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Stability and Accuracy of a Non-Linear Mode! for the Wave Resistance Problem A. J. Musker Admiralty Research Establishment Haslar, England Abstract A non-linear Rankine-source method for predicting the wave resistance of a surface ship in calm water has recently been published by the author. In the present paper, an investigation into the stability and accuracy of the method is presented. The method is shown to be quite insensitive to changes in the various parameters which are necessarily associated with the numerical procedure but which do not feature in the mathematical formulation of the problem. A stable parameter regime is identified which has been found to give excellent agreement with towing tank data for the Series 60 hull. 1. Introduction During the last decade there has been a great deal of activity in the field of numerical ship hydrodynamics. New methods for simulating a wide range of phenomena have been proposed; such methods invariably involve the generation of a grid covering either the boundaries of the domain (as in a panel method) or the complete interior of the domain (as in an Euler or a Navier-Stokes code). Although it is usual to compare any numerical predictions with available experiment data (or, where appropriate, analytic data) it is somewhat rare to find examples of the effect of systematic variations in grid geometry on the accuracy of the solution. Equally, there may be factors other than the grid geometry which affect the outcome of a simulation and these too are often overlooked. In addition to the accuracy of a method, its stability also needs to be assessed. This may be achieved with the aid of theoretical treatment but ultimately the degree of stability of a method can usually only be gauged by its actual behaviour in trial calculations. This paper addresses both the accuracy and stability aspects of a panel method due to 629 Musker [1] designed to calculate the potential flow past a ship hull in steady translation in calm water. The method employs non-linear forms of the free-surface boundary conditions and on the basis of some initial tentative calculations it has been found to give good agreement with experiment data for both the Wigley hull and the Series 60 hull (see Musker [2]). Although the above problem can be formulated in terms of an integral of a source density distributed on the hull and the free surface, in practice the domain boundaries are made discrete by representing them using a finite number of panels. In addition, the free surface is not known a priori because it is part of the solution. Consequently panels must be associated either with the calm free-surface (with the actual free-surface represented by means of a Maclaurin expansion of the prevailing boundary conditions) or with the approximate free-surface relating to the most recent iterate in a convergent sequence of surfaces. The present method employs the first of these two options and clearly falls into the category of so-called Rankine source methods. Rankine source methods have been the subject of intense development in recent years in connection with a particular formulation of the Neumann-Kelvin problem (see, for example, Raven [3]). The Neumann-Kelvin (NK) problem models the wave-making of a hull by a solution of Laplace's equation subject to a linearised form of the free-surface condition and an exact form of the kinematic condition on the hull surface. Despite the much disputed problems of uniqueness of solution for the NK problem a large number of computer codes have been prepared based on this theory. The beauty of the Rankine source method lies in its inherent ability to provide engineering solutions to the NK problem (usually in a modified form using double-body linearization) whilst at the same time providing the potential for addressing the

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non-linear problem. On the other hand, the more conventional approach to solving the NK problem, using a Havelock or Kelvin wave-making source distribution, has the important advantage of requiring fewer panels which automatically satisfy the important radiation condition that waves should travel downstream. Staying with the linear problem for the moment, it is interesting to note the great diversity of solutions quoted in the literature. Figure l shows a collection of computer predictions of wave resistance for the Series 60 hull (CB = 0.6, model fixed) drawn from the two Workshops on Ship Wave-Resistance Computations held at the David Taylor Research Center (see references [4] and [5]). All these predictions use linearised theory for treating the free surface but differ in their method of solution (NK using Rankine source technique, NK using Havelock source technique, thin ship theory - in which the hull boundary condition is applied at the centre-plane etc). It is interesting to note that a mean curve drawn through the data would be significantly above the tank results. More recently, Chen and Noblesse [6] have remarked on the considerable scatter evident in published predictions of wave resistance based on similar theory. As noted by Bat [4] if no algebraic or computer truncation errors are incurred then results of numerical computations based on the same mathematical formulation should be the same. Clearly this is not the case. If the possibility of algebraic or coding error is dismissed (itself a dangerous assumption) then the explanation for the disparity in the predictions most likely lies in the mechanism whereby truncation errors appear. The obvious source of such errors is the manner in which the hull is divided into facets or panels. The