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Adequacy of Free Surface Conditions for the Wave Resistance Problem H. Raven (Maritime Research Institute Netherlands, The Netherlands) ABSTRACT This paper compares three different lin- earized free surface conditions for steady potential flow around a ship, viz. the Kelvin condition and the slow-ship conditions of Dawson and of Eggers. All are implemented in a Rankine-source method of the type proposed by Dawson. The comparisons concern the predicted wave resistance and wave patterns, the magni- tude of the nonlinear terms neglected in the FSC, and the remaining errors in the dynamic and kinematic conditions at the predicted free surface. Rather substantial errors are found for all linearized methods. Dawson's condition does not perform any better than the Kelvin condition except for full hull forms. The occasional prediction of a negative wave re- sistance for full ships is explained in terms of an energy flux through the free surface. The condition of Eggers, though theoretically preferable, is found to lead to non-con- vergence of the basic perturbation expansion near the bow in one of the cases and so to give worse results than Dawson's. NOMENCLATURE D downstream plane up to y=h Do downstream plane up to y=0 E energy flux En Froude number FS free surface Y=h FSo undisturbed free surface y=0 g gravity acceleration n n n linearized wave elevation double-body wave elevation (23) h-nr wave elevation including nonlinear * terms (18,25) wave elevation including nonlinear and transfer terms (19,26,28) p fluid density total velocity potential ~double-body potential §' f-. 1. INTRODUCTION intersection of plane D with free surface normal vector n longitudinal component of n px pressure Rw wave resistance Rwf wave resistance according to (7) Rwp wave resistance according to (9) U ship speed Vn normal velocity component of control surface x,y,z Cartesian coordinate system; x astern, y upward, z to port; origin at ~ L un less stated otherwise H.C. Raven, MARIN, P.O. Box 28, 6700 AA Wageningen, The Netherlands 375 One of the oldest and most extensively studied problems in ship hydrodynamics is the determination of the wave pattern and wave resistance of a ship sailing in still water. This is partly related with its visible importance for the efficiency of a ship and the fact that it is the primary quantity derived from towing tank tests; but also the rather obvious formulation of an appropriate mathematical model for the potential flow with free surface boundary conditions has played a role. Particularly in the sixties and seven- ties much work has been done to simplify this basic model to a method, simple and yet accu- rate enough in practical applications. The most important result of all this activity was probably the formulation of the slow-ship theory, based on linearization with respect to the flow about the hull at zero Froude number, i.e. the "double-body flow". At the Second International Conference on Numerical Ship Hydrodynamics two papers applying this condi- tion, by Baba [1] and by Dawson [2], attracted much attention due to the realism of the pre- dictions shown. Additionally Dawson's method raised much interest owing to its very origi- nal and apparently straight-forward implemen- tation differing substantially from current methods of that time. Since then, Dawson's method or a very similar one has been implemented by many others. One member of this family is our pro- gram DAWSON, which follows the same basic procedure but differs in several details from

the original method. More about it can be found in [3]. This program is being used extensively as a design tool at MARIN, giving detailed information on the pressure and streamline direction on the hull, the wave pattern, the resistance and other forces, etc. With careful interpretation this allows to optimize a design before model tests are being performed. As a matter of fact this has led to a few quite successful designs. It appears that in some respects the ideal role of Numerical Ship Hydrodynamics integrated in the design procedure is being approached here al- ready. Still, in practical applications to current ship forms certain problems and doubts on the reliability of the predictions may arise. In studying these it appears that some aspects of slow-ship theory have not yet been settled entirely. The principle of Dawson's method is generally taken for granted, as is illustrated by the fact that even the alge- braic mistake [3] in the derivation of the free surface condition is copied without comment in most papers! Current work on wave resistance calculations is mostly concerned either with further extensions of Dawson's method to new applications, or with methods solving the nonlinear problem, and does not address the basics of the methods so widely used now. In a certain sense this paper makes a plea for a renewed critical look at the wave resistance calculation methods. It questions the assumed superiority of the slow-ship con- dition by showing that for a large class of ships the predictions are nothing better than if the Kelvin condition is used instead. The paradoxical occurrence of negative wave re- sistance predictions, perhaps more widely known but ignored, is studied in Section 3. The validity of the linearization assumptions is investigated by estimating the nonlinear terms and by directly evaluating the velocity field at the predicted free surface. Some of the conclusions motivate and direct the de- velopment of a method to satisfy the nonlinear boundary conditions; Section 6 briefly dis- cusses the prospects for such work. 2. WAVE RESISTANCE PREDICTIONS 2.1. Background The mathematical model pertinent to cal- culation of the wave pattern and wave re- sistance of a ship is that of a potential flow subject to kinematic and dynamic free surface conditions (FSC's). Solving this problem in its exact form is quite complicated, not only owing to the nonlinearity of the dynamic con- dition but also since both conditions must be applied at the unknown free surface. The general approach is therefore to linearize the FSC. This linearization can be performed with respect to either the undisturbed uniform flow or the double-body flow. In the former case, the Kelvin FSC is obtained: fax by O (1) which, if combined with the exact form of the hull boundary condition gives rise to a "Neumann-Kelvin" problem. The other approach results in the slow-ship FSC, which can e.g. have the form chosen by Dawson: Fn2~ t~ _ + Ott a ]~`,2 + t2) (.x ax + (z az) (.x fx + (zfz ~ ax ~ {z) + + by = 0 (2) where ~ is the double-body potential and ~ is the total potential. Present methods also impose that the perturbation potential ¢'=~-. satisfies the exact hull boundary condition. Since the appearance of the slow-ship FSC this has become by far the most popular boundary condition in wave resistance calcula- tion methods. Its assumed superiority is probably based to a great extent on the results of the First Workshop on Ship Wave Resistance Calculations in 1979 [4]. Here many different methods were applied to common test cases, viz. the Wigley hull, an Inuid form, the Series 60 block 60 hull, a fast naval vessel and a tanker model. For all these cases Dawson's predictions were consistent and acceptably close to the experimental values. On the other hand, Fig. 1 illustrates the results of methods solving the Neumann-Kelvin problem. The only conclusion to be drawn from this figure is that large numerical errors must be present in most of them, since solu- tions of the same problem predict resistances differing by a factor of 2 in some cases. The magnitude of these errors forbids all conclu- sions on the relative merits of the free surface conditions used, and the results are certainly no reason to do away with the Neumann-Kelvin approach! Still this is what happened: from about 1980 onwards, the majori- ty of the effort in the wave resistance field concerned Dawson's method including the FSC (2). It seems that the motivation for the general preference for the slow-ship condition is not very sound. Actually a comparison of predictions by Neumann-Kelvin methods and slow-ship methods is only allowed if one is very careful on the point of numerical accuracy. This is because these two classes of methods use an entirely different approach and contain therefore numerical errors of a different origin. Neumann-Kelvin methods generally exploit the known form of the Green function for the problem. So-called Kelvin source panels are distributed over the hull and along the water 376

