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Near-Field Nordinearities and Short Far-Field Ship Waves F. Noblesse, D. Hendrix (David Taylor Research Center, USA) ABSTRACT The short divergent waves in the steady wave pattern of a ship are analyzed on the basis of a linear far-field flow representation and a nonlinear near-field flow approximation. More precisely, the far-field wave spectrum is determined in a simple and practical manner by means of a waterline- integral approximation obtained from a modified Neumann-Kelvin integral representation; and the nonlinear near-field flow along the ship waterline is determined via a nonlinear correction defined by a simple analytical expression. Numerical calculations for the Wigley hull predict short divergent waves too steep to exist in reality within a significant sector in the vicinity of the ship track. The predicted waves also exhibit a well-defined peak at an angle from the ship track equal to about 1° to 2°. INTRODUCTION The ongoing search for explanations of the features displayed by remote-sensing images of ship wakes has prompted the formulation of various alternative theoretical hypotheses. One such hypothesis is that some features of ship wake radar images might be attributable to characteristics of the steady Kelvin wave pattern. Efforts to determine whether the pattern of steady far-field waves generated by a ship does in fact exhibit any notable property capable of causing a corresponding identifiable feature in remote- sensing images have motivated a number of recent studies of the Kelvin wake, including Scragg [1i, Barnell and Noblesse [2i, Keramidas and Bauman [3i, Milgram t4], and Trizna and Keramidas [5~. All these numerical studies are based upon a highly- simplified hydrodynamic model, namely the zeroth-order slender-ship approximation proposed in Noblesse [6~. It is shown in Baar t7i, Andrew, Baar and Price t8i, Lindenmuth, Ratcliffe and Reed [9], and Noblesse, Hendrix and Barnell t10] that the first-order slender-ship approximation provides fairly realistic predictions of the wave profile, the 465 near-field wave pattern and the long waves in the wave spectrum. However, the zeroth-order slender-ship approximation is a poor approximation to the Neumann-Kelvin theory in the short-wave limit, as is indicated by the numerical results depicted in Figure 2a in [10] and is confirmed by the results obtained further on in this study. The wave calculations reported in [1] through [5] therefore are unlikely to provide realistic representations of the short waves in the wave pattern of a ship. Earlier numerical calculations of the wave spectrum or the wave pattern of a ship may be found in the literature, e.g. Sharma [11] and Tuck, Collins and Wells [12~. However, these calculations are also based upon a highly-simplified hydrodynamic model, namely the Michell thin- ship approximation. Similarly, the far-field wave calculations of Ursell [13] correspond to an elementary free-surface pressure singularity. The steady wave pattern and wave spectrum of a ship thus have been relatively little studied, and are ill known. In particular, the asymptotic behavior of the wave-spectrum function in the short-wave limit is not known, and it is not known whether this asymptotic behavior might explain the common observation that the wake of a ship in the vicinity of the track exhibits no short divergent wave. The short waves in the wave-spectrum function and the far-field wave pattern of a ship are examined in the present study within the context of a somewhat more realistic hydrodynamic model than the Michell thin-ship theory and the slender- ship approximation used in the previously mentioned numerical studies. More precisely, the theoretical framework adopted in the present study is that provided by the linearized Neumann-Kelvin flow representation. This theory defines, via well- known formulas, the steady wave pattern (and the wave resistance) of a ship in terms of the wave- spectrum function, which is defined in terms of the flow at the ship mean wetted-hull surface.
Accurate theoretical predictions of the steady wave spectrum of a ship therefore are necessary for obtaining reliable wave-signature (and wave- resistance) predictions. However, accurate numerical calculations of the wave-spectrum function cannot readily be obtained because this function is defined as the sum of two integrals, namely a line integral around the ship mean waterline and a surface integral over the ship mean wetted-hull surface, which very nearly cancel out in the manner recently shown in Noblesse, Lin and Mellish t14~. Inaccuracies which inevitably occur in the numerical evaluation of the waterline and hull integrals cause imperfect cancellations between these two integrals and correspondingly large errors in their sum. The foregoing cancellation phenomenon and the resulting numerical inaccuracies are especially acute for the short waves in the wave spectrum. A mathematical remedy to this fundamental numerical difficulty is presented in [14~. The remedy consists in an alternative mathematical expression for the wave-spectrum function. This alternative expression defines the wave spectrum as