J Mar Sci Teclinol (2010) 15:331-344 D O I 10.1007/S00773-010-0102-1
O R I G I N A L A R T I C L E
Validation of an approach to analyse and understand
ship wave making
Hoyte C. R a v e n
Received: 2 January 2010/Accepted: 16 June 2010/Published online: 7 August 2010 © J A S N A O E 2010
Abstract This paper discusses a rational and systematic procedure f o r understanding and analysing steady ship wave patterns and their dependence on h u l l f o r m . A step-wise procedure is proposed i n w h i c h the pressure distri-bution around the hull is invoked to provide a quahtative understanding o f the connection between h u l l f o r m and wave making. I n a recent publication it was shown how this understanding explains various k n o w n trends and, i n combination w i t h wave pattern computations by free-sur-face potential flow or Reynolds-averaged Navier-Stokes ( R A N S ) methods, can often be exploited to reduce wave maldng by m o d i f y i n g the h u l l f o r m . The present paper provides support f o r the guidelines given, validates the decomposition into different steps and indicates the con-nection w i t h previous theoretical approaches.
K e y w o r d s Ship waves • Slow-ship linearisation • H u l l f o r m design
1 Introduction
This is perhaps a somewhat unusual paper. W e address one o f the fundamental issues i n ship hydrodynamics: the relation between ship h u l l f o r m ' and the wave pattem i t generates. However, rather than proposing a new prediction method or a specific application, we o f f e r an approach that helps to understand this relation, w h i c h is essential to design a ship hull f o r m such that it has l o w wave
H . C. Raven ( E l )
Maritime Research Institute Netherlands ( M A R I N ) , P.O. Box 28, 6700 A A Wageningen, The Netherlands e-mail: h.c.raven@marin.nl
resistance. The paper discusses and validates a way o f w o r k i n g and general insights. W e go back to simple linear potential theory f o r steady ship waves, and base ourselves on w e l l - k n o w n textbook material. However, to the best o f the author's knowledge, that material has rarely, i f ever, been presented i n this context, and the proposed procedure based on it does not seem to be widely k n o w n . Therefore, we hope that the present paper may be o f practical use f o r many.
Obviously, f o r ship performance, wave m a k i n g is j u s t one o f the aspects; viscous resistance, wake field and propulsion efficiency also play important roles. However, separate consideration o f wave making is still quite rele-vant, as reduction o f wave resistance can often be achieved without unduly affecting the viscous resistance or wake field. Moreover, while wave resistance is usually not the largest resistance component f o r a merchant ship, i n many cases i t is the component that is most easily reduced by proper modifications to the h u l l f o r m . This is an opportu-luty and a challenge at the same time. H o w should these modifications be determined?
In the past, resistance reductions were targetted i n a model testing program. A model o f an i n i t i a l h u l l f o r m was towed, its resistance measured, and the wave pattem was observed by an experienced naval architect. H u l l f o r m modifications were then proposed based on experience, on i n t u i t i o n and to some extent on 'trial and eiTor'. Several models were often built f o r a single project, m a k i n g the procedure rather time-consuming and costly.
Extensive research i n the last century has led to several theoretical wave resistance prediction methods intended f o r use i n ship design. A variety o f linearised methods have been proposed, i n w h i c h originally also the h u l l boundary conditions were linearised (thin-ship, flat-ship, slender-body methods), but later only the free-surface boundary
conditions ( N e u m a n n - K e l v i n , slow-ship). A strong impetus was provided b y the introduction of various slow-ship theories, which appeared to provide better resistance pre-dictions than earlier methods. One practical and widespread method of this class was that proposed by Dawson i n 1977 [1], w h i c h has been used i n ship design f o r some time.
Around 1990 a major step was made i n the theoretical development, w i t h the introduction o f complete nonlinear free-surface potential f l o w codes [ 2 - 5 ] . These methods produce much more accurate predictions, and their use i n design is widespread now. One of these codes w i l l be used in this paper f o r studying trends and validating guidelines.
Nevertheless, these codes do not design a ship, but rather enable evaluation o f a given design. I t is the designer who decides on the required h u l l f o r m modifications, and the problem is still how to derive proper design adjust-ments that lead to a reduction o f wave resistance.
I f one has a tool that predicts the wave resistance o f a ship h u l l f o r m exactly, the best approach to design f o r m i n i m u m wave resistance could be a f o r m a l optimisation technique. This requires that a wave resistance prediction tool be linked to an optimiser and a parametric hulJ f o r m variation tool. Several examples o f such procedures have been published, e.g. [ 6 - 8 ] . However, i n this context a very similar issue arises: how to choose parametric h u h f o r m variations o f f e r i n g the best prospect f o r wave resistance reduction. W i t h o u t an answer to this, a vast number o f hull f o r m parameters need to be addressed, m a k i n g the opti-misation time-consuming and unclear.
To improve ship h u l l f o r m s f r o m a wave-making point of view, and to assess the quahty of predictions and opti-misations con-ectly, it is desired to understand how wave making is connected w i t h features o f the h u l l shape. This is a d i f f i c u l t question to answer i n general. There are several simple guidelines on the best h u l l f o r m f o r given speed ranges, rules o f thumb f o r bulbous bow dimensions or transom height, indications on frame shapes etc. However, without further insight i t is hard to understand these guidelines, to k n o w when they apply or to move beyond what has been tried before.
Recently, an alternative approach has been described i n the new issue of the 'Ship Resistance and F l o w ' volume o f the Principles of Naval Architecture series [ 9 ] . This is a formalisation of a w o r i d n g procedure we have developed at M A R I N i n the course o f many projects f o r shipyards over the years. I t has been used i n essence since 1988, i n i t i a l l y using predictions f r o m the DAWSON code [10] and later predictions f r o m RAPID [ 5 ] . W e assume that a computed wave pattern and flow field are available, and we analyse these to deduce what design changes are desired to reduce certain wave components. The analysis is based on decomposing the relation between h u l l f o r m and wave making into simpler steps, and t r y i n g to understand those
separate steps. This understanding is loosely based on linear theory, is not always quantitatively precise, but i f combined w i t h computational predictions can lead to a successful and sound hull f o r m improvement process.
The present paper briefly summarises the essence o f this approach, but mainly it focusses on a discussion and vali-dation o f the trends and guidelines described, and on an inspection o f the validity o f the approach overall. Thus, i t provides support f o r the simplified analysis, and i n addition casts an interesting light on the validity o f certain older slow-ship wave-making theories.