51 ~ 3 2 O_ RESIDUAL RESISTANCE IHUANG & von KERCZEK, 1972) RESIDUAL RESISTANCE (TODD, 1963) WAVE RESISTANCE (LONG. CUT, TSAI & LANDWEBER, 1975) --- WAVE RESISTANCE (XY-METHOD, WARD, 1964) ~ ~ ~ WAVE RESISTANCE (MODEL FIXED AT ZERO TRIM AND SINKAGE, CAl ISAL, l9B0) 1 ~ X _ _1'~ ) art ~ . '? A//' . /' in'/ ~ _ 0.15 0.20 0.25 0.30 a_ FROUDE NUMBER Fig. 1 Wave resistance coefficient for Series 60 model. Symbols indicate results of various Neumann-Kelvin methods [4] line; the potential induced by these panels automatically satisfies the Kelvin free sur- face condition. Numerical errors are present in the hull panelling, the treatment of the singularities of the Green function, the numerical integrations over the panels and the waterline integral. On the other hand, most slow-ship methods use a distribution of Rankine sources on both the hull and a part of the free surface sur- rounding the hull. Discretization errors are made in the panelling of both surfaces, but also in the difference scheme used to imple- ment the velocity derivatives in the free surface condition. Additionally, the trunca- tion of the free surface domain and the "numerical" imposition of the radiation condi- tion may introduce errors. However, a fairer comparison of different FSC's is possible. The flexibility of Dawson's numerical implementation allows to treat all sorts of linearized FSC's in basically the same way. Instead of using Kelvin sources to solve the Neumann-Kelvin problem, we now use a distribution of Rankine sources on the hull and a part of the free surface for both FSC's; and the velocity derivative is implemented by a difference scheme. Actually, the only change needed in the program DAWSON is to replace the double-body flow terms in the free surface condition by a uniform flow, as appears from the formulations given above. Due to the very similar implementations of the FSC's the numerical errors will now be of comparable magnitude; and using the current experience on the required discretization we can make sure that these errors have little influence on the predictions. It is true that near the stagnation points, in the limit for zero panel size, numerical errors could locally again dominate the comparison due to singular behaviour. But in practice these singularities are always "discretized away". Thus the comparison tells us how important the double-body flow contribution to the FSC is for practical discretizations, which have been checked to be adequate. 2.2. Results This methodology has been applied to a number of ships. First the standard test cases were attempted. For the Wigley hull, a discre- tization with 20*6 hull panels and 10*38 free surface panels was used in the calculation for Fn=0.40. As expected in view of the near- uniformity of the double-body flow for this slender hull, the difference between the Neumann-Kelvin and Dawson resistance predic- tions amounted to only 1%. A more realistic test case is the Series 60, block 60 hull. 24*20 hull panels and 10*128 free surface panels were used. Fig. 2 displays the predicted resistance curves, compared with some of the experimental data. Here again the differences between both methods are negligible, except perhaps above Fn=0.32. This contradicts the results of the Workshop mentioned before, where the Neumann- Kelvin predictions were, on the average, sub- stantially higher. Which FSC is more accurate cannot be deduced from comparison with experi- mental data, not only because of the small differences but also due to the disappointing scatter of all available measurements. 6 5 0.20 0.25 FROUDE NUMBER , I _ I ~ - RESIDUAL RESISTANCE (HUANG & von KERCZEK, 1972) RESIDUAL RESISTANCE (TODD, ig63) WAVE RESISTANCE (LONG. CUT, TSAI & LANDWEBER, 1975) --- WAVE RESISTANCE (XY-METHOD, WARD, 1964) 3' *( ~ WAVE RESISTANCE (MODEL FIXED AT ZERO TRIM AND SINKAGE, CALISAL, 1980) Calculated (Dawson FSC) I I ___ Calculated (Kelvin FSC) ~,y,/',/' ,~.. . a,,' __ A_ ._ / ,// 1 // i/ // // 1 0.35 Fig. 2 Wave resistance coefficient for Series 60 model, predicted with Kelvin and Dawson's FSC 377

Fig. 3 compares the predicted wave profiles for both methods. Both are in close agreement with each other and with the data. Generally, Dawson's FSC leads to a small forward shift of the bow wave and a somewhat more pointed wave shape. The largest differ- ences are found aft of the ship's stern: The Neumann-Kelvin predictions often have a sub- stantially larger wave amplitude here. This can probably be explained by the damping effect of the base flow acceleration in the slow-ship FSC. The same sort of comparison has been made for several practical cases. The fact that there is remarkably little difference between the Dawson and Kelvin predictions is a re- current feature for all ships with a block coefficient up to about 0.60 or 0.70. This is in marked contrast with the general convic- tion' Since in commercial projects the present use of the predicted wave resistance values is generally qualitative or comparative rather than quantitative, we can safely state that the Kelvin FSC performs just as well as the slow-ship FSC. unto Fig. 3 Wave profiles along the hull predicted Fig. with Kelvin and Dawson's FSC; Series 60, Fn = .35 and Fn = .2Z - Dawson - Kelvin There are, however, exceptions. Since the difference between both FSC's increases with the nonuniformity of the double-body flow, there will be a limit on the hull fullness for which the above statement is valid. Fig. 4 shows what happens for very full hull forms, in this case a 55000 tdw tanker with a block coefficient of 0.82. At higher speeds (in excess of the service speed) both predictions are similar (though not identical) again, but for decreasing speed quite substantial dif- ferences aDDear. The K~1 v; n Per ~;11 yields a experi- predicts a rapid decrease of the resistance. Even so the predicted wave patterns are very much alike (Fig. 5) and give no indication of such drastic resistance differences. Again Dawson's condition typically results in a forward shift of the bow wave. Due to the large curvature of the waterlines of such full hull forms this brings about a large resistance difference, concentrated at the forebody. large resistance far exceeding the mental value, while the slow-ship Fort For the same ship at ballast draught the resistance predicted using Dawson's condition is in good agreement with the experimental data, while the Kelvin resistance exceeds this by a factor of 3. At full draught the Kelvin result is 4 times as high as the experimental result at the service speed, but Dawson's condition yields a negative re sistance! This physically unacceptable behaviour has been found for several slow, full-formed ships. 11 10 q 8 6 378 calculated wave resistance for a tanker model . . _. tow Kelvil ~_ Cw Dawson . . _ . , ,' ,_ / / 0 0 0.: 2 0. I- _ r r ~ o 4 Calculated wave resistance for a tanker model 0 Experiment

Fig. 5a Calculated wave patterns of tanker model, with Dawson's (left) and Kelvin's FSC (right), at Fn = 0.177 (service speed) o at_ N ID - - 1 i. I I I I o O- to 1 0 o o 16 32 Summarizing the results of these compari- sons, for a large class of ships there is no reason to prefer Dawson 's FSC to the Kelvin FSC; for practical discretizations the double- body flow effect is of minor importance. In extreme cases however, appreciable differences in resistance may occur, but none of the FSC's gives an accurate and reliable result. The next section first tries to resolve the mystery of negative wave resistance. Then, we shall assess the adequacy of the FSC's by other means. 3. THE PARADOX OF NEGATIVE WAVE RESISTANCE 3.1. General The fact that a negative value of the wave resistance is sometimes predicted by Dawson's method is known to more people that Dawson Kelvin 1 11 l,\: 1,\. 1/` ~ .1/~6 ^\\ /e l: 4{ 1`, ~ ~ /° i\ i\~ll2 ''\-/1~ ~144~/lm Fig. 5b Calculated wave profiles along the hull 379 apply this method to real commercial ship hull forms. Generally this phenomenon is attributed to an insufficient resolution of the large pressure gradients on the hull. As a matter of fact we often find large differences in the pressure on adjacent panels, and the simple pressure integration over the hull could well be numerically inaccurate. The accuracy of the pressure integration and of alternative formulations of the re- sistance based on Lagally's law has been discussed in [3]. It has been shown there that, provided the pressure integration is corrected for the zero-Froude number pressure integral, these formulations are all of about the same level of accuracy, determined by the accuracy with which the hull boundary condi- tion is satisfied and so by the density of the hull panel distribution. Therefore, for the present test case the hull panelling has been

