*4.1. Earlier Proposals*

As has already been noticed, the existing IMO manoeuvrability standards [1,33] had been devised without account for external disturbances.

The authors are not aware of any reasons for such a decision and of any related discussions. In fact, it is possible to avoid including requirements into official standards presuming that the environmental factors can be taken into account for each design individually using available prediction methods whenever it is considered necessary by the customer. In particular, this viewpoint was discussed and recognized with some hesitation as preferable by the first author 25 years ago [48].

However, the current trend is exactly to complement the existing IMO set of standards with additional environment-dependent requirements [9,30,49,50].

Even though the existing IMO standards are formulated for calm water, the importance of accounting for environmental factors had been discussed in the literature more than once. These factors, first of all, are disturbances caused by wind, sea waves and current although in a more general sense variations of the ship draught and trim, shallow water effects and hydrodynamic interaction with other borders and bodies can also be interpreted as perturbations.

Norrbin [17] underlining the importance of accounting for the wind action pointed out that "some passenger ferries have the relative windage area comparable to that of the old windjammers". Spyrou and Vassalos [51] remarked that while a ship may have good manoeuvring performance in calm sea and even keel, its behaviour may become unacceptable under external disturbances and/or in a trim-by-the-bow condition.

Lowry [52] formulated a general definition that "a vessel is controllable if the forces and moments generated by the vessel can overcome the [external] aero/hydrodynamic forces acting on the vessel". As the "forces generated by the vessel" are also hydrodynamic, this statement may look not very exact but it is clear that the author meant here disturbances from sea waves.

Li and Wu [37] also paid attention to the necessity of the introduction of criteria related to environmental factors. Koyama and Kose [18] stated that in developing manoeuvring standards the current should not be considered at all as long as strong tidal currents presumably can be avoided while the wind effects can not and must be taken into account. It was proposed to consider the wind speed 15 m/s for the course keeping at 5 kn speed of

advance. A similar steering-in-the wind formulation was also discussed in [40]. Zaky and Yasukawa [30], studying the effects of the load condition for the virtual tanker KVLCC2, discovered that the wind balancing rudder angle becomes smaller in ballast condition with a stern trim than in full load in spite of a substantially larger relative windage area. Some researchers, including the first author of the present paper, see [48], tended to believe that the capacity of a ship to withstand external factors in the sense of manoeuvrability should not be standardized but, in any necessary case, must be subject to design specifications and special analysis.

Latest studies on manoeuvrability in adverse weather conditions mainly focused on second-order wave excitation effects including added resistance in waves [13]. The latter is really important in the formulation of minimum-power requirements which include the ability of the ship to keep controllability under adverse conditions characterised by the significant wave height varying from 4.5 to 6.0 m and for the wind speed from 19 to 22.6 m/s depending on the ship's length. The controllability must be demonstrated using some trusted prediction method in the sense that at some small speed, e.g., 2 knots, the ship is capable to maintain heading advancing into head seas [13]. Regarding that "trusted method," it is clear that at present the only reliable method for predicting a ship's behaviour in waves is an application of CFD codes which are, however, prohibitively slow for performing systematic calculations, especially in irregular seas. However, as will be discussed below, there is a possibility to bypass this difficulty.

The latest case of interest in manoeuvring criteria accounting for external factors is associated with expected underpowering caused by EEDI requirements while "low-powered" tankers designed to minimize the costs of construction, operation and maintenance had been already mentioned by Norrbin [17]. It still does not mean, however, that EEDIcompliant ships require some special sets of criteria and standards but they are more likely to become vulnerable to external factors and the existing standards are supposed to be revised and extended in such a way that this vulnerability is limited to acceptable levels.

Of course, any kind of hydrodynamic and aerodynamic perturbations must be somehow checked and counteracted and the effectiveness of the ship control device must be sufficient for that purpose. The hydrodynamic interaction issues (such as the ship-to-ship interaction) may be quite important per se but they are hardly associated with the underpowering as in this case all categories of forces will be simultaneously proportional to the squared ship speed.

If the minimum-power requirements lead in this or that case to the reduction of the installed power and of the ship design speed as compared to those selected without the influence of such requirements, it may happen that either the IMO standards must be somehow modified or complemented with standards based on additional criteria.

In general, the following external factors can be considered:


It is interesting to note that all (!) these factors had been considered during the development of earlier Russian national manoeuvring standards as presented by Mastushkin [19] who led and supervised that development. As those national standards, see [8], in no way influenced the development of the IMO criteria, the issue of the external factors practically had never been discussed on the international level before studies on the STANAG standards [15]. These standards, however, were driven by a desire to account for all possible navigational situations and missions sensitive to the manoeuvring performance of naval combatants and auxiliary ships. While this approach is understandable, it is hardly rational as long as the simplicity principle is definitely neglected.

### *4.2. Standards Accounting for Disturbances from Sea Waves*

The task of developing manoeuvrability criteria accounting for the wave action is more complex and contains more uncertainties than it could seem at first sight. It is only evident and known from the seamanship practice that less powered (slower) ships are more sensitive to the action of the sea. The scenario approach followed by some researchers [49,50] looks quite natural and suitable. However, the selection of critical scenarios is not easy. Probably, an adequate understanding of the complexity of the task of handling a ship in rough sea waves can be obtained from the following abridged quotation [53], (pp. 118–121):

"The great wave flung the bows up, pushed the *Ulysses* far over to starboard, then passed under. The *Ulysses* staggered over the top, corkscrewed wickedly down the other side, her masts, great gleaming tree trunks thick and heavy with ice, swinging in a great arc as she rolled over, burying her port rails in the rising shoulder of the next sea.

'Full ahead port!' . . . 'Starboard 30!'

