NUMERICAL WASH PREDICTION USING A FREE-SURFACE FANkeetild3

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1 Delft University of Technology Ship Hydromecimnics Lahoratori Library Mekelweg CD Delft NUMERICAL WASH PREDICTION USING A FREE-SURFACE FANkeetild3 1,1,one ax: ',7 Hoyte C. Raven Maritime Research Institute Netherlands (MARIN) SUMMARY Possibilities for numerically predicting wave wash are discussed. A distinction is made between the generation of the ship wave system, and its propagation to the far field. The former is computed using a steady potential flow method; some validations for cases relevant to wash are shown, and the applicability is discussed. For uniform water depth the computed wave pattern can be extended to the far field using the analytical decay rate or a spectral representation. For cases including wave transformation and refraction, a method is being developed in cooperation with WLIDelft Hydraulics based on coupling the calculation of the ship's wave making with a Boussinesq-type model for wave propagation. AUTHOR'S BIOGRAPHY Dr. H.C.Raven is Senior Researcher at MARIN and coordinates the research in the field of Resistance and Flow at that institute. He is specialised in Numerical Ship Hydrodynamics, with emphasis on ship wave making aspects, and in the past has developed the RAPID code. 1. INTRODUCTION In recent years there has been increasing attention for the effects of ship waves, or wave wash. Ship waves impinging on shores incidentally have been found to cause bank or bottom erosion, damage or nuisance to moored vessels or small craft, to endanger people bathing or walking along the coast, or to harm vulnerable natural environments. Often such effects occurred due to operation of fast ferry services; but even for conventional ship types, increasing inland waterway traffic or the use of larger units or higher propulsion power may cause concerns about wash effects. For the planning of fast ferry services, for the design of vessels, waterways and harbours, and to establish operational guidelines, it is important to be able to predict wash effects in an early stage. The present paper is concerned with such predictions. The occurrence of wash problems connected with fast ferry services was partly unexpected, since their wave pattern in deep water often is relatively small. However, problems caused by ship wash in practice appear to be often related with sailing at critical or supercritical speeds; with wave amplification due to propagation from deep to shallow water; with moored vessels responding more strongly to longer waves; therefore, related with water depth and wave length in addition to wave amplitude. Today it is recognised that it is the combination of characteristics of the vessel and the waterway that determines the occurrence of wash effects. Therefore, ideally a prediction of wash effects should be made for a certain ship design in a certain waterway. In its most general form, this would require a method to compute the flow around the vessel and the wave pattern in a waterway of arbitrary geometry. This entails some practical difficulties: The dimensions of the waterway are often large. The ship wave height may be requested at a distance of many ship lengths from the vessel. In a wave pattern prediction the computational domain thus becomes much larger than usual, which causes practical difficulties. Often, variations in the waterway width or depth cause changes in the far wave pattern that need to be taken into account and resolved. An example is the prediction of the increase of the wave amplitude due to decreasing water depth. If the depth, width or cross section of the waterway vary along the path of the ship, the flow field in principle is unsteady; the same is true for penetration of ship waves into a harbour. Other unsteadiness may arise due to variation of the ship's speed, which is common in areas where wash problems are imminent, such as estuaries. Raven Page

2 The general approach thus would require solving an unsteady problem in a three-dimensional domain of large dimensions, at the same time requiring a high resolution in order to incorporate the waves themselves and all local effects. In many cases this is unfeasible, and simplifications are applied. In order to solve problem (A), we make a distinction between predicting the wave generation by the ship (near field) and the resulting far-field wave pattern. The wave generation is predicted by usual means, in a domain of moderate extent (Section 2). The far field then is computed from propagation of the waves generated in the near field. If there are no effects of the bank outline or bottom topography upon the wave propagation, it is rather easy to extend the near field by analytical techniques (Section 3). Otherwise, the evolution of the ship waves can be predicted using a wave propagation model; e.g. a Boussinesq-type wave model that allows to include irregularly-shaped river banks, arbitrary bathymetries and wave breaking. Here, we use the Boussinesq-type model developed by WLIDelft Hydraulics and present some results of calculations done by that institute. When coupled with the wave generation prediction, this still leads to a computational problem of substantial size, but much more feasible than the original