refined, from 545 to 1090, and then to 1415 panels on one half of the hull. Table 1 shows that this does reduce the zero-Froude number pressure integral (which has the exact value zero due to d'Alemberts paradox). However, the wave resistance (according to all three ex- pressions) converges to a negative value! Similarly, increasing the free surface panel density did not give any substantial change in the predicted wave resistance. Therefore, contrary to the general belief discretization errors can probably be rejected as cause for the negative resistance, at least in some of the case investigated. A negative value of the wave resistance in the presence of a physically realistic wave pattern radiated by the ship as shown in Fig. 5 is, of course, paradoxical. Apparently such a wave pattern need not have the correct energy budget. The wave pattern represents radiation of wave energy; since this energy travels with the group velocity which in harmonic deep-water waves is half the phase speed, it lags behind the wave crest propaga- tion, hence it must be supplied at the wave origin, i.e. at the ship. But a negative re- sistance means that instead the ship extracts energy from the waves. A steady flow can only exist if some other source of energy is present. 3.2. Energy Conservation To investigate what the source of energy can be we consider the energy fluxes through the boundaries of a control volume surrounding the hull and moving with the hull, in a sta- tionary frame of reference (Fig. 6). We define the x-axis to point astern, the y-axis ver- tically upward with y=0 at the undisturbed water level. As derived in [5] the general expression for the energy flux through a surface, ex- pressed in a general unsteady potential I, is: E = - rip { P.t( En - Vn) - p Vn}dS (3) where Vn is the velocity of the surface itself in the normal direction, and the energy flux is defined positive in the sense consistent with that of the normal. For a steady flow, at = Uf , where now V) is the disturbance of the uniform flow and U the ship speed. The energy fluxes out of the control volume then become: through the hull surface: EH = -U l~ P . nx . dS = -U . R (4) (where n is the inward normal on the hull); through the upstream plane: EU = -U l~ pay dS through the downstream plane: ED = +U l~ pay dS - ~pU l~ (¢x - by - gads (5) (6) where the lateral and upstream boundaries (but not the downstream plane) have been assumed to recede to infinity. It is noted that no energy flux is present through the free surface, since the pressure is equal to zero and ~ -V , for exact satisfaction of the free surface conditions. Table 1 Calculated resistances for tanker model, En = 0.1765 A) influence of hull panel density; Dawson's FSC. I Sumner or nu' ~ panels per side 1 545 ~ 1090 | | Zero-Fn pressure integral | .00071 | .00042 | Wave resist. pressure integration ~ -.00082 ~ -.00068 ~ Wave resist. Lagally integr. over FS ~ -.00054 ~ -.00048 ! Wave resist. Lagally integr. over hull ~ -.00068 ! - ~ 00054 ~ 1 .. 1 _ B) influence of free surface condition; 1090 hull panels ~ Free surface condition 1415 .00020 -.00067 _.00047 -.00052 ~ Kelvin ~ Dawson ~ Eggers | - 1 1 1 Wave resist. pressure integration ~ .00270 ~ -.00068 ~ -.00158 Wave resist. Lagally integr. over FS ~ .00282 ~ -.00048 ~ -.00146 Wave resist. Lagally integr. over hull ~ .00288 ~ -.00054 -.00156 ! Waterline integral (10) I .00062 1 .00054 .00015 380

f ~ ~ 1 1 Rwp = up Or (-< + ~ + 2)dS + P [l fx by dS (9) - t Energy conservation now demands that . -EH = U.R ED wf _. Eu ._ EFso _ I_ Do ~ EDO U. Rwp ~ Eu Fig. 6 Control volumes for energy balance; above: exact case; below: linearized case Conservation of energy then requires that Rwf = Up l~ (~¢x ~ by + (z~dS + + Peg ~ n2 dz T (7) The first integral is over a transverse plane astern of the hull and the second one along its intersection IT with the free surface; the latter line integral basically takes into account the contribution of the potential energy, as it stems from the pay term in the pressure. This is a well-known result that is the basis of wave-pattern analysis methods. However, the derivation may not be entirely relevant to our calculations. The linearized FSC is imposed on the undisturbed free surface, not on the actual wave surface. The fluid domain considered thus extends only to y=O, and so should our control volume. Therefore we now redefine the control Volume by taking not the exact free surface as its upper boundary but the y=0 plane. The energy fluxes must then be slightly modified. An energy flux through the un- disturbed free surface may now be present, according to the general expression (3): O FSo (8) ~ y It is important to note that here no FSC has been substituted yet. It is noted in passing that the energy supply through the hull surface, and thereby the resistance found, now only concerns the y<0 part of the hull. This introduces a dif- ference that can be approximated by: = + UPS ~ n nx do (10) WL However, this waterline integral is often ignored, following Dawson [4], since it gener- ally does not improve the results; its magni- tude for the present case is included in Table 1 and found to be not negligible, but not affecting the conclusions on the negative re- sistance either. Similarly the integral over the downstream plane extends only to y=O. In any case, Rwf may be considered as the resistance that is physically associated with the generation of the wave pattern that is predicted in the far field, since (7) has been derived without making any simplification of the boundary conditions. On the other hand, the pressure integration over the hull in our linearized calculation need not be equal to Rwf because of the simplifications of the FSC and the different control volume, but it will give a result in agreement with Rwp. There- fore, the origin of the paradox, being a contradiction of the calculation and our physical insight, must be sought in the dif- ference between both expressions. Substitution of the potential of a free harmonic wave in Rwf shows that the wave re- sistance so obtained is positive definite, in agreement with our physical observation of the radiated wave pattern. Also its value is independent of the position of the aft plane, since in harmonic waves there is a constant horizontal energy transport and a pure ex- change between kinetic and potential energy. In expression (9) however, the horizontal energy flux is not constant: the potential energy is absent as the control volume extends now only to y=O, and it thus cannot compensate the variations of the kinetic energy flux. But still the resistance from (9) is independent of x, since for harmonic waves the variations of the kinetic energy flux are now compensated by the flux through the undisturbed free surface, the integral over FSO. More generally, we can evaluate (9) by substituting the Kelvin condition: 381

Of' a by = unX ; ~ = - g''; - P fir ox by dS = +Pg l~ hex dS = Y2pg ~ n2 dz + pug ~ h2 nx do (11) T WL where n is now the inward normal on the water- line. So Rwp and Rwf are equal to leading order, except for a waterline integral which is of opposite sign to (10); this difference in sign has been called "Gadd's paradoxon" and is dealt with in [6]. Apart from this, if the Kelvin condition is imposed (7) and (9) give the same result at least asymptotically; so the pressure integration over the hull in our linear problem, which satisfies (9), is equal to the resistance deduced from the far-field wave pattern corresponding to (7), hence it is positive and independent of x if a system of harmonic waves is present at the aft plane. Again the energy flux through FSO just takes into account the variations of the potential energy. It appears that this correspondence relies on the precise relation between the velocity components at the free surface and the wave elevation, so on the FSC. However, for free surface conditions other than Kelvin's this equality may be lost. E.g. for the slow-ship condition we find from (9): Rwp = ~kp [[ (_~2 + f2 + ~z)dS + - P ·~r (X (~xnX + (ZhZ + fxbrx + + ~ZnrZ)dS (12) which does not correspond with Rwf generally. If a realistic system of waves is present behind the ship, (7) will again give a positive resistance. But Rwp may be different as part of the wave energy may have been supplied through FSO instead of by the ship. The intuitively expected correspondence between a radiated wave pattern and a re- sistance acting on the hull is thus not verified for all FSC's. In the extreme case the pressure integration over the hull can give a negative resistance: the ship rides on waves generated by the free surface condition, in this case governed by the double-body flow nonuniformity. In order to obtain a more realistic wave resistance prediction for such extreme cases, two approaches now suggest themselves. Since Rwf seems to have the desired property of positive definiteness, we could try to deduce the resistance from the far-field wave pat tern. Otherwise, we could perhaps modify the FSC so as to eliminate the excess energy flux through the free surface. 3.3. Far-Field Resistance Calculation Evaluation of (7) is not a straight- forward matter in a Rankine-source method, since it requires to calculate the velocities in a dense grid of field points in a trans- verse plane astern of the ship. The integral over D thus found must then be supplemented by the Line integral and a contribution of the difference D-Do. An alternative would be to apply one of the formula's used in wave pattern analysis methods, which are based on (7). Before undertaking such a study we better first consider its chance of success. Since we want to get a resistance independent of the x-position of the transverse plane, this plane should be located outside the region where there is still an excess energy flux through the free surface. We know that for vanishing double-body flow disturbance the slow-ship condition boils down to the Kelvin condition, for which we have shown the absence of leading-order energy flux. Therefore we must choose the downstream plane far enough from the hull to avoid the double-body flow disturbance. In practice a logical choice would be a position at or beyond the aft edge of the free surface panel distribution. However, since no singularities are located aft of this plane, d'Alemberts paradox is valid for the 'body' generated by the collection of sources on the double body and free surface: the momentum flux through the transverse plane will be always zero! The wave resistance is an internal force inside this virtual body and is compensated by an opposite force exerted by the hull on the free surface panels. and Therefore the integral over Do is zero, Rwp = ~P l~ fix By dS (13) FSo which is precisely the expression obtained from Lagally's law. Similarly the first term of (7) is dominated by this effect and looses the properties it has in harmonic waves; thus Rwf cannot be supposed to be positive definite and independent of x any more. This must be caused by the truncation of the free surface domain locally distorting the potential field corresponding with harmonic waves. Rw (1) will only give a reliable result if the free surface panelling is continued a large distance beyond the aft plane D, which means a considerable extra expense in computer time. 382