The next sea, passing beneath, merely straightened the *Ulysses* up. Next, at last, Nicholls understood. Incredibly, because it had been impossible to see so far ahead, Carrington had known that two opposing wave systems were due to interlock in an area of comparative calm: how he had sensed it, no one knew, would ever know, not even Carrington himself: but he was a great seaman and he had known. For 15, 20 s, the sea was a seething white mass of violently disturbed, conflicting waves—of the type usually found, on a small scale, in tidal races and overfalls—and the *Ulysses* curved gratefully through. Then another great sea, towering almost to bridge height, caught her on the far turn of the quarter circle. It struck the entire length of the *Ulysses* for the first time that night—with tremendous weight. It threw her far over on her side, the lee rails vanishing. [ . . . ]

And still, the great wave had not passed. It towered high above the trough into which the *Ulysses,* now heeled far over to 40◦, had been so contemptuously flung, bore down remorselessly from above and sought, in a lethal silence and with an almost animistic savagery, to press her under. The inclinometer swung relentlessly over—45◦, 50◦, 53◦ and hung there an eternity, while men stood on the side of the ship, braced with their hands on the deck, numbed minds barely grasping the inevitable. This was the end. The *Ulysses* could never come back. [... ] ... the *Ulysses* shuddered, then imperceptibly, then slowly, then with vicious speed lurched back and whipped through an arc of 90◦, then back again. [... ] 'Slow ahead both! Midships!' 'Steady as she goes!' The *Ulysses* was round."

The quoted fragment is an artist's description of a turn in heavy seas of a cruiser participating in a WW2 arctic convoy. However, this description is based on personal experience of its author and in fact it well agrees with recommendations given in seamanship manuals, see, e.g., [54]. This indicates that the selection of ship handling scenarios and formulation of explicit criteria for manoeuvrability in sea waves may become a prohibitively difficult task with plenty of uncertainties. However, even if some reasonable scenarios are established, it will be very difficult to verify compliance of a given ship with the corresponding criteria: there is no chance to perform full-scale trials, scaled model experiments of this nature are very complex and the existing theoretical and numerical methods are not matured enough to provide reliable estimates. Of course, there are some ways to bypass those difficulties.

The first way was de facto followed in several later proposals for manoeuvrability criteria in adverse conditions and consists, first, in lifting the requirement of direct verifiability of each criterion in the spirit of the philosophy proposed in [26] and, second, in simplifying the model for the wave action, which is very complex. Considering the hull forces, it is possible to separate the first-order and second-order effects of sea waves. The first-order excitation forces are of primary importance in ship seakeeping and also are the

main cause of the ship's yawing in a seaway. In regular waves, they result in ship responses with the wave encounter frequency and are roughly proportional to the wave amplitude at least when the nonlinear effects are not very pronounced. From the point of view of manoeuvrability, the most unfavourable is the case of long following waves. Possible loss of controllability in extreme seas and situations similar to that described in the quotation above are also caused by first-order effects. The second-order reactions result in regular waves in time-averaged steady forces and moments which, of course, will affect the motion of the ship in the horizontal plane. In particular, they will result in the added resistance in waves but also in the additional constant sway force and yaw moment. These forces and moment are roughly proportional to the square of the wave amplitude and reach their maxima in relatively short waves. In irregular seas the second-order excitation is more complex and, besides the steady component, results in low-frequency excitation which can cause resonant motions of moored or anchored ships.

Most latest proposals on manoeuvring criteria and standards are based on the methods predicting only second-order wave forces possibly combined with the steady current and wind action while the first-order effects are accounted for indirectly by setting some sea margin to the maximum rudder deflection angle, see, e.g., [9,11,49,50,55]. Apparently, for the power standardization problem, this approach is completely justified but it is less evident for the ship controllability in waves. In particular, the extreme situation described above cannot be captured in this way.

However, there are reasons to believe that it is possible to bypass the difficulties related to accounting for wave disturbances. Such alternative is based on two rather evident observations concerning possible outcomes of the analysis of each possible sea waves scenario:


It is clear that in the latter case any additional criterion turns out excessive and can be ignored but in the former case, a more effective rudder should be required.

Another evident observation is that such a more effective rudder will also improve the indices of the turning ability and directional stability. This means that this additional requirement can be accounted for implicitly, through appropriate tightening of the main manoeuvring standards, i.e., decreasing the allowed values of the advance, tactical diameter and zigzag overshoot angles. Does that mean that no additional separate or explicit criteria accounting for the seas and hydrodynamic interaction would then be necessary? The answer would be definitely "no" if and only if it were possible to reduce the impact of the mentioned external factors without enhancing the effectiveness of the steering device. This looks, however, highly unlikely.

Of course, the shape of the hull and especially its fullness may influence the interaction and wave excitation forces but this shape is almost entirely determined by resistance, propulsion and other design considerations. Similarly, it is practically impossible to alter the gyration radius in yaw motion through redistribution of mass along the hull. The only realistic modification consists of increasing the size of the stabilizing skeg or installation of additional stabilizers which improves the directional stability but inevitably at the expense of the turning ability. An objection may arise that, for instance, ships with different values of the block coefficient are subject to different levels of excitation loads and would require different requirements to the rudder effectiveness but this difference can be accounted for both by explicit or implicit criteria.

Hence, it can be assumed that the resistive capacity of a ship with respect to the action of hydrodynamic exogenous factors (sea waves) can only be augmented by increasing the effectiveness of the rudder which, in its turn, is guaranteed if the turning ability and the directional stability are increased simultaneously. It means that the existing standards of the turning ability and stability are to be revised and maybe tightened but no additional special explicit standards are necessary. It must be emphasized that ships with a good turning ability alone can easily show poor resistance against external factors including sea waves if these ships are not directionally stable or even stable but with insufficient stability margin. This may happen with full-bodied vessels and, as long as the external factors are ignored, can be even viewed as beneficial because a less effective small rudder is then sufficient for good turning qualities but such rudder will not be able to appropriately counteract external forces especially at low speed.

#### *4.3. Standards for Controllability in Wind*

The same principle of increasing the rudder effectiveness through tightening the calm-water standards could be also applied to the aerodynamic (wind) factors but here the situation looks somewhat different because, contrary to hydrodynamic loads, it is possible to define several very evident and significant parameters affecting the level of sensitivity of a ship to wind. Since long ago, it has been established with certainty, see, e.g., [12,56–58], that the manoeuvring sensitivity to the wind depends on the following four fundamental dimensionless parameters: the relative density of the air *ρ<sup>A</sup>* = *ρA*/*ρ*, where *ρ<sup>A</sup>* is the air density and *ρ* is the water density; the relative lateral windage area *A <sup>L</sup>* = *AL*/(*LT*), where *AL* is the lateral windage area, *L* is the ship waterline length, *T* is the ship draught; the relative longitudinal position of the centroid of *AL* defined as *x <sup>A</sup>* = *xA*/*LOA*, where *xA* is the abscissa of the centroid and *LOA* is the length overall; and the relative wind speed *Vw* = *Vw*/*V*, where *Vw* is the absolute wind speed and *V* is the actual speed of the ship with respect to water. The air relative density is practically constant and its variability can be ignored while the remaining parameters can vary in a wide range and are extremely important as they influence the mentioned sensitivity substantially. This fact is quite transparent intuitively and well familiar to all seafarers. In particular, it has always been known that slow vessels with a large windage area and small draught are especially difficult for handling in wind and even the controllability can be lost completely.