complete 3D unsteady problem (Section 4). Thus problem (B) can be solved. If the wave evolution in the far field is solved as an unsteady problem, most of problem (C) is solved as well. Still, however, in our approach the generation of waves by the vessel is supposed to be a steady (or quasi-steady) process unaffected by any unsteady events in the far field. This imposes some restrictions on the applicability of the method. The paper is set up as follows. Section 2 describes the prediction of ship wave generation, and discusses some examples for cases relevant to wash: a fast catamaran ferry, a low-wash waterbus in a channel, and a vessel at supercritical speed. Section 3 reviews some analytical techniques to extend the near-field wave pattern to the far field for constant-depth or deep-water cases. Section 4 describes the coupled method mentioned above, and shows some initial results. Main conclusions are formulated in Section PREDICTING THE WAVE MAKING 2.1. NONLINEAR FREE-SURFACE POTENTIAL-FLOW METHODS We first consider the prediction of the wave pattern generated by a ship sailing at constant speed and heading in a waterway of constant cross section. In a frame of reference moving with the ship this is a steady problem. While various simplified solution methods have been proposed in the past (e.g. thinship, flat-ship, slow-ship linearised methods, Neumann-Kelvin methods), today the state of numerical algorithms and computer speed permits to solve the full nonlinear inviscid flow problem with little effort, which is far more accurate and general. Some of such codes exist and are in regular use; a review can be found in [1]. The principal methods have some common features, and the description below applies to most of them. The flow model used is that of a steady 3D incompressible potential flow around the ship hull. The hull boundary condition requires there is no flow throueh the shin hull surface. Moreover, the shin must be in equilibrium with the hydrostatic and hydrodynamic pressure forces, i.e. squat is computed and incorporated. On the wave surface, which is initially unknown, a kinematic boundary condition (no flow through the wave surface) and a dynamic boundary condition (pressure equal to atmospheric pressure) are imposed. These are the exact inviscid free-surface boundary conditions, in principle including all nonlinear effects on wave generation and propagation. In case of shallow water, a condition of zero normal velocity must be satisfied on the bottom of the waterway as well, and the same applies to channel walls. Since the free-surface boundary conditions are nonlinear and are to be imposed on a wave surface that is unknown, the problem cannot be solved directly. An iterative procedure is applied, which starts with Raven Page

3 a flat water surface and a guessed trim and sinkage. In each iteration a particular form of the freesurface boundary condition is imposed on the current guess of the wave surface. To solve the intermediate problem in each iteration, a boundary integral or panel method is used. The ship hull surface is subdivided in a large number ( ) of small surface elements, which each bear an unknown 'source density', i.e. they emit or absorb fluid at an unknown rate. Similar 'source panels' ( ) are distributed over a plane at a small distance above the wave surface. The entire flow field is now defined to consist of the uniform inflow from ahead (equal to minus the ship speed) plus the sum of the induced velocity fields of all N source panels (which, at this stage, still contain the unknown source strengths as a proportionality factor). The hull and free-surface boundary conditions are equations for the velocity components, and can thus be expressed in terms of the unknown source strengths. They are imposed in a set of N discrete points ('collocation points), which are the centres of the hull panels, and specially arranged points on the actual guess of the wave surface. Thus we obtain a system of N equations in N unknowns. This can be solved for the new source strengths, and the new velocity and pressure field are easily found. Updates of the wave surface, the velocity distribution at the wave surface, and the trim and sinkage are thus obtained, which form the basis for the next iteration. The final solution that satisfies all nonlinear boundary conditions has been reached when no further change occurs, usually after 5 to 20 iterations THE RAPID CODE A well-known nonlinear free-surface potential flow code is RAPID (Raised-Panel Iterative Dawson), developed at MARIN in the period [2] and subsequently further extended and refined. Since 1994 it is in routine use in practical ship hull design at MARK and is licensed to a number of shipyards, institutes and universities. While it is based on the general approach outlined above, it has some distinguishing features: The location of free-surface collocation points, the panel elevation and the formulation of the difference schemes in the free-surface boundary condition are based on a theoretical analysis of numerical stability, dispersion and damping [1,2]. Careful modelling and detailed experimental validation of the flow off a transom stern has been carried out. An efficient iterative solver for the system of equations has been developed (preconditioned