distribution of such panels around the hull, particularly near the bow, stern and waterline is bound to have an effect on the solution. In the case of a Havelock source the numerical evaluation of the Green function poses its own problems. If a Rankine source rather than a Havelock source is used then perhaps an even greater problem presents itself: how large an expanse of the free surface in the vicinity of the hull needs to be modelled using additional panels and how should the resolution of this region be chosen? The manner in which the source density is distributed algebraically on each panel will also affect the solution. Recent work by Ni [7] has shown that the use of higher order curved panels with linearly varying source density is computationally more efficient compared with constant density panels. Finally, Rankine source methods invariably employ Dawson's approach [8] to satisfy the radiation condition that waves should not travel upstream of the hull. This approach involves the use of an upwind finite difference advection scheme for the wave disturbance and can take many forms. Clearly the operator of any Rankine source code has a great deal of freedom in the manner in which a particular problem is posed. The purpose of the present paper is to investigate the effect of making certain decisions about the geometry of the mesh and the type of advection scheme used for the particular case of the method described in Reference [l]. This method is first outlined for completeness and the 'degrees of freedom' at the operator's disposal are identified and minimised to manageable proportions. These are then systematically adjusted and the effect on the accuracy of prediction and the stability of solution is recorded. All the calculations were performed for the Series 60 hull (CB = 0.6). 2.- }~E~al~L~LE A Cartesian coordinate system is used in which the xy plane is in the calm free surface with the x axis positive upstream and the z axis positive downwards. The fluid is assumed to be inviscid, incompressible, irrotational and infinitely deep. A scalar velocity potential is defined such that -grad. is the local fluid velocity vector. This potential satisfies Laplace's equation everywhere within the fluid subject to the following boundary conditions. (grad ¢) · n' = 0 on the hull surface, [aZ]O ([aZ2]0 dl [dl]o + ~ 4: [[82¢ 1 dx + [82¢ 1 dv1 (2) dl ltaxBz]O dl [Byaz]O dlJ and 630 r ~ r ~ laxio laxazio ' [ ] [ ''-] + [~] [a2~--] 1 ay O ayaz O az O aZ2 ol = 2 [U~ - [3xjO [8YjO i8Z]o] (3)

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on the calm free-surface. In the above equations, n' (nx, ny, nz) represents the outward normal vector on the hull surface, 1 represents a distance measured along a double-body streamline measured positive in the locally upstream direction, Up is the ship speed, g is the acceleration due to gravity and ~ represents the wave elevation. The last two boundary conditions are non-linear and result from Maclaurin expansions of the exact boundary conditions prevailing on the free surface [1]. For the purpose of comparison, the above set of boundary conditions reduces to the Neumann-Kelvin problem if the spatial velocity gradients are set to zero. The ship speed is defined such that grad ~ = (- U-, 0,0) (4) at ~ except where waves are present. Dawson's approach [8] is used to ensure that waves are adverted downstream. The velocity potential is prescribed by means of a source density distribution associated with the hull (and its optical image in the z = 0 plane) and the free surface. Constant, planar, source density panels are used throughout the analysis and in the case of the free surface panels these are raised very slightly above the calm water-plane to improve the modelling of the spatial velocity gradients [9], [10]. The latter are calculated analytically and great care has been taken to avoid singular regions (in the numerical sense) by employing suitable asymptotic expressions within the program. To avoid leakage, an additional system of vertical panels (referred to loosely as 'deck panels') Joins the water-line on the hull to the neighbouring system of free-surface panels; a Neumann condition is imposed at the centroids of these panels. Successive application of the boundary conditions at a large number of control points on the hull and calm water-plane leads to a system of non-linear simultaneous algebraic equations for the unknown source densities and wave elevations. The associated Jacobian matrix is calculated analytically and the equations are solved iteratively using Newton's method without relaxation; each associated linear system is solved directly using a Crout factrisation. The criterion adopted for complete convergence is that the root mean square of the sum of the residuals (comprising the most recent corrections to the source densities and wave elevations) is less than 0.002. Although strictly dimensionally inconsistent, in practice this criterion relates to a correction in wave elevation of 1 mm for a hull of length 100 m. Further details of the non-linear algorithm used can be found in Reference [1~. Having solved the equations, the pressure at a particular control point on the hull is found from: + 1 (U2 - Grad . l 2 ) The wave resistance is found by integrating the pressure around the hull taking due account of the hydrostatic correction associated with the predicted water-line (note that for consistency with the applied boundary conditions this correction is not performed for the linear case): ITS where S is the wetted area. As is often found in panel methods, a residual resistance is observed at zero speed because of the manner in which the hull is made