It seems that contrary to the criteria driven by sea waves excitation it is easier to introduce a separate standard based on the criterion of controllability in the wind than to establish some additional rules for tightening calm-water criteria, which must capture dependence on the windage area parameters mentioned above. In fact, two additional wind-controllability requirements were developed by Mastushkin [19] and these are up to present embedded into the Rules of the Russian Maritime Register of Shipping [8], see also [12,16]. Multiple test calculations performed with more than 100 ships showed that after appropriate adjustment this criterion became non-trivial: some existing ships that did not satisfy it did really suffer from controllability problems in wind. Of course, for the majority of ships that criterion looked excessive, as the requirements of turning ability/stability in calm water were for them more strict. A similar situation will be with any consistent wind criterion. In particular, it will be certainly redundant for ships with high draught and low windage area: for such ships, second-order wave loads may often exceed aerodynamic ones. On the other hand, many ships in ballast condition, passenger ships and ferries will be more prone to aerodynamic loads than to any other type of excitation and then that additional requirement turns out significant. For instance, it was found by Zaky and Yasukawa [30] that while the second-order wave forces for a typical tanker in full load exceeded by 50% the wind forces, for the same ship in ballast the wave force was 10% lower than the wind force.

#### *4.4. Modification of Calm-Water Standards*

#### 4.4.1. Simplified Approach

Recognizing that simultaneous improvement of the turning ability and directional stability definitely improves the ship's capability to counteract external factors, it can be sufficient to modify the required (standardizing) values of the turning ability and directional stability criteria. Regarding the turning ability, as a zero approximation the existing values (5 ship lengths for the tactical diameter and 4.5 lengths for the advance, both independent of the reference time) can be conserved or somewhat tightened regarding that they are satisfied with the majority of existing ships with a tangible margin [29]. However, the values of the measures of directional stability apparently must be revised. Given the previous discussion in Section 3, it goes about the dependence of the width of the hysteresis loop or of the zigzag 1st overshoot on the ship reference time *T*ref. Again, as a zero approximation, it could be recommended to eliminate this dependence by introducing flat standards, see Figures 1 and 2. This modification seems quite appropriate as most existing ships already satisfy the thus modified standard: a sampling of existing vessels presented in [40] demonstrated compliance with the "flat" standard for most ships and did not show definite dependence on *T*ref while the scatter is substantial.

It is remarkable that such deep revision and tightening of IMO standards apparently would not result in any serious problems regarding compliance difficulties as multiple sources (see [40,59–61] and Figure 2 where full-scale data from [40] are plotted as circle symbols) indicate that while the scatter of full-scale data is substantial, the dependence of the overshoot angles and/or loop width on *T*ref is weak and often even absent. In the majority of cases, the standards are satisfied with a considerable margin. Exceptions are mainly associated with very specific vessels, such as hopper dredgers [62] but these hardly can be treated in a common way.

While Gong et al. [59] detected a larger scatter of trials data at low Froude numbers, the data in the cited source indicate that OS1 rarely exceeds 10–11 degrees. The same authors have also found by means of interactive simulation that reduction of OS1 from 13 to, say, 8 degrees with simultaneous reduction of the tactical diameter from 4.3 to 3.7 lengths definitely resulted in safer and more accurate steering.

Even though such modifications of the IMO standards would be, most likely, beneficial even in this form, it would be hardly the best possible solution.

#### 4.4.2. Improved Modifications

Before an extended numerical study, see the next Subsubsection, is undertaken some heuristic improvements can be made. In particular, the same value of the reference time can correspond to larger and faster vessels or to smaller and slower ones. It is, however, evident that in the first case sensitivity to waves at the same sea state will be weaker, which means that the directional stability requirement can be eased (but not necessarily!) for larger ships and probably the ship length can then be introduced as additional parameter although for each *L* the requirement may remain flat as far as it is stricter than the existing IMO standard. This is illustrated by additional dash-dot lines in Figures 1 and 2 where it is supposed that the base requirement described by the lowest dash-dot line must be applied for all ships with the length not exceeding some value *L*1, which must be specified, while the two remaining dash-dot lines show possible looser requirements for some larger ship lengths *L*<sup>2</sup> and *L*3. These eased requirements can be viewed as a compromise between the existing and completely "flat" standards. The flatness of the standard, at least above some value of the reference time, makes less critical selection of the speed *V*: the IMO standards define it as 0.9 of the speed corresponding to 0.85 of the maximum continuous rating of the engine(s) and the same definition can be kept although the actual reachable speed in waves may be lower than that.

#### 4.4.3. Supporting Studies

Regarding a better account for sea waves, development of more sophisticated and better substantiated requirements to the measures of directional stability, multiple simulations in waves must be performed for a set of generic ship mathematical models with varying indices of the turning ability and inherent directional stability must be performed and followed by appropriate analysis. Moreover, any representative set of mathematical models must embrace not only various values of the shaft power but also different types of propulsors (fixed- or controllable-pitch propellers) and various typical engine characteristics as, e.g., the loss of actual power will be very different for a DC electric motor and a

common turbocharged 2-stroke Diesel engine. It does not mean that the resulting standards must be differentiated for all possible configurations but differences in behaviour must be captured and analysed. Such numerical study must be, however, preceded by the development of a sufficiently accurate, fast and commonly recognized program for simulating the arbitrary motion of a ship in waves. Existing codes of this kind can hardly be applied as they suffer from various limitations and uncertainties. Application of Reynolds-Averaged Navier–Stokes Equations (RANSE) CFD codes or model tests looks unrealistic because of unacceptable time requirements: at least real-time or even accelerated-time simulations must be feasible. Unfortunately, no one of faster existing available codes can guarantee capturing all substantial effects accompanying manoeuvring in waves. For instance, one of the latest and relatively consistent developments [63] is fully linear (except for the account for 2nd-order excitation forces) and neglects many coupling effects, let alone variations of the hydrodynamic characteristics of propellers and rudders and interaction coefficients. The 2nd-order wave forces combined with the linear seakeeping model were used by Chillece and el Moctar [55] and by Kobayashi [64]. Only second-order forces were modelled for a tanker by Zaky and Yasukawa [30] who combined the sea with the wind action.