GMRES), making the code particularly fast. Tools have been developed to derive the wave resistance from a 'transverse cut' analysis of the calculated wave pattern [3]. The entire computation, for a case with e.g panels, may take about half an hour on a PC, or 2 minutes on a CRAY C90 vectorcomputer. Validations (e.g. [1,2,4]) have shown that the predicted flow and wave pattern generally are accurate. Being based on inviscid flow theory, the method disregards the effect of boundary layers, dead water zones behind a transom, flow separation, or wave breaking. Consequently, the principal shortcoming is the fact that the amplitude of the stem wave system is usually overestimated; very little for slender transom stem vessels, more for fuller hull forms EXAMPLES OF WAVE PATTERN PREDICTIONS FOR RELEVANT CASES Wash-related problems in practice pose some additional demands to wave pattern prediction codes; e.g. they feature: typical fast ferry hull forms, such as ships with large transom sterns, multihulls, SES, hydrofoils; high Froude numbers; effects of restricted waterway depth and/or width; supercritical/transcritical speed regimes; sometimes, accelerating or decelerating vessels. Below, some examples are given of what can and cannot be predicted today. Raven Page

4 2.2 (a). Catamaran wave making Since the demihulls of catamarans usually are very slender, thin-ship linearised methods would seem ideally suited for predicting the wave making However, additional approximations are then needed for transom flow modelling and incorporation of trim and sinkage, and predictions contain little detail. Instead, most nonlinear free-surface panel codes can deal with multihulls just as well [5,6] without any fundamental change to the methods. By adding a treatment of the additional free-surface segment between both demihulls, a method is obtained that incorporates the hull and wave interference effects, the dynamic trim and sinkage,, and the flow off the transom sterns without any empirical input or correction. To assess the accuracy of RAPID for such applications, wave pattern measurements have been carried. out in the MARIN towing tank for a large passenger catamaran in deep water. Longitudinal wave cuts were measured at different distances off the demihull centreplane. To identify the wave interference, corresponding cuts were measured for a single demihull Fig. 1 shows that the calculated and measured wave cuts at 36 Im (Fn = 0.53) are in close agreement, both for the single demihull and the catamaran, except for some shorter wave components further aft. The computations appear to represent most of the physics of the wave making and wave interference. catamaran(rapid), 36 knots. z =44.25'm danihull(rapid) catamaran(experiment),demihull(experimem),.. : j 1.00 /00 X / L :00 Figure 1. Computed and measured longitudinal wave cuts for demihull and catamaran, at 0.23 L off demihull centreline. 36 knots, deep water. As an illustration of how such computations may help to investigate and reduce wave making and wash problems, we noticed that the catamaran had a much larger trim by the stem and transom immersion than the demihull. This was caused by a wave trough of the bow wave system of one demihull reaching the other close to the stern. This suggested that much of the wave interference could be eliminated by restricting the dynamic trim; which was confirmed by additional computations. In general, predicting the near-field wave pattern of many high-speed craft, including its precise, dependence on hull form and main parameters, is quite possible today by standard methods. Such computations can provide support for important design decisions on main dimensions, demihull slenderness and separation, transom stern area, and the usefulness of devices to counteract the dynamic trim; all of which are relevant for wash generation as well. 2.2 (b). Transcritical and supercritical speeds Incorporating shallow water effects in a free-surface panel code is easily done by including a mirror image of the entire hull and free-surface panel distribution in the bottom of the waterway. The bottom 'thus becomes a symmetry plane of the solution,, which guarantees satisfaction of the boundary condition. However, for nonuniform water depth mirroring usually is impractical; in such cases we. add a panel distribution on the bottom and impose a boundary condition of zero normal velocity there Raven Page

5 Pronounced shallow-water effects occur at critical speed, at least if the water depth is small compared with ship length or draft. In agreement with observations, computations show a large increase of the resistance, sinkage and trim and the generation of large transverse waves unbounded by a Kelvin wedge. Fig. 2 shows an example, for a vessel running at 27 knots in 20 m of water (FnL=0.38, FriH=1.0). Figure 2. Computed wave pattern at critical speed. However, for strong critical-speed effects the calculation is difficult and the flow field displays a large sensitivity to all kinds of disturbances. There are reasons to view such solutions with suspicion: the sensitivity of the flow to e.g. trim and sinkage will lead to larger residual errors in the solution; if the propulsion is not taken into account, a substantial deviation might occur; at critical speed the