discrete. Accordingly, the result for the double body calculation (where no waves are present) is first subtracted to yield the final wave resistance. The free surface grid uses double-body streamlines generated by Runge-Kutta integration to within a lateral error of 10-5 of the ship length. These streamlines are separated laterally by equal intervals at the upstream edge of the calm free-surface and for each streamline the control points are positioned at equal intervals in arc-length; this process is facilitated using cubic splines. The free surface panels are constructed automatically such that their stream-wise edge projections onto the z = 0 plane follow the streamlines and have edges along the water-line which are equal in length. An additional constraint is imposed in that the panel aspect ratio away from the hull is typically unity. A four-point upwind finite difference scheme is used to advect disturbances in the downstream direction. The boundary condition imposed on the first three control points at the upstream end of a given free-surface streamline is different from equations 2 and 3. Instead, the vertical fluid velocity in this region is forced to be zero. In addition, the last three control points at the downstream end are treated differently. Here, the coefficients in the advection scheme are reduced gradually to 25 per cent of their normal value to dampen the waves at the edge of the domain. 631

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It should be noted that the method does not employ any clustering of panels in the vicinity of the bow and stern - in the author's opinion such clustering provides too many extra degrees of freedom to allow a sensible study to be undertaken. In addition, the highly variable step-length for the finite difference advection operator in such a scheme will almost certainly affect the stability of the solution algorithm. The hull panels are arranged to mesh exactly with the free surface nodes along the water-line. Below the water-line, panels are constructed automatically in one of two ways. Firstly, they can be constructed such that an equal number of quadrilateral panels is used at each longitudinal station ('single-patch method' - see [1]). Secondly, a mixture of quadrilateral and triangular panels is used to allow the number of panels at a particular section to be reduced in the ratio of local section depth to keel depth. This is done in such a way as to maintain a reasonably constant panel area and aspect ratio (which are thought intuitively to be important from the point of view of numerical stability) whilst at the same time lifting the restriction imposed by the single-patch method that the hull panels below the water-line must be generated on the basis of strictly vertical stations. This is known as the 'multi-patch' method and was developed to extend the capability of the panel code to hulls with more complicated geometry - particularly in the bow and stern regions. The BLINKS computer-aided design system [11] is used for interpolation purposes. 3. Stability and Accuracy Study The following list defines the most important factors which need to be addressed before a calculation can be performed for a particular hull geometry: a. Length of domain. b. Half-width of domain (ie excluding image in xz plane). c. Distance between the bow and the leading edge of the domain. d. Free-surface grid resolution. e. Hull grid resolution. f. Free-surface panel elevation (above z = 0 plane). g. Type of advection scheme. Experience with the code suggests that there are significant differences in predicted resistance between large and small domains. Not surprisingly, however, at any particular Froude number and for a fixed grid resolution, the resistance approaches an asymptote as the size of the domain increases. Accordingly, in order to decrease the number of degrees of freedom for the present study, a large domain was used throughout the investigation. This was to 3 ship lengths in the longitudinal direction and extended to 1.5 ship lengths transversely measured from the centre-plane. The distance between the bow and the leading edge of the domain was set to one half of the ship-length; this was based on previous experience with the method. The hull and free-surface grids are related in the sense that the water-line nodes are forced to coincide prior to the construction of the vertical deck panels. In addition, an optional facility exists to position extra hull panels between water-line stations defined by the free-surface nodes so that the hull panel density (defined as the number of stream-wise hull panels per free-surface panel) can assume any integer value. If this option is invoked then the grid generator positions these extra panels such that their stream-wise edges form equal divisions along the water-line. In this way the grid generating code can automatically calculate the mesh on the hull once the hull panel density and free-surface resolution have been set since only an integer number of panels is allowed along the water-line (in practice the latter condition is met within the computer program by a very minor adjustment to the absolute length of the domain). It can be seen, therefore, that there are now only four degrees of freedom which need to be considered: the grid resolution, the hull panel density, the free-surface panel elevation and the type of advection scheme. It should be noted that no other operator intervention is called for once these factors have been set. 4. Numerical Experiment The Series 60 (CB = 0.6) hull was used for all the computer runs described in this investigation. The hull was fixed and no allowance was made for sinkage or trim (in the author's opinion this in no way detracts from the usefulness of the study). Unless otherwise stated, the following default conditions were used for all the runs: hull panel density = 1 and 632 panel elevation = 15 per cent of average panel diagonal