#### 4.4.4. Design Implications for Steering Arrangement

Even though most ships apparently will satisfy even modified and tighter manoeuvring criteria, situations in which design decisions aimed at strengthening the ship's controllability are required may happen and then various solutions are possible. It is important to emphasize one more time that the directional stability must be augmented together (concurrently) with the turning ability of the ship design in consideration as only in this case the ship capacity to resist adverse factors will be improved. Augmentation of the rudder area is the most evident and straightforward solution. Simulations under wave and wind disturbances performed by Ohtagaki and Tanaka [44] demonstrated that the mean course deviation could be reduced from approximately 10 to 5 degrees with simultaneous reduction of the square-root mean rudder deflections from 13 to 8 degrees when the rudder relative area was increased from 1.2% to 2.5%. Norrbin [17] suggested using skegs or fixed fins to improve the directional stability. Haraguchi and Nimura [42] demonstrated direct reduction of the OS1 and OS2 overshoot angles by the factor of 3 and reduction of the OS3 angle by the factor of 2 when the relative rudder area was increased from 1% to 1.5%.

It must be said that in a rather typical situation of a normal rudder working in the propeller slipstream augmentation of the rudder area will definitely increase the turning ability while improvement of the directional stability may become much less pronounced. In this case, simultaneous augmentation of the rudder and the skeg areas may be recommended [38]. Similar solutions can be very promising in the case of poddriven ships [42,65]. However, it makes sense to keep in mind the natural improvement of controllability in some defining loading conditions. For instance, Zaky and Yasukawa [30] obtained that a typical ballast condition often results in simultaneous growth of the turning ability (due to larger relative rudder area) and of the directional stability (due to trim by the stern). Finally, the so-called high-lift rudders and other steering devices can be applied. For instance, Yamada [44] indicated that the application of the high-lift Schilling rudder permits considerable reduction of the OS2 angle. Eda and Numano [66] demonstrated the effectiveness of twin flap rudders.

### **5. Manoeuvrability Criteria in Wind: Theoretical Support**

*5.1. Ship Wind Resisting Capacity Criterion and Standardization Scheme*

#### 5.1.1. Preliminary Comments

As was discussed above, although in theory the approach based on adjustment of the existing IMO criteria can also be applied to account for wind perturbations, the special nature of aerodynamic exogenous factors and, in particular, the existence of few well-defined parameters characterizing aerodynamic peculiarities of ships makes reasonable the introduction of an additional criterion. Even though a straightforward approach has inclined many researchers to require direct demonstration of the capability of a ship to perform arbitrary manoeuvres in a given wind [67], since long ago it has been understood that the ship remains controllable in wind of some specified velocity if it is capable to maintain any required course in straight run [56]. It was also demonstrated with the help of direct computations [58] that the ability to maintain an arbitrary course is equivalent to the ability of maintaining an arbitrary heading or arbitrary air drift angle. This scenario had been used by Mastushkin [19] for developing wind controllability criteria implicitly contained in the Rules [8], see also their concise description in [12]. Those criteria were applied using some simplified manoeuvring models under the assumption of the presence of constant wind with a certain specified speed. In general, the scheme turned out reasonable and practical although certain additional simplifications, such as the assumption that the rudder effectiveness is unconditionally higher for rudders with a higher aspect ratio, somewhat impair the quality and consistency of the standard. The approach followed here is similar in some respects but the authors tried to avoid evident drawbacks of earlier suggestions and to leave more flexibility in the standardizing scheme. Any wind controllability standard is supposed to be applied indiscriminately to all ships but in the case of underpowered ships, the corresponding criterion is more likely to become critical, i.e., leading to augmentation of the rudder effectiveness. The following material assembles practically all theoretical elements necessary for devising a rational and relatively simple standard for ship resistance to wind action, but the corresponding standard itself presumes specification of the wind speed value through a statistical adjustment to some representative subset of the existing fleet of vessels is left beyond the scope of the present article.

#### 5.1.2. Formulation and Main Relations

The gustiness of the wind can be neglected in manoeuvring problems as the ship in the horizontal plane represents an aperiodic system with relatively large time lags and possessing no resonant frequencies. Similarly, self-sustained oscillations of the loads observed on non-steamlined bodies even in a sready flow will not have any significant effect. Assuming also that the wind-induced roll does not affect significantly the behaviour of the ship in the horizontal plane, the most general modular mathematical model for the steady ship motion in wind is described by the following set of nonlinear algebraic equations:

$$\begin{aligned} X\_H(\mu, \upsilon) + X\_P(\mu, \upsilon, n) + X\_R(\mu, \upsilon, n, \delta\_R) + X\_A(\mu\_A, \upsilon\_A) &= 0, \\ Y\_H(\mu, \upsilon) + Y\_R(\mu, \upsilon, n, \delta\_R) + Y\_A(\mu\_A, \upsilon\_A) &= 0, \\ N\_H(\mu, \upsilon) + N\_R(\mu, \upsilon, n, \delta\_R) + N\_A(\mu\_A, \upsilon\_A) &= 0, \\ Q\_P(\mu, \upsilon, n) + Q\_E(n^\*, n) &= 0, \end{aligned} \tag{1}$$

where *X*,*Y*, *N*, *Q* stand for the surge force, sway force, yaw moment and the shaft torque, respectively; the subscripts *H*, *P*, *R*, *A*, *E* correspond to the hull, propeller, rudder, air (or aerodynamic) and engine, respectively; *u* and *v* are the velocities of surge and sway; *n* is the actual propeller rotation frequency (rps), and *n*∗ is the ordered rotation frequency.

The kinematic parameters present in the set (1) are most convenient for the description of involved forces and are related to alternative parameters, which may be more suitable for human perception:

$$\begin{aligned} u &= V \cos \beta\_{\prime} \ v = -V \sin \beta\_{\prime} \\ u\_A &= V\_A \cos \beta\_{A\prime} \ v\_A = -V\_A \sin \beta\_{A\prime} \end{aligned} \tag{2}$$

where *<sup>V</sup>* <sup>=</sup> <sup>√</sup>*u*<sup>2</sup> <sup>+</sup> *<sup>v</sup>*<sup>2</sup> <sup>+</sup> is the ship speed relative to the water, *β* is the drift angle, *VA* = *u*2 *<sup>A</sup>* + *<sup>v</sup>*<sup>2</sup> *<sup>A</sup>* is the airspeed of the ship or the relative wind speed and *β<sup>A</sup>* is the air drift angle or the apparent wind angle.