The paper is organised as f o l l o w s . First we give a brief outline o f the wave resistance prediction code that has been used to generate the examples provided, and we briefly recall a relevant part of basic ship wave theory. I n Sect. 4 we outline the separate steps made and the guidelines f o r each. Section 5 then provides a discussion of each step, and gives illustrations o f the guidelines. Section 6 discusses the relation o f the present stepwise procedure w i t h earlier ship wave theories, then some examples are given of the validity of the procedure i n practice. Finally, we discuss some examples o f the application of the approach to aspects o f h u l l f o r m design, and draw conclusions.
2 W a v e pattern computations
For the methodology discussed below the particular flow code used is rather immaterial, but f o r completeness we briefly indicate the method we have used to generate the wave patterns and flow fields shown. This is the free-sur-face potential flow code RAPID [5, 11], w h i c h solves the problem o f the steady free-surface flow around a ship at constant f o r w a r d speed i n still water. It imposes exact inviscid, f u l l y nonhnear free-surface boundary conditions, and takes into account the dynamic t r i m and sinkage o f the vessel. The solution is obtained i n an iterative procedure i n w l i i c h the wave elevation, flow field, t r i m and sinkage are repeatedly updated u n t i l a converged solution o f the non-linear problem has been found. I n each iteration a hnear problem is solved using a panel method. Source panel distributions are used on the hull surface and on a plane at a smaU distance above the wave surface. B o t h panellings are repeatedly updated during the iterative process. T y p i c a l l y we use 2000-3500 panels on one half o f the h u l l surface and 4000-20000 on one half o f the wave surface. For a complete calculation the computation time usually varies f r o m a f e w minutes to h a l f an hour on a standard single-processor personal computer, RAPID has been used i n h u l l f o r m design on a daily basis at M A R I N since 1994, and i n addition is used b y several licencees w o r l d w i d e (shipyards, navies and universities).
J Mar Sci Teciinol (2010) 15:331-344 333
3 Ship waves
I n our process to assess and minimise wave resistance, we w i l l focus on the wave pattern generated by the ship i n the first place, as it is far more informative than just a predicted wave resistance value. W h i l e the basic properties o f ship wave patterns are described at length elsewhere, e.g. [9, 12], f o r reference we briefly recall some o f the essential relations here.
W e consider a ship i n steady f o r w a r d motion i n still water. W e use a frame of reference m o v i n g with the ship, w i t h X pointing aft and z upward. I n this system the wave pattern and the flow around the h u l l are steady. W e make the usual assumption that wave making is dominated by inviscid-flow properties. Evidently, the generation o f stern waves is affected by viscosity, but their propagation less so, and the general approach sketched below remains relevant. A potential-flow assumption can then be made, at least f o r the wave properties away f r o m the h u f l surface. The velocity field is the gradient field o f a scalar potential (j), w h i c h for incompressible flows satisfies the Laplace equation V^</) = 0. The pressure is f o u n d f r o m B e r n o u l l i ' s equation and consists of a hydrostatic pressure and a hydrodynamic contribution ;;hd- The latter is the meaning-f u l quantity i n our case. I t is nondimensionalised using the fluid density p and the ship speed V as:
Phd V ( / ) - V ( / ) , . ^ = 1 ^ = 1 — v ^
-Steady surface potential flow must satisfy free-surface boundary conditions, demanding that the pressure at the water surface is atmospheric, and that the wave surface is a streamsurface. These two boundary conditions are nonlinear. I n the development of prediction methods f o r ship wave patterns, linearisations f o r small wave steepness have long been used, and w h i l e the resulting methods were usually not quite accurate, linear theory is most appropriate f o r providing an understanding of ship waves. Linearisation relative to the undisturbed u n i f o r m flow leads to the so-called K e l v i n boundary condition:
F « ^ ? ^ + ^ = 0, (2)
i n w h i c h Fii = V/^/gL. This boundary condition is linear and homogeneous, and therefore admits superposition of solutions, just l i k e the Laplace equation.
SpecificaUy, a ship wave pattern, at least i n the f a r field, can be approximated as a superposition o f sinusoidal waves propagating over an undisturbed flow, so-called free waves. Properties of sinusoidal waves are easily derived f r o m the Laplace equation and K e l v i n condition. W e readily find that the wave propagation speed c and the wave length X are related by the w e l l - k n o w n dispersion relation:
F i g . 1 Direction, phase speed and length of wave components i n three-dimensional cases
8
For a two-dimensional steady case, e.g. a submerged cylinder at right angles to the flow moving w i t h constant speed V through still water, the phase velocity o f the steady two-dimensional (2D) waves must be equal to V, and f r o m Eq. 3 we find X = InV^lg = InFn^L. So, far enough aft of the body, there is just a single wave component present. I n three-dimensional cases such as ships, there is an additional degree of freedom, which is the wave propa-gation direction, indicated as 0 i n F i g . 1. A local distur-bance, such as the bow o f a ship, can generate an infinite set of wave components propagating i n various direc-tions - 7 i / 2 < 0 < +n/2. I n actual ship wave patterns, components w i t h angles up lo 6 - 6 0 ° - 7 0 ° can often be observed.
I n order f o r waves to be steady i n the coordinate system m o v i n g w i t h the ship, the phase velocity must be c = y cos Ö, and according to the dispersion relation
A(0) = 2nc-/g = InFirL cos^ 0. (4)
The longest waves i n the pattern are therefore the trans-verse waves, which have length XQ = InFn^L, the funda-mental wave length. A l l other steady waves are shorter by a factor cos^ Q. The distinction between transverse and diverging waves is often chosen at (? = 3 5 ° , but below we shall frequently denote waves w i t h Ö « 0 ° as 'the trans-verse waves'.
The f a r - f i e l d wave pattern is the sum o f a f l wave com-ponents generated by different parts o f the h u l l , progressing i n various directions and interfering. The assumption of undisturbed base flow and small wave amplitudes is cer-tainly satisfied i n the far field. However, different potential fields are needed i n the near field, since the superposition of wave components cannot be expected to satisfy the boundary condition on the ship h u l l . Moreover, the wave components propagating away f r o m the hull pass through the curved flow field w i t h variable velocity that prevails close to the hull, and thereby are affected by refraction, causing changes i n wave amplitude, length and direction. Therefore, the pure superposition of sinusoidal waves can
only represent the wave pattern i n the far field. Neverthe-less, f o r general understanding, these refraction effects are of secondary importance.