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The latter figure was based on previous experience with the code and careful analysis of the ability of constant source density panels to model spatial gradients of fluid velocity [10]. Four numerical experiments were performed with the aim of determining the effect of varying any one of the above-mentioned degrees of freedom. The runs were performed at the following Froude numbers: 0.239, 0.271, 0.287, 0.303, 0.319, 0.335 and 0.351 based on ship-length at the water-line. arithmetic was performed using 64 bit precision. The experiments are labelled A, B. C and D and are described below. Experiment A (effect of free-surface grid resolution) All This was performed in two stages. Firstly (experiment Al), three free surface grids were generated each of which was fixed, in relation to the ship-length, for all Froude numbers (henceforth such grids will be termed 'mode 1' grids). They are referred to as coarse (1166 free-surface panels), medium (1904 free-surface panels) and fine (2673 free-surface panels). The number of free-surface panels distributed along the water-line of the hull was 17, 23 and 27 respectively. Single-patch hull meshes were used for this experiment. Secondly (experiment A2), grids were generated such that their resolution related not to the ship-length but rather to the wave-length of the transverse waves expected to be shed by the hull. In this instance the grid resolution was described by a parameter, n, where n is the number of panels per transverse wave-length. The total number of panels positioned along the length of the free-surface domain is thus: n x domain-length 2~Fr2 x ship-length This meant that for each Froude number a new grid (henceforth termed a 'mode 2' grid) had to be calculated to conform to the chosen value of n. This exercise was carried out for values of n ranging from 4 to 12 in intervals of 2 (it was not anticipated that a value of n as low as 4 would perform very well but for the sake of completeness it was included in the experiment). Multi-patch hull meshes were used for this experiment. Experiment ~ (effect of hull panel density) This experiment compared the performance of the method using two different values of the hull panel density; the values used were 1 and 3. Mode 2, multi-patch grids were employed throughout, with n set to 10. Experiment C (effect of panel elevation) This experiment was designed to investigate the effect of elevating the free-surface panels above the calm water-plane. As for Experiment B. mode 2, multi-patch grids were employed throughout, with n set to 10. Three different elevations were tested: zero, 15 and 30 per cent of the average panel diagonal. Experiment D (effect of choice of advection scheme) The above three experiments were performed with two different upwind advection schemes for the stream-wise wave-slope: Taylor Scheme: Slopei = - ~` x [1.667 OCR for page 629
Experiment A A comparison of the coarse and fine grids is shown in Figure 2 and a summary of the results is presented in Figure 3. The method converged for all speeds using the coarse grid. For the medium grid the method failed at a Froude number of 0.335 whilst for the fine grid the method failed for the three highest Proude numbers within the Range used. It should be recalled here that Experiment Al was designed around the original method described in Reference [1] employing mode 1 free-surface grids and single-patch hull grids. It is perhaps surprising that the coarse grid should have led to such good agreement with experiment data particularly at the lower Froude numbers where shorter wave-lengths are prevalent. It is similarly curious to observe the closer agreement between the coarse and fine meshes at the lower Froude numbers compared with the medium mesh results. The results for experiment A2 (mode 2, multi-patch hulls) are summarised in Figure 4. It can be seen that all the runs converged successfully for values of n from 4 to 10. Par n = 12, however, the method diverged for Froude numbers of 0.319 and 0.335 (note that the lowest Froude number case was not processed because it exceeded the capacity of the computer - this was the only run in the whole investigation which could not be processed because of size limitations). Not surprisingly, values of n = 4 or 6 appear to be quite inadequate to resolve the flow-field and hence the wave resistance. The data for n = 8 define the shape of the curve well but slightly under-estimate the resistance for the higher Froude numbers. Excellent agreement is obtained for n = 10 and no improvement in accuracy is achieved for the stable runs when n = 12. Experiment B In this experiment mode 2 free-surface grids and multi-patch hull grids were used with a hull panel density of 3 (see Section 3). Figure 5 shows a perspective view of one of the meshes from a point below the water-line looking towards the hull. The effect of the increased resolution of the hull can be seen in Figure 6 to be quite weak. At the higher Froude numbers the increased hull panel density consistently lowers the resistance by a very small amount whilst little or no effect is observed at the lower Froude numbers. All the runs performed were completely stable. Experiment C Figure 7 illustrates the effect of elevating the free-surface panels above the calm water-plane. The absence of data points for the case of zero panel elevation indicates the instability inherent in