) = |o [K(t)] p dt · (6) The wave-spectrum function K(t) in Eqs. (3) and (6) may be expressed as the sum of two terms [9], as follows: it(t) = Ko(t) + K¢(t) , (7) where Ko represents the (zeroth-order) slender-ship approximation and Rip the Neumann-Kelvin correction term in the Neumann-Relvin approximation Ray + K¢. More precisely, the function JO + Kid corresponds to the usual linearized Neumann-Kelvln approximation, in which the nonlinear terms in the free-surface boundary condition are neglected. These nonlinear terms yield an additional term in the expression for the spectrum function K, defined by an integral over tile mean free surface [9,13], which is ignored here. The slender-ship approximation Ko is defined explicitly in terms of the value of the Froude number and the hull shape, whereas the Neumann-T~elvin correction K~ also depends on the value of the potential at the mean bull surface. The functions X~ and Kit are considered in turn, beginning with the stender-ship approximation Do. in THE SLENDER-SHIP APPROXIMATION The slender-ship approximation Ko(t) to the wave-spectrum function K(t) is defined in [9] as the sum of a line integral Kw(t) along the ship's mean waterline w and a surface integral Rh(t) over the ship's mean wetted- hull surface h, as follows: Ko(t) = KW(t) + Kh(t) , (8) where the waterline and hull integrals are given by K = J (E++E )nx ty dQ , (9a) Kh = v |h exp(v p z)(E++E )n da . (9b) In these expressions, E+ represent the trigonometric functions defined as E+ = exp[-iv p (ux + vy)] , (10) where u and v are given by u = 1/p and v = tip ; (lla,b) it may then be seen from Eq. (4) that we have 1 > u > 0 and O < v < 1 (12a b) _ _ _ _ , for O < t < ~ , with u + V2 = 1 (13) Furthermore, w and h represent the positive halves of the mean waterline and of the mean wetted-hull surface, respectively, as is depicted in Fig. 1 where h = s + b with s - hull side and b = hull bottom. Also, do is the differential element of arc length of w and da the differential element of area of h. Finally, n = (ox, ny, no) is the unit vector normal to h and pointing outside the ship, and t = (tx, ty, tz = 0) is the unit vector tangent to ~ and pointing toward the bow, as is shown in Fig. 1. The hull bottom of a typical ship is a nearly horizontal surface, so that we have nx ~ O on b, but nx is usually significant on the hull side in the bow and stern regions. However, the bull side of a typical ship is a nearly vertical surface, i.e. we have nz ~ O on s. It is therefore convenient to express the hull integral as the sum of integrals over the hull side and the hull bottom, and to modify the bull-side integral into a form involving tile source density no instead of nx by using Stokes' theorem in tile manner sI~own in [13]. The slender-ship approxi~a- tion Ko(t) may then be expressed in the form Ko(t) = K *(t) + K ,(t) + Kh*(t), (14) where tile functions K *(t), KW,(t) and K~l*(t) are det ined as low* = is (E +E ) (nx -u )ty di , ( 1 5a ) 147

K , = u2 J , exp(v2p2z)(E++E )t do , Kh* = -iv u J exp(v p z)(E++E )n da 1 ~2 ~ ~ 2 2~= 1~ >~ AN (15b) I v Jb =~`v ~ ~~+ =- MANX ~~= e (15c) In the foregoing modified expression for the slender-ship approximation Ro(t), the function Kw*(t) represents a modified water- line integral, with source density (nX2-u2)ty instead of nX2ty in Eq. (9a). Furthermore, the function Kw'(t) corresponds to a line integral along the waterline-like curve w' separating the hull side s and the hull bottom b, as is shown in Fig. 1; the unit tangent vector t = (tx, ty, tz) to the lower waterline w' points toward the bow. Finally, Rh*(t) represents a modified hull integral consisting of the sum of an integral over the hull bottom b and the hull side s, with source densities given by nx and -iunz , respectively. The latter source density is null for a wall-sided ship and, more generally, vanishes in the limit t ~ ~ , as may be seen from Eqs. (4) and (lla). The hull-side integral therefore is generally less important in the modified expression (14) than in the original expression (8). In particular, the hull-side and hull-bottom integrals in Eq. (15c) are null for a wall- sided ship with a flat horizontal bottom (i.e., a strut-like form), for which Eq.(14) thus expresses the slender-ship approximation Ro(t) as the sum of two line integrals. For large values of v2p2 = (sec26)/F2 , the trig- onometric functions E+ defined by Eq. (10) oscillate rapidly. The dominant contribution to the modified waterline integral Rw* in Eq. (14) therefore stems from the points, if any, where the phases v2p2(ux + vy) of the trigo- nometric functions E+ are stationary. These points of stationary phase are defined by the conditions udx + vdy = 0, which yield the relations ut + vt = 0 , tx = v , ty = + u ; (16a,b,c) the latter two relations can be obtained from Eq. (16a) by using Eq. (13) and the identity tX2 + ty2 = 1. The term u2 in the integrand of the modi- fied waterline integral Kw* defined by Eq. (15a) stems from the integral on the