Abstract: Practical issues about wave forces to be included in the ship manoeuvring mathematical model (MM) are formulated. A current status of research in this field is thoroughly investigated in aspects of data availability and validity. The impact of first order wave forces is briefly characterised, the major interest is turned upon the second order forces. A strong effect of the latter is proved through a simulation of a ship turning test in regular wave. Despite a rather theoretical meaning, the ship manoeuvring simulation in regular wave is however questioned.

　

 The ship manoeuvring behaviour in waves, like any other aspect of the ship manoeuvring is under steady development. Though a big progress has been made so far, the consistency and completeness of the data for second order wave forces are still being looked for by means of the theoretical, experimental and empirical (regression, analytical) methods.

　

 The bibliography concerning the ship manoeuvring forces due to waves (of first and second order), as compared to references dealing with ship motions (heave, pitch, roll) and structure loads (shear forces and bending moments) in a seaway seems to be like a drop in the ocean. A major reason (among many) for such a situation (also strictly connected with e.g. a financial support for the research) is that the ship manoeuvring is not so important i.e. the ship service safety and/or economy is directly little dependent upon the ship manoeuvring to justify such an interest. In rough weather conditions, the adopted practise is to suspend manoeuvring (approaching, berthing, mooring, etc.) due to uncertainty in random (unpredictable) conditions and because of the uncertain/unknown knowledge of the ship's response (though some mathematical models try to put some light hereto). Sometimes, a qualified pilot (master) is employed. A knowledge, experience, or intuition of the latter is often restrained only to local conditions and some ships, thus being very narrow, and moreover, kept in his mind ('not available to a wider audience'). A handover of the ship's con to such experts is sometimes risky as they are still humans and their knowledge, though hoped to be a thorough one, is not perfect for objective reasons - as always reflecting the current status of marine hydrodynamics and control engineering science. It should be emphasised here that the advantage of an expert over pure algorithms comes from his real intelligence and 'a mental database' of real-world experiences as being hard to get a quantitative and systematic form. The point is that advances in the science, among others pertaining to wave forces and ship manoeuvring response estimation, could greatly improve the current manoeuvring practise.

　

 Though in harbour areas (as sheltered ones) the wave action does not seem to be of a destructive magnitude, where the wind influence is just considered to have a higher impact, there are plenty manoeuvres to be performed in open sea or less sheltered areas. Moreover, the role of wave effect (see also most text-/handbooks on ship manoeuvring) is frequently diminished, probably due to the fact that the wave propagation coincides with the wind direction and 'augments' the latter. However, relying barely in such circumstances on ship aerodynamic characteristics obtained e.g. through experiments, we could underestimate the resulting linear and angular ship motion. It is true that estimation of the wave impact meets some difficulties (theoretical and/or experimental), but limiting oneself to a relatively easier wind effect is not excused, at least in view of latest progress in research. A typical dispute among scientists is provided e.g. by R.A. Barr contribution to a discussion accompanying [20]. A fact that the wind effect is sometimes weak is raised e.g. in [2]. The aim of the present study is to examine the state of the art in estimation of the wave second order forces from practical point of view (the availability and validity of data), and their importance (sensitivity) in the manoeuvring simulation. Though many references pretend to solve the problem, they do not really owing to different reasons. Since this is a preliminary study, the case of turning test is analysed.

　

 Having a look at typical values of the wave second order sway force and comparing it with the hull hydrodynamic forces during manoeuvring, the significance of the former is obvious. A good example (very easy but thus more convincing) is shown below. The hull hydrodynamic sway force during a pure lateral manoeuvring motion is given by:

　

　

while the wave second order sway force reads:

　

　

 Assuming the non-dimensional coefficients of both expressions as equal (e.g. of order ca. 0.8, what is easy to be deduced from available model tests and in the case of low wave/ship length ratio - λ/L), one could arrive at the following relationship:

　

　

 Taking further a ship draught about 10[m], the ship sway velocity is nominally reaching the wave amplitude - vy[m/s]≈ζo[m]. Hence, even at relatively small wave amplitudes of order 〜1[m], the sway velocity (and thus the drift angle) is able to double (or more) its value as usual in calm water manoeuvres. Therefore at least in some cases, the wave action through second order forces is not negligible in ship manoeuvring.

　

 The ship manoeuvring motion differential equations are as follows (under wind and wave conditions):

　

　

where

　

　

　The wave forces (forces and moments) in a general 6DOF form could be divided basically into two components (able to be examined independently) -the first order forces (at the earliest discovered and mostly investigated so far) and the second order forces:

　

　

 In the ship manoeuvring (a planar motion in 3DOFs) the attention is paid on the surge, sway forces and yaw moment. However, the ship heave, pitch and roll motions are accounted for indirectly in (5), as the wave manoeuvring forces to a great extent rely upon characteristics of those motions. Though in a calculation of the first order forces (theoretical methods have been rather properly validated in this case already), the heave, pitch and roll motions could be omitted, they are absolutely significant in the second order forces. A reference is made here to e.g. the widely known theory of the added resistance calculation in waves (actually the second order surge force) and the problems associated thereto.