In the case of absence of current, the following relations are valid:

$$\begin{aligned} u\_A &= u + V\_w \cos(\chi\_w - \psi), \\ v\_A &= v - V\_w \sin(\chi\_w - \psi), \end{aligned} \tag{3}$$

where *Vw* is the absolute wind speed, *χ<sup>w</sup>* is the absolute wind direction angle, *ψ* is the ship heading angle.

The aerodynamic surge force *XA*, sway force *YA* and yaw moment *NA* are represented in the standard way:

$$X\_A = \mathbb{C}\_{XA}(\beta\_A) \frac{\rho V\_A^2}{2} A\_{T\prime} \ \ Y\_A = \mathbb{C}\_{YA}(\beta\_A) \frac{\rho V\_A^2}{2} A\_{L\prime} \ N\_A = \mathbb{C}\_{NA}(\beta\_A) \frac{\rho V\_A^2}{2} A\_L L\_{OA\prime} \tag{4}$$

where *CXA*, *CYA*, *CNA* are the force/moment coefficients, *AT* is the transverse projection of the above-water part of the hull, *AL* is its longitudinal projection and *LOA* is the length overall.

The dependencies of the forces and moments in (1) are closing the ship mathematical model and can be specified in various ways. In general, these dependencies can be very complicated and highly nonlinear. So far, the only simplifying assumption was that the propeller sway force and yaw moment could be neglected. A comprehensive nonlinear mathematical model is suitable and even desirable for simulating the ship manoeuvring motion in wind but seems to be excessively complicated for standardizing purposes. The following simplification seems to be adequate for that task: the speed of the ship with respect to water *V* is fixed and specified. In this case, the equations for the surge forces and shaft torques can be separated from the remaining two and are often ignored.

The assumption about the fixed speed *V* is sometimes doubted as in reality it also depends on the wind and in more detailed mathematical models used in simulation systems, it is continuously estimated and depends not only on the relative wind but also on the sea state, type and limiting characteristics of the main engine, type of the propulsor and on the throttle settings while the latter may be conditioned by voluntary reduction of speed to avoid resonances, reduce slamming and green water on deck. However, in simplified models oriented at the definition of manoeuvring criteria not only the assumption of a fixed ship speed is acceptable but the specific value of this speed is less important than it could seem because what matters is its magnitude relative to the wind speed *Vw* and the standardizing value for the latter is supposed to be adjusted to the existing fleet.

Most empiric methods use linearized or relatively easily linearizable rudder models [56,68,69] which is sufficiently appropriate for a standardizing problem. Then the sway–yaw equilibrium equations can be represented as

$$\begin{array}{l} \mathcal{N}'\_{H}(\boldsymbol{v}') \cdot \frac{1}{2} \rho V^2 \mathcal{L}T - \mathcal{C}^{\underline{u}}\_{RN} \cdot \frac{1}{2} \rho V^2\_{R} A\_{R} \boldsymbol{a}\_{R} \cos \delta\_{R} + \mathcal{C}\_{AY}(\boldsymbol{\beta}\_{A}) \cdot \frac{1}{2} \rho\_{A} V^2\_{A} A\_{L} = 0, \\\ N'\_{H}(\boldsymbol{v}') \cdot \frac{1}{2} \rho V^2 L^2 T - \mathcal{C}^{\underline{u}}\_{RN} \cdot \frac{1}{2} \rho V^2\_{R} A\_{R} \boldsymbol{a}\_{R} \boldsymbol{x}\_{R} \cos \delta\_{R} + \mathcal{C}\_{AN}(\boldsymbol{\beta}\_{A}) \cdot \frac{1}{2} \rho\_{A} V^2\_{A} A\_{L} L\_{OA} = 0, \end{array} \tag{5}$$

where *Y <sup>H</sup>* and *N <sup>H</sup>* are the hull sway force and yaw moment coefficients, *v* = *v*/*V* is the dimensionless sway velocity, *VR* is the magnitude of the rudder velocity with respect to water, *AR* is the rudder area, *α<sup>R</sup>* = *δ<sup>R</sup>* − *κβ* is the rudder attack angle, *κ* is the flow straightening factor, *xR* is effective rudder abscissa in the body frame, *ρ* and *ρ<sup>A</sup>* are the water and air density, respectively. It is assumed here that the linear rudder model is characterized by the normal force coefficient gradient *C<sup>α</sup> RN*. As the minimum value of the function cos *δ<sup>R</sup>* is 0.85 in most cases, this factor can be dropped. In addition, it can be assumed that *LOA* ≈ *L*.

The set (5) can then be represented in the following non-dimensional form:

$$\begin{cases} Y\_H'(\boldsymbol{\nu}') - \mathbb{C}\_{RN}^u \overline{V}\_R^2 A\_R' (\boldsymbol{\delta}\_R + \boldsymbol{\kappa} \boldsymbol{\nu}') + \mathbb{C}\_{AY}(\boldsymbol{\beta}\_A) \overline{\boldsymbol{\rho}}\_A \overline{V}\_A^2 A\_L' = 0, \\ N\_H'(\boldsymbol{\nu}') - \mathbb{C}\_{RN}^u \overline{V}\_R^2 A\_R' (\boldsymbol{\delta}\_R + \boldsymbol{\kappa} \boldsymbol{\nu}') \mathbf{x}\_R' + \mathbb{C}\_{AN}(\boldsymbol{\beta}\_A) \overline{\boldsymbol{\rho}}\_A \overline{V}\_A^2 A\_L' = 0, \end{cases} \tag{6}$$

where *VR* = *VR*/*V*, *A <sup>R</sup>* = *AR*/(*LT*) is the relative rudder area, *ρ<sup>A</sup>* = *ρA*/*ρ*, *VA* = *VA*/*V* and *A <sup>L</sup>* = *AL*/(*LT*).

The difference between *VR* and *V* is caused by the influence of the wake behind the hull and by the influence of the propeller slipstream. The total effect can be positive (more typical for merchant ships) or negative depending on which part of the rudder is working in the propeller race and on the configuration of the stern. The straightening coefficient *κ* depends on the same factors but never exceeds 1.0.