hull side in Eq. (8), as may be seen by comparing the alternative expressions for the function Ko given by Eqs. (8), (9a,b) and Eqs. (14), (15a,b,c). We have nx = -t along the top waterline of a wall-sided ship; Eq. (16c) therefore shows that the term nX2-u2 in the integrand of the modified waterline integral Kw* vanishes at a point of stationary phase for a wall-sided ship. This result indicates that the waterline integral and the hull-side integral in Eq. (8) cancel out in a first approximation (specifically, within the stationary-phase approximation) for a wall- sided ship. The major contributions to these two integrals thus are combined into the modified waterline integral Kw* in the modi- fled expression (14), and the modified bull- side integral in Eq. (14) is less important than the original hull-side integral in Eq. (8) as was already noted. The modified expression for the slender- ship approximation Ko(t) defined by Eqs. (14) and (lSa,b,c) thus is better suited for accu- rate numerical evaluation than the usual expression defined by Eqs. (8) and (9a,b) for large values of v2p2, that is for small values of the Froude number and/or large values of t = tans. However, significant can- cellations may be expected to occur between the line integrals Kw* and Kw' in Eq.(14) for relatively large values of the Froude number and small values of tans. More precisely, the term -u2(E++E_)ty in the integrand of the top-waterline integral Kw* defined by Eq. (15a) and the integrated u2exp(v2p2z)(E++E_)ty of the lower-waterline integral Kw' defined by Eq. (lSb) may nearly cancel out if exp (v2p2z) ~ 1, that is for small values of v2p2d where d is the ship draft. It therefore is useful to express Eq. (14) in the following form: K~(t) = K*(t) + K'(t) , (17) where the functions K*(t) and K'(t) are defined as lo'* = 1 (E +E ) w ~ _ [nx -u +u exp(v p z)]ty dQ , (18a) K' = u |w' exp(v p z)(E++E )t do - u | exp(v p z)(E++E )t dI - iv u | exp(v p z)(E++E )n da + v |b exp(v p z)(E++~ )n da . (18b) In the integrals along the top waterline w in Eqs. (18a,b), z is to be taken equal (or, more generally, approximately equal) to the vertical coordinate of the point (x,y,z) on the lower waterline w1 , in such a way that the integrals along the lower waterline w' and the top waterline w in Eq. (18b) nearly cancel out. In the simple case of a strut-like hull form we have nx = 0 on the hull bottom b and no = 0 on the hull side s. Furthermore, the lower waterline w' is identical to the top waterline w except for a vertical translation equal to the ship draft d, and z in the integrals along the lower and top waterlines w' and w in Eq. (lab) is equal to -d. For such a simple strut-like hull we then have K'(t) = 0 and F.q. (17) yields Ko(t) = K*(t). The modified waterline integral K* defined by Eq. (18a) thus provides an exact expres- sion for the slender-ship approximation To in the special case of a strut-like hull form. For a simple hull in the shape of a strut the alternative expressions for the slender- ship approximation TO defined by Eqs. (8) and (17) become 148

Y`o(t) = K (t) + ~ (t) = Y~*(t) , (19) where the hull integral Ah in Fq. (8) was replaced by the hull-side integral Ks since we have nx = 0 on the horizontal bottom of a strut. The real and imaginary parts of the functions Kw(t) , KS(t) and fit) _ K*(t) are depicted in Fig. 2 for 0 < t = tans < 5 (corresponding to 0 < ~ < 79°) for a specific strut-like hull form at three values of the Froude number, namely O.1 (top row), 0.2 (center row) and 0.3 (bottom row). The strut considered for the calculations presented in Fig. 2 has beam11ength and draft/length ratios equal to 0.16 and 0.07, respectively, and consists of a pointed bow region 0.2 < x < 0.5 with parabolic waterlines, a straight middle-body region -0.3 < x < 0.2 and a rounded stern region -0.5 < x < -0.3 with elliptic waterlines. The top row of Fig. 2, corresponding to the small value of the Froude number F equal to 0.1, shows that the function K~ is signif- icantly smaller than the waterline and hull- side integrals Kw and Us in Eqs. (19) and (8). This numerical result is in accordance with the previously-established theoretical result that the major contributions to the integrals Kw and Rs cancel out for small val- ues of the Froude number. The function Ko , especially its read part represented by a solid Line, is also appreciably smaller than the functions Kw and Ks in the center row of Fig. 2 corresponding to F = 0.2 and, to a reduced degree, in the bottom row correspond- ing to the fairly large value 0.3 of F. The integral KW (t) along the lower waterline w' in Eqs. (14), (15b) and (18b) is also depicted in Fig. 2. The top row of this figure, corresponding to F = 0.1, shows that the lower-waterline integral KW'(t) is negli- gible in comparison with the function,Ko(t) _ K*(t) for all values of t due to the expo- nential function exp (v2p2z) in the integrand of the lower-waterline integral Kw . How- ever, this integral is significant for