In view of their relation with (7) and the assumptions on the potential field, the same is likely to be true for wave pattern analysis methods. Besides, these could be subject to errors caused by the inherent damping of the numerical approximation. The experiences of Maisonneuve [7] with such a resistance evaluation do show that this is not a good alternative for the pressure integra- tion. 3.4. Energy Flux Through The Free Surface Another, more fundamental remedy for the occurrence of a negative wave resistance could be to modify the FSC so as to eliminate the excess energy flux through the free surface. Whatever the FSC imposed this energy flux into the fluid domain can be written as FS l~ (P U (x EVn - U · Ap · nx)dS (14) where the integration is over the predicted wave surface, AVn is the remaining normal velocity and Ap the pressure prevailing there. To eliminate the energy flux at every point of the free surface both AV and Ap should be zero, so we have to satisfy the exact non- linear FSC's; the best that can be achieved in a linearized method is a reduction of the energy flux to higher order in the perturba- tion parameter. For the slow-ship condition this param- eter is the square of the Froude number Fn. Since the resistance coefficient is su~posed- to be of O(Fn ), dE/dt should be O(Fn ). fix and ~ contain the double-body disturbances which are 0(1), hence both AV and Ap must be O(Fn ). Now Dawson's FSC does contain terms up to O(Fn ) and ought to satisfy this require- ment; but as has been pointed out in [3] and will be shown in the next section, Dawson's FSC is inconsistent due to the absence of terms incorporating the transfer of the veloc- ities from the actual to the undisturbed free surface. Including these terms, ~ ~ + I' , yields an FSC that can be writienY{s: YY Fn {~(+xx + fizz + (x ax + Liz a - z) (.x + Liz) + (.xx + fizz + ax ax + it a-z) (.x tx + If + ~ fix ~ {Z)) BY (15) This in fact reduces AV and Ap to O(Fn6) and thus should suit our purpose. Now this is exactly what has been derived in a different way by Eggers [8], in a paper that seems to have been given little atten- tion. From the analogous requirement that the far-field resistance found from a slow-ship calculation should be invariant to leading order for the choice of control volume, he derived this same FSC (15). Although in his paper this is suggested to form an additional proof of the correctness of the order assump- tions made in the linearization, the above consideration shows that actually it is subject to these same assumptions. Therefore, Eggers's FSC does provide equality of Rwf and Rwp up to higher order in Fn, just like the Kelvin condition does up to higher order in the wave steepness. This could solve the problem of negative wave resis- tances. The FSC (15) has therefore been imple- mented in our program and applied to a few cases. For the Wigley hull the predicted re- sistance differed by 3% from the Dawson pre- diction, and the wave profile was almost iden- tical. For the Series 60 model the resistance curve is included in Fig. 7 and turns out to be similar in shape but consistently lower than those obtained with the two other FSC's. Surprisingly however, the predicted Rw for the tanker form (Table 1) is even more negative than that found with Dawson's FSC. The wave profile along the hull is hardly different; the wave pattern (Fig. 8) shows a remarkable reduction of the Kelvin angle, but is otherwise similar to Dawson's. These re- sults suggest that only more energy has been supplied through FS. Moreover, the resistance curve goes down for increasing speed, and at somewhat higher speeds flow reversal at the free surface is predicted! The results were found to be extremely sensitive to details of the implementation. It thus seems that for large double-body flow disturbances this FSC is not adequate, notwithstanding its theore- tical consistency. An explanation of this will be given in Section 5. 6 . . ~t ~ .35 0 .40 Fig. 7 Predicted wave resistance coefficient using Kelvin, Dawson's and Eggers's FSC Kelvin - Dawson - - Eggers 383

Fig. 8 Calculated wave pattern for tanker model, with FSC of Eggers 3~5. Summary Summarizing some of the findings of this section, the paradoxical occurrence of nega- tive wave resistance in the presence of a plausible radiated wave pattern can be ex- plained by the possibility of an energy flux through the undisturbed free surface differing from the amount needed to represent the poten- tial energy variations. This energy flux can be expressed in the remaining normal velocity and pressure at the calculated free surface. Reduction to higher order of both quantities leads to the FSC derived by Eggers. Although this gives reasonable predictions for slender ships, for the tanker hull an erroneous flow field is sometimes obtained and the resistance is again negative. Therefore the higher order contributions to the energy flux are probably significant here. The next section attempts to shed some light on the magnitude and origin of these higher order terms in general. 4. HIGHER-ORDER TERMS 4. lo Approach In the derivation of the linearized free surface conditions several linear and nonlin- ear terms have been dropped based on the fact that they are of higher order in the pertur- bation parameter adopted. Now the notion of higher order just means that these terms be- come insignificant asymptotically for a van- ishing value of the parameter; no information is available a priori on the actual magnitude of these terms in practical cases. Therefore the range of applicability of a linearized method can in general only be determined in practice. It is clear, however, that agreement with experimental data is not a very suitable yard- stick for the accuracy of wave resistance cal- culations and has therefore not been used in Section 2. The residual resistance is often affected by uncertainty on the viscous rem Distance, and wave pattern measurement tech- niques give results consistently lower than the residual resistance and with a consider- able amount of scatter. Therefore a method to assess the magnitude of the neglected terms directly could give a much better idea of the adequacy of FSC's. In addition it might pro- vide directions for setting up a method to solve the exact nonlinear problem. Evaluation of the nonlinear terms would in principle be possible from velocity mea- surements at a very dense grid of points near the free surface; but no such measurements seem to be available. A better way would be to compute these terms from solutions of the non- linear problem; but these exist only for un- realistic test cases, or in the form of numerical solutions of insufficient accuracy. For these reasons another approach has been chosen here: to evaluate the nonlinear terms 'a posterior)', from the flow field cal- culated by a linear method. Of course this technique does have its own restrictions: How the predictions would change if the nonlinear terms were included in the FSC is not clear; nor is there any certainty on how the terms themselves would change in magnitude. Additionally, the singular behaviour at the stagnation points will cause some of the higher-order contributions to blow up. Hence also the comparison of the higher-order terms is limited to practical discretizations, although, as we have found, within fairly wide margins: approaching the stagnation points magnifies the relative magnitude of the non- linear terms but does not immediately affect the comparison of FSC's. 4.2. Derivation Of Higher-Order FSC's To define the neglected terms we must first of all expand the FSC's to higher order. The combination of the kinematic and dynamic free surface conditions that is actually im- posed demands that the flow at the undisturbed free surface has a direction parallel to the isobar planes. Only afterwards the free sur- face elevation is retrieved by using the dy- namic condition. In view of this, in the fol- lowing I have chosen the approach to start from the kinematic condition and to derive the higher order terms in it. Now the prime difficulty here is that the exact FSC should be applied at Y=h, not at the undisturbed free surface y=O. To derive the error in the FSC would require the calculated velocity field at Y=n; but this cannot be eva- luated since for To, the singularities gener- ating the flow lie inside the fluid domain. Therefore we have to resort to Taylor expan- signs to express flow quantities at Y=h in those at y=O. But precisely the validity of these expansions has been a point of debate in the derivation of the slow-ship FSC. The wave 384