The ideal propulsor model typically used by empiric methods, see e.g., [69,70], for modelling the influence of the propeller slipstream on the rudder will yield:

$$\begin{array}{l} \overline{V}\_{R}^{2} = \overline{A}\_{R0} + \overline{A}\_{RP}(1 + \mathcal{C}\_{TA}),\\ \kappa = \kappa\_{H} \frac{\overline{A}\_{R0} + \overline{A}\_{RP}\sqrt{1 + \mathcal{C}\_{TA}}}{\overline{A}\_{R0} + \overline{A}\_{RP}(1 + \mathcal{C}\_{TA})} \end{array} \tag{7}$$

where *AR*<sup>0</sup> = *AR*0/*AR*, *AR*<sup>0</sup> and *ARP* are the parts of the rudder area *AR* outside and inside the slipstream, respectively, *κ<sup>H</sup>* is the hull straightening factor varying from 0.3 to 1.0, *CTA* is the propeller loading coefficient defined as:

$$\mathcal{C}\_{TA} = \frac{8T\_P}{\rho \pi D\_P^2 V^2} \tag{8}$$

where *TP* is the propeller thrust and *DP* is the propeller diameter.

The hull straightening factor is assumed following the general recommendations [57,69] or it can be adjusted. The propeller thrust and the ship speed in (8) should correspond to the actual ship motion in wind and be obtained from the surge equation but, in first approximation and for standardizing purpose, it is possible to assume that the thrust corresponds to the free run with the speed *V*, which is then a free parameter.

As the drift angle of the ship advancing along a straight path in wind practically never exceeds 30 degrees, polynomial approximations for the functions *Y H*(*v* ) and *N H*(*v* ) will be adequate. It makes sense to apply the simplest approximations used in the Pershitz empiric method [12,56]:

$$\begin{array}{l} \mathcal{Y}'\_{H}(\upsilon') = \mathcal{Y}'\_{\upsilon}\upsilon' + \mathcal{Y}'\_{\upsilon\upsilon}\upsilon'|\upsilon'| ,\\ \mathcal{N}'\_{H}(\upsilon') = \mathcal{N}'\_{\upsilon}\upsilon' \end{array} \tag{9}$$

where the coefficients ("hydrodynamic derivatives") can be estimated using the procedure described in [57,69]. This model exploits the fact that the drift angle nonlinearity is much weaker for the yaw moment than for the sway force and is the nonlinear model of minimum complexity. A fully linearized model is also possible and was used in [58] but, as the substantially more accurate model (9) still permits analytical solutions, its application may be preferable. If the primary mathematical model for the sway force and yaw moment has a different structure, it always can be reasonably approximated with (9). For instance, suppose that the coefficients of a very common cubic model

$$\begin{array}{l} \mathcal{Y}'\_{H3}(\upsilon') = \mathcal{Y}'\_{\upsilon3}\upsilon' + \mathcal{Y}'\_{\upsilon\upsilon\upsilon}\upsilon'^3, \\ \mathcal{N}'\_{H3}(\upsilon') = \mathcal{N}'\_{\upsilon3}\upsilon' + \mathcal{N}'\_{\upsilon\upsilon\upsilon}\upsilon'^3. \end{array} \tag{10}$$

are known. Then, requiring direct matching at the values of the dimensionless sway velocity *v* <sup>1</sup> and *v* <sup>2</sup> (for instance, they may correspond to 10 deg and 20 deg values of the drift angle) for the sway force and only at *v* <sup>2</sup>. it is possible to obtain the approximating formulae:

$$\begin{array}{l} \mathcal{N}'\_{v} = \mathcal{Y}'\_{v3} - \mathcal{Y}'\_{vvv} \upsilon'\_{1} \upsilon'\_{2}, \\ \mathcal{N}'\_{vv} = \mathcal{Y}'\_{vvv} (\upsilon'\_{1} + \upsilon'\_{2}), \\ \mathcal{N}'\_{v} = \mathcal{N}'\_{v3} + \mathcal{N}'\_{vvv} \upsilon'\_{2} \end{array} \tag{11}$$

Of course, the approximation can be performed in different ways including the application of the least-square method.

Substituting (9) into (6) and performing obvious transformations it is possible to bring the equilibrium equations to the following form:

$$\begin{array}{l} \mathsf{C}\_{Y}^{\upsilon}\upsilon' + Y\_{\upsilon\upsilon}'\upsilon'|\upsilon'| + \mathsf{C}\_{Y}^{\delta}\delta\_{R} = f\_{Y}(\overline{\mathsf{V}}\_{A\prime}\beta\_{A})\_{\prime} \\ \mathsf{C}\_{N}^{\upsilon}\upsilon' & + \mathsf{C}\_{N}^{\delta}\delta\_{R} = f\_{N}(\overline{\mathsf{V}}\_{A\prime}\beta\_{A})\_{\prime} \end{array} \tag{12}$$

where

$$\begin{array}{ll} \mathbf{C}\_{Y}^{\upsilon} = \mathbf{Y}\_{\upsilon}^{\prime} - \kappa \mathbf{E}\_{\mathsf{R}\prime} & \mathbf{C}\_{Y}^{\delta} = -\mathbf{E}\_{\mathsf{R}\prime} & f\_{Y}(\ ) = -\mathbf{K}\_{A} \mathbf{y}(\boldsymbol{\beta}\_{A}) \overline{\mathbf{V}}\_{AL\prime}^{2} \\ \mathbf{C}\_{N}^{\upsilon} = \mathbf{N}\_{\upsilon}^{\prime} - \kappa \mathbf{E}\_{\mathsf{R}} \mathbf{x}\_{R\prime}^{\prime} & \mathbf{C}\_{N}^{\delta} = -\mathbf{E}\_{\mathsf{R}} \mathbf{x}\_{R\prime}^{\prime} & f\_{N}(\ ) = -\mathbf{K}\_{AN} (\boldsymbol{\beta}\_{A}) \overline{\mathbf{V}}\_{AL\prime}^{2} \end{array} \tag{13}$$

where

$$E\_R = \mathbb{C}\_{Rn}^a \overline{\nabla}\_R^2 A\_R'; \ K\_{AY}(\beta\_A) = \mathbb{C}\_{AY}(\beta\_A) \overline{\rho}\_A A\_L'; \ K\_{AN}(\beta\_A) = \mathbb{C}\_{AN}(\beta\_A) \overline{\rho}\_A A\_L' \tag{14}$$

and where *ER* has the meaning of the rudder effectiveness index.