small and moderate values of t = tans in the center and bottom rows of Fig. 2 corresponding to F = 0.2 and 0.3, respectively. For typical hull forms Eq. (17) expresses the slender-ship approximation Ko(t) as the sum of the modified waterline integral K*(t) defined by Eq. (18a) and the remainder ~'(t) defined by Eq. (18b). The remainder K'(t) may generally be expected to provide a rela- tively small correction to the dominant waterline integral K*(t). In particular, the integrals along the lower and top waterlines w' and w and the hull-bottom integral in Eq. (18b) decay exponentially due to the exponen- tial function exp(v2p2z) in their integrands. These three integrals thus are negligible for sufficiently large values of v2p2 , for which the major contribution to the remainder R' stems from the upper part of the hull side in the hull-side integral in Eq. (18b). It may thus be useful to divide the bull side into an upper part -d < z < 0 and a lower part z < -d , where ~ is solve fraction of the depth of the hull side s. The upper hull side can be approximately defined by tile parametric equations x = a(Q) + z a(Q ) and y = b(Q) + z 3(Q) for -(A) < z < 0 , where represents the arc length along the top waterline w defined by x - a(Q) and y = b(Q), and a = ax/az and ~ = ay/a' are the slopes of the hull surface at the waterline. The con- tribution of the upper part of the hull side to the hull-side integral in Eq. (18b) can then be expressed as an integral along the top waterline w, which can be grouped with the top-waterline integral K* defined by Eq. (18a). In this manner the do~lnant waterline integral it* is modified by including the con- tribution of the upper part of the hull side to the hull-side integral in Eq. (18b), whereas the remainder K' is modified by restricting the integration in the hull-side integral to the lower part of the hull side. This modified remainder K' thus is expo- nentiaLly small for large values of v2p2 = (sec20)/F2 and can only be significant for small and moderate values of v2p2. The hull bottom b, the lower part of the hull side s and the lower and top waterlines w' and w in expression (18b) for the remainder it' may then be approximated by using a relatively coarse discretization, whereas a finer discret izat ion may be used for representing the top waterline w in expression (18a) for the dominant waterline integral K* . The modi f led f orm of the top-waterline integral ( 18a ) including the contribution of the upper hull side to the hull-side integral in Eq. (18b) can easily be derived from Eq. (18b). THE NEUMANN-KELVIN APPROXIMATION The correction terra K<, in Eq. (7) for the Neumann-Y~elvin approximation Ko + K¢, to the wave-spectrum function K is defined by the sum of a waterline integral and a l~ull- surface integral [ 9,13]: K<~(t) = KW(t) ~ KW'(t) + KH'(t), (20) where Kw(t) and Kw'(t) are the waterline integrals and KH' (t) the hull-surface integral def ined as A Iw (E/E_) (tX4)t+sx~s)ty dQ , (21a) Kw ' = iv p | (E++E )¢ t do, ( 2 1 b ) KH' = (v p ) |h exp (v p z ) (E+n++l3: n )¢ da . ( 21 c ) In the foregoing equations E+ are the trigo- nometric functions defined by Eq. (10). The functions n+ in Eq. (21c) are defined as n = -n + i(un + vn ) . (22) + z x y In Eqs. (21a-c) and (22), t = (tx, ty, 0) is the Ullit vector tangent to the waterline w and pointing toward the bow, as was already defined, s = (sx, sy, sz ) is a unit vector 149

tangent to the hull surface h and pointing downwards and n = (ox, ny, nz) is the unit vector normal to ~ and pointing outside the ship, as is shown in Fig. 1. Finally, it and Is in Eq. (21a) represent the components of the velocity vector V) in the directions of the unit tangent vectors t and s to h, respectively. Numerical evaluation of the waterline and hull integrals in Eq. (20) is a seemingly simple task, given the value of the potential on the mean hull surface h + w ; in partic- ular, the integrands of the integrals defined by Eqs. (21a-c) are continuous functions. Nevertheless, accurate and efficient numeri- cal evaluation of these integrals requires careful analysis because the trigonometric functions E+ defined by Eq. (10) oscillate rapidly for large values of v2p2, as is the case for typical values of the Froude number F = 1/v and of the Fourier variable p2 = 1 + t2 = sec2O, and because the potential ~ in the integrands of the waterline and hull integrals Kit' and KH' in Eqs. (21b,c) is mul- tiplied by tile large numbers v2p and (v2p)2, respectively. More precisely, we have 102 < v4 < 104 for 0.1 < F < 0.32 and p2 > 10 for > 72°; values of (v2p)2 as large as 105 thus are possible. The waterline and hull integrals Kw' and KH' in Eq. (20) may then be expected to be dominant and to largely cancel out, as is shown in Fig. 3. More precisely, Fig. 3 depicts the func- tions Kid , KW + Kw' and KH' for O < t < 10 (corresponding to 0 < ~ < 85°) for the simple bull form considered previously in Fig. 2 with an assumed simple mathematical expres- sion for the value