elevation including the double-body contribu- tion is O(Fn ); it has been argued that the perturbation I' has a wavy character with wave number O(Fn ), so all terms in the Taylor expansion for the transfer would be of the same order; neither the analytic continuation of the flow field, nor the truncation of the expansions would then be permitted. We have, however, adopted the basic order assumptions of Eggers [8] as discussed in [3] and conse- quently ignored the possible order reduction by differentiation. The nonlinear terms can be formulated consistently, including only the contributions of next-higher order, or inconsistently in an attempt to approximate the exact FSC as close- ly as possible. There is a quantitative dif- ference between both forms but the conclusions are the same for most cases. The best estimate of the wave elevation is that which includes the nonlinear terms and the transfer terms; it is noted n below. For the Kelvin condition, the perturbation param eter is the wave steepness ~ = ~ . The ve- locities at Y=n are: Fn Ax = (+x)y=o + n fxy + 0(~3), etc. (16) Substituting this into the dynamic FSC gives the following expressions for the wave height n linear: n = -Fn fix + 0(£2) nonlinear: n = n-~l2 Fn2 :¢X2 + (,2 + +,2] + o(~2) (18 including transfer: n* = -n- Fn2 n fxy + 0(~3) (19) The consistent decomposition of the kinematic FSC is then: by= Ox+ (§xbX+ ~znZ) + (nx- Ax) + term 1 term 3 term 4 (nX - -ax) - n my + 0(~3) (20) term S term 6 while an inconsistent form is: n* = -n- Fn2 n (+x fxy + Ty~yy + Adz) (21) (y = Ax + (¢xnX + ~znZ)+~lx aaX + by aaZ](n-n term 1 term 3 term 4 (¢x ax + at az](n --n) + term 5 * * * n ( lynx + ~yznZ - EYE) (22) term 6 For the consistent form of the slow-ship FSC the wave height expressions are: double-body: or = Y2 Fn (1 ~ (x - (z) (23) linear: n = Y2 Fn2 (1 - (x - t2z _ 2.Xfx - 2tzfz) (24) nonlinear: n = h - Y2 Fn2 (+,2 + ¢,2 + ¢,2) (25) including transfer: n* = -n- Fn2<n2~ ~ + ~ ~. 4t,2 r x My z zyy yy Cry x~xy + (ztzy - tyy~y]) + O(Fn8) (26) The resulting expression for the FSC is getting excessively complicated. The higher- order terms now contain third derivatives of the double-body potential. These are rather hard to calculate and quite susceptible to numerical oscillations. A simplified form was therefore derived by dropping the transfer terms associated with the double-body poten- tial. This is not entirely a matter of convenience; as shown in Fig. 9 the extension of the double-body flow field above the un- disturbed waterplane, based on its assumed symmetry, only reduces the accuracy of the approximation for hulls with strongly flared sections at the waterline. With this simplifi- cation the consistent form becomes: = An + An + fin + fan + y x x z z x rx z rz term 1 term 2 fxhx + And + (.X aaX + {z aaZ](-n-n) + term 3 term 4 (.x ax + it aaz)(n --a) + term 5 + br(hrxtxy + nrz~yz) - /2 or tyyy term 6 n'f~yy - brTyy + O(Fn8) (27) term 7 385

where ~ = ~ - En hr(.xfxy + (z¢YZ) (28) and the inconsistent form is identical to (22). Actual flow Doubln-bodv flow l Fig. 9 Transfer terms for double-body flow may reduce the accuracy In all above expressions, the notation of the terms is as follows. Terms 1 and 2 are linear and are included in the linearized FSC. Terms 5 and 6 result from the transfer of the FSC to the undisturbed free surface, while terms 3 and 4 are the other nonlinear con- tributions to the FSC. Term 7 is a linear transfer term for the vertical velocity. If, here, we would not neglect the terms connected with the double-body flow transfer, its form would be Arc yy - ~ tyy = 0(Fn2) (29) which we shall denote by 'term 8' in the fol- lowing. This is the transfer term alluded to in the previous section; it is neglected in Dawson's FSC but is present in that of Eggers. It may be noted that for the double-body flow replaced by a uniform flow, the in- consistent expressions become equal, but the consistent odes (19,20) and (27,28) do not for term 5 and ~ . 4.3. Results Evaluating all these terms was restricted mainly to the Wigley hull, the Series 60 block .60 model and the tanker form dealt with before. For clearness of presentation we shall sum up the main conclusions drawn, grouped under the test case concerned. The figures illustrating these results are set up as follows: the quantity plotted is indicated in the caption; as all terms are contributions to the vertical velocity, they are non-dimensionalized by the ship speed and compared with either the vertical velocity itself or the dominant contribution to it (term 1), as indicated. The abscissa is the panel number on the longitudinal strip of free surface panels along the hull and the centre line. The panel lengths are uniform. The loca- tions of bow (left) and stern(right) need no further indication. 1. Series 60 block .60 at Fn = 0.35 and 0.22 * Of the linear terms included in the FSC, only term 1 is significant (Fig. 10). Only at bow and stern term 2 may give a modest contribution, but due to the considerable phase shift between ~ and Or this contribu- tion is often of the wrong sign. Probably term 2 could well be deleted from the FSC but computationally this gives no simplifi- cation at all. * The linear term 8 neglected in Dawson's FSC is generally larger than term 2 and is most- ly concentrated at the bow (Fig. 10). For the Series 60 hull its maximum contribution is about 20%, at Fn=0.35. Including this term reduces the wave resistance by 20% in this case. * The nonlinear terms (i.e. the error in the linearized FSC) are dominated by the trans- fer terms. The effect of imposing the FSC at the correct location is often far more important than to include all squares and cross-products. * The sum of the higher-order terms is shown in Figs. 11 and 12 for two Froude numbers. It appears that at Fn=0.35 it is of quite substantial magnitude, locally as large as the linear terms. That still the predictions are acceptable must be due to some sort of 386

o.2l O . 1 I -O. 1 - -0.2 Fig. 10 Linear terms in free surface con dition; Dawson's FSC, Series 60, Fn = .35 Term 1 ------ Term 2 - - Term 8 error cancelling, or to the fact that the evaluation a posterior) is too pessimistic. Y 0 At Fn=0.22 all methods have fairly small error terms, confirming that linearization is adequate in this case. The sum of the neglected terms (terms 3 to 8 for Kelvin and Dawson, 3 to 6 for Eggers), which indicates the adequacy of the linearized FSC, is of comparable magnitude for all FSC's; all are oscillating with similar amplitude and mean value. No signi ficant improvement for the slow-ship condi tions is observed. * Eggers's FSC gives results closely corres ponding with those of Dawson's; although term 8 is now included in the FSC, the sum of neglected terms is not substantially less than for Dawson's FSC, since term 8 has a much smaller wave number than the other terms; therefore the mean value of the error is changed but the result is not visibly improved in general. z. Tanker model, Fn - 0.1765 Similar conclusions were drawn for this test case, except for the following: * The nonlinear terms for the slow-ship FSC's * are of moderate magnitude here, so the linearization is not particularly inadmis sible (Fig. 13). 387 n 1 Y o Y o- TV ~ ,1~\ ~ 1 \ He |,/ I if ~! IN ,'. ,~,,t,ilkil~l ~ Al V ',3iJI ~I~~N`l T -0.1- 1) \~J 0.1 - Eggers ~ Jilt -G.l ~ (Vl/ Fig. It Nonlinear terms in free surface conditions; Series 60, Fn = .35 --- Term 1 _ Terms 3 + 4 + 5 + 6 1160 1' 1\ ~0 ~1 ~0 The transfer term 8 is now quite large at the bow, about 70% of the vertical velocity (Fig. 14). It is understandable that this causes the substantial difference between the predictions using Dawson's and Eggers's condition. The nonlinear terms are significantly larger for the Kelvin FSC than for both slow-ship FSC's now (Fig. 13). The sum of the neglected terms is somewhat larger for Dawson's FSC than for that of Eggers in this case, which is entirely due to term 8.