introducing spatial derivatives of fluid velocity calculated using panels of constant source density if the control point lies in the plane of the panel [10]. With the possible exception of the two runs at a Froude number of 0.271 it can be seen that there is little difference between the results for 15 per cent and 30 per cent panel elevations. All the runs performed with elevated free-surface panels were stable. Experiment D Attention here will be focussed on three results of significance. In the first two cases the data refer to mode 2, multi-patch grids with n = 10. Figure 8 shows the very small effect the spline advection scheme has on the accuracy of the predicted wave resistance compared with the Taylor scheme; the results are very nearly identical. Figure 9 shows the data for Experiment ~ with zero panel elevation replotted using the results for the spline advection scheme. The results are compared with the single Taylor result from Figure 7. Two important conclusions emerge from these data. Firstly, the effect on the accuracy of using a spline scheme is minimal. Secondly, the results for zero panel elevation are not only inherently unstable using the Taylor scheme, as observed above, but also inherently inaccurate when they can be made stable by employing a spline scheme. This again reinforces the findings of Reference t10]. Further evidence of the more stable nature of the spline scheme is offered in Figure 10, which shows the spline results for Experiment A2 with n = 12, compared with the Taylor results with n = 10. The results for Froude numbers 0.319 and 0.335 when n = 12 are now stable - thus allowing a comparison in accuracy between n = 10 and n = 12 to be made throughout the whole speed range. It can be seen that there is little advantage to be gained by increasing n above 10. The stability of panel methods for the linear problem has been analysed recently using Fourier techniques by Sclavounos and Nakos [133. They found that the numerical damping associated with upwind finite difference schemes decreases as the grid becomes finer or as the number of upstream nodes included in the scheme at a particular control point increases. Although the present technique is non-linear this nevertheless probably explains why the method has a tendency to diverge for the finest grid tested (n = 12) using the Taylor scheme. Their work may also explain the stability problems experienced by the only other non-linear methods which, to the best of the author's knowledge, have appeared in the 634

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literature since 1985 (Maroo and OgIwara [14], Xia [15], Ni [7] and Rong et al [16]). Marno and Ogiwara used a two-point formula but still experienced quite severe convergence problems, whilst Xia and Ni both used three-point formulas. Xia found that this method diverged for all cases tested if a 4-point scheme was adopted but much better convergence characteristics were observed when this was abandoned in favour of a (less accurate) 3-point scheme. Ni employed a sophisticated higher order panel scheme very successfully using a three-point scheme but again found stability a problem using an extra upstream node. In the present method, using mode 2 free-surface grids and multi-patch hull grids, stability only became a problem at the highest resolution (for the highest Proude number only) and this was completely alleviated by utilising a four-point spline rather than a four--point Taylor scheme. In the method of Rang et al a two-point scheme was used near the stern region to provide sufficient damping to ensure convergence. Mention should be made of the practice of elevating the free-surface panels above the calm water-plane. It might be argued that the ensuing system of equations will be less well conditioned in the sense that a control point on the z = 0 plane would not be so strongly associated with its corresponding panel if the panel were raised above it. On the other hand, the author has shown that the modelling of the spatial gradients of fluid velocity which are required in the non-linear formulation is inadequate if the control points are contained in the plane of the panels. Indeed, it is so inadequate as to cause the method to diverge. It should be noted that the amount by which the panels need to be raised is very small - typically between 0.8 per cent and 1.5 per cent of the ship length, depending on the Froude number, when n = 10. In addition, Figure 7 demonstrates that the predicted resistance appears not to be particularly sensitive to the panel elevation. Xia [15] also tried raising his free-surface panels and found improved convergence at some expense in accuracy. Panel elevations used by Xia, however, were much larger - typically between 2.5 per cent and 10 per cent of the ship-length - and in the author's opinion these values are too large. Ni [7] has circumvented the problem altogether by adopting higher order panels with linearly varying source density which follow the calculated free surface as it progresses towards its final converged solution. The present work suggests that such sophistication may not be necessary. Finally, a synopsis of the published data for the above non-linear methods is presented in Figure 11. The Series 60 data for the method of Marno and Ogiwara is taken from Reference [17]. The present results using the default conditions of 15 per cent panel elevation (based on average panel diagonal, not ship-length), four-point Taylor advection scheme, mode 2, multi-patch grids with n = 10 are included for comparison. 