of the velocity potential at the hull surface. Specifically, the potential in Eqs. (21a-c) is taken as ~ - F2 exp(v2z) cos[v2(x-1/2)-3~/8], which corre- sponds to an elementary plane progressive wave. This simple hull form and assumed sim- ple expression for the potential at the hull surface are used for tile calculations pre- sented in Fig. 3 because they permit accurate numerical calculations (the required integra- tions can be partially performed analytical- ly) and they are adequate for the purpose of numerically illustrating the essential prop- erties of the several alternative mathemati- cal expressions for tile Neumann-Kelvin correction Kid examined in this study. Figure 3 corresponds to a value of the Froude number equal to 0.15. It may be seen from Fig. 3 that the function ~¢ is considerably smaller than the waterline and hull integrals Kit + Kw' and KH' . In particular, the waterline and hull integrals rK~ + Kw' and KH' do not appear to vanish in the limit t ~ ~ (D ~ 90°). Significant cancellations therefore occur between the waterline and hull integrals in Eq. (20). These significant cancellations occur for all values of ~ but are especially notable for large values of 3, corresponding to the snort divergent waves in the spectrum. The errors which inevitably occur in the numerical evaluation of the integrals Kw + Kw' and KH' cause imperfect numerical cancel- lations between these components and corre- spending large errors in their sum. Numerical errors in the sum Kid can be especially diffi- cult to control because the errors associated with the numerical evaluation of the hull integral KH' and the waterline integral Kw + Kw' are not necessarily comparable (due to differences in the errors associated with numerical integration over hull panels and waterline segments). The usual expression (20) for the Neumann-Kelvin correction Rip in Eq. (7) thus is ill suited for accurate numerical evaluation. A modified mathemati- cal expression in which the cancellations between the waterline and hull integrals KW + Kw' and KH' depicted in Fig. 3 are automati- cally and exactly accounted for, via a mathe- matical transformation, is presented below. By using Stokes' theorem in the manner explained in [13] we can combine the water- line and hull integrals Kw' and KH' defined by Eqs. (21b,c) into a modified hull integral ~~ , as follows: KH(t) = KW'(t) + KH'(t) , (23) where the modified hull integral Kit is given by XH = iv P Ah exp(v p z)(E+a++E a ) da (24) with a+ = nzal/ax - nxa¢/a~ + iv(nxa¢/3y - nya¢/3x) . (25) By substituting Eq. (23) into F.q. (20) we may then obtain the following modified expression for the Neumann-Kelvln correction T<¢ (t): K¢(t) = KW(t) + KH(t) . (26) The functions Kit' , KH' and KH are depic- ted in Fig. 3. This figure shows that the waterline and hull integrals Kw' and KH' are considerably larger titan the modified hull integral KH . Although the latter integral is identical to t'ne sum of the integrals Kw' and KH' , it clearly is preferable to evalu- ate XH directly by means of Eqs. (24) and (25) rather titan as the sum of the integrals KW' and KH' defined by Eqs. (21b,c). The modified expression for the Neumann-Kelvin correction R+(t) given by Eqs. (26), (21a), (24) and (25) therefore represents a signifi- cant improvement in comparison with the usual expression given by Eqs. (20) and (9la-c). It is shown in [13] that the cancellations between the waterline integral Kw' and the hull integral Ad' depicted in Fig. 3 can be explained mathematically for a wall-sided ship form by performing an asymptotic analy- sis in the limit v2p2 ~ ~ similar to that presented previously in this study for the slender-ship approximation `~ . The functions Kw , KH and K~ in the modi- fied expression (26) for the `,leumann-Relv~n 150

correct ion are depicted in Fig. 3. It may be seen that the waterline integral KW and the modified hull integral R}I are appreciably larger than their sum K`t,, especially for large values of t. Signif icant cancellations therefore still occur between the waterline and hull integrals in Eq. (26 ). Further rnod- ifications of the expression for the function K<, defined by Eqs. (26), (21a), (24) and (25) are then desirable for numerical calculat ions. These Audi f icat ions are now presented. By making use of Stokes' theorem and a classical formula in vector analysis we can obtain [ 13] the following alternative expres- sion for the Neumann=Kelvin correction K<p: K<,(t) = KW(t;C) ~ KH(t;C), (27) where (w(t;C) and RH(t;C) are the modified nxa¢/ay-nyal/ax = waterline and hull integrals def ined as Kit = |W (E+a++E_a_) do, (28a) KH = iv |h exp (v p z ) (E+A+-E A ) da . ( 28b ) The amplitude functions a+ and A+ in Eqs. ( 2 8a, b ) are gi ven by a+ = (tX¢t + sxts)ty + u(v - Cu)a¢1at, (29a) A+ = (Cu - v)(n al/az - nzal/ay) _ Y + (Cv + u)(nzal/aX - nxa¢/a~) + iC(nxal/ay - n al /ax) . (29b) y In the foregoing expressions, C is an arbitrary complex function of t which may be selected at will. Equations (27), (28a,b) and (29a,b) thus define a one-parameter fa~n- ily of alternat ive ~natl~ematically-equivalent expressions for the Neumann~elvin correction K¢, in Eq. ( 7 ). The velocity components ad /a x, al lay and at/az in Eq. (29b) defining the amplitude functions A+ in Eq. (28b) can be expressed in terms of tile components ~ t and ~ s of the velocity vector V) along two unit vectors t = (tx, ty, to) & s = (sx, s, s ) (30a,b) tangent to the hull surface. Note precisely, we have Vl = n a¢/an + t it + s is, whicl, yields a¢/ax = nxa¢/an + that + Seas ' a¢/ay = nyal/an + tyTt ~ Syl a¢/az = nzal/an + tort + Sz¢S ~ where a ~ /a n is the velocity component along the unit outward normal vector n to the hull surface def ined as (31a) (31b) (31c) n = (t x s) / | t x s | . (32) The vectors t and s to the ship bull surface are tangent to curves which approximately correspond to waterlines and f ramelines and they point toward the bow and the keel, respectively. The vectors t and s thus are roughly (but not necessarily exactly) ortho- gonal. At the mean free surface, the vector t is tangent to the top waterline (and we thus have tz = 0 ) in agreement Title our pre- vious de f ini t ion. Equations (31a-c) yield nya~laz-nza¢/ay = (n t -n t )¢ + (n s -n s )f (33a) y z z y t y z z y s nza¢/ax-nxa¢/a~ = (n t -n t )¢ + (n s -n s )l , (33b) z x x ;: t z x x ~ s (nxty-nytx)¢t + (nxsy-nysx)ls · (33c) By using F.qs. (33a-c) in Eq. (29b) we may express the amplitude functions A+ in the form | t x s | A+ = [ (v-Cu)tx + (u+Cv)ty - iCt~] al /a s - [ (v-cu)sX + (u+cv)sy - ices] al /a t . (34 ) The co nponents ~ s and ~ t of V) along the unit tangent vectors s and t and tile veloci- ties a¢/as = Vies and a¢/at = V¢.t are related as follows a¢/as = is ~ ~ it ' a¢/at=¢t+£~;,' AS = (a¢/aS - £ 3~/3t)/(1 _ c2 ) , (35C) = (a¢/at - ~ a¢/3s)/(1 - c2 ), (35d) t where ~ is def ined as (35a) (35b) (36) An asymptotic analysis [ 13] indicates that the modified waterline and hull inte- grals Kw(t;C) and KH(t;C) in Eq. (27) vanish more rapidly titan the corresponding integrals KW(t) and KH(t) in Eq. (26) if tile function C(t) vanishes in the limit t ~ =, that is if we have C(t) ~ O as ~ ~ ~ . (37) An obvious choice for tile arbitrary function C(t ) that satisfies this condition is C(t) = 0 . (38) [he corresponding expression for the Neumann- Kelvin correction it;, may be we itten in the fort 151

K<p(t) = KW"(t) + KH"(t), (39) where the waterline and hull integrals Kw" and KH" are defined by Eqs. (28a,b), (29a), (34) and (38). The functions YW and XH in Eq. (26) and the functions Kw" and KH" in Eq. (39) are depicted in Fig. 3. It may be seen that the functions Kit' ' and KH" vanish more rapidly than the functions KW and KH for increasing values of t, in accordance witl, condit ton (37). The cancellations occurring between the waterline and hull integrals KW and KH for large values of t thus are signif icantly reduced in the alternative expression (39 ), which is therefore preferable to expression (26 ) for large values of t. However, the functions Kw" and KH" are appreciably larger than the functions Kw and KH for small values of t, and significant cancellations thus occur between the waterline and hull integrals Kw" and KH'' for small values of t. Expression (26 ) therefore is preferable to expression (39 ) for small values of t, where- as the reverse holds for large values of t. The amplitude funct ions in the integrands of the waterline and hull integrals in Eqs. (26) and (27) can be shown [ 13] to be nearly identical for small values of t if tl~e-arbi- trary function C(t) satisfy the condition v - u C(t) << 1 as t ~ 3 . (40) This condition ensures that the waterline and hull integrals in Eq. (27) are nearly identi- cal to the corresponding integrals in Eq. (26 ) in the limit t ~ 0. The large-l and small-l conditions (37) and (40) are satis- fied if the function C(t) is selected in the form C(t) = u v N(t) . (41) Condition (40) then becomes v(l-u A) << 1 as t ~ 0. Equat ion ( 13 ) shows that we have v(l-u2X ) ~ v3 as t ~ O if the arbitrary func- tion N(t) satisfies the condition A (t) ~ 1 as t ~ 0 . (42) An obvious choice f or the funct ion ~ ( t ) satisfying condition (42) is N(t) = 1 . (43 ) The corresponding expression for the Neurrann- lkelvin correction K,~, may be written in the form K`,(t) = KW (t) + KH (t), (44) where the waterline and hull integrals KW* and KH* are defined by Ens. (98a,b), (29a) and (34 ) in which C is replaced by uv , that is C(t) = u v = t / (1 + t2 ) . (45) The functions KW and KH in Eq. (26) and the functions Kit* and KH* in Eq. (44) are depicted in Fig. 3. It may be Sean that the functions Kw* and KH* vanish more rapidly than the functions Kit and KH for increasing values of t. In this respect, the funct ions Kw* and KH* are comparable to the functions Kw" and OH" also depicted in Fig. 3. However, the functions By* and KH* and the f unc t to ns KW " and AH " are s igni f icant ly di f ferent for small and moderate values of t. More precisely, the