lateral truncation of the free-surface domain may well introduce a much stronger effect than otherwise, due to the large transverse wave crests that are generated. The existence of a steady solution sometimes may be questioned. Soliton generation in channels or other unsteady phenomena are absent in the computations. For cases near critical speed we are not aware of much validation for steady free-surface potential flow methods, but we suspect that for small water depth, when critical-speed effects are dominant, the predictions are of qualitative value only. An example in [6] shows large deviations from experiments at critical speed. For ships in channels in the transcritical range, special methods like [7], which do predict the soliton generation, may be a better option. For supercritical conditions, however, chances are much better. A computation can be made without much difficulty. A greater width of the free-surface domain may be needed: in the near-supercritical range the leading waves in the pattern are at a much wider angle than the Kelvin angle, and the decay of the wave amplitude with distance is much smaller or absent, causing a possibly larger truncation effect. Validation of such supercritical flow predictions is desired, but reliable data are scarce. Ref. [6] shows a quite favourable comparison for a hull wave profile at Fnh = A qualitative comparison can be made with observations by Whittaker et al [9], who point out that the typical supercritical wave pattern of high-speed ferries starts with two continuous wave crests; the first angle some degrees larger. The separation between both crests thus increases with distance and may suggest extremely large wave lengths in the far field. Fig. 3 shows the computed pattern for a monohull at 40 knots, at Fnh = 1.1. The free-surface domain extends 7.5 L aft of the stern and 6 L laterally, and had 9164 panels per half. The results are in qualitative agreement with observations. The crests have the expected concave shape, the relatively weak first crest is at the correct angle of 25 degrees, the much higher second crest at around 40 degrees. The contour lines show that decay with distance is minimal, as it should for shallow-water waves. Reflections occur at the outer boundary and disturb the downstream part of the domain. Raven Page

6 eq. t-..1* - Figure 3 Computed wave pattern at supercritical speed, for fast monohull at 40, knots (FnL=0.55, FnH=1.1). 2.2 (01Bottom and channel effects Wave pattern predictions for a vessel in a channel formed an important part of the EU-project LIUTO [10], which concerned the development of a new low-wash waterbus for the city of Venice. The new vessel had to generate minimal waves at a speed of 5.94 kn in the Canal Grande in Venice. Part of the project was the extension of RAPID for ships in shallow channels, and a validation of the predictions. The channel is typically 45 m wide, and the water depth of 4.5 m in the central part decreases linearly to a depth of 1.0 m at the vertical banks, over a width of 15 in. Panels were put on the sloping side parts of the bottom and on the vertical banks, and mirroring in the horizontal part of the bottom was used. To represent the most critical situation for erosion, computations were made for the vessel out of the centre of the channel. The asymmetry of the flow, the rather low speed (Fn=0.196), the required resolution of the wave steepening near the banks and of the reflection at the channel walls all added up, to a large-size calculation with panels.. Fig. 4 shows the computed wave pattern for a speed of 8 knots. While propagating into the shallower area near the bank at the port side the wave pattern strongly deforms, and the wave amplitude increases to more than twice the amplitude it has in a channel with rectangular section. This illustrates how important the effect of wave propagation into shallow water can be for wash problems. The predictions have been compared with data from experiments in MARIINT's shallow water basin, in which a sloping bank was constructed along one side over a part of the length of the tank. Wave probes were mounted at 4 lateral positions, both in the rectangular and in the trapezoidal part of the tank. Fig. 5 shows that the wave pattern and its deformation in shallow water is quite well predicted; the reflection, however, is somewhat incompletely resolved, which explains the underestimation in the downstream part; and some phase differences occur. While these results concern a displacement-type of vessel at rather low speed, they illustrate the current potential of prediction methods to represent the complicated physics playing a role for ships in channels or natural waterways. However, the resolution of small-scale features is somewhat imperfect due to the restriction on the number of panels imposed by the memory capacity of the CRAY C90, machine then used. If the relative size of the waterway is larger, the number of panels required soon becomes excessive. A less 'brute-force' approach is then desired.. Possibilities for this are discussed in Sections 3 and 4. Raven Page

7 Figure 4 Computed wave pattern for LIUTO waterbus at 8 knots. The deformation of the wave pattern along the shallow bank is clearly shown. e.nerunent upenment ill (charmei vr.11) Si X/L Figure 5. Calculated and measured longitudinal wave cuts for LIUTO waterbus in shallow channel with sloping bank. Speed 8 kn. Transverse distances L, L and L (at the channel wall). Raven Page