#### 5.1.3. Useful Analytical Solutions

Standard analysis based on the Equation (12) presumes computation of equilibrium values of *v* (or *β*) and *δ<sup>R</sup>* as functions of *β<sup>A</sup>* at some fixed value of *VA*. The solution always exists for this model but when the values of the parameters *A <sup>L</sup>* and *VA* are large enough, the required equilibrium rudder deflection angle can exceed the maximum possible value *δ<sup>m</sup>* within a certain interval of the air drift angle *β<sup>A</sup>* and this is then interpreted as a loss of controllability in wind. This analysis can be complemented by an investigation of the inherent local stability of attainable equilibrium regimes, see, e.g., [71–74] and such analysis is of considerable theoretical interest. However, it is not relevant for estimation of controllability in the sense formulated above as the steered ship represents a closed-loop system and can be stabilized on any reachable course with the helmsman or autopilot. Necessary allowance for the rudder oscillatory motions (sea margin) is accounted for in setting the value of *δ<sup>m</sup>* which is assumed to be smaller by 5–7 degrees than the maximum deflection angle of the rudder.

The set (12) can be solved analytically. First, physical considerations and computations performed with the linearized model show that at positive *β<sup>A</sup>* the drift angle *β* must be negative, which corresponds to *v* > 0 which makes possible substituting *v* |*v* <sup>|</sup> with *<sup>v</sup>*<sup>2</sup> as, due to symmetry, it is sufficient to consider only non-negative values of the air drift angle. Then, expressing the rudder angle from the second equation and substituting it into the first one the following quadric equation is obtained:

$$\mathbf{Y}'\_{vv}\mathbf{v}'^2 + \left(\mathbf{C}^v\_Y - \mathbf{C}^v\_N/\mathbf{x}'\_R\right)\mathbf{v}' + f\_N/\mathbf{x}'\_R - f\_Y = \mathbf{0}.\tag{15}$$

The assumption that always *x <sup>R</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>2</sup> will not lead to substantial errors and is justified in approximate analysis. Then, the non-negative solution to this equation is given by the formulae:

$$\begin{array}{l} v' = \frac{-2\mathcal{C}\_N^v - \mathcal{C}\_Y^v - \sqrt{\left(\mathcal{C}\_Y^v + 2\mathcal{C}\_N^v\right)^2 + \mathcal{Y}\_{lv}^l \left[f\_Y(\mathcal{G}\_A) + 2f\_N(\mathcal{G}\_A)\right]}}{\delta\_R = \left[f\_N(\mathcal{B}\_A) - \mathcal{C}\_N^v v'\right] / \mathcal{C}\_N^\zeta} \,\tag{16}$$

These formulae should be applied for a set of values of *β<sup>A</sup>* ∈ [0, *π*] to check whether the controllability can be lost or not. However, the equilibrium Equation (12) can also be interpreted in other ways.

First, assuming that the ship sails at the controllability limit, i.e., *δ<sup>R</sup>* = *δ<sup>m</sup>* and that the value of *VA* is fixed, it will be possible to solve the set (12) with respect to *v* and *βA*. However, due to nonlinear dependencies *CAY*(*βA*) and *CAN*(*βA*) this can only be performed numerically and it is clear that the solution does not exist if at a given *VA* the ship is controllable with some margin.

In addition, under the same assumption *δ<sup>R</sup>* = *δ<sup>m</sup>* it is possible to eliminate *v* and to obtain a single equivalent equation which can be represented in one of the following two forms:

$$
\mathbb{C}\_{VVVV}\nabla\_A^4 + \mathbb{C}\_{VV}\nabla\_A^2 + \mathbb{C}\_{V0} = 0\tag{17}
$$

or,

$$
\mathbb{C}\_{EE}\mathrm{E}\_R^2 + \mathbb{C}\_E\mathrm{E}\_R + \mathbb{C}\_{R0} = \mathbf{0},
\tag{18}
$$

where,

$$\begin{aligned} \mathbf{C}\_{VVVV} &= Y'\_{vv} \mathbf{K}^2\_{A\mathcal{V}} \cdot \mathbf{C}\_{VV} = A\_{VVVE} \mathbf{E}^2\_R + A\_{VVE} \mathbf{E}\_R + A\_{VVV} \cdot \mathbf{C}\_{V0} = A\_{EE} \mathbf{E}^2\_R + A\_E \mathbf{E}\_R \\ \mathbf{C}\_{EE} &= A\_{VVEE} \mathbf{V}^2\_A + A\_{EE} \cdot \mathbf{C}\_E = A\_{VVE} \mathbf{V}^2\_A + A\_{E'} \cdot \mathbf{C}\_{R0} = \mathbf{C}\_{VVVV} \mathbf{V}^4\_A + A\_{VVV} \mathbf{E}^2\_{A\prime} \\ A\_{VVEE} &= \mathbf{x}^2 \mathbf{x}^{\prime}\_R (K\_{AY} \mathbf{x}^{\prime}\_R - K\_{AN}), \\ A\_{VVE} &= \mathbf{x} K\_{AN} (\mathbf{N}^{\prime}\_{v} + \mathbf{Y}^{\prime}\_{v} \mathbf{x}^{\prime}\_R) - 2 \mathbf{x}^{\prime}\_R (\mathbf{Y}^{\prime}\_{vv} K\_{AN} \delta\_m + \mathbf{x} \mathbf{N}^{\prime}\_v K\_{AY}), \\ A\_{VV} &= \mathbf{N}^{\prime}\_v (K\_{AY} \mathbf{N}^{\prime}\_{v} - K\_{AN} \mathbf{Y}^{\prime}\_v), \ A\_{EE} = \mathbf{x}^{\prime}\_R \delta\_m (\mathbf{Y}^{\prime}\_{vv} \mathbf{x}^{\prime}\_R \delta\_m - \mathbf{x} \mathbf{Y}^{\prime}\_v \mathbf{x}^{\prime}\_R + \mathbf{x} \mathbf{N}^{\prime}\_v), \\ A\_E &= N^{\prime}\_v \delta\_m (\mathbf{Y}^{\prime}\_{v} \mathbf{x}^{\prime}\_R - N^{\prime}\_v). \end{aligned} \tag{19}$$

The bi-quadric Equation (17) can be easily solved analytically with respect to the relative critical airspeed *VA*(*βA*), which corresponds to the maximum ship air speed at which no loss of controllability occurs for the current ship configuration. It is often assumed [19] that the maximum sensitivity to the wind is observed at *<sup>β</sup><sup>A</sup>* <sup>≈</sup> <sup>3</sup> <sup>4</sup>*π*. However, while on average this assumption is justified, the most "dangerous" air drift angle can be substantially different and it makes sense to search for *β<sup>A</sup>* = argmin*VA* which is rather simple in the one-dimensional problem and can be performed, e.g., by means of a dichotomy algorithm. Once the critical values of *VA* and *β<sup>A</sup>* are determined, it is possible to determine the minimum value of the absolute wind speed *Vw* at which controllability can be lost.