 An implementation of the wave action in (4) does not encompass only the wave external excitations (or disturbances) in form of first or second order forces. Moreover, the resulting oscillating motions in surge, sway and yaw change 'a constant nature' of added masses and hull hydrodynamic forces (referred to as damping forces in linear models of the ship motions in waves) - e.g. [3], [7], [25]. Those are experienced as steady during calm water manoeuvring (zero frequency case) and being functions of the wave encounter frequency while manoeuvring in waves. Some limited research results exist so far with reference to a few hull shapes. It is very interesting to test the influence of the sway added mass and some hull linear derivatives (as the mostly sensitive in ship manoeuvring), being actually frequency functions, upon the final manoeuvring prediction. However, an exact (adequate and accurate in view of all other effects/interactions) model of ship manoeuvring is required to carry out such a sensitive simulation.

　

2.1 Brief synopsis on first order forces

　

 The first order forces are being decomposed into Froude-Krylov (F-K) and diffraction terms:

　

　

 The former originates from the wave pressure on the hull if this one is fixed and transparent for waves (amazingly simple in integration), the latter makes a regard to the interference (disturbance) the ship hull, by her presence in the water, produces to the wave system. The ship, in case of the diffraction terms, could be treated as fixed one moving with a constant forward velocity (no other motions than in surge direction) and/or a free floating one also with the forward velocity. However, the advance velocity problem is capable to be solved in many more or less approximate ways. The F-K terms are dominant (not only with regard to diffraction terms but over the second order forces as well) in some applications e.g. with regard to the course keeping in following waves - [4] and [23] for instance, anyhow the diffraction part generally makes an essential contribution. The problem of sailing in following waves (low encounter frequency) is rather a specific region of the open-sea ship manoeuvring and more prone to smaller ships at higher advance speeds e.g. fishing vessels. Some other important issues about the first order forces are presented below.

　

 If the ship is subject to a harmonic (regular) deep water wave of a profile given by:

　

　

all the first order forces (forces and moments in all directions), of components (7) either taken totally or separately, and denoted hereafter symbolically as F1, are harmonic functions of the encounter frequency:

　

　

where the encounter frequency in a general form is:

　

　

 The wave incidence angle γWVrel stands at 0[°] for head waves. The amplitude of first order forces is always proportional to the wave amplitude. However, both the amplitude and phase angle in (9) (strictly their transfer functions) are further the functions of the hull shape and dimensions, wave incidence angle, absolute wave frequency, encounter wave frequency and the ship surge velocity (represented by Froude number, the sway and yaw velocities are disregarded):

　

{f10, ψ1} = ζo・f('hull shape', γWVrel, ω,ωE, Fn)　(11)

　

 The available plenty monographs on wave forces e.g. [25] could be consulted for details of (11). The most important is that YWVrel, ω and Fn exist independently from ωE, though the latter contains also all the former variables. Taking into account the hydrodynamic similarity laws (the model test results could be directly and easily used), the best representation of first order forces (and not only) for ship manoeuvring purposes (lookup tables), are given by the following alternative relationship:

　

{f10, ψ1} = ζo・f('hull shape', γWVrel, λ/L, Fn)　(12)

　

 The encounter frequency is not included exclusively and the absolute frequency is replaced by the wave-ship length ratio. In case of the F-K components only, the Froude number in (12) does not usually occur.

　

 The mean of first order force over encounter wave period equals to zero thus not producing 'a long-term' steady force able to change e.g. 'average' ship manoeuvring track and/or other motion states. In such a situation any shift e.g. in a simulated turning circle should be properly investigated against possible ship course dynamic instability, especially at high wave heights - some studies do not have such an analysis e.g. [10], [21]. Normally the first order forces cause high frequency (equal to the encounter frequency) oscillations imposed upon manoeuvring kinematic data, but keeping their original average values as in calm water. Though the forces are rather huge, they do not make big 'jumps' in kinematic data due to a short acting time. A very good example of the first order force magnitude, roughly estimated through F-K parts (without Smith effect), is based on a rectangular barge of length L and beam B as next.

 Considering the average difference of the wave profile on opposite ends of the barge as Δζ, both in longitudinal and transverse directions, the surge and sway force are as follows:

　

　

or approximately:

　

　

 Even at relatively medium wave heights, (15) and (16) lead to force values comparable with normal manoeuvring forces.

　

 In the irregular wave (real-world conditions), a commonly adopted and validated approach is to use a linear superposition of regular wave force contributions according to the wave energy spectrum. In other words, the irregular wave manoeuvring analysis is always dependent upon regular wave impact sub-analyses. Therefore a good simulation algorithm should consist of data related to regular waves.