8 2.2 (d) Unsteady flow effects A basic limitation of the free-surface panel methods discussed here is the fact that they solve a steady flow problem, in a coordinate system moving with the ship. It is, therefore, impossible to predict the instantaneous waves generated when a vessel passes through its hull speed or critical speed, unless a quasi-steady approximation is sufficiently representative. Moreover, unsteadiness caused by variations in the waterway cross-section, or near critical speed, cannot be incorporated either. These aspects may be important in some practical wash problems. A prediction technique would then have to solve the unsteady free-surface problem by a time-stepping approach. While some such methods exist, most are still rather remote from practical applicability to the complicated problems concerned. As explained in the Introduction, a partial solution is to admit unsteady effects only in the far field, and to consider the ship wave generation as a steady phenomenon. This does cover effects of bottom topography if remote from the ship, but not the effects of acceleration or deceleration, water depth variations under the path of the vessel, etc. Section 5 provides an example of this approach. 3. PREDICTION OF THE FAR-FIELD WAVE PATTERN The methods described typically are confined to rather near-field wave pattern and flow predictions, since they are based on the use of simple Rankine (1/R) source distributions on the boundaries of the flow domain. Unlike Havelock sources these sources themselves do not imply any wave behaviour; the wave pattern is obtained by a wavy distribution of the free-surface source strengths. Consequently, outside the part of the water surface covered with panels no waves are computed. If the far-field wave pattern is desired, the free-surface panelling would need to extend to far from the vessel, while retaining a good resolution of the waves. In practice, it is rarely feasible to go beyond a few ship lengths distance. Moreover, the discretisation of the free-surface boundary conditions introduces small numerical errors, which take the form of a numerical wave damping and numerical dispersion. These errors can be well controlled in the near field, but tend to accumulate with distance from the ship, thus reducing the accuracy of the predicted far-field waves. Even if the requested distance can be reached, a direct calculation thus may not be the best option. Therefore, for predicting wash a free-surface panel code often needs to be supplemented with a suitable far-field representation or a model for the wave propagation. The possibilities for this differ according to the speed regime DEEP WATER OR SUBCRITICAL CONDITIONS Some analytical methods permit to extend the wave pattern to the far field for uniform water depth. Various tools are available, such as simple decay rates, Kelvin source distributions, spectral representations. All of these, however, depend on the assumption of linearity in the far field. This assumption is justified in view of the decreasing wave amplitude and the absence of near-field flow perturbations outside the vicinity of the vessel. The simplest idea, applicable for deep water waves, is to use the known asymptotic decay of a linear (Kelvin) ship wave system to estimate the maximum wave height and wave directions/periods in the far field. For the waves inside the Kelvin wedge the decay is proportional to (distance)-'4, but the waves on the Kelvin wedge decay with (distance)45 and therefore are dominant in the far field. It is remarked here that the decay rate applies to the wave envelope: The Kelvin wedge is made up of a series of isolated short crests and troughs, and the maximum crest or trough height cannot be read from a single record or computed wave cut. However, the maximum peak-to-trough height is a rather smooth function of the lateral distance. Based on the computed near field, wave heights can thus be extrapolated to any distance using the given decay rate. However, to avoid arbitrariness the computed wave pattern must also display the Raven Page

9 same decay rate at a sufficient distance from the vessel. Experience tells that in numerical computations usually a faster decay is found, unless care is taken to minimise the numerical damping. Fig. 6 shows the maximum peak-to-trough wave heights on the Kelvin wedge of a bow wave pattern computed with RAPID, on a double logarithmic scale. A dense free surface panelling of 48 * 218 panels was used (for Fn=0.3), without increase of the panel sizes away from the hull. In addition, the code contains a special difference scheme designed to minimise numerical damping [1]. The dotted line with slope -1/3 shows that the computed wave pattern displays precisely the correct decay in the far field (0.4 < z/l < 1.1). For z/l < 0.4 (log z,/l < -0.4), a different behaviour is found due to nonlinear and near-field effects; at the outer edge of the free-surface domain, reflections and truncation effects disturb the behaviour. Extrapolation of wave heights based on analytical decay is easily done here; but with free-surface panels it does not come for free. Figure 6. analytical un f 10log z/l Decay of peak-to-trough wave height with distance, as computed; compared with (-1/3) power decay. Quite recently we have developed a far more convenient approach to predict the far field. The far field is described as a ship wave spectrum, which (by virtue of the dispersion relation) is a one-dimensional spectrum F(u), G(u) in terms of a single wave number or wave direction: 1 c(x, y) = f F(u)sin(sx) + G(u)cos(sx) cos(uy) du 27t 0 sin where u s = cos u2 2 This spectrum can be derived from the computed wave pattern, using a 'wave pattern analysis' approach as originally developed for towing tank tests [11]. 