The unique relation between wave direction, wave speed and wave length leaves only the amplitude and phase o f the wave components i n the f a r field as unknowns. I n an expression f o r the free-wave pattern
A ( ö ) COS g ( x c o s (? - f y s i n i y 2 cos2 9
~B{e) sin- . - c o s ö - f }'sin(
y 2 , d0, (5)
the amplitude and phase are determined by the fimctions A{Q) and S(0), w h i c h represent the free wave spectrum. The wave resistance can then be expressed by;
K=\npV^ j [A{Of+B{ef )d0. (6)
Therefore the wave resistance depends quadratically on the wave amplitude, and there is a weighting factor cos"* 0. The last result means that transverse waves are far more important f o r wave resistance than divergent waves, a significant finding f o r ship h u l l f o r m design.
4 Outline of the approach
The procedure to analyse a ship wave pattern and to understand how it depends on the ship h u l l f o r m is explained and illustrated more extensively i n [ 9 ] . For completeness, i n this section we summarise i t .
W e suppose that a computed wave pattern and flow field f o r an i n i t i a l h u l l f o r m at the desired speed are available, e.g. f r o m a free-surface potential-flow or free-surface R A N S computation. Alternatively, an observed or mea-sured wave pattern w o u l d be useful, although, as w i l l appear, availability o f the (computed) hydrodynamic pressure distribution plays a central role i n the analysis.
The analysis o f the wave pattern and flow field evidently must start w i t h identification o f the dominant wave com-ponents. Referring to Eq. 6 we see that, f o r the wave resistance, 'dominant' means a large amplitude and a not too large angle 0, i n view o f the cos^ 0 weighting factor. Because o f their greater wave length, transverse waves may often be visually less apparent than the steeper, shorter diverging waves, but nevertheless they may be donfinant for wave resistance.
To decide how to m o d i f y the h u l l f o r m to reduce those dominant waves, one needs to i d e n t i f y w h i c h h u l l f o r m aspects generate these, e.g. bow, stern, shoulders or
perhaps bow + fore shoulder, or a detail o f a bulbous bow etc. Then those aspects can be adjusted i n a way that reduces the amplitudes o f these waves and improves their interference. However, i n this step one needs to understand the connection between h u l l f o r m and wave maldng. H o w can one guess w h i c h o f the many different possible wave components is generated preferentially by a certain h u l l f o r m feature? A n d what change should be applied to that feature?
To obtain an insight into the complicated relationship between h u l l f o r m and wave making, it has been found useful i n practice to make certain simplifications. These can be f o r m a l l y represented as a virtual distinction between two separate steps:
1. Consider the relation o f the h u l l f o r m w i t h the hydrodynamic pressure distribution over the hull, and consider the association w i t h the resulting pressure disturbance at the still-water surface
2. Estimate the wave-making properties o f that still-water surface pressure distribution
For each o f these steps, general guidelines can be derived f a i r l y easily, and these w i l l be summarised below.
Step 1: From hull form to free-surface pressure distri-bution The quanfity to be considered first is the distridistri-bution over the h u l l surface o f the hydrodynamic pressure coef-ficient. This pressure distribution is deterinined by the h u l l f o r m and the ship speed, water depth etc. However, as we try to go step by step, wave effects are disregarded i n this first step; this makes the hydrodynamic pressure coefficient independent o f ship speed.
The relation between h u l l f o r m and h u l l pressure dis-tribution is described more completely i n [9] and is only summarised here, as the f o l l o w i n g guidelines.
H i g h pressures occur:
• Near stagnation points, i n the b o w area. I n most cases there is also a pressure rise towards the stern, dependent on stern type and viscous effects.
• A t concave streamwise curvatures: the centrifugal acceleration acting on the flow is balanced b y a pressure rise av/ay f r o m the centre o f curvature. • A t large streamwise slopes: a large angle o f the
streamlines relative to the longitudinal direction (out-ward on the forebody, in(out-ward on the aftbody) leads to a somewhat elevated pressure, as can be explained f r o m the concavity o f streamlines further f r o m the h u l l .
L o w pressures occur:
• A t convex streamwise curvatures, again due to centri-f u g a l acceleration.
• Due to displacement effect, the volume taken b y the h u l l requiring a small speedup o f the flow next to i t .
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Except f o r the stagnation point effect, the effect o f streamwise curvature is by f a r the strongest i n most prac-tical ship hull cases.
These are, o f course, w e l l - k n o w n relations. Quantitative results can easily be obtained by making a double-body potential-flow computation.
The aspect o f the pressure f i e l d around the hull that is most important f o r wave generation is the pressure distri-bution at the still-water surface. Therefore \Ye have to assess that pressure distribution, again i n connection w i t h the h u l l shape. Suppose that a h u l l f o r m feature causes a pressure variation on the h u l l i n longitudinal direction at a certain depth, having a length scale Lp i n that direction. Evidently this is accompanied by a pressure disturbance i n the flow away f r o m the h u l l and at the still-water surface— but w i t h what properties? Here the f o l l o w i n g guidelines hold:
9 The amplitude o f the pressure disturbance decreases quickly w i t h increasing distance/L^; the length scale i n longitudinal direction o f the pressure disturbance increases w i t h distance/Lp.
e A short pressure variation on the hull thus has just a local effect on the pressure field; a longer pressure variation is felt at larger distances.
® Therefore, the pressure distribution on the still-water surface is caused by all h u l l pressure variations along the waterfine, and by the larger-scale pressure varia-tions farther beneath the water surface.
» Conversely, pressure variations at the water surface w i t h small length scales can only be generated by foiTn features close to the water surface.
Some illustrations w i l l be provided i n the next section.
Step 2: From free-suiface pressure distribution to wave pattern W e consider the wave pattern as driven by this hydrodynamic pressure distribution at the location of the still-water surface. F r o m the dynamic free-surface boundary condition the wave elevation is ( = ^Fn'^'L • Cp, but here we cannot simply use the double-body pressure coefficient; i f , e.g. Cp is h i g h on a part o f the h u f l just below the water surface, this w i l l generate a local wave crest plus a trailing wave system behind i t . The question is what that wave system w i l l l o o k like; W h i c h of the wave components generated w i l l be significant and w h i c h wiU not?
The answer lies i n the guideline;
• A wave is most e f f e c t i v e l y generated by a pressure variation o f comparable length and shape.
Explanation and interpretations o f this guideline are given below.