6. Conclusions .. . _ _ . .. A panel method based on a boundary integral approach to solving the non-linear free-surface wave-making problem has been thoroughly tested against tank data for the Series 60 hull. The number of degrees of freedom at the disposal of the operator of the computer code has been minimised with the aim of investigating systematic changes to the input parameters which may affect the numerical solution but which do not feature in the mathematical formulation. Two types of grids were tested. The first type, labelled mode 1, single-patch, uses panel geometries on the calm free surface which are independent of Froude number but which are kept constant with respect to the ship-length. The hull panels are organised so that the same number of panels is used at each station. The second type, labelled mode 2, multi-patch, uses panel geometries which depend on the Froude number in the sense that a fixed number, n, of free-surface panels per transverse wave is selected. In addition, the hull panels are organised into quadrilateral or triangular panels automatically such that the size and shape of panels do not change significantly from station to station. The following conclusions can be drawn: a. As expected, the method was unstable for zero free-surface panel elevation. The method was invariably stable, however, for non-dimensional panel elevations (based on average panel diagonal) of 15 per cent and 30 per cent. Furthermore, the solution was observed to be reasonably insensitive to the choice of non-zero panel elevation. b. Accurate results were obtained using a coarse, mode 1, single-patch grid for all Froude numbers. This approach became unstable as the grid became finer. c. The mode 2, multi-patch grids exhibited greater stability and were used for the remainder of the investigation. The results approached the tank data asymptotically as n increased from 4 to 12, with very little difference in the accuracy between 10 and 12. The n = 12 results diverged at 635

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two of the higher Froude numbers using 5. the four-point Taylor advection scheme. d. No case of instability or significant change in accuracy of prediction was recorded for any value of n using the four-point spline scheme. e. A threefold increase in hull panel density in relation to the free-surface grid resulted in no significant effect on the stability or accuracy of the method. f. Comparison with other non-linear methods described in the recent literature shows the method to be very stable and accurate using mode 2, multi-patch grids with n set to 10. Having identified a stable and accurate parameter regime, it is hoped to perform a similar investigation in the near future to determine the effect of decreasing the extent of the free surface surrounding the hull. Tests with different hull geometries are also planned. A^L-n~wl ",la.=m~r~t" This project would never have been completed without the skill and dedicated support of Mrs P R Loader of the Admiralty Research Establishment, Haslar. The author would like to thank her for her sustained effort and enthusiasm. Or G Michel of Aerobel Defence Technology (formerly Principia Mechanica) provided the interface software for BLINKS and Mrs C Patis attended to many of the computer runs. Their help is gratefully acknowledged. 8. References 11. 12. 1. Musker, A J. "A Panel Method for Predicting Ship Wave Resistance", 13 Proceedings of the 17th Symposium on Naval Hydrodynamics, The Hague, August 1988. 2. Musker, A J. "A Solution of the Non-Linear Wave Resistance Problem", Proceedings of the International Symposium on Ship Resistance and Powering Performance, Shanghai, April 1989. 3. Raven, H C, "Variations on a Theme by Dawson", Proceedings of the 17th Symposium on Naval Hydrodynamics, The Hague, August 1989. 4. Bai, K J and McCarthy, J H. "Proceedings of the Workshop on Ship Wave-Resistance Computations", DTNSRDC, November 1979. 636 Noblesse, F and McCarthy J H. "Proceedings of the Second DTNSRDC Workshop on Ship Wave-Resistance Computations", DTNSRDC, November 1983. 6. Chen, C Y and Noblesse, F. "Comparison between Theoretical Predictions of Wave Resistance and Experimental Data for the Wigley Hull", Journal of Ship Research, Vol 27, No 4, December 1983. 7. Ni, S Y. "Higher Order Panel Methods for Potential Flows with Linear or Non-Linear Free Surface Boundary Conditions", Chalmers University of Technology, 1987. 8. Dawson, C W. "A Practical Computer Method for Solving Ship-Wave Problems", Proceedings of the Second International Conference on Numerical Ship Hydrodynamics, Berkeley, 1977. 9. Musker, A J. "A Note on Free-Surface Flow Prediction", ARE Technical Memorandum TM(UHR)86306, March 1986. Musker, A J and Loader, P R. "A Modified Boundary Element Method for Predicting Ship Wave Resistance", Proceedings of the 7th International Conference on Finite Element Methods in Fluid Flow Problems, The University of Alabama, April 1989. Catley, D, Okan, M B and Whittle C, "Unique Mathematical Definition of a Hull Surface, its Manipulation and Interrogation", WENT, Paris, July 1984. Kim, Y H and Jenkins, D, "Trim and Sinkage Effects on Wave Resistance with Series 60, CB = 0.6", DTNSRDC Report SPD-1013-01, September 1981. Sclavounos, P D and Nakos, D E, "Stability Analysis of Panel Methods for Free-Surface Flows with Forward Speed", Proceedings of the 17th Symposium on Naval Hydrodynamics, The Hague, August 1989. 14. Marno, H and Ogiwara, S. "A Method of Computation for Steady Ship-Waves with Non-Linear Free Surface Conditions", Proceedings of the 4th International Conference on Numerical Ship Hydrodynamics, Washington DC, 1985. 15. Xia, F. "Numerical Calculations of Ship Flows with Special Emphasis on the Free Surface Potential Flow, Chalmers University of Technology, 1986.