the sum of modified waterline and hull integrals. No significant cancellation occurs between these modified integrals, which are of the same order of magnitude as the wave-spectrum function defined as their sum. The large cancellations occurring between the waterline integral and the hull integral in the usual expression for the spectrum function thus are automatically and exactly accounted for, via a mathematical transformation, in the alternative new expression given in [14~. This new expression for the wave spectrum thus is considerably better suited than the usual expression for accurate numerical calculations, notably for the short waves in the spectrum. Nevertheless, the integral representation for the wave spectrum given in [14] is not suitable for evaluating the very short divergent waves. For instance, the use of this (or indeed any similar) integral representation for the numerical calculation of ship waves with wavelength between 5 cm and 40 cm, corresponding to backscattering of the electromagnetic waves in typical systems used for remote sensing of ship wakes, would require an exceedingly large number of extremely small panels for representing the hull surface. In fact, such short waves can only be evaluated analytically, via a short-wave asymptotic approximation of the integral representation of the spectrum function given in [14~. Reliable predictions of the short waves in a ship wave spectrum are quite difficult to obtain because of the significant numerical difficulties mentioned in the foregoing, as well as for yet another reason. This second source of difficulties stems from the fact that short far-field ship waves are closely related to the velocity distribution along the ship mean waterline, especially the fluid velocity at the ship bow and stern, as is shown further on in this study. However, existing near-field-flow calculation methods (so-called nonlinear methods included) are unable to provide realistic velocity predictions at a ship bow and stern because they cannot model the strongly nonlinear flow in the immediate vicinity of these points, as was recently shown in Noblesse, Hendrix and Kahn [15~. The nonlinear analytical/experimental and analytical/numerical velocity distributions along the mean waterline of the Wigley hull defined in [15] are used in the present study for predicting the short divergent waves generated by the Wigley hull. FOURIER REPRESENTATION OF THE WAVE PATTERN The wave potential owed at any point ~ = (t, it, ~ < 0) behind the stern of a ship advancing at constant speed in calm water can be defined in terms of a Fourier representation, as is well known. Specifically, (20) in [14] yields away = (2/11) .1; exp~v2(p2)cos~v2llpt) Im exp~iv24p) K(t) aft, (1) where the wave potential few and the coordinates I,, 11, ~ are nondimensional with respect to the ship length L and speed U. and v is the inverse of the Froude number F; we thus have v = 1/F with F=U/(gL)~/2, (2a,b) where g is the acceleration of gravity. Furthermore, p in (1) is related to the Fourier variable t as follows: p = ~l+t2~/2. (3) Finally the (nondimensional) function K(t) in (1) is the wave-spectrum function. The wave potential is defined by (1) in terms of a familiar Fourier superposition of elementary plane waves propagating at angles ~ from the ship track (the x- axis ~ given by . tang = t. (4) The amplitudes of these elementary plane-wave components are essentially given by the spectrum function K(t), which thus contains essential information directly relevant to a ship's signature (and wave resistance). This study is concerned with the numerical/analytical evaluation and the behavior of the spectrum function K(t) in the short wave limit t ~ so (0 ~ 90°~. It is convenient and useful to express the spectrum function K(t) as the sum of two terms as follows: K(t) = Knit) + Knit), 466 (5)
where Ko represents the zeroth-order slender-ship approximation and Kit the Neumann-Kelvin correction term in the Neumann-Kelvin approximation Ko+ Kit . More precisely, the spectrum function Ko+ Kit corresponds to the usual linearized Neumann-Kelvin approxima- tion, in which the nonlinear terms in the free- surface boundary condition are neglected. These nonlinear terms result in an additional term in (5), defined by an integral over the mean free surface [6,14], which is ignored in this study. The slender- ship approximation Ko is defined explicitly in terms of the value of the Froude number and the hull shape, whereas the Neumann-Kelvin correction Kit also depends on the value of the tangential fluid velocity at the hull [14~. For a ship with port and starboard symmetry, as is considered here, the slender-ship approximation Ko and the Neumann-Kelvin correction Kit can be expressed in the form Ko = Ko + Ko, Kq, = Kq,+ + K<,, where the superscripts + and - correspond to the contributions of the port and starboard sides of the ship, respectively. Modified mathematical expressions for the functions Ko and K,~ are given in tip. Approximate forms of these expressions valid in the short-wave limit are now given. THE WAVE SPECTRUM: APPROXIMATE INTEGRAL REPRESENTATION The expression for the slender-ship approximation Ko defined by (30) in [14] involves the exponential function exp(P2z), where p2 is given by p2 = v2p2 = (sec20~/F2 (7) For negative values of the vertical coordinate z, the exponential function exp(P2z) is negligibly small in the short-wave limit p2 ~ 00 . We then have Ko+ ~ iw E+ (nX2-u2) ty dl -iv2u is exp(P2z)E+nzda (8) asP2 moo The expression for the Neumann-Kelvin correction K'¢, defined by (73) and (74a,b) in [14] can likewise be approximated by restricting the integration over the hull surface h in (73) to the hull side s. We then have Kit+ ~ iw E+ Aw+ dl + iv2 is exp(P2z) E+ Ah+ da/ ~ t x s as p2 ~ no, where the amplitude functions A and Ah+ are defined as (9) Aw+ = (tXOt+sx~s~ty+u~v~u) (°t+~°s), (lOa) Ah+ = [(V-CU)tx+(U+cv~ty-ictz] (¢S+£¢t) - t~v-Cu~sx+(u+Cv~sy-iCsz] (Ot+£0S) . (lOb) The functions E+ and E_ in (8) and (9) are the trigonometric functions defined by (19) in [14], that is we have E+ = expi-iP2(ux+vy)], (11) where u and v are given by u = 1/p and v = t/p; (12a,b) It may then be seen from (3) that we have 1 > u > 0 and O < v < 1 for O < t < so, with u2+v2 = 1 Furthermore, w and s in (8) and (9) represent the positive halves of the mean waterline and of the mean wetted-hull side and dl and da the differential elements of arc length of w and area of s, respectively. Also, t = (tx, ty, tz) and s = (sx, sy, sz) are unit vectors tangent to the hull side along `6a' curves which roughly correspond to waterlines `6b' and framelines, respectively. The vectors t end s are roughly (but not necessarily exactly) orthogonal and point toward the bow and the keel line, respectively. At the mean free surface, the vector t is tangent to the mean waterline w and we thus have tz = 0. The normal vector n = (nx, ny, nz) to the hull side is defined by n = (txs)/|txs| . The term ~ in (lOa,b) is defined as and at and Us represent the components of the velocity vector V) along the unit vectors t and s tangent to the hull side; we thus have Vo - nx n = at t + Us s, where the hull boundary condition 3~/3n = nx was used. The components of and as of Vo along the tangent vectors t and s and the velocities Jo/Ot = V¢.t and aq,/Os = Voles are related as follows: 34/3t = At + ENS, 3~/3s = US + cot, at = (OO/&t - £3O/8s)/(l- £2), (14c) Us = (30/3s - £~/3t)/~1- £2) (14d) Finally, C in (lOb) is an arbitrary complex function of t. Equations (9) and (lOa,b) thus define a one- parameter family of mathematically-equivalent expressions for the Neumann-Kelvin correction K¢. It may be seen from (30) and (73) in [14] that an estimate of the error associated with the approximate expressions (8) and (9) is provided by the exponential function exp(P2z) where z is taken equal to the negative of the ship draft d. This error (13) (14a) (14b) 467
estimate is smaller than a prescribed error £ for p ~ Pe, or equivalently for ~ ~ He = sec -lapel, where Pe is given by pe(F; £, d) = F [ln~l/£~/d]~/2 . The values of Pe and Be corresponding to values of the error £ and the ship draft d equal to 0.01 and 0.05, respectively, and to five values of the Froude number between 0.1 and 0.5 are listed in Table 1. F 1 0.1 1 02 1 03 1 04 1 05 Pe 1 1.9 2.9 3.8 4.8 He 0 59o 70° 75o 78° Table 1. Values of Pe and qe for £ = 0.01 and d = 0.05. The exponential function exp(P2z) in the integrands of the integrals over the hull side s in (8) and (9) decays rapidly with decreasing (negative) values of z if p2 >> 1, that is for small values of the Froude number and/or large values of t = tang. The major contributions to the hull-side integrals in (8) and (9) therefore stem from the upper part of the hull side s in the vicinity of the waterline w. These hull-side integrals can in fact be approximated by single (one-fold) integrals along the waterline. These waterline-integral approximations for the spectrum functions Ko and K`p are now given. THE WAVE SPECTRUM: WATERLINE INTEGRAL APPROXIMATION The upper part of the hull side s can be defined by the following parametric equations: x = (~1) + xlkl~s + x2~1)s2/2 + .... y = 11~1) + ye ills + y2(l~s2/2 + .... -z = z] flus + z2(l~s2/2 + .... where s 2 0 and the curve s = 0 corresponds to the waterline w. The waterline is then defined by the parametric equations x = (~1) and y = Al), (16a,b) where 1 is the arc length along w. The previously- defined unit tangent vectors t and s to the hull side s are given by t = (tx' ty, tz) = 3~x/3l, s = (sx' sy, sz) = IS In particular, at the waterline w, we have (tx, ty, tz) = (k,', it', 0), (17a) (sx, sy, sz) = (xl, Y1' -Z1) ~(17b) where the notation ~ )' denotes differentiation with respect to the arc length 1 along w. By using the foregoing representation of the upper hull side s, we can approximate the integrals on the hull side in (8) and (9) as integrals along the mean waterline w. Details of this short-wave asymptotic approximation are given in Noblesse and Hendrix 1161, where the following waterline- integral approximation to the spectrum function K(t) is obtained: K ~ [w {A+ expl-iP2(u4+v~)l + A_expl-iP2(ut-vll~l} dl . The amplitude functions A+ are defined as A+ [-sz+i~usx+vsy)] = Ao-+ S+¢s+ Thy, (19) where Ao+ corresponds to the slender-ship approximation and is given by Ao+ = -szty~nX2-u2) +ittynX2(usx+vsy)-syu2(utx+vty)~. (20) The