5 Discussion of the steps
The relation between hull f o r m and wave maldng has thus been decomposed i n two separate and f a i r l y understandable steps. N o w not only some more support may be needed f o r the guidelines given, but one may also wonder whether i t is allowed at a l l to make this decomposition. Therefore, i n the present section the different steps wUl be considered i n more detail, iflustrations w i l l be given and restrictions and approximarions indicated, whUe subsequent sections address the validity of the decomposition i t s e l f
Step L From Indl form to free-swface pressure distri-bution Figure 2 shows the pressure field f o r the K V L C C 2 tanker i n double-body flow. It illustrates w e l l the various trends described; a high pressure around the bow and stern stagnation points, l o w pressures at all convex streamwise curvatures (fore and aft shoulder, transition to the bottom i n the forebody, sides o f stern gondola) and a small underpressure along the parallel midbody caused by the displacement effect. W e also observe that pressure variations close to the waterline are continuous f r o m h u l l to s t i l l -water surface, and that smaller-scale pressure variations further beneath the water surface, such as the low-pressure area at the stern gondola, have no visible effect at the s t i l l -water surface.
To support the guidelines f o r assessing the free-surface pressure field f r o m the h u l l pressure distribution, let us first consider a p o i n t source of strength Q at position (XQ, 0, zo) i n an undisturbed i n f l o w V. As is wefl k n o w n , this pro-duces a flow and pressure field coiTesponding w i t h that around a semi-infinite body of revolution, a 'Rankine h a l f - b o d y ' . The dimensions of that body, and the length
Fig. 2 Double-body pressure distribution f o r K V L C C 2 tanker
point source
Fig. 3 Pressure field induced by a point source in a u n i f o r m inflow in
-Y-direction. The source induces a pressure variation at the still-water surface w i t l i a maximum and m i n i m u m pressure. The three planes show the pressure pattern (starboard half only) at the still-water surface f o r submergence depths o f the source equal to Lp, 3Lp and 5Lp [9]
scale o f the pressure variation over i t , are proportional to Lp = \ / < 2 / i^nV). The resulting pressure variation at the still-water surface i n double-body flow can be approxi-mated (linearised) as:
X - XQ
R (7)
f o r a point at the still-water surface at a distance R f r o m the source. So, the pressure amplitude at the still-water surface decays proportional to {LplRf, w h i l e the length scale increases linearly w i t h RILp. Figure 3 illustrates this behaviour. Wave patterns generated b y such submerged point sources w i l l be shown i n the next subsection.
Therefore, f o r this isolated source, the first two guide-lines given are confirmed. For other perturbations the decay of the pressure amplitude can be different, but i n the far field i n thi-ee dimensions (3D) the decay is still proportional to {LplRf' or even quicker.
To elaborate on the second guideline, i.e. ' A short pressure variation on the h u l l has just a local effect on the pressure field; a longer pressure variation is felt at larger distances'. F i g . 4 shows two examples o f the pressure field around an array o f 10 equal point sources on the x-axis. The first graph is f o r closely spaced sources, g i v i n g a short pressure variation; i n the second graph the sources are spread out over the interval [—4.5, 4 . 5 ] , and the length scale o f the pressure variation is larger. The source strengths are such that an equal pressure amplitude is generated at the lower boundary z — 2.2 f o r both cases. However, due to the greater length scale, at larger distance the pressure amplitude is twice as large f o r the second case, illustrating that longer pressure variations extend to larger distances.
\ •.
-0 5 X
Fig. 4 Pressure field around source array. Left short pressure
variation, right longer pressure variation with equal amplitude
Fig. 5 Hydrodynamic pressure distribution f o r K V L C C 2 tanker, i n
double-body flow {top) and free-surface flow {bottom)
Thus, the guidelines provided f o r step 1, relating the pressure field with the h u l l f o r m , are f a i r l y evident. H o w -ever, the objection could be raised that we have not con-sidered the effect o f the free water surface at a l l i n addressing the relation between h u l l f o r m and pressure distribution. W e have simply used a pressure distribution i n double-body flow, but is that still relevant to explain wave maldng? Or is the pressure field including the waves entirely different?
The answer evidentiy depends on the Froude number, since the double-body flow can be considered to be the l i m i t o f free-surface flow f o r Fn -> 0. W e show some examples to illustrate the level o f correspondence. Figure 5 compares the double-body and free-surface pressure dis-tributions, on hull and water surface, f o r the K V L C C 2 tanker model. The free-surface case is f o r Fn = 0.1423. Obviously there is a large degree o f correspondence between the h u l l pressure distributions. However, there is an increase o f the underpressures close to the water suif ace as a result o f free-surface effects, clearly observable at the
J Mar Sci Teciinol (2010) 15:331-344 337
Fn=0.0
Fn=0.18
Fn=0.25
Fn=0.316
F i g . 6 Hydrodynamic pressure distribution f o r Series 60 CB = 0.60 hull, in double-body flow (top) and at different Froude numbers
f o r e and aft shoulder. There is no apparent change o f the pressure at the stern gondola. This increase o f the pressure extremes, i n particular the underpressures, close to the water surface is a very common observation. Nevertheless, i n general the distributions remain similar.
Figure 6 makes a similar comparison f o r the Series 60 CB = 0.60 h u l l , at Froude numbers 0.0 (double-body), 0.18, 0.25 and 0.316. W e see that many o f the features o f the hull pressure distribution are kept f o r increasing speed. However, there is a clear increase o f the overpressure at the bow, which also shifts aft w i t h increasing speed. The underpressure at the fore shoulder is also augmented by the free-surface effects, and is i n addition affected by inter-ference w i t h the bow wave system, i n particular at Fn = 0.18 and Fn = 0.316. The aft shoulder underpres-sure shows similar effects and tends to move gradually aft w i t h increasing speed, interference again playing a role. We notice that further aft there is a greater Fn dependence, due to the cumulative effect o f waves generated further upstream. L i k e i n the previous case, the pressure distribu-tion i n double-body flow stifl gives a good impression o f the pressure distribution i n free-surface flows, as long as the wave making and wave interference are not too pronounced.
I n general, the connection that step 1 makes between the h u l l f o r m and pressure distribution, regardless o f free-sur-face effects, seems a u s e f u l approximation f o r most mer-chant vessels at usual Froude numbers. However, the free surface usually increases the pressure extremes and causes a slight aft shift o f these. Besides, the wave pattern gen-erated at some point along the h u l l causes an oscillating pressure field further aft.