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16. Rong, H. Liang, X and Wang, H. "A IOOO OCR for page 629
1000 x Cw 4 3 4 1,............ 1.8 2.2 Or-se Med~iu'~' Fire Exp-~-~e'~t Mode 1, Si'~gle-Patch ._ Panel Eleva-t-ior~ Taylor ~dvec-tion Scene X X X/ X W~ _ ~~ i B He 2 al 6 ~ ' ~ · · l · Fr Fig 3 E-F-Fect of Gr-id Resolution 1000X CW ire-= r~=8 ~n=lO On=12 _ Experdr~,ent Mode 2, Multi-Patch 15% Panel Elevation Taylor- Advection scheme X10 -1 t~ ¢ V + ~ Hi/ ~ Off , . . . . . / V Fr F1g ~ Effect of Grid Resolution 638 ~ , X10 -1 8

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Fig 5 Perspective Ark - w from Men - sty Hall (Panel D - nudity = 3) t000 ICE ~ Hull Panel Density = 1 ~ Hull Panel Density = 3 4 - Experiment Mode 2, Multi-Patch 1~% Panel Elevation Taylor Advection Scheme 2 . 1.8 /! xt Fig 6 Effect of Hull Panel Density 639 3.8

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\090 of ~ X - Panel Elevat-8on Panel Elevation Panel Elevation Experiment = 0 = 15= = 30% Mode 2, Multl-Patch Taylor Advection Scheme 1 ~ X ~ 81.1.18',8~' ]8~8 2#28 to 4 - 3 2 1 X ,. .~.,.~ ,8~ . 2.9 5.0 3.4 3.8 X10-t F r F8g 7 Effect of Free-Surface Panel Elevat8on 1000~8C ~ Taylor W X spine -Experiment Mode 2, Multi-Patch 15% Panel Elevat80n eX/ ~_r / , ~ , ~ ~ 3.0 8r F"8g 8 Effect of Advect-8on Scheme 640 X10 18

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MOW ~ Spilne Advection Scheme 4.00 0 Taylor Adveictlon Scheme 3.SO 3r 00 2eSO 2.00 \.50 \.00 O.S~ D.00.- \,80 Mode 2, Multi-Patch Zero Panel Elevation /d / ~ / / 2 I ~ ~ ~ r 2~;0 3~{JO F r Fig 9 Saline Calculation for Zero Panel Elevation t000 ~ C ~ Splir~le Schel~le In ~ ~ Taylo'~ Scl-lerrie In = 4 3 2 ~ to 3.40 3.8U X10 -1 = Mode 2, Multi-Pa-Uch 15~o Panel Elev~t~ion NO insufficient computer memory to lowest Denude number- when n = 1 2r my/ 3.0 3,4 F." -fig -10 E-F-Fec-t of Spl-ilne Michelle For In = 12 641 X10 -1 3.e

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1000 x Cw 4 3 2 1 Ogiwara 1987 t17] Via 1986 C15] Hi 1987 C7] ~ hong 1987 C16] · Present Method --Experiment Mode 2, Multi-Patch 15= Panel Elevation Taylor Advection Scheme o ~ a/ ,~ Q /` a _ 1.8 2.2 2.6 3.0 F r F1g 11 Comparison with Other Non-LinQar methods Copyright (C) Controller HMSO London 1989 3.4 3.8 X10-1 642