We thus conclude that for the Series 60 model even from this evaluation of the non- linear terms no advantage for the slow-ship condition over the Kelvin condition can be observed. Only for very full hulls the slow- ship condition is again more appropriate than Kelvins condition for practical free surface discretizations. The large differences in neglected terms are largely in agreement with the wide spread in the predictions for the tanker case; but the slightly greater accuracy of the linearization underlying Eggers's con- dition is not reflected in a better resistance prediction. Further study of this is therefore needed and will be performed in the next sec- t;on -O. y O -O. 1J Fig. 12 Nonlinear terms in free surface conditions; Series 60, Fn = .22 Term 1 - - Terms 3 + 4 + 5 + 6 - o in. 13 Terms neglected in FSC; terms 3 to 6 for Kelvin and Eggers, terms 3 to 8 for Dawson; Tanker model, Fn = 0.176C . . Fig. 14 Importance of linear transfer term; Dawson's FSC, Tanker model, Fn = .1765 V _ Term 8 388

In general it was found that at least for the more severely nonlinear cases the Taylor expansions do not converge as quickly as one might hope, as testified by considerable dif- ferences between consistent and inconsistent formulations of some of the nonlinear contri- butions. Thus any method solving a nonlinear problem must apply the FSC right at the actual free surface, otherwise the most important nonlinear effects are missed or poorly re- presented. This is a fact not properly recognized in some of the methods proposed up to now [9,10]. Another fact learnt from this exercise is the appearance of double and triple wave numbers in the higher-order terms as could be foreseen from the theory. Ac- cordingly a nonlinear calculation only makes sense if the discretization is fine enough to resolve their contributions! 5. DIRECT EVALUATION AT THE FREE SURFACE Our doubts on the validity and accuracy of the Taylor expansions partly apply as well to the calculation of nonlinear terms per- formed here. The dominance of the transfer, the difference between consistent and incon- sistent forms and the oscillatory character of the higher derivatives of the calculated ve- locities make the comparison somewhat unsafe. Therefore another approach has been developed. As mentioned above, a direct evaluation of the velocity field at the actual free sur- face is prohibited in the usual method by the singularities inside the fluid domain. This can be avoided by generating the flow field by singularities not on the undisturbed free sur- face, but above it at a distance sufficient to keep clear of the highest waves. In the course of another study such a method had been devel- oped. Source panels are located above the un- disturbed free surface at a fixed distance, while the collocation points where the FSC is satisfied remain on the undisturbed free sur- face. Once the solution has been obtained, it is an easy matter to compute the velocities generated by these sources in points on the calculated free surface. From these, the residual errors in the exact kinematic and dynamic FSC can be found. For the same test cases we then draw the following conclusions; 1. Series 60 block .60 model, Fn = 0.35 and 0.22 * There are significant differences between the velocities on the undisturbed free sur- face and those on the actual free surface, particularly for the vertical velocity. Including the transfer terms from Taylor expansions makes the tangential velocities fairly accurate except at the stern wave; but the difference in vertical velocity is not well represented by the consistent term 8 (Fig. 15). 389 The remaining errors in the dynamic FSC are represented by the difference between the "linearized" and the "exact" wave elevation (Fig. 16), and provide little surprise. The Kelvin linearization consistently gives slightly larger errors. ~ , the wave height approximation employed in the previous sec- tion, appears to be almost exact in this case. The error in the kinematic condition (Figs. 17 and 18) is generally more important than that in the dynamic FSC. In the case con- sidered the advantage of Dawson's condition compared with Kelvin's is now slightly more pronounced than with the method of Section 4. The error at the bow for Fn=0.35 is 55% of the vertical velocity for Kelvin and 30% for Dawson. At lower Fn the difference is reduced. 1.0 y 0.9 0.1 ~ Vertical vel oc i ty i ~ 1 ll {bl 40 1\ 80 -0.1 It ~ Fig. 15 Velocities at undisturbed free surface, with transfer term and evaluated at actual free surface; Dawson's FSC, Series 60, Fn = .35 U*, V* (y = 0) ------ U , V (including transfer) - - Uex, Vex (y = h)

lo2 1.0 .0 n n Kel vin 80 Fig. 16 Wave profile along the hull, and error in dynamic free surface condition; Series 60, Fn = .35 Olin (linearized) n ( including transfer) - ~ (evaluated at Y = Olin) * The FSC of Eggers leads to slightly smaller errors than that of Dawson at both speeds. * It was verified that the evaluation of the nonlinear terms as made in the previous sec- tion was qualitatively right and indicative of the accuracy of the FSC, although unduly oscillatory. Thus we find that in this evaluation the advantage of slow-ship linearization is rather more pronounced. The reduction of the errors for decreasing Fn is slow. Although term 8 does not quite well approximate the transfer effect, including it seems to increase the accuracy somewhat. 2. Tanker model, Fn = 0.1765 Some remarkable conclusions could be drawn from these calculations. 1 * Again the error in the dynamic condition is quite small for the slow-ship approach, And well represented by the transfer terms (h ). The Kelvin linearization again leads to a somewhat larger error in In. * The error in the kinematic condition (Fig. 19) is very much larger for the Kelvin FSC than for Dawson's FSC, and amounts to 0.33 times the ship speed, which is 1.57 times the vertical velocity ahead of the bow. For Dawson's condition it is 0.152 times the ship speed. This agrees with the bad wave resistance prediction obtained with the Neumann-Kelvin method. * The same error with Eggers's condition has sharp positive and negative peaks near the bow, of +0.178 and -0.192 times the ship speed, so this method is now suddenly worse than Dawson's, which matches its bad re- sistance prediction. * This large and irregular error is explained by the excessive transfer effect on v and, in particular, u (Figs. 20 and 21). There are extreme differences between the veloc- ities on y=0 or those including transfer terms and those evaluated on the free sur- face, if the condition of Eggers is used. Even the error in v including term 8 (i.e. including part of the transfer effect) is larger than the error in v (without any transfer correction) with Dawson's FSC. * A negative u, i.e. flow reversal, was found at the free surface just ahead of the bow. The explanation of this lies just in the fact that the transfer term 8 is included in the FSC. The FSC can then be written as (15), and the coefficient of fax is ~ (x ~ 'k which is zero for (x = ~ = 0 577 In the present case (x = 0.658, and the coefficient is quite small. As a result, an excessive value of fax is not controlled by the FSC. Now Vex - v = h) + Y2 h2¢ + ... = term 8 + + n' any + I/2 h2fyyy + ... (30) Hex - u = boxy + ok n any + . . . ( 31) which shows that exactly the consistent inclu- sion of transfer effects increases the higher order contributions to the transfer, to the extent of invalidating the FSC. This is, therefore the explanation for the unrealistic results obtained with Eggers's free surface condition for full hull forms, in contrast with its theoretical preferability and acceptable results for less extreme cases. 390

o. ~ ~ :~,'l. ~ v Dawson It ~ N iK _ Eggers o A ~ ~ ~ to o o ~ ~ 'J'-~;f-/~4J:6\4\ '//60 -~1\li/-'1~' ~ Fig. 17 Error in kinematic free surface con- dition, evaluated at predicted free surface; Series 60, Fn = .22 V ___ AV -_ 1 o 8_ o o Da O- >_ - _ to o . o 8 Kelvin I\ ~ ///~~\\ ~ 1\ ~ 16 \4 ~32t-~ 48 _q6 Dawson / \ 8 a R- ~R \~\~---~\l t'- ~ --~L 11 '-- ~ o o o - o Eggers Jew ~ / ~ \,~ , ~ ~ ~ / I)\ I I ~ ' )(' r '~ ' ~ ~ -~' \' ~ '_ 7f [5 ~\,~j~ 48 ~ ~7 \ / I Fig. 18 Error in kinematic free surface con- dition, evaluated at predicted free surface; Series 60, Fn = .35 V - AV :112\8 ~160 Fig. 19 Error in kinematic free surface con- dition, evaluated at predicted free surface; Tanker model, Fn = .1756 It is fair to point out here that the possible vanishing of the coefficient of ~ has al- ready been given due attention before. Brandsma and Hermans [11] thought this fact a reason not to trust the order assumptions underlying the Taylor expansions for the transfer and proposed an alternative with imposition of the FSC on the double-body wave surface. Eggers [6,121 interpreted the change of sign of the coefficient near the bow as a change of character of the mathematical problem and tried to explain this in physical terms. But the vanishing coefficient is an artefact of the Taylor expansion used for transferring the boundary condition; and as shown here, this expansion tends to diverge if the coefficient vanishes. Therefore I believe that no physical interpretation may be given to the change of sign, and that it simply imposes a bound on the hull fullness for which this free surface condition is applicable (and perhaps even theoretically preferable). 391