Evidently, at higher Fn and more pronounced wave maldng the agreement w i l l be worse, and the pressure dis-tribution is not as easily explained i n terms o f the h u l l f o n n only. Moreover, there are some hull f o r m features f o r which double-body flow is substantially different f r o m free-surface flow. For example, f o r a bulbous bow extending just beneath the still-water surface, i n double-body flow the flow has to pass hoiizontally over the bulb and through the narrow gap between the bulb and its m i r r o r image; this can generate extreme pressure variations that are largely absent in free-surface flow as the flow direction is different. A l s o for immersed transom sterns, there are locally substantial differences i n pressure distribution between double-body and free-surface flow. I n such cases the double-body pres-sure distribution is still connected w i t h the h u l l f o r m as usual, but is less relevant f o r the wave making.
Therefore, i n practice we normally analyse the h u l l pressure distribution f r o m a free-suiface potential-flow computation, rather than that f r o m double-body flow, keeping i n m i n d the effect o f the free surface on the magnitude and position o f pressure extremes. W i t h expe-rience this somewhat less f o r m a l procedure works w e l l .
Step 2: From free-swface pressure distribution to wave pattern The guideline given above was: ' A wave is most effectively generated by a pressure variation o f comparable length and shape'. This tells us that a w a v y pressure dis-tribution m o v i n g over the stiU-water surface w i l l prefer-entially generate waves that have a comparable wave length. The condition o f coiTesponding shape applies to both the longitudinal and transverse directions.
To illustrate what this means. Figs. 7 and 8 show a variety o f wave patterns generated by synthetic pressure distributions acting on the free surface o f an i n f l o w w i t h speed V. The imposed pressure distributions are sinusoidal in longitudinal direction, w i t h a single period over a length Lp, and constant w i t h smoothed sides i n transverse direc-tion, w i t h w i d t h Bp. The pressure is given by:
^ 2 7 I(A - - A O ) '
p{x~xo,y - yo) = P • sin • f i y - yo) f o r -0.5Lp<x-Xo<0.5Lp, i n w h i c h ƒ()') = 1.0 f o r \y\ < 0.25Bp and
f { y ) = cos^ 2n{y - Q.25Bp) Ï0ïQ.25Bp<\y\<Q.5Bp
For a given speed, the length Lp and w i d t h Bp o f the pressure distribution have been varied to study the resulting wave pattern. The dimensions o f the pressure patch are here given as fractions o f the transverse wave length XQ f o r the speed considered.
Figure 7 shows h o w the wave pattern responds to vari-ations o f the length o f the pressure patch ( w i t h a sinusoidal shape over that length), at constant w i d t h Bp — X.Q. The top
V/L: O.C0S5 0 00? -0.0015 -0.001 -0.0005 0 D.Q005 0.Q01 O.Q015 O.O02 0 0025
-3 -2 -1
X / L
Fig. 7 Wave patterns generated by travelling pressure patches. Top
width Bp = XQ, length Lp = 2/lo; Second graph width XQ, length XQ,
Third graph width /IQ, length 0.5/lo; Bottom graph width 0.5.'.o, length
0.5io
figure is f o r a pressure patch length Lp = 2XQ. A S the transverse wave is the longest possible wave i n the steady pattern, this pressure distribution fits no wave w e l l , and the wave amplitudes are very l i m i t e d , waves being mostly generated by the sides o f the pressure patch. The second graph is f o r a pressure distribution w i t h Lp = /Q, which fits a transverse wave precisely and thus appears to generate a system w i t h pronounced transverse waves and a l i m i t e d amount of somewhat more diverging components. I n the third graph, f o r Lp — O.SAQ, the transverse components have decreased strongly and diverging waves are more
-3 -2 -1
X / L
X / L
F i g . 8 Wave patterns generated by travelling pressure patches, w i t h
length Lp = and widths Bp = 2Ao, and 0.25Ao (top to bottom)
pronounced, as these fit the longitudinal distribution w e l l , being shorter hy a factor o f cos 6 i f measured i n l o n g i t u -dinal direction. W h e n next also the w i d t h o f the pressure patch is halved to Bp — 0.5 XQ (fourth graph), the d i v e r g i n g waves fit even better and are still more dominant.
Figure 8 shows the wave patterns of pressure patches w i t h fixed length Lp — XQ and varying widths. One m i g h t expect dominance of transverse waves i n a l l cases, but the w i d t h o f the pressure distribution appears to matter as w e l l . I n the first graph, f o r Bp = 2Xo, there are j u s t transverse waves and very littie else. H a l v i n g the pressure w i d t h to ?iO still gives dominant transverse waves, but somewhat weaker. I f the w i d t h of the pressure patch is reduced to 0.25^0 (Fig- 8 third graph), transverse waves are still there but much lower; diverging components are dominant i n amplitude, as they fit well this small w i d t h .
Therefore, the guideline given above w e l l indicates the wave patterns generated by free-surface pressure distribu-tions. B o t h the transverse and the longitudinal extent matter, and the more localised a pressure disturbance is, i n length and w i d t h , the more i t tends to generate d i v e r g i n g
J Mar Sci Teclinol (2010) 15:331-344 339
waves. Anticipating qualitatively the type o f wave pattern f r o m a simple free-surface pressure distribution is rela-tively straightforward. Moreover, owing to the l i n k hetween the flow around the h u l l and the free-surface pressure distribution, anticipating the wave pattern f r o m a ship h u l l shape has got a rational basis, w h i c h is the main objective o f the analysis procedure.
Of course the guideline f o r step 2 is i n itself nothing new. There have been several studies o f wave m a k i n g caused by surface pressure disturbances, e.g. [13, 14]. Linear theory provides analytical expressions f o r the wave pattern [15] w h i c h indicate that the amplitude o f waves generated i n direction 6 is proportional to the 2 D Fourier transform of the distribution o f the longitudinal pressure derivative at the wavenumber o f those waves. Referring to the wave height expression f o r the far field (Eq. 5), the expression is: A ( ö ) + i 5 ( 0 ) npV^JJ 9A-FS exp ig(A-cos 9 +ysm( y2 cos21 d,Td)'. (8)
This confirms that a pressure distribution with a length scale and shape matching the wave components produces the largest wave amplitudes. Therefore, the simple rule given above is actually precise i n case o f pressure distri-butions o f infinite extent.