functions Kit" and KH" are appreciably larger than the functions KW and KH, as was already noted, whereas the functions Kw* and KH* are comparable to, indeed somewhat smaller than, the functions KW and KH for small and moderate values of t. Figure 3 shows that the cancellat ions which occur between the waterline and hull integrals 'CW and KH in Eq. (26) are reduced signifi- cantly for the ~nodif fed waterline and hull integrals 1~* and KH* in Eq. (44). The lat- ter expression for the Neumann - kelvin correc- tion K<, therefore is preferable to the former one for accurate numericn1 calculations. In summary, the waterline and hull integrals KW + Kw' and KH' in the original expression (20 ) for tile Neumann-Relvin cor- rection R<> and the Modified waterline and hull integrals Kw* and KH* in the alternative expression (44 ) are depicted in Fig. 4 for the simple case considered previously in Fig. 3 and for three values of the Froude number, namely F = 0.1 (top row), 0.2 (center row) and O.3 (bottom row). The function K<> is also represented in Fig. 4. This f igure shows that the spectrum function R`1, is sig- nificantly smaller and vanishes much more rapidly with increasing values of t than its components Rig + Kw' and Kit' in the usual expression (20), as was already observed in Fig. 3. Large cancellations therefore occur among these components and Eq. (20) is ill suited for accurate numerical calculations, notably for evaluating the short divergent waves in the wave spectrum corresponding to large values of t. However, Fig. 4 also shows that the modified waterline and hull integrals Kit* and KH* in Eq. (44) are appre- ciably smaller than the original waterline and bull integrals KW + Kw' and KH', and are in fact comparable to the function'. Although the alternative expressions (20) and (44) are mathematically equivalent, the modi- fied expression (44) clearly is considerably better suited than the usual expression (20) for accurate numerical calculations. CONCLUSION In expression (a) for the velocity poten- tial of steady flow about a ship, the wave potential () at any point ~ = (:,n ,` < 0) behind the stern of the ship is defined by Eq. (3), where v and p are given by Eqs. (l) and (4), respectively, and K(t) represents the wave-spectrun function. The latter func- tion is defined by Eq. (7) as the sum of the slender-ship approximation , which is defined explicitly in terms of the Froude 152

,nurnber and the shape of the hull, and the Neumann-Kelvin correction term Kd>, which also involves the potential at the hull. The wave resistance of the ship is def ined in terms of the wave-spectrum function by means of the Havelock formula (6 ). The Fourier variable t in Eqs. (3 ) and (6 ) is related to the angle ~ of propagat ion of the f ree waves in the ship wave pattern by the relation t = tans, as is given by Eq. (5). The slender-ship approximation TO in Eq. ( 7) is def ined by the usual expression given by Eqs. (8) and (9a,b), or by the alternative modified expression defined by Eqs. (17) and (18a,b). The latter expression defines the spectrum function Ko as the sum of a modified integral K* along the ship waterline w and a remainder K' . In the special case of a strut-like hull form, the remainder K' is null and the modified waterline integral K* provides an exact expression for the slender- ship approximation Ho. The Neumann-Kelvin correction R`b in Eq. ( 7) is def ined by the usual expression given by Eqs. (20) and (21a-c) or by the alterna- tive modified expression given by Eqs. (27), (28a,b), (29a) and (34). This alternative expression involves an arbitrary complex function C(t), and thus defines a one- parameter family of mathematically-equivalent expressions for the Neumann-Kelvin correction K¢, in F~q. (7). In particular, the first modif fed expression, def ined by Eqs. (26 ), (21a), (24) and (25), obtained in this study is a special case of the general expression given by Eqs. (27), (28a,b), (29a) and (34) corresponding to the choice C(t ) = t. Analytical and numerical considerations led to the particular expression given by Eq. (44), which corresponds to the choice C(t) = uv def ined by Eq. (45) where u and v are given by Eqs. (lla,b). Figure 4 shows that the mathematical expression corresponding to E q. (44 ) is considerably bet ter suited than the usual expression (20 ) for accurate numer- ical evaluation because the large cancella- tions which occur between the waterline integral Kw + Kw' and the hull integral KH' in the usual expression (20 ) are automati- cally and exactly accounted for, via a mathe- matical transformation, in the new expression (44 ) involving the modif fed waterline and hull integrals KW* and KH* ~ Another interesting feature of the new expression for the Neumann~elvin correction K~, given by Eqs. (27), (28a,b), (29a) and (34 ) is that it only requires the tangential velocity at the hull, not the potential, whereas the usual expression given by Eqs. (20) and (21a-c) requires the values of both the