terms S+ and T+ in (19) are defined as S+ = -sz~sXty + uv~sxtx + syty)] -i [usz2ty - (utx + vty)(sxsy + uv)], (21a) T+ = (tXty + uv) (-sz + ivsy) + iutx~utx + vty) (usy + VSx) (21b) For sufficiently large values of p = secO, the Neumann-Kelvin approximation K thus is defined by the waterline-integral approximation (18), (19), (20), and (21a,b). These equations provide a simple and practical basis for numerically evaluating the short divergent waves in the steady wave spectrum of a ship, given the value of the fluid velocity components as and of at the waterline, by dividing the waterline w into a large number of straight segments within which the amplitude functions A+ are assumed to vary linearly. Figures la,b,c depict the real and imaginary parts (15a) of the Neumann-Kelvin approximation to the `15b' spectrum function K(t) for the Wigley hull at three values of the Froude number equal to 0.1, 0.25 and (15c) 0.4. The dashed-line curves in these figures correspond to the waterline-integral approximation (18) obtained in this study; the solid-line curves correspond to the exact mathematical expression for the Neumann-Kelvin approximation to the spectrum function given by (21), (22), (73) and (74a,b) in [14~. The velocity potential on the Wigley hull in these exact expressions, and at the waterline in the waterline-integral approximation (18) obtained in the present study, is taken as the (first order) slender-ship potential defined in [6~. Figures la,b,c show that the waterline-integral approximation (18) does in fact become quite accurate for sufficiently large values of t = tang. The waterline-integral approximation (18) may then be used henceforth in this study. Figures la,b,c also show that the general numerical method based on the exact expressions for the wave-spectrum function given in [14] can provide reliable predictions if a sufficiently large number of panels is used for representing the ship hull form. 468
WAVE-SPECTRUM FUNCTION WIGLEY HULL AT F=0.10 Exact {98000 panels} o ~Waterline integral x ~ .O 0 lo o xr ~ Hi WAVE-SPECTRUM FUNCTION WIGLEY HULL AT F=0.40 Exact {16000 panels) - - - Waterline integral 4 5 6 tanS Fig. 1a. Real and imaginary parts of the wave- spectrum function king. WAVE-SPECTRUM FUNCTION WIGLEY HULL AT F-0.25 Exact (32000 panels) ~--- Waterline integral 611111~1~14~ 6 8 10 tang 12 14 Fig. 1b. Real and imaginary parts of the wave- spectrum function king. Y 0 a) Cal -1 , c~ !' i' r~> ,, me, . . 1 6 10 14 to tend Fig. 1c Real and imaginary parts of the wave spectrum function kilo). THE WAVE SPECTRUM: STATIONARY-PHASE APPROXIMATION An analytical approximation to the spectrum function K(t) can in principle be obtained by applying the method of stationary phase, since the trigonometric functions expt-iP2(u4+vll)] in (18) oscillate rapidly for large values Of p2 = v2p2 = (sec20~/F2. This method shows that the major contributions to the waterline integral (18) stem from the end points of the integration range, that is the ship bow and stern, and the points where the phases of the trigonometric functions expt-iP2(uE,+v~] are stationary. These points of stationary phase are defined by the conditions ud;+vd~ = 0, which yield the relations utx+vty = 0, tx = v, ty = +-u . (22a,b,c) by virtue of (17a), (13), and the identity tX2+ty2 = 1. By using (12a,b) and (13) in (22b,c) we may obtain tan) = ty/tX = Flit = -+cotanO, where o is the angle between the x-axis and the unit tangent vector t to the waterline. We thus have the relation 1 0 1 = ~/2 - ~ , (23) which shows that the very short divergent waves in the steady wave spectrum of a ship primarily stem from the central (midship) portion of the ship 469
where the waterline is almost parallel to the centerplane, as well as the ship bow and stern. It is shown in [16] that application of the method of stationary phase to (18) fails to provide a simple, practically useful analytical approximation because the second terms in the asymptotic expansions for the contributions of both the end points (i.e., the ship bow and stern) and the points of stationary phase are of the same order of magnitude (in the limit t ~ or) as the first terms, and thus cannot be neglected. Unfortunately, the second terms in the asymptotic expansions are extremely complex. It can nevertheless be shown that we have K ~ KB,S + KPhaSe, where KB S and KphaSe correspond to the contributions from the ship bow and stern and from the interior plinths) of stationary phase, respectively. Furthermore, we have KB S ~ 1/t3 and Kphase ~ 1/t4 as t ~ ~ . (24) NEAR-FIELD FLOW AND NONLINEARITIES The tangential velocity components as and ¢~ in (19) are merely taken equal to 0 in the zeroth-order slender-ship approximation to the spectrum function defined in [6~. More generally, the values of these velocity components at the mean waterline w may be predicted numerically using any near- field-flow calculation method, including the relatively simple first-order slender-ship approximation defined in [6i, in the Neumann- Kelvin approximation to the spectrum function. However, it was already noted that existing near- field-flow calculation methods, including the so- called nonlinear