As a first illustration o f the results o f the combination o f both steps, we inspect the wave pattern generated by the still-water surface pressure distribution as i n E q . 7, thus approximating the wave pattern o f a submerged point source. Figure 9 shows that pattern f o r submergences o f 5, 4 and 3 times the length scale associated w i t h the point source. Defining the length scale o f the pressure distribu-tion at the still-water surface as twice the distance hetween m a x i m u m and m i n i m u m pressure, f o r a submergence o f 5Lp this is close to i o f o r the speed considered. The result is the generation o f a weak ti-ansverse wave system. I f the submergence is reduced, the amplitude o f the pressure disturbance increases q u i c k l y , but also the length scale o f the surface pressure distribution gradually decreases. Consequently, the amplitude o f all waves increases, but the rapid increase o f the diverging wave components is more evident. This supports the guideline that:
• Deeply submerged pressure disturbances can only generate longer waves; shallow pressure disturbances can generate also shorter waves.
Because o f its analytical background, also the second step i n the procedure may seem absolutely sound, but again there is an important possible objection to its use i n the present context: the waves generated by the surface pressure field are
-3 -2 X / L -1 r -1 -2 X / L
F i g . 9 Wave patterns generated by a submerged point source, at submergence o f 5, 4 and 3Lp (top to bottom)
computed w h i l e disregarding the presence o f the ship h u l l . I n reality the h u l l boundary condition must m o d i f y the waves i n the near field. Moreover, the waves do not propagate through still water, but through the curved flow field w i t h variable velocity around the h u l l . This produces changes i n the dis-persion relation, and refraction effects that m o d i f y the wave lengths, directions and amplitudes. One result is the apparent f o r w a r d shift o f the K e l v i n wedge o f a ship's b o w wave system, but also the strength o f diverging wave components is affected hy these effects [ 1 6 ] . However, the resulting differences are not too substantial f o r the dominant waves i n the pattern, so f o r the qualitative analysis and understanding that we a i m at, the approximation is applicable.
6 Relation with ship wave theories
W h i l e the discussions above may be physically plausible, some stronger theoretical foundation is desired. I n fact, related theoretical two-step approaches have already been proposed i n the past, albeit w i t h a different objective. Specifically the procedure is very much i n the spirit o f slow-ship theory, as demonstrated below.
Let us first indicate the f o i m o f the free-surface boundary conditions f o r a potential flow w i t h an imposed free-surface pressure field. The kinematic and dynamic free-surface boundary conditions, nondimensionalised w i t h ship speed and length, read:
(/>,C.+ 0 . = 0 a t 2 = av,3'), (9)
c P l - <\>] - < \ > ] ) - ^ F/ » ^ C , ( A -3 0 - C = 0 ^ ^ ^ ^
a t z = C(A-,)')>
i n which Cp(x,y) = p{x,y)/{}^pV'^) is the nondimensional pressure distribution imposed at the free surface. A combined f o r m o f both conditions is thus obtained:
W e see that the imposed pressure distribution provides a f o r c i n g term on the right-hand side i n the combined free-surface boundary condition, representing the streamwise derivative of the imposed free-surface pressure.
Next, we consider wave making by a ship, without any free-surface pressure distribution. The fiow is determined by the nonlinear boundary conditions 9 and 10 w i t h o u t the Cp contribution, and by a body boundary condition. As men-tioned, i n the past various linearised formulations have been proposed, and one o f the more successful ones was the so-called slow-ship linearisation or double-body linearisation, used i n e.g. Dawson's method [ 1 ] . I t is based on the assumption that the flow w i t h free surface is a small pertur-bation o f the flow without free surface around the h u l l , i.e. the double-body flow. The potential is then decomposed as:
4>{x,y,z) = ^x,y,z) + (p'{x,y,z), (12) where Vd* is the double-body flow velocity, and we
assume that the perturbation velocity V(p' = (9{Fn'^) < 1 at the free surface. W h i l e there has been some debate on the proper f o r m o f the free-suiface boundary condition, a usual version as derived e.g: i n [17], is
= ( $ , ( , . ) . , - + ( f M o (13)
w h i c h is to be imposed atz = C, ( A , Z ) =^Fu^{l - V O - V<1>). Therefore, the right-hand side can be written as
2 V dyJ^Fn^-J
I f we compare this expression w i t h Eq. 11, we see that, save f o r the second term which is a 'transfer t e r m ' approximating Q)^ at z — Cr> it conesponds precisely w i t h
the f o r c i n g by an imposed pressure distribution. Comparing the f o r m of the free-surface boundary conditions we see that, i n slow-ship theory, the wave pattern is considered as generated by a free-surface pressure distribution
c.(.,3') = - p r , ,
w h i c h is the negative o f the pressure f o u n d i n double-body flow at the still-water surface. The minus sign can be understood f r o m the fact that a high pressure at the still-water surface i n double-body flow generates a positive local wave elevation; to get the same wave elevation without the h u l l being present, we have to impose the opposite o f that pressure on the water surface.
The distinction o f the two steps, and the use of the free-surface pressure distribution f r o m double-body flow to determine the wave generation, thus agrees w i t h slow-ship theory, and just like i n our procedure, t w o steps have to be made to apply slow-ship theory methods. However, an essential feature o f slow-ship theory was the fact that i n that second step the wave propagation over the curved and variable double-body flow field (which determines the vari-able coefficients at the left hand side) was taken into account. This is an aspect that we disregard i n our approximate pro-cedure, but again, our a i m is understanding not predicting.
Moreover, i n most o f the slow-ship methods also the h u l l boundary condition was taken into account i n the second step. On the other hand, i n earlier methods more analytical techniques were attempted, and the h u l l bound-ary condition was sometimes disregarded i n the wave pattern prediction. Baba and Takekuma [18, 19] actually derive the same slow-ship boundary condition, and solve the wave pattern as i f generated by the opposite o f the double-body free-surface pressure distribution i n the absence o f the ship h u l l . I n the derivation the nonunifor-m i t y o f the double-body flow is taken into account, but their expression f o r the far-field wave pattern is actually equal to that f o r a pressure distribution i n an undisturbed fiow [15], the n o n u n i f o r m i t y o f the flow only remaining i n the expression f o r the free-surface pressure i t s e l f
Another comparable procedure is discussed by L i g h t h i U [20]. Again, the flow field is thought to be composed o f a double-body fiow field and a resulting wave disturbance. The double-body flow field causes a pressure variation over the still-water surface, which again generates the wave pattern. However, i n this reference there is no derivation, and no discussion o f the disregarded effects of the non-u n i f o r m i t y o f the flow; the pnon-urpose i n his book again is understanding rather than prediction.