8 Transverse cuts at some distance aft of the ship are used [3]. Using a special Fourier analysis and solving the resulting overdetermined system we obtain the spectrum functions F and G. Subsequently the far-field wave pattern can be reconstructed from the formula given above. Raven Page ,

10 interpolated in data, large domain from spectrum, small domain = X/L Figure 7. Longitudinal wave cut at z/l = 1.125; computed directly on large domain, and reconstructed from spectrum derived from small-domain computation. Fig. 7 compares longitudinal wave cuts at a distance of ship length from the centreline. The full line was computed directly on the free-surface grid used for Fig. 6 (10464 panels), the dotted line was reconstructed from the spectrum derived from a standard computation with a smaller domain and coarser panelling (2112 free-surface panels). Some deviations are observed, primarily resulting from the different resolution on the two grids. By reducing the integration and aliasing errors in the conversion from transverse cuts to spectrum to longitudinal cut we hope to improve the agreement. In any case the approach seems quite promising. Fig. 8 illustrates how a small-domain computation (the small box around the ship) can be extended to a very large domain, in just seconds of calculation time. At present this far-field prediction is for deep-water waves only , 0.25 Figure 8. Far-field wave pattern from spectrum derived from small-domain computation. x/l An alternative technique based on an analytical representation of the wave pattern outside the domain of the panel code, matched with the near field, has been proposed by Yang et al [12]. Their method is based on the same linearity assumptions but is perhaps more flexible and numerically accurate CRITICAL OR SUPERCRITICAL CONDITIONS As long as the near-field computation of the wave pattern near critical speed has not been validated, there is probably no point in extending it to a far field. In any case, the dominant (transverse) waves at Raven Page

11 _ critical speed are pure shallow-water waves: the group velocity is equal to the phase speed, there is no dispersion, and if a steady condition is reached the decay with distance is not governed by inviscid effects but by e.g. bottom friction and breaking. At supercritical speeds, the outer waves bounding the wave pattern also propagate at critical speed. A conservative assumption is to assume no decay at all, and to suppose a far-field wave height (in uniform water depth) to be equal to that in the near field. At the same time, various subcritical wave components are present in the pattern, some of which may also be relevant to wash effects. For these components decay will be present due to dispersion. 4. COUPLING OF WAVE GENERATION AND WAVE PROPAGATION MODELS The methods to predict the fax field discussed above are limited to rather simple situations: uniform or large water depth, unrestricted width, steady conditions, no nonlinear effects in the far field. In practice, however, many wash problems are affected by the local bottom or bank topography, e.g. wave amplification and refraction due to variable water depth; wave focussing due to reflection and refraction; wave penetration into harbours. Often these cannot be incorporated in a steady description in a ship-fixed frame of reference. As opposed to the simple situations addressed in the previous section, the prediction of the wave generation by the ship now needs to be coupled with a separate model for wave propagation. Again, in general cases the wave propagation problem is a 3D unsteady nonlinear free-surface problem, and for extensive domains large computational resources would be needed. However, simplified approaches may be adequate, and various levels of modelling have been applied. Kofoed-Hansen et al [13] use a spectral representation of the wave field. The ship wave pattern is derived either from measurements or from empirical data, and is replaced by an equivalent spectrum of long-crested waves, the evolution of which is computed. The phase information and transient character of the ship waves are thereby lost; therefore there is no Kelvin wedge' and, in uniform water depth, no decay with distance from the ship; empirical corrections are supplied to include the latter effect. Also it is pointed out that nonlinear effects in shallow water reduce the accuracy of the shoaling predictions. Still, useful predictions have been obtained with limited effort. Whittaker et al [9,8] have used a combination of various coastal wave transformation models, mostly in the spatial domain, to predict the ship