methods, cannot provide accurate predictions of the velocity components as and (~ in the immediate vicinity of a ship bow and stern. The velocity components as and of along the wave profile of a ship can be defined in terms of the nondimensional elevation e = Eg/U2 of the wave profile and its slope en = dE(L)/dL in the direction of the unit tangent vector t to the mean waterline by means of analytical expressions given in [15~. More precisely, (27) and (23) in t15] define the velocities bo/3t = tx30/OX+ty3~/3y and Jo/Os = sx34/3x+sy O¢/3y+szOo/3z at the wave profile of a ship as follows: bo/3t = tx - (l-2eyl/2/ [l+`l+~2ye~2~l/2 (25a) arias = SX - t£+~1+~2)sze~(l-2e~l/2 /[l+~1+~2ye~2~1/2 (25b) where £ and p2 are given by £ = s · t = sxtX+Syty ~ p2 = n 2/~1_nz2) = (Sxty-Sytx)2/Sz2. The tangential velocity components As and of can then be determined from the velocities Jo/3s and Jo/3t by means of (14c,d). The analytical expressions (25a,b) can be used in conjunction with either experimental measurements or numerical predictions of the wave profile, corrected at the bow and the stern in the manner specified by the nonlinear local solution given in [15~. In the special case of a wall-sided hull like the Wigley hull, we have ~ = 0 since nz = 0 at the waterline, and £ = 0 since we may choose the tangent vector s to the hull at the waterline as s= (0,0,-1~; (25a,b) thus become a¢/at = tx- (l-2eyl/2/`l+et2yl/2 (26a) 30/3s = et (l-2eyl/2/`l+e~2yl/2 (26b) It is shown in [15] that the steady wave profile at the ship bow must be tangent to the stem, and likewise at the stern. This tangency condition shows that we have en = -or at the bow and the stern of the Wigley hull. It then follows from (26a,b) that we have Jo/3t = tx ~ 1 , (27a) 34/3s =-(l-2e~l/2~ 1 (27b) at the bow and the stern of the Wigley hull. The velocities 34/3t and JO/3s along the horizontal and vertical tangent vectors t = (tX,ty,0) and s = (0,0,-1) to the Wigley hull at the waterline are depicted in Figs. 2a and 2b for a value of the Froude number equal to 0.25. The dashed-line curves in these two figures were determined using the nonlinear expression (26a,b) in which the wave- profile elevation e is taken as the experimental profile obtained at the University of Tokyo and corrected at the bow and stern in accordance with the previously mentioned tangency condition [15~. The solid-line curves in Fig. 2a correspond to numerical predictions obtained using the slender- ship approximation defined in ted. Figure 2a shows that discrepancies between these linear numerical predictions and the corresponding nonlinear analytical/experimental predictions are quite large in the vicinity of the bow and the stern, where nonlinear effects indeed are important. The solid- line curves in Fig. 2b were obtained from the nonlinear expression (26a,b) in which the wave- profile elevation e is taken as the profile predicted numerically using the slender-ship approximation [6], corrected at the bow and stern in the manner specified in [15] and used also for determining the analytical/experimental dashed-line curves in Figs. 2a and 2b. Thus, both the solid-line curves in Fig. 2b and the dashed-line curves in Figs. 2a and 2b correspond to the nonlinear analytical expression (26a,b). The discrepancies between these nonlinear analytical/experimental and analytical /numerical predictions clearly are much smaller than the discrepancies corresponding to the linear numerical predictions shown as solid-line curves in Fig. 2a. The oscillations in the latter curves correspond to the divergent waves in the wave pattern. These 470
VELOCITY AT WATERLINE WIGLEY HULL AT F=0.25 slender-ship approximation -- nonlinear analytical/ experimental prediction lo ANEW.' to _ 0 1 1 - (~,,'\ , . . . . . . . . . stern bow Fig. 2a. Velocity components Abet and 3~/3s. oo _ o _ d. ~o _ no> _ o oh \ ~ ~ 0 VELOCITY AT WATER LINE WIGLEY HULL AT F=0.25 slender-ship approximation with nonlinear correction nonlinear analytical/ experimental prediction ~1 stern bow Fig. 2b. Velocity components Jo/3t and Jo/3s. Oscillations do not appear in the corresponding numerical predictions corrected for nonlinear effects depicted in Fig. 2b because the number of numerical data points used for defining the curves in this figure is fairly small (only 26 data points are used for both the experimental and the numerical results). THE WAVE SPECTRUM CORRESPONDING TO FOUR NEAR-FIELD-FLOW APPROXIMATIONS The modulus, ~ K I, of the wave-spectrum function K(t) of the Wigley hull is represented in Figs. 3a,b and 4a,b for values of t = tang in the range 7 ~ t < 19, which approximately corresponds to values of ~ in the range 82° < ~ < 87°. Figures 3a and 4a correspond to a value of the Froude number F equal to 0.25, while Figs. 3b and 4b correspond to F = 0.4. The dashed-line curves in Figs. 3a and 3b correspond to the zeroth-order slender-ship approximation, so that the velocity components of and as in (19) are merely taken equal to 0. The solid-line curves in these two figures correspond to the Neumann-Kelvin approximation defined by (18), with the near-field velocity components ¢~ and (PS in (19) determined from the first-order slender ship potential