velocity potential and its gradient at the hull. the new expression for K<, obtained in this study thus def ines the wave-spectrum function Ro + K<p in terms of the speed and the size of the ship, the shape of the mean wetted-hull surface and the tangential veloc- ity at the mean hull surface. This expression is suitable for use in con junction with any near-f ield f low-calculat ion method, inclucling boundary-integral-equation methods based on source distributions and other numerical methods in which the velocity vector is determined directly rather than derived f rom the potential. It provides a practical alla robust method for coupling a far-f ield Neumann~elvin flow representation with any near-field flow-calculation Tnethod, including methods based on the use of Rankine sources or f inite dif Eerences. For large values of t = tanO, the ma jor contributions to the integrals over the ship hull surface in Eqs. (18b) and (28b) stem f rom the upper part oE the hull surface in the vicinity of the mean waterline w due to the exponent ial funct ion exp (v2 p2 z ) . These hull integrals, and consequently the spectrum functions Ko and K`,, may then be approxi- mated by waterline integrals for large values of tan0, as has indeed been shown in this study for the special case of a wall-sided hull. This asymptotic approximation can be extended to ship forms having flare, and ref ined by retaining the f irst few tercas in the asymptotic approximation. A detailed short-wave asymptotic analysis has been per- f ormed and will be reported elsewhere as it is important for evaluating the short diver- gent waves of interest for applications to remote-sensing of ship wakes. Simpliffed approximate expressions for the wave spectrum functions Ro and K~ defined in terms of sin- gle (one fold) integrals along the ship axis (or waterline) can also be obtained in the long-wave limit by expanding the exponential function exp (v2p2z) and the trigonometric function exp (iv2pty) in Taylor series. These long-wave approximations will also be reported elsewhere. The short-wave and long-wave asymptotic analyses mentioned above may also suggest alternative choices for the arbitrary function C(t) in Eqs. (27), (29a) and (34) to that defined by Eq. (45). ACKNOWLEDGMENTS This study was funded by the Office of Naval Research under the Applied Hydrodyna- mics Research program at the David Taylor Research Center. z _ ~ Y n ! /~ f _ ~---. x Fig. 1 - Definition Sketch 153

K Kit Ko=K* Kw, 0.04 0.02 0.00 -0.02 0.04 n na n nn 0.04 0.08 0.12 0.06 0.00 -0.06 -0.12 _ 11 O o \\ Jl l It ., , ~ dt ~,,.N _ / , \ /% /'l,j \~_ \ , 11 O o I_ ~ O / o CO l 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 tans tans tans tans Fig. 2 - Comparison of the usual expression'<o ~ Kw + Ks and the alternative modified expression Ko = K* for tile slender-ship approximation Ko to the wave-spectrum function '(it) in the special case of a simple strut-like hull [form. The real (solid line) and imaginary (dashed line) parts of the functions Kw (first column on left), Ks (second column) and Key = K* (third column) are depicted for O < tans < 5, corresponding to 0 < ~ < 79°, and for three values of the Froude number F. namely 0.1 (top row), 0.2 (center row) and 0.3 (bottom row). The integral Kw' along tile lower waterline w' in Eqs. (14) and (18b) is also depicted in the column on the right. REFERENCES 1. Lindenmutl,, I}.T., T.J. Ratclil5fe and A.M. Reed, "Comparative Accuracy of Numerical Kelvin Wake Code Predictions - Wake off," David Taylor Research Center Report OTRC/SHD-1260-01 (1988). 9. Coleman, Roderick M., "Nonlinear Flow About a Three-l~imensional Transom Stern," Proceedings of the Fourth International Conference on Numerical Ship Hydrodynamics, Washington, DC, pp. 234-244 (1985). 3. Miyata, Hidealci and Sl~inicl~i Nishimura, "Finite-Difference Simulation of Nonlinear Ship Waves," Journal of Fluid Mechanics, Vol. 157, pp. 327-357 (1985). 4. Gadd, (ME., "A Method of Computing the Flow and Surface Wave Pattern Around Hull Forms," Transactions of the Royal Institute of Naval Architects, Vol. 118, pp. 207-215 (1976). 154 5. Dawson, Cal., "A Practical Computer Method for Solving Ship-Wave Problems," Proceedings of the Second International Conference on Numerical Sllir) Hydrodynamics, Berkeley, CA, pp. 30-38 (1977). 6. Daube, ()livier, "Calcul non lineeire de la resistance de vagues d'un navire," Comptes Rendus de 1'Acadernie des Sciences, Paris, France, Vol. 290, pp. 235-238 (1980). 7. Baar, Job J.tI., "A Three-l)imensional Linear Analysis of Steady Ship Motion in Deep 'hater," Ph.D. thesis, Brunel University, U.K., 182 pp. (1986). 8. Andrew, R.~., ~J.J.M. Baar and IJ.G. Price, "Prediction of Shir) Wavemalcing Resistance and Other Steady Flow Parameters Using Neur~ann- Relvin Theory," Transactions of the Royal Institution of Naval Arch sects, Vol. 130, pp. 119-129 (1988). 9. Noblesse, Francis, "A Slender-Ship Theory of Wave Resistance," Journal of Ship Research, Vol. 97, pp. 13-33 (1983).