wave propagation in an estuary. Much of the wave propagation was computed using a parabolic mild-slope approximation. Essentially a single component of the ship wave system could be considered at a time. Measured wave patterns were used as input. Chen [14] presents a method that predicts both the wave generation and propagation, for a ship in a channel. The wave generation is modelled by a slender-body approach, including some nonlinear terms. The wave propagation is based on a reduced form of the Boussineso eouations, valid for only gradual variations of the depth in longitudinal direction and for near-critical speed. Both models are coupled by a matched asymptotic expansions technique. Consequently, also the wave generation in principle contains unsteady effects now. Soliton generation near critical speed is predicted by this model. Good results are shown for a ship in a channel with irregular cross section. A drawback of the method could be the simple representation of the ship wave making. A different coupled wave making / wave propagation model has been set up in a joint project of MARIN and WLIDelft Hydraulics, for the Public Works Department (Rijkswaterstaat) of the Dutch Ministry of Transport Aims of the study are the development of a prediction tool for ship-induced water motion along banks of a waterway, and prediction of motions of moored ships caused by the Raven Page

12 passage of other vessels. The method should be able to deal with irregular lateral boundaries such as spurs or harbour basins; and should handle a variety of ship types, from push barges to fast ferries. The method is based on coupling a prediction of wave generation using a steady free-surface potential flow code, with a prediction of wave propagation using nonlinear Boussinesq-type equations. In the latter method the vertical coordinate is eliminated from the system; still it includes all horizontal variations in the flow field and offers good prospects to retain the character of a ship wave system, if that enters the Boussinesq domain through the boundary conditions. The ship is supposed to be unaffected by any variations in the waterway depth or width. Therefore, the wave generation is assumed to be a steady process in a ship-fixed frame of reference, and is computed using RAPID. Overlapping with this inner' free-surface panel domain is an outer domain in which the propagation of the ship waves is computed, solving unsteady Boussinesq-type equations as developed by WL1Delft Hydraulics in an earth-fixed frame of reference. These computations have been done by WLIDelft Hydraulics using their Boussinesq-type model TRITON [15]. TRITON does not pose any restrictions to the shape of the outer domain, so the effect of a variable bottom topography can be included. Due to the applied Boussinesq-type approximation, the modelling of wave dispersion and shoaling is restricted to relatively long waves, and in the current version of TRITON accurate as long as the wave lengths are larger than about twice the water depth. The wave signal is imposed along the edges of the outer domain where the wave is incoming. The inner, ship-fixed domain slides along the edge of the outer, earth-fixed domain and imposes a wave signal of constant shape that moves along that edge during the passage of the ship. This boundary condition is determined by making a longitudinal wave cut of the steady wave pattern in the inner domain at the position of the interface, and transforming it at every time step to the earth-fixed coordinate system. Depending on the type of boundary, weakly-reflecting or totally reflecting conditions are imposed elsewhere. It is remarked that in the inner domain, no incoming waves through the side boundaries are allowed; therefore the two flow models do not interact, there is just a one-way coupling from wave generation to wave propagation. Therefore, if variations in the waterway cross section or other phenomena taking place in the outer domain are so strong that they significantly affect the wave generation, that effect is not incorporated. This may be the case e.g. for narrow channels of variable width, or near critical speed. As before, also other unsteady effects originating in the inner domain are disregarded. A basic check of the coupled system has been made in a computation for a 110 m Rhine cargo vessel at a speed of 4.7 m/s in 4 m water depth. The free-surface panel domain in the wave pattern computation extended from the centreline to a distance of 75 m, the outer domain from 37.5 to 75 m off the centreline. The ship waves were imposed at the longitudinal boundary at y = 37.5 m, as well as at the 'upstream' transverse boundary of the outer domain. The composite plot in Fig. 9 shows the wave pattern computed by RAPID, below the horizontal line; and its extension computed by TRITON, above the line. The two solutions match well; the bow and stern wave Kelvin wedges are continuous through the interface, indicating precise satisfaction of the dispersion relation. Fig. 10 compares a lonqitudinal wave cut computed by the panel method alone, and computed with the coupled system. The agreement is fair, with some overestimation of the wave amplitude by the coupled system, until at x=130m waves reflected at the outer edge reach the cut. Raven Page