N I ~ 1F 1 a3 a_ l ~ ~ I: t2~ 128: \; Fig. 20 Vertical velocities, at undisturbed free surface, with transfer term and evaluated at actual free surface; Tanker model, En = .1765 V* (y= 0) U* ------ V (including transfer) _____- U _ VeX (y = A) ___ u 6. PROSPECTS FOR SOLUTION OF THE NONLINEAR PROBLEM To my knowledge only very little informa- tion of this kind on the adequacy of linear- ized free surface conditions has been published up to now. Although the consistency of the FSC of Eggers, as opposed to that of Dawson, is well-known, it seems as if nobody had tried yet what results the modification has. The first results published here seem already to cast an entirely new light upon this issue. This suggests that there is still much more work to do on linearized methods. On the other hand, all the studies reported in this paper at best add to our insight but give no indication on how current methods can be improved: the widely used FSC of Dawson has turned out to be not markedly superior for most ships, but to be a fairly 392 Dawson's FSC ~ l' ID I_ to -1- 1 1 0 16 I i' - 1 ll ll ll ll ll ll 4 96 1 12 128 Fig. 21 Horizontal velocities, at undisturbed free surface, with transfer term and evaluated at actual free surface; Tanker model, Fn = .1765 (Y = 0) (including transfer) (Y= If) reliable choice throughout the range of practical hull forms. Even the occurrence of negative wave resistances could not be cured within the framework of linearized methods. From the practical point of view, therefore, this study contributes nothing at all. Although one may thus disagree on the question whether or not the linearized methods are well enough established to make the step towards a nonlinear method, it is true that computationally current methods require only little effort, and the present computer capacity allows commercial use of far more complicated programs. Moreover, a nonlinear method could well resolve some of the problems met with current methods. The negative re- sistance predictions dealt with in Section 3 will be eliminated by solving the exact problem with sufficient numerical accuracy.

Additionally, the magnitude of the nonlinear terms9 even in cases where the predictions are generally realistic asks for a method that includes these terms just for reasons of reliability of the predictions. Furthermore, certain physical effects, known to be important and sometimes forming the main difference between variations of a design, are entirely eliminated by the lin- earization. Examples of this are: - Slightly or partly submerged bulbous bows; the phenomena occurring in the vicinity of the bow are strongly nonlinear and cannot well be represented by a linear approach. In addition, the increased submergence due to the sinkage and the bow wave elevation sub- stantially alters the effect of the bulb, a phenomenon poorly represented in a linear calculation. - The flow off a very flat stern, as often found on ferry hulls. In this case the waterline shape changes considerably due to the wave elevation; the experience is that linearized methods overpredict the stern wave height. - The gradual transition from the flow off a stern with flat sections to the flow off a transom stern; in a linearized approach both regimes are reasonably well representable, but one has to choose beforehand which one is more appropriate. This is not a very satisfactory approach. Hence there are several practical incentives to develop a nonlinear method even though there may still be more work to be done on linearized methods. Such a development has therefore recently been undertaken at MARIN. Abandoning the linearizations adds several complications to the problem. The following aspects should in principle be taken into account: - The free surface condition should be applied at the actual free surface; this is probably the dominant effect, as concluded from Sec- tion 4. - The hull boundary condition must be applied on the actual wetted surface, instead of on the surface below the undisturbed waterline. - All nonlinear terms (squares and cross products of disturbances) should be in- cluded. - The dynamic trim and sinkage of the hull should be taken into account. Without restricting oneself from the outset to a direct extension of current linearized methods one has a wide variety of methods to choose from. In the first place, the nonlinear steady problem could be solved either by iteration or by a transient ap- proach. Both methods have already been pro- posed. The iterative approach may have a greater efficiency if successful, but meets problems in the convergence. Secondly, in each time step or iteration the Laplace equation for the velocity potential has to be solved (provided that the potential flow model is retained). Here again several alternatives are possible. Panel or boundary integral methods appear to be the most popular choice. Whatever the choices made, it is clear that any advance compared with linearized methods is only possible if the utmost care is taken in the numerical solution. The fact that the nonlinear terms generally have higher wave number already illustrates this necessity. But also the behaviour near the hull/free surface intersection could be far more difficult to deal with than in a linearized method. All these aspects deserve separate studies. In any case most of the nonlinear solutions published up to now are, in my opinion, not more accu- rate than linear solutions of state-of-the-art numerical accuracy. Even if this can all be solved satis- factorily, the theory remains limited to potential flows, without any viscous or wave- breaking effects. This could prohibit conver- gence of the solution in the limit of zero discretization spacing near the waterline, and additional methods to deal with this region might become necessary. Nevertheless I believe that the develop- ment of a method to solve the problem of potential flow with nonlinear free surface boundary conditions is the best next step for further enhancing the role of Computational Fluid Dynamics in the optimization of the wave-making characteristics of ships. 7. CONCLUSIONS This paper has provided more detailed information on the adequacy of linearized free surface conditions for the wave resistance problem. All of them were implemented in a Rankine-source method of the type proposed by Damson. In particular the Neumann-Kelvin formulation and two free surface conditions from the slow-ship theory, that of Dawson and of Eggers, have been compared. These compari- sons concerned the wave resistance and wave profile predictions, the magnitude of the terms neglected in the FSC, and the remaining errors in the dynamic and kinematic FSC at the predicted free surface. The main conclusions are summarized below. 1. For practical discretizations Dawson's FSC gives results not significantly different from solutions of the Neumann-Kelvin prob- lem, for all ships with a block coefficient not exceeding about 0.60 or 0.70. Also from the magnitude of the terms neglected in the linearization no significant advantage for Dawson's condition is found. This is at variance with the general preference for the slow-ship approach. 393