0.08 0.00 .08 0.16 n l6 0.16 0.16 0.08 0.00 0.08 0.16 0.16 0.08 0.00 -0.08 0.16 0.16 0.08 * ~ 0.00 -0.08 0.16 1 2 3 4 5 6 tans 7 8 9 1 2 3 4 5 6 7 8 9 tans - - 0.16 0.08 0.00 .~. 0.08 0.16 0.16 0.08 0.00 0.08 0. 16 0.16 0.08 0.1 6 0.16 0.08 0.00 0.08 0.16 0.16 0.1 6 _ ~ _ - - * -4 Fig. 3 - Comparison of the usual expression R¢, =KW +KW' +KH' and the three alternative modified expressions K,~> = 'Kw + Kit, K,~> = Kw~ + KH" and K,p, = 'POW* + KH* for tile Neumann-Kelvin correction term K<> in the expression for tile wave spectrum function K(t) for a simple strut-like bull form and an assumed simple expression for the potent tat at the hull surface. The real and i Imaginary parts of the ten functions K¢, 'Kw ~ Kw', Kw', KH', Kw ~ KH ~ KW ~ KH , Kw* and KH are depicted for O < tang < 10, corresponding to O < ~ < 85°, and for a value of the Froude number equal to 0. 15. 155

KW+KW KH Kin KW KH 1i,.' In 1' -in n4 -n ns -n 08 n no 0~24 ll . \ l - l ~ ~ l - - 1 2 3 4 1 2 3 4 tans tang ,, . ,~ 1 2 3 4 1 2 3 4 1 2 3 4 tang tans tans C: _ . I ~ ~ I Fig. 4 - Comparison of the usual expression Kid = Kw + Kw' + KH' and the alternative modified expression Rip = Kw* + KH* for the Neumann-T~elvin correction term Rip in the expression for the wave-spectrum function it(t) for a simple strut-like hull form and an assumed simple expression for the potential at the hull surface. The real and imaginary parts of the five functions KW + Kw' (first column on left), KH' (second column on left), Kid (center column), Kw* (second column on right) and KH* (first column on right) are depicted for O < tans < 5, corresponding to 0 < < 79°, and for three values of the Froude number F. namely 0.1 (top row), 0.2 (center row) and 0.3 (bottom row). Large cancellations occur between the waterline integral Kw + Kw' and the hull integral KH' in the usual expression for Rip which is then ill suited for accurate numerical calculations, notably for evaluating the short waves in the spectrum corresponding to large values of tans. The modified waterline integral Kw* and hull integral KH* in the alternative new expression for Kid are significantly smaller than the usual waterline and hull integrals, and are comparable to the function Kit . Although the alternative expressions Kid = 'KW + Kw' + KH' and Kid = Kw* + KH* are mathematically equivalent, the latter expression is considerably better suited than the former one for accurate numerical evaluation. REFERENCES 10. Scragg, Carl A., Britton Chance Jr., John C. Talcott and Donald C. Wyatt, "Analysis of Wave Resistance in the Design of the 12-Meter Yacht Stars and Stripes," Marine Technology, Vol. 24, pp. 286-295 (1987). 11. Noblesse, F., D. Hendrix and A. Barnell, "The Slender-Ship Approximation: Comparison Between Experimental Data and Numerical Predictions," Proceedings of the Deuxiemes Journees de l'Hydrodynamique, Ecole Nationale Superieure de Mecanique, Mantes, France, pp. 175-187 (1989). 156 12. Barnell, A. and F. Noblesse, "Numerical Evaluation of the Near- and Far-Field Wave Pattern and Wave Resistance of Arbitrary Ship Forms," Proceedings of the Fourth Conference on Numerics Ship Hydrodynamics, Washington DC, pp. 324-341 (1985). 13. Noblesse, F., and W.M. Lin, "A Modified Expression for Evaluating the Steady Wave Pattern of a Ship," David Taylor Research Center Report No. 88/041 (1988). 14. Noblesse, F., "Alternative Integral Representations for the Green Function of the Theory of Ship Wave Resistance," Journal of Engineering Mathematics, Vol. 15, pp. 241-265 (1981).