defined in ted. The numerical results depicted in Figs. 3a,b show that the predictions corresponding to the Neumann-Kelvin approxi- mation (first-order slender-ship approximation) are significantly larger than those corresponding to the zeroth-order slender-ship approximation. The latter approximation thus appears unlikely to provide realistic predictions of the short waves in the wave spectrum of a ship, as was already noted. The predictions corresponding to the foregoing Neumann-Kelvin approximation are also depicted in Figs. 4a and 4b. These predictions correspond to the thick solid-line curves located much below the other two sets of curves represented in Figs. 4a and 4b. The latter two sets of curves correspond to the Neumann-Kelvin approximation defined by (18), with the near-field velocity components ¢~ and as in (19) determined from the nonlinear expression (26a,b). These two sets of curves thus correspond to nonlinear near-field-flow predictions, whereas the thick solid-line curves in Figs. 4a,b correspond to linear near-field-flow predictions. More precisely, the thin dashed-line and solid-line curves in Figs. 4a,b correspond to the nonlinear experimental and numerical, respectively, near-field-flow predictions depicted in Fig. 2b. It may be seen from Figs. 4a,b that discrepancies between these predictions of the wave spectrum corresponding to the nonlinear experimental and numerical near-field-flow predictions are relatively small, whereas the prediction of the wave spectrum corresponding to 471
~ ---- cv~ 1A 1 1 WAVE SPECTRUM OF WAVE SPECTRUM OF WIGLEY HULL AT F=0.25 0 WIGLEY HULL AT F=0.40 x Cal o -- Zeroth-order slender-ship approximation - Neumann-Kelvin theory (first-order slender-ship approximation) 7 10 13 tang (a) o 16 19 Fig. 3. Modulus of the wave spectrum function king. WAVE SPECTRUM OF WIGLEY HULL AT F=0.25 | A 7 Linear Neumann-Kelvin theory Nonlinear analytical/numerical Nonlinear analytical/experimental ^) -. ~ 13 16 tang (a) Fig. 4. Modulus of the wave spectrum function kite). the linear near-field-flow calculations are much smaller. These numerical results indicate that the major contributions to the waterline integral (18) clearly stem from the ship bow and stern, and that the simple nonlinear correction of the linear numerical predictions of the wave-profile elevation presented in [15] and depicted in Fig. 2b thus can be used effectively for predicting the short-wave tail of the wave-spectrum function. THE FAR-FIELD DIVERGENT WAVES It is appropriate to analyze the far-field wave pattern of a ship in terms of the nondimensional far-field coordinates (x,y,z) = v2~t,q,() = (X,Y,Z)g/U2, where (X,Y,Z) are dimensional and ((,ll,() = (X,Y,Z)/L are the nondimensional near-field coordinates used in (1), the nondimensional 10 13 16 19 tend (b) WAVE SPECTRUM OF 0 WIGLEY HULL AT F=0.40 x 0 co Y 0 -cot 0 . ~ . to 19 7 10 13 tang (b) 16 19 potential ~ = v2q, = ~g/U3 and the wave-spectrum function k = v2K. By using these far-field variables in (1) we may obtain the equivalent alternative expression (x,y,z) = Im J; [E+(t; x,y,z) + E_(t; x,y,z)] kits aft, (28) where the functions E+(t; x,y,z) are defined as E+(t; x,y,z) = exp~zp2+i~x+yt)p] with p given by (3~. The nondimensional free-surface elevation e~x,y) = Eg/U2, where E is dimensional, is given by e = 3~/Ox, where the function 3~/Ox is evaluated at the mean free-surface plane z = 0. By differentiating (28) we may obtain ~e~x,y) = Re [e+(x,y)+e_(x,y)], (29) 472
where the functions e+ are defined as e+(x,y) = J; exptix0+(t;cs)] kits p aft; (30) the phase functions 0+(t;6) in (30) are given by 0+(t;6~= (1+6t~p, with 6 defined as 6 = y/~-x)=tana. We have x < 0 and we may assume y > 0 since the wave pattern is symmetric about the ship track y=O. We thus have c'> OandO<a<~/2. Asymptotic approximations, valid in the limit x ~ - so, to the functions e+ defined by (30) and (31) can be obtained by using the method of stationary phase, as is well known. The phase 0_(t;6) = (l+at~p is monotonic increasing for t > 0. The phase 0+(t;63 = (l-at~p is monotonic decreasing for t > 0 if 6 > 1/23/2, whereas it is stationary at t = t_(c') and t = the) if O < 6 < 1/23/2, as is shown in detail in [2~. It may then be seen from (30) and (31) that we have 7re~x,y) ~ Re e+(x,y) asx - -A if 0<6<1/23/2_0.35. The points of stationary phase t_~6) and the) are defined by i+ = [1+~1-862~1/2~/~46) and correspond to the transverse and the divergent waves, respectively, in the wave pattern. Only the divergent waves are considered here. Let the stationary-phase value the) be denoted ha), which may be expressed in the form ~ = T/~26) with the function T(6) defined as T = [1+~1-862~1/2~/2 The corresponding value of p = ~l+t2yl/2 = ~l+~2yl/2 then is given by p = P/~26) with the function P(6) defined as p2= ~l+462+(l-862)l/2~/2. (37) At the stationary point t = ~6), the phase function 0+(t;6) defined by (31) takes the value 0~6) given by ~ = Q/~4cs), where the function Q(6) is defined as Q = Pt3-(l-862yl/2~/2 By using (32) in the foregoing expression for ~ we may then obtain x~ = -Qx2/~4y) By applying the method of stationary phase to the integral e+ defined by (30) we may then obtain (Tc/2~1/2~-X)l/2(l_862yl/4 e ~ Re p3/2tk/~26~3/2] expt-i {Qx2/~4y)-~/4~] (40) as x ~ -A if O < 6 < 1/23/2 _ 0.35, where k represents the value of the wave-spectrum function kits at the stationary point t = ~6) defined by (34) and (35~. It may readily be seen from (35), (37), and (38) that the functions T(6), P(c,j, and Q(6) are nearly equal to 1 for small values of 6 = y/~-x). (31) The asymptotic approximation (40) shows that the phase Qx2/~4y) _ x2/~4y) of any divergent wave is nearly constant along a parabola having the y-axis (32) as axis and the origin x = 0, y = 0 as vertex. The divergent waves in the steady far-field wave pattern of a ship thus consist of a family of such parabolas, as is well known. It may be seen from (40) that the amplitudes, a say, of these waves are given by (-x)~/2a = (2/~/2 p3/2 ~ ~ k ~ /~26~3/2~/~1 862yl/4 `41' The corresponding wavelengths, ~ say, are given by = 2~/ I VQx2/~4y) 1. We may then obtain ~ = 8~62/p2 (42) The steepness s = a/\ of the divergent wave defined by (41) and (42) thus is given by `_Xy~/2s = `2~3~-~/2 p7/2 ~ ~ k ~ /~2~7/2~/~1-862~/4 (43) The angle ~ between the ship track and the `33' direction of propagation of the divergent waves is defined by the relation cotanO = 2(tana)/T, as may readily be verified from (4), (32) and (34~. The corresponding interior poisons) of stationary phase on the ship waterline is (are) defined by the condition ~ ty ~ = 26/P, which readily follows from (22c), (12a) and (36), or by the alternative equivalent condition (23~. In summary, we thus have X/ (2~) = 8(tan2oc)/ [1 +~1-8tan2or)~/2+4tan2oci, (44a) cotanO = tan ~ 0 ~ = 4(tana)/~1+~1-8tan2a)~/2] . (44b) (34) These equations define the wavelength X, the wave propagation angle ~ and the waterline tangent angle I ~ I as functions of the ray angle or in the wake. Equivalent simple expressions defining X, ~ and a as functions of ~ ~ ~ can also be obtained from (36) (44a,b). Specifically, we have \/~2~) = sin2¢, ~ = n/2 - ~ 0 I, (45a,b) land = (tan ~ 0 ~ ~/(2+tan24) . (45c) The following equivalent expressions explicitly define \, ~ ~ ~ and a in terms of O: \/~2~) = cos20, ~ ~ ~ = ~/2 - 0, (38) land = (tan0~/~1+2tan20) . Finally, the equivalent expressions sin ~ ~ ~ = ~ \/~2~/2, tang = (2~/~-1~/2, (47a,b) `39' land = (2~/~-1~l /2/ (4~/~-1) (47c) explicitly define ~ ~ I, ~ and a in terms of \. By using (3) and (4) in (46a,b,c) we may obtain the following expressions for the values of the wavelength X, the wave propagation angle 0, the waterline tangent angle l o l, and the wake ray angle or corresponding to a given value of p: (46a,b) (46c) 473
\/(211) = 1/p2, cosO = sin I 0 I = 1/p, (48a,b) tang = (p2 1)l/2/(2p2-l) (48c) In summary, the far-field pattern of divergent waves is defined by (40), and the waves' amplitude, wavelength, and steepness by (41), (42), and (43). These expressions involve the value of the wave spectrum function k - v2K at the stationary point t = ~(c') defined by (34) and (35). The corresponding values of p - (l+t2)~/2, the wavelength X, the wave propagation angle 0, the wake ray angle or, and the waterline angle ~ ~ ~ are related by (48a,b,c). A practical approximate expression for the spectrum function K(t) is given by the waterline approximation (18). This approximation is valid in the limit p ~ or, although Figures la,b,c indicate that it may be used in practice for moderate values of p. The near-field flow-velocity components ¢ and as at the mean waterline in (19) can be determined from the wave-profile elevation e by using the analytical expressions (25a,b) obtained in [15~. The foregoing expressions provide an approximate but complete theoretical basis for predicting the short waves in the steady wave spectrum of a ship. Figure 5 depicts the steepness (-x)~/2s, defined by (43), of the divergent waves in the steady wave pattern of the Wigley hull for values of the wave ° angle or in the range 0 < or < 4°, which approximately corresponds to values of t in the range 7 ~ t < to for which the waterline approximation (18) was previously shown to provide fairly accurate predictions. The top and bottom parts of Figure 5 correspond to values of the Froude number F equal to 0.25 and 0.4, respectively. The solid-line and dashed-line curves in this figure correspond to the nonlinear near-field-flow predictions defined by (26a,b), where the wave- profile elevation e is determined from experimental measurements and numerical calculations, respectively, in the manner explained previously. These analytical/ experimental and analytical/numerical steepness predictions are in reasonable agreement with one another. Both predictions exhibit a fairly well-defined peak at values of of approximately equal to 1.9° and 1.4° for F = 0.25 and 0.40, respectively. Let ~ denote the value of (_x)~/2 s. We thus have s = 6/(-x)~/2 = 6F/(-X/L)~/2 since x = Xg/U2. We then have s = F6/10 at a distance of 100 ship lengths behind the ship. It may be seen from Fig. 5 that the predicted peak values of the wave steepness at 100 ship lengths thus are approximately equal to 0.3 and 0.6 for F = 0.25 and 0.40, respectively. These predicted peak values are extremely large and the corresponding waves could not exist in reality. 