2. For full hull forms the Kelvin FSC predicts REFERENCES a resistance far exceeding the experimental value. On the other hand, both slow-ship FSC's predict, paradoxically, a negative wave resistance while the predicted wave pattern is physically plausible. The magni tude of the neglected terms is much larger for the Kelvin FSC than for Dawson's FSC here. 3. 4. The use of a linearized free surface condi- tion imposed on the undisturbed free sur- face changes the energy balance in such a way that a negative resistance is not ruled out. Wave energy can locally be supplied through the free surface which has no coun- terpart in a wave resistance acting on the hull. If the linearized free surface condition is consistently formulated, the possible nega- tive contribution from the free surface energy flux is reduced to a higher order in the perturbation parameter than the wave resistance itself. This is true for the Kelvin condition and for that of Eggers, but not for Dawson's FSC due to the absence of transfer terms. S. The FSC of Eggers yields a resistance con- sistently lower than Dawson's, for the Series 60 model. For the tanker model how- ever, an even more strongly negative resis- tance and an erroneous flow field were ob- tained. The cause of this was found to be the near-vanishing of the coefficient of fxx in the FSC near the bow; thus an exc- essive value of ~ is not controlled by the FSC. As a result the Taylor expansion underlying the linearization locally does not converge, and the reduction of the energy flux to higher order does not pre- vent a negative resistance here. 6. The transfer term neglected in Dawson's FSC and included in that of Eggers was found to 9. be of substantial magnitude even for the Series 60 hull. 7. The magnitude of the neglected terms in the FSC and of the errors in the exact FSC at the predicted free surface has turned out to be quite significant even if fair re- sistance predictions are obtained. E.g. for the Series 60 model at Fn=0.35, the error in the vertical velocity using Dawson's FSC amounted to 307. 8. Although more work would be needed to make the foundations of linearized methods sounder, part of the problems and uncer- tainties might be eliminated by solving the exact nonlinear problem. In this respect the present study has shown the importance of imposing the FSC at the actual free surface and the higher resolution required for accurately incorporating nonlinear effects. 394 1. Baba, E. and Hara, M., "Numerical Evalua- tion of a Wave-Resistance Theory for Slow Ships " Proceedings of the Second Int. Conf. on Numerical Ship Hydrodynamics, Berkeley, 1977, pp. 17-29. 2. Dawson, C.W., "A Practical Computer Method for Solving Ship-Wave Problems," Proceed- ings of the Second Int. Conf. on Numerical Ship Hydrodynamics, Berkeley, 197i, pp. 30-38. 3. Raven, H.C., "Variations on a Theme by Dawson", Proceedings of the 17th Symposium - on Naval Hydrodynamics, The Hague, Nether- lands, 1988, pp. 151-172. Proceedings of the Workshop on Ship Wave Computations, DTNSRDC, Bethesda, Md., USA, 1979. 5. Wehausen, J.V., and Laitone, E.V., "Sur- face Waves," Encyclopaedia of Physics, Vol.IX, Springer Verlag, 1960, pp. 446- 778. 6. Eggers, K., "A Method for Assessing Numerical Solutions to a Neumann-Kelvin Problem," Proceedings of the Workshop on Ship Wave-Resistance Computations, Sup plementary Papers, DTNSRDC, Bethesda, Md., USA, 1979, pp. 526-527. 7. Maisonneuve, J.J., "Resolution du Probleme de la Resistance de Vagues des Navires par une Methode de Singularites de Rankine," Thesis, ENSM, Nantes 1989. 8. Eggers, K., "On the Dispersion Relation and Exponential Variation of Wave Compo- nents Satisfying the Slow-Ship Differen- tial Equation on the Undisturbed Free Surface," Research Report 1979, Study on Local Nonlinear Effect in Ship Waves, pp. 43-62. See also: Schiffstechnik Bd. 28, 1981, pp. 223-252. Maruo, H., and Ogiwara, S., "A Method of Computation for Steady Ship Waves with Nonlinear Free Surface Conditions," Pro- ceedings of the 4th Int. Conf. On Numeri- cal Ship Hydrodynamics, Washington D.C., 1985, pp. 218-230. 10. Musker, A.J., "A Panel Method for Predicting Ship Wave Resistance," Pro- ceedings of the 17th Symposium on Naval Hydrodynamics, The Hague, Netherlands, 1988, pp. 143-150. 11. Brandsma, F., and Hermans, A.J., "A Quasi- Linear Free Surface Condition in Slow-Ship Theory," Schiffstechnik, Bd. 32, 1985, pp. 25-41. 12. Eggers, K., "A Comment on Free Surface Conditions for Slow Ship Theory and Ray Tracing," Schiffstechnik, Bd. 32, 1985, pp. 42-47.

DISCUSSION Paul D. Sclavanous Massachusetts Institute of Technology, USA I would like to congratulate Dr. Raven for yet another thorough study on the effect of the free-surface linearization upon the evaluation of the ship wave resistance. Having read the article, I would like to offer a conjecture on why the wave resistance from pressure might be negative and invite the author to discuss it. The potential flow near the ship bow and stern, subject to either the Neun~ann-Kelvin or a double-body condition, most likely develops a singular behavior associated with the finite entry angle of the waterline or from the singularity of gradients of the double body flow at its stagnation point. This singular behavior may be sufficiently strong that a localized contribution to the resistance may arise from the waterline due to a local singularity of the hydrodynamic pressure. This contribution would be directly analogous to the leading-edge suction force in linearized hydrofoil theory, which cannot be captured by direct pressure integration over the mean chord position. Should such a singularity exist and be of sufficient strength, it will contribute to an Of 1 ) component to the resistance arising from the waterline, which cannot be accounted for by integrating the pressure over the mean position of the ship hull. This effect, if it exists, may shed more light into the occurrence of negative wave resistance reported by the author. ~ . AUTHORS' REPLY This is a good point; the possible occurrence of singularities at the stagnation points should indeed be a matter of concern. However, in my opinion, the comparison with a leading-edge singularity is not entirely valid. At a zero-thickness leading-edge, a finite force contribution arises by the pressure going to minus infinity, while in the present case a positive resistance contribution could only arise from a positive pressure which, however, is bounded by the stagnation pressure. The latter seems to be numerically fairly well- resolved here. Furthermore, it seems reasonable to assume any force contribution from the waterline to have a vertical extent scaling with the wave length or stagnation height. It can then be expected to be similar to the waterline integral (10), which turns out not to eliminate the negative resistance in all cases. Hence, a localized force contribution is expected to be at least of O(Fn4), and to be already approximately included in the present results. A more probable effect of the singularity seems to me the occurrence of a localized energy flux through the free-surface. This, too, may contribute to the wave generation without being found in the pressure integration. Recently, I have calculated this energy flux for the tanker model according to Eq. (14). Its distribution has a large spike quite close to the bow, strongly suggesting the presence of a singularity. For the FSC of Eggers, the spike occurs at the point where the coefficient of ¢,,` vanishes; for the other FSCs it is found slightly further aft. This spike almost entirely determines the total energy influx, which, if expressed as a contribution to the wave resistance coefficient, is of the same order of magnitude as the pressure integral but may be severely grid-dependent. The localized energy flux through the free-surface due to this singularity thus largely explains both the large differences between the three FSCs and the large negative resistance values even if the energy flux is formally of higher order. These results therefore support the explanation given in my paper but stress the possible role of singularities at the bow in this respect. DISCUSSION J. Nicholas Newman Massachusetts Institute of Technology, USA One thought is prompted by the case shown in Figure 4 where, for the lowest Froude number, the Kelvin free-surface condition overestimates the wave resistance whereas the Dawson condition yields a negative value. Obviously the average of these two would be an improvement! This suggestion is not entirely facetious. It is common in perturbation solutions to find higher-order approximations oscillating about the correct result and diverging to an increasing extent. An example is the high-aspect-ratio lifting surface (lifting- line) theory as described by Van Dyke and reproduced in Figure 5.22 of my book. If this analogy has any relevance, it implies that we should look for a different asymptotic approximation about the zero- Proude-number limit and construct a composite approximation in the manner described by Van Dyke. I admire the spirit of this paper and look forward to further contributions from the author. AUTHORS' REPLY Thank you for pointing out this interesting analogy. Although the Kelvin condition and the slow-ship FSC are based on different perturbation parameters, one could loosely regard slow-ship theory as an approximation to higher order in flow nonuniformity (so, in some slenderness parameter). Some of my results in fact suggest that for increasing nonuniformity it diverges (or rather, it produces unrealistic results) more quickly than the Neumann-Kelvin approach. In the case of lifting line theory, the similar behavior indicates that systematic expansions to higher order will not bring us any further, and a different basic approach is needed to get a higher accuracy. It appears to me that a strict analogy would imply here that we should construct a different approximation about the limit for zero flow nonuniformity rather than for zero Froude number. Alternatively, the present behavior might indeed suggest the need to revise the zero- Froude-number limit. Some possibilities for this have been proposed in the past but seem not to have been pursued. 395