I n any case, the two-step procedure proposed here appears to be related w i t h slow-ship theory i n general, and w i t h some o f the earlier versions, such as that by Baba and Takekuma, i n particular. Here one m i g h t suppose that this
J Mar Sci Teclinol (2010) 15:331-344 341
w o u l d l i m i t the applicability of our approach to l o w Froude numbers only, since slow-ship theory is asymptotically valid f o r Fn 0. However, f r o m experience with e.g. Dawson's method it is k n o w n that, i n practice, this linearisation works fairly w e l l f o r most merchant vessels and is not confined to low speeds only. Even so, the s i m -p l i f i e d slow-shi-p methods mentioned, and the additional simplifications we make, e.g. disregarding the hull bound-ary condition f o r the waves and neglecting the effect of the double-body fiow n o n u n i f o r m i t y on the wave propagation, may ask f o r some further validation.
7 Validation of the two-step approach
As we have pointed out, the two steps distinguished i n our procedure separate physical effects that i n fact are occur-ring together and interact. Evidently, mutual infiuences are thus omitted. One may wonder whether this still leads to the right conclusions on the relation of h u l l f o r m and wave pattern.
The previous section has shown that there is a strong relation o f the two-step approach advocated here w i t h early slow-ship linearised ship wave-making theories. Therefore, a global validation is possible by evaluating the wave making o f ships by those methods. However, we do not need to f o l l o w the methods very precisely, as some of their simplifications were required by the computational possi-bilities at the fime o f development. Therefore, what we w i l l do is the f o l l o w i n g :
« Compute the double-body fiow around a ship h u l l , using a usual panel code essentially f o l l o w i n g the method o f Hess and Smith [ 2 1 ] .
• Evaluate the resulting pressure distribution pix, y) at the still-water surface around the h u l l , f r o m the velocities f o u n d i n a dense net o f ' o f f b o d y points' on the still-water surface.
• Impose the opposite of that pressure distribution, —p(,x, y), on the free surface and compute the wave pattern generated by i t , i n the absence o f the ship h u l l . W e simply use here the nonlinear potential flow solver, not any linearised formulation, as it is available and very fast.
The question arises of what to do w i t h the waterplane area o f the ship. This part o f the water surface i n the wave pattern computation is internal to the h u l l i n the double-body flow computation, so we do not have a pressure distribution here. Various choices have been made i n the past: Baba and Takekuma impose no pressure i n this area, while LighthiU suggests to extrapolate the pressure field next to the ship inside i t , i.e. f o r a symmetric case the pressure is largely constant i n transverse direction across
the waterplane area. I n some first evaluations we f o u n d that the latter m o d e l leads to exaggerated transverse waves. This can he understood i f we imagine a very shallow, very wide ship h u l l : any local pressure at the waterline w o u l d then be modelled as a wide pressure strip across the beam of the h u l l , w h i c h w o u l d have much too large an effect on transverse wave maldng. I n the examples shown below we have therefore not imposed any pressure i n the area o f the ship waterplane itself, just the pressure field surrounding i t . I n this way we obtain a prediction o f the wave pattern subject to the same assumptions and simplifications as underlie the analysis approach. W e w i l l compare that wave pattern w i t h the ship wave pattern predicted directly by the nonlinear free-surface potential flow code RAPID, i n w h i c h these approximations are not made; thus we get direct i n f o r m a t i o n on the importance o f the approximations. Moreover, we compare w i t h experimental data.
The first case is the Series 60 = 0.60 model at Fn = 0.316. Figure 10 compares the wave pattern so obtained f o r the pressure distribution w i t h that o f the ship h u l l itself. The agreement is remarkable. The wave cuts at y/L = 0.2067 shown i n F i g . 11 show that the approximate procedure actually predicts the same type o f waves f o r this case, but w i t h small differences i n phase and amplitude.
Figure 12 shows a similar comparison f o r the Kriso Container Ship, a benchmark case w i t h bulbous b o w and transom stern. The differences are perhaps somewhat larger here, m a i n l y i n the amplitude of diverging bow waves, but the m a i n features o f the wave pattern, and the essence o f what is needed i n order to understand the relation between h u l l f o r m and wave pattern, are again w e l l represented. The wave Cut i n F i g . 13 shows good coixespondeiice w i t h the result o f the complete nonlinear prediction, w h i c h is i n very good agreement w i t h the experimental data.
The last comparison we show is f o r the 'Hamburg Test Case', a containership, at Fn = 0.238. Figure 14 shows the
Fig. 10 Computed wave patterns, o f Series 60 = 0.60 at
Fn = 0.316 (left side), and o f its double-body pressure field moving
at the same speed over the still-water surface (right side), in a perspective view f r o m ahead
0.01
0.005
-0.005
-0.01 h
X/L
Fig. 11 Computed wave cuts at ylL = 0.2067, f o r Series 60
Cg = 0.60 at Fn = 0.316 (full Une), and of its double-body pressure
field moving at the same speed over the still-water surface (dotted
line), and experimental data [22] (markers)
A .
Fig. 12 Perspective view o f computed wave patterns, o f the Kiiso
Container Ship (KCS) at Fn = 0.26 (left side), and of its double-body pressure field moving at the same speed over the still-water surface
(riglit side)
wave patterns and F i g . 15 the wave cuts, again w i t h experimental data. W h i l e there is no doubt that the pre-dictions f r o m the complete free-surface panel method are better, the ability o f the two-step approach to represent the main features o f the wave pattern is evident.
W e note i n passing that the Froude numbers f o r these examples d i f f e r , but there is no indication that the slow-ship-like method used works better f o r l o w e r than f o r higher Fn; other aspects o f the particular case dominate. F r o m a number o f such comparisons we have observed that larger deviations occur i n cases w i t h large bulbous bows very close to the water surface, or w i t h immersed transom sterns. I n both cases the double-body flow gives no adequate representation o f the local pressure field. Therefore, f o r qualitative understanding of the wave making, inspection o f the h u l l pressure distri-bution f r o m a free-surface potential flow computation may actually be preferable, at least i n such cases.
0.006 h 0.004 h 0.002 -0.002 -0.004 h -0.006 -0.008 -0.5 0.5 X / L
Fig. 13 Computed wave cuts at y/L = 0.1509, f o r KCS model at
Fn = 0.26 (full line), and of its double-body pressure field moving at
the same speed over the still-water surface (dotted line). The markers represent experimental data
1 - 1 1 1 1 1 1 1 1 1 II
Y/L -OOI -0C03-O.0C5 -0.004 -0 0Q2 0 0.003 ODM DOCS 0OÜ3 0.01 1
Fig. 14 Computed wave patterns, of the Hamburg Test Case at
Fn = 0.238 (upper half), and o f its double-body pressure field
moving at the same speed over the still-water surface (lower half) Overall w e believe that the agreement o f the wave patterns f o u n d by the simplified, two-step approach w i t h those f r o m a complete wave pattern prediction is striking and supports the adequacy o f the simple analysis procedure proposed here.