　

2.2 Review of available second order force data

　

 Since early 60s, the theory of wave second order forces has reached a great progress, though there is still much to do. It is still behind the theory of first order forces (and ship motions), in which case the numerical calculations take the place of very expensive and time-consuming model tests. In spite of some encouraging results, the theory of second order forces requires still plenty model tests for validation, particularly in the whole range (0°-180°) of the wave incidence angle as of importance in ship manoeuvring. Studying the available references, the point is that the calculated second order sway forces are comparatively less validated than those in surge direction (so called 'added resistance' as mentioned before). The situation is more and more critical in case of the second order yaw moment, actually constituting the arm of the sway force and being very sensitive to the under-water hull. For all above reasons, the scientists are very careful while presenting the theoretical results in the area lying far from the region verified already. In some applications, notably in offshore engineering design and analysis, the theoretical methods are well accepted, as relating to rather simple shapes. In case of larger structures e.g. ship hulls, the numerical methods provide always a qualitative adequacy and only to some extent a quantitative one, For design purposes both of them may be sufficient as 'a safety margin' is commonly adopted. In view of acquiring the theoretical data for precise manoeuvring analysis and/or manoeuvring control, more model tests and improvements to the theory are required.

　

 The second order forces are proportional to the square of wave amplitude. They could be considered in regular waves (simple but mostly theoretical case) and in irregular waves (complex but true case). Under a regular wave, the forces always assume constant values i.e. independent from time. In an irregular (random) wave of a given spectrum, they could be treated as a linear superposition of 'regular' responses. The latter procedure (normally a non-linear transformation is substantial) has been validated practically to some extent (mainly for added resistance) by model tests and theoretical analyses (so called quadratic transfer functions) e.g. [26], [28], [30]. Due to non-linear interactions between adjacent parts of sea spectrum, the irregular wave produces also the low frequency second order forces, which were given a special interest in e.g. [22], [26]. These kinds of forces oscillate around the mean second order forces (obtained by the mentioned above superposition) and according to e.g. [22] are not rather significant in ship manoeuvring, though are very essential in e.g. SPM moored vessels analyses. For these reasons at least, the low frequency forces will not be dealt with later in the paper. A good review of theoretical methods and efforts for calculating the second order forces was provided e.g. in [26].

　

 It is worthwhile to note that many second order force estimation algorithms are present, though they could be assigned into a few categories, each one has pluses and minuses (none of them is a best remedy for all situations). In relation to the added resistance as mostly representative and meaningful, the lack of decidedness is seen e.g. in [18], [9]. Anyhow, the theories give here a fairly good prediction (despite of the method used) for head/bow waves and poorer one for other wave directions (especially in case of following/quartering waves- e.g. [19], [12], [9]). The theories for calculating other second order forces (of similar numerical nature) - the sway force and yaw moment - are awaiting a further progress to eliminate the sources of their mutual discrepancies. It means they require a detailed and wide examination of the background assumptions sensitivity and experimental validation of different phenomena known up to now through theoretical calculations mainly e.g. [6], [16], [24]. Other aspects, which are seldom being risen (though are discussed in some works) are computational errors and/or the inaccuracy of experimental measurements.

　

 For practical purposes, an analogous relationship to (12) should be available for the ship under consideration in the whole range of regular wave incidence angle, its length and ship Froude number:

　

{FxWV2, FyWV2, MzWV2} = ζO²・f(γWVrel, λ/L, Fn)　(17)

　

though in case of the added resistance some approximate empirical/graphical methods e.g. [30] (γWVrel from 0°to 180°) or theoretical/tabular ones e.g. [9] (γWVrel from 0°to 90°) exist for irregular waves. Though the Froude number dependence is absolutely significant for the surge force, e.g. [11], it seems to have a smaller impact in the sway force estimation but again very strong in the yaw moment-e.g.[14].

　

 A review of published ready-for-use regular wave relationships (17) has been carried out as aimed to get some typical charts of them for any ship type, thus being able to be tuned e.g. through full scale manoeuvring trial data. Some remarks are presented below.

　

 The regular wave data for the surge second order force (added resistance) in the whole range of independent variables in (17) are however hardly available in the literature. They relate to one specific ship and are mostly theoretical ones without an extensive validation e.g.: [19] and [27] (γWVrel up to 90°, Fn=0.2, theoretical), [31] (γWVrel up to 90°, Fn=0, theoretical), [30] (complete chart obtained through the experiment but reprinted from other source, which is not available to the Author), [24] and [5] (Fn=0, theoretical). A huge contribution in this field seems to have been analytical method [29] as supplying easy and comprehensive empirical formulas for an arbitrary ship hull. Anyhow, a closer look at the background original experimental data allows to say that the method is not reliable (too great simplification, particularly pertaining to the wave incidence angle dependence and the propeller thrust increase measurement as a source of the added resistance estimation).

　

 In relation to the both second order sway (drift) force and yaw moment, it is worthwhile to refer also to the sources mentioned in the previous paragraph (concerning the added resistance) of the same limitations also, namely: [19] and [27] (sway force only), [31], [24], [5]. Additionally, a thorough theoretical analysis was performed in e.g. [16] (Fn=0, theoretical). Though being limited to a few specific cases, [8] (Fn=0, λ/L=0.35, 0.5, experiments on a fixed model) and [14] (Fn=0 and Fn≠0, λ/L=1.0, theoretical) bring e.g. some new patterns of the second order yaw moment. Finally a very interesting approach (requiring anyhow a deeper look inside) is applied e.g. in [17] supplying some semi-empirical practical formulas for the second order forces.