474 WAVE STEEPNESS (WIGLEY HULL) --- Analytical/numerical Analytical/experimental F=0.25 1_ 0 1 F=~.40 2 3 4 ~ (degrees) Fig. 5. Steepness of the short divergent waves. CONCLUSION A simple practical analytical/numerical method for calculating the short divergent waves in the steady wave spectrum of a ship has been presented. The method is based on a waterline-integral approximation obtained from a modified Neumann-Kelvin integral representation of the wave-spectrum function. A comparison of numerical predictions obtained using the exact integral representation and the waterline-integral approximation showed excellent agreement for sufficiently short waves. This agreement demonstrates both the validity of the waterline- integral approximation and the robustness of the more general numerical method based on the exact integral representation. The waterline-integral approximation to the spectrum function shows that the short divergent waves in the wave pattern of a ship are defined in terms of the near-field flow along the ship waterline, and may indeed be regarded as an image of the flow along the ship waterline.
The foregoing linear far-field flow representation has been used in numerical experiments seeking to determine the sensitivity of the far-field waves to the near-field flow along the ship waterline. The simplest approximation is the trivial approximation corresponding to the zeroth-order slender-ship approximation, in which the disturbance potential is merely ignored. The next level of approximation for the near-field flow is that corresponding to the first-order slender-ship approximation. Significant discrepancies were found between the wave-spectrum predictions corresponding to these two near-field-flow approximations. The other two near-field-flow approximations were determined by applying a simple analytical expression defining the fluid velocity along the waterline in terms of the elevation of the wave profile, corrected at the bow and the stern according to the nonlinear local solution given in t15~; both wave profiles measured experimentally and predicted numerically using the slender-ship approximation were used. Discrepancies between the wave-spectrum predictions corresponding to the latter analytical/experimental and analytical /numerical near-field-flow approximations were found to be much smaller than the discrepancies between the wave-spectrum predictions corresponding to these nonlinear near-field-flow approximations, on one hand, and to the linear near-field flow prediction given by the slender-ship approximation on the other hand. This result indicates that the short divergent waves in the wave pattern of a ship stem mostly from the ship bow and stern; a result that is not surprising but points to the fundamental difficulty of predicting the short divergent waves since the flow at a ship bow and stern is strongly nonlinear and quite difficult to compute. Numerical calculations for the Wigley hull showed that the steepness of the short divergent waves predicted on the basis of the foregoing linear far-field/nonlinear near-field flow analysis is too large for the waves to exist in reality within a sector of several degrees in the vicinity of the ship track. The wave steepness was also found to exhibit a well-defined peak at an angle of approximately 1° to 2° from the ship track. It is unclear whether or not these numerical predictions have real physical implications. On the one hand, the physically- unrealistic predicted wave steepness might be regarded as a theoretical indication for the common observation that the wake of a ship in the vicinity of the track does not contain divergent waves. On the other hand, these physically-unrealistic numerical predictions might be regarded as an indication that the assumptions of linear far-field waves and of steady nonlinear near-field flow underlying the mathematical model and the numerical results may not be correct. In particular, assumptions about the flow in the immediate vicinity of the ship bow and stern clearly are crucial for the prediction of the short divergent waves in the wave pattern. Indeed, the present study underscores the need for a better understanding of the flow at a ship bow and stern. It should be recognized that the assumptions underlying the short-wave analysis presented in this study are also adopted for numerical predictions of the longer waves in the steady wave pattern of a ship. In this respect, numerical experiments analogous to those performed in this study for the short divergent waves also seem useful for determining the sensitivity of the longer waves in the wave pattern to the assumptions used in the calculation of the near-field flow at a ship hull. Such numerical experiments will be presented in [16~. ACKNOWLEDGMENTS This study was performed at the David Taylor Research Center with support from the Surface Ship Wake Signature Task, the Independent Research Program, and the Applied Hydrodynamics Research Program funded by the Office of Naval Research. The authors also wish to thank Dr. Robert Hall at SAIC and Dr. Patrick Purtell at DTRC for discussing the paper. 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