8 Applications
The general philosophy outlined above enables us to understand and foresee a variety of, parfly w e l l - k n o w n , properties o f ship wave making and design trends. W e mention just a f e w points below to illustrate the use o f the approach; R e f [9] gives several additional examples.
J M a r Sci Teclinol (2010) 15:331-344
0.008
0.004
-0.004
\--0.008
F i g . 15 Computed wave cuts at y/L = 0.184, f o r the Hamburg Test Case at Fti = 0.238 (full Une), and o f its double-body pressure field moving at the same speed over the still-water surface (dashed line). The markers represent experimental data
• Transverse waves have the largest weighting factor i n the resistance expression. Transverse waves are stron-gest i f the free-surface pressure distribution has a length scale matching those waves, i.e. Lp x XQ. For the significant, overall pressure variation along the hull spanning f r o m b o w to stern, this is the case at Fn !V 0.4, which is the explanation o f the primary resistance hump.
• The overpressure around the bow stagnation point is rather extensive f o r a blunt b o w and induces a pressure at the still-water surface over a wider area, whereas f o r a sharp bow i t is more localised i n length and w i d t h . Therefore, a sharp b o w generates more diverging bow waves than a blunt bow.
• A wide, flat transom stern tends to generate transverse waves due to the w i d t h o f the pressure disturbance i t creates. However, the corners o f a transom stern are local pressure disturbances creating sharply diverging waves. • A catamaran generates stronger transverse waves than
one w o u l d expect f r o m the slenderness o f its side hulls, due to the larger transverse extent.
• A submarine at some distance under the surface cannot generate diverging waves, as there are no short scales i n the surface pressure disturbance.
• A t l o w Fn, all possible waves are relatively short, and respond to short surface pressure disturbances w h i c h can only be induced by near-surface h u l l f o r m features. Therefore, i t is the waterline shape that counts. However, at higher Fn the waves are generated by larger length scales that are also affected by f o m i features farther beneath the surface, and the sectional area curve becomes more important.
343
Localised h u l l f o r m features far beneath the water surface (e.g. stern bulbs) are generally insignificant f o r wave making.
W h i l e one wants to avoid matching o f a pressure length scale w i t h a wave to minimise wave making, the opposite is the case i f one wants to design a h u l l f o r m modification to suppress a given wave. I f a certain dominant wave is observed i n a computed or measured wave pattern and i t needs to be reduced, one should aim at making a h u l l f o r m modification that creates a pressure change w i t h a length scale comparable to that wave.
Design trends f o r bulbous bows can be understood f r o m the desired matching o f length scales, the guidelines on the relation o f the suiface pressure distribution w i t h the hull pressure distribution etc.
Furthermore, the general notions f o l l o w i n g f r o m the approach described play an important role i n the improvement o f h u l l designs i n practical applications. The practice we use is that first a computation is made f o r an i n i t i a l h u l l f o r m , normally using a free-surface potential flow code. I n a careful analysis o f the com-puted wave pattern, i n combination w i t h the h u l l pres-sure distribution f o u n d f r o m the same (free-surface) computation, the guidelines and insights summarised i n Sect. 4 and several others are used to relate the dominant wave components w i t h h u l l f o r m features, and to decide on h u l l f o r m modifications that w o u l d reduce those waves. Next, these modifications are again evaluated b y a f u l l y nonlinear free-surface potential flow computation, i f not a free-surface viscous-flow computation. A f t e r that, a next analysis and improvement step can be made. I n just a f e w directed design steps a substantial reduction o f wave m a k i n g and wave resistance can often be achieved. Sinfilarly, the same insights have proven successful i n the context o f a systematic h u l l f o r m variation procedure to achieve wave resistance reduction [7, 8 ] .
9 Conclusions
The main question addressed i n the present paper is: What is the relation between h u l l f o r m properties and ship wave patterns? The approach advocated here, w h i c h is a f o r -malisation o f a way o f w o r k i n g developed at M A R I N i n the course o f many years, decomposes this into two steps: the relation between ship h u l l f o r m and free-surface pressure distribution, and the relation between that pressure distri-bution and the wave pattern i t generates. The separate steps have been discussed and illustrated. M a i n conclusions can be summarised as f o l l o w s :
• The advantage o f the decomposition into t w o steps is that each o f these is ruled by a f e w simple and understandable relations as Usted i n Sect. 4. A n t i c i p a t -ing the wave pattern generated by a h u l l - i n d u c e d surface pressure distribution is m u c h more straightfor-ward than anticipating the wave pattern o f a ship. • The two-step approach proposed here is closely related
w i t h the f o r m e r 'slow-ship linearised theory', but has additional approximations as i t does not pay attention to the propagation o f waves over the f l o w f i e l d around the h u l l , and to the e f f e c t o f the h u l l boundary c o n d i t i o n on the waves. These simplifications are, however, essential f o r conceptual s i m p l i c i t y , and have been f o u n d to be permissible f o r the present puipose. The analysis approach has been f o u n d to be applicable to ships at normal displacement speeds, not o n l y slow ships. • I n computational studies we have observed that a strict
computational equivalent o f the general approach predicts wave patterns that coiTespond at least qualita-tively w i t h the wave pattem according to a f u l l nonlinear theory, and w i t h experimental data. W h i l e i t is not meant to propose an alternative p r e d i c t i o n method, this lends support to the use o f the method f o r analysing ship wave patterns and their dependence on h u l l f o r m .
Evidentiy, applying these insights to i m p r o v e ship h u l l f o r m s f r o m a wave-making point o f v i e w stiU remains an art, and substantial experience and f e e l i n g are s t i l l needed. However, a sound qualitative understanding o f the physics allows one to make directed steps towards an i m p r o v e d h u l l f o r m . Thus, a s i m p l i f i e d analysis helps to achieve good results i n a process based o n advanced computational tools. T o advocate this combination was an objective o f this paper.
Acknowledgments The permission f r o m the Society of Naval
Ai-chitects and Marine Engineers ( S N A M E ) to reproduce some material f r o m [9] i n this paper is acknowledged.
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