13 Figure 9: Wave pattern predicted by coupled model. Near field computed by potential flow code, extension by Boussinesq model. Longitudinal section at distance of 60. m Figure 10. Comparison of longitudinal wave cut 60 m off centreline. Full line is coupled model, dotted line is RAPID alone. The second test concerned a case with genuine unsteady effects, caused by a step increase of the width of a channel. The calculations clearly represented the waves reflected at the channel wall, a rapid propagation of the primary wave system into the wide part of the channel, and the weak diffracted wave pattern passing around the corner into the wide part of the waterway. Validation of these observations could not be carried out yet for absence of data. While the study so far addressed conventional ships only, the coupled approach is not restricted to those, and can handle all ship types that can be dealt with by the panel code, including high-speed craft. The results obtained hold promise for its use to study, understand and predict transient phenomena on ship waves during propagation towards shores, into harbours etc. Further validations and extension of the model will soon be undertaken. 5. CONCLUSIONS The present generation of prediction methods for ship wave making consists of practical, efficient and accurate codes that are applied on a large scale in ship design. The examples show that they can represent much of the physics relevant for wash; they can be very useful for comparing ship designs, and may give indications for minimising wash of a certain vessel by selection of trim, speed, etc. There may, however, be serious doubts regarding their validity in case of strong critical speed effects (for small water depth / ship length ratio), while predictions for supercritical speeds are plausible but deserve further validation. Since these methods are confined to rather near-field predictions, for predicting the far field they need to be supplemented. In case of subcritical speed and uniform water depth, the most practical method seems to be representation as a ship wave spectrum, as proposed in this paper; generalisation to shallow water cases is desired. In more general cases, the coupling of the steady wave pattern prediction with an unsteady nonlinear Boussinesq-type equations model permits to address a variety of practical problems. While a first verification of the coupling was favourable, further refinement, generalisation and validation are desired. Raven Page

14 ACKNOWLEDGEMENTS The development of the coupled RAPID/TRITON model is carried out in close cooperation with WL1Delft Hydraulics, in particular with Jr. H. Verheij, Jr. N. Doom and Dr. M. Borsboom; their contribution is gratefully acknowledged. The Dutch Public Works Department 'Rijkswaterstaat' has sponsored the work described in Section 4, and is thanked for their permission for this early publication. REFERENCES RAVEN, H.C., Inviscid calculations of ship wave making --- capabilities, limitations and prospects,' 22nd Symposium on Naval Hydrodynamics, Washington DC, RAVEN, H.C., 'A solution method for the nonlinear ship wave resistance problem', Doctor's Thesis, Technical University of Delft, RAVEN, H.C., and PRINS, H.J., 'Wave pattern analysis applied to nonlinear ship wave calculations', 13th hit. Workshop on Water Waves and Floating Bodies, Alphen aid Rijn, Netherlands, RAVEN, H.C., and VALKHOF, H.H., 'Application of nonlinear ship wave calculations in design', 6th PRADS symposium, Seoul, Korea, BERTRAM, V., and HUGHES, M.J., 'Practical predictions of wash using fully nonlinear wave resistance codes', HIPER99 symposium, Zevenwacht, South-Africa, LEER-ANDERSEN, M., CLASON, P., OTTOSON, P., ANDREASSON, H., and SVENSSON, U., Wash Waves Problems and Solutions', SNAME Annual Meeting, , CHEN, X.-N., and SHARMA, S.D., 'A slender ship moving at a near-critical speed in a shallow channel', Jnl Fluid Mechanics, Vol. 291, p , 1995 MARITIME AND COASTGUARD AGENCY, Investigation of high-speed craft on routes near to land or enclosed estuaries', Research Project 420, WHITTAKER, T., BELL, A., SHAW, M., and PATTERSON, K., 'An investigation of fast ferry wash in confined waters', RINA Conf. Hydrodynamics of High-Speed Craft, London, '10. RAVEN, H.C., VAN HEES,M., MIRANDA, S., and PENSA,C., 'A new hull form for a Venice urban transport waterbus: Design, experimental and computational optimisation,' 7th PRADS symposium, Den Haag, Netherlands, EGGERS, K.W.H., SHARIvIA, S.D., and WARD, L.W., 'An assessment of some experimental methods for determining the wavemaking characteristics of a ship form', SNAME Transactions, Vol. 5, YANG, C., LOEHNER, R., NOBLESSE, F., and HENDRIX, D., 'Fourier-Kochin extension of fully nonlinear near-field ship waves', 7th Int. Conf. Numerical Ship Hydrodynamics, Nantes, France, KOFOED-HANSEN, H., JENSEN, T., KIRKEGAARD, J., and FUCHS, J., Prediction of wake wash from high-speed craft in coastal areas', RINA Conf. Hydrodynamics of High-Speed Craft, London, CHEN, X.-N., 'Ship wave making over a natural topography,' 7th Int. Conf. Numerical Ship Hydrodynamics, Nantes, France, BORSBOOM, M., DOORN, N., GROENEWEG, J., and VAN GENT, M., ' A Boussinesq-type wave model that conserves both mass and momentum,' 27th ICCE 2000, Sydney. Copyright MARIN, 2000 Raven Page