A solution to the quadric Equation (18) at fixed values of *VA* and *β<sup>A</sup>* will provide the minimum required rudder effectiveness index *ER* which is always positive. Once the required value of *ER* is fixed, the designer can adjust the actual rudder effectiveness mainly through variating the effective rudder area *A RE* or the rudder normal force gradient *Cα RN*. The latter can be increased by increasing the rudder aspect ratio but this must be performed with care as the higher is the aspect ratio, the lower is the stall angle and this highly undesirable stall may then occur at moderately large rudder attack angles.

To close the schemes described above it is necessary to define the dimensionless aerodynamic characteristics of the ship hull *CXA*(*βA*), *CYA*(*βA*), *CNA*(*βA*).

#### *5.2. Aerodynamic Characteristics of Ships for Standardizing Purposes*

While ship mathematical models used in ship handling simulators require the most accurate prediction of aerodynamic loads to obtain a more realistic reaction of the model to the simulated wind, the logic of developing manoeuvring standards often permits some conservative approximate estimates which somewhat facilitates the task. In this section, methods for estimating the aerodynamic characteristic of ships are mainly discussed from this viewpoint.

In contrast with the submerged part of the hull of a surface displacement ship, the above-water part is rarely streamlined and very often possesses a rather complicated, edged and peculiar shape especially bearing in mind that not only it includes the hull per se but also superstructures, deckhouses and, possibly, other structures and equipment.

As result, during several decades the aerodynamic characteristics of ships were mainly determined using wind-tunnel tests although lately application of CFD methods is gaining popularity and it is recognized that the credibility of physical and numerical modelling has become comparable [75].

Fortunately, the dependence of the aerodynamic characteristics on the angular velocity of yaw can be neglected and, besides the Reynolds number, the aerodynamic force coefficients will only depend on the air drift (or attack, or sideslip) angle. As most above-water shapes are edged, the separation lines will be fixed and dependence on the Reynolds number is then insignificant. On the other hand, the following circumstances lead to various complications and uncertainties:


It can be assumed, that as long as results of appropriate wind-tunnel tests or CFD computations for a given configuration are available, they can be effectively applied for simulating the manoeuvring motion of the corresponding ship in wind or for checking compliance with some wind related criteria. The experimental option is nowadays relatively costly not because of wind-tunnel tests per se, which are in this case simple and not excessively time-consuming, but because of the necessity of manufacturing required scaled models although maybe rapid development of the 3D printing will reduce this cost considerably.

If the above-water configuration of a ship is close to one of the models tested before and the corresponding aerodynamic coefficients are available, these can be used with a rather high degree of certainty. However, the prototype approach is not always possible and several attempts to devise "universal" methods were undertaken. For instance, the method developed by Melkozerova and Pershitz, see [57], provides approximations for the sway force and moment coefficients for single-island and twin-island superstructures whose relative lengths and positions of the centroids serve as defining factors.

Later two alternative empiric methods were developed on the basis of systematic wind-tunnel tests: the method of parametric loading functions developed by Blendermann and the method of trigonometric series. Both methods are described in detail by Blendermann [75] and are only outlined below.

The method of loading functions is a specific approximation of tabulated aerodynamic characteristics performed over 15 ship types and allowing corrections for the longitudinal position of the centroid of the lateral windage area. The method of trigonometric series was developed by Fujiwara and it presumes approximation of the aerodynamic coefficients with 3 or 4-terms trigonometric polynomials over the whole range of the sideslip angle. The coefficients of these trigonometric polynomials are algebraic polynomial regressions on a set of geometric parameters. Specifics of the latter is the separate treatment of the geometry of the above-water hull per se and of the superstructures. Recently, an interesting and promising method based on the combination of elliptic Fourier descriptors with artificial neural networks was proposed [76] but still not brought to the practical tool level.

The methods outlined above are more suitable for simulation applications requiring accurate predictions of aerodynamic loads. As was mentioned earlier, standardization tasks can be often effectively handled using some approximate or upper-end conservative estimates and a simple method aiming at possessing such properties is proposed below.

A simple generic model for the surge and sway force coefficients is based on the evident fact that all loads are 2*π*-periodic. Then, accounting for the evident symmetry properties and retaining only the first non-zero terms in the Fourier expansion of the coefficients it is obtained:

$$\mathbb{C}\_{AX} = -\mathbb{C}\_{AX0}\cos\beta\_{A\prime}\ \mathbb{C}\_{AY} = \mathbb{C}\_{AY0}\sin\beta\_{A\prime} \tag{20}$$

where *CAX*<sup>0</sup> and *CAY*<sup>0</sup> are some positive constant parameters representing the magnitude of the aerodynamic force in head/stern or beam apparent wind, respectively.

The average recommended values of these parameters are [57]:

$$
\mathbb{C}\_{AX0} = 1.0, \; \mathbb{C}\_{AY0} = 1.05. \tag{21}
$$

Of course, these values are approximate, suggested many decades ago and reflect the averaged properties of the then existing hull shapes. In addition, an alternative and even more obsolete value *CAY*<sup>0</sup> = 1.2 is mentioned in the book [56].

To estimate possible uncertainties associated with approximations (20), it makes sense to check them against recent experimental data. The richest available database of this kind is that collected by Blendermann on the basis of his own wind tunnel tests. Blenedermann's data can be found in 3 sources: in the reference book [77], in the report [78] and the manual [75]. The cited report describes the most complete database containing data for 48 configurations (in some cases two different configurations correspond to the same ship but at different loading conditions). A thorough comparison carried out by the authors showed that 21 of these configurations were also presented in [77], and 21—in [75]. All data from the report [78] were checked against alternative sources and digitized by the authors using the available tables of values. The book [77] contained one additional configuration not present in the report and it was decided to ignore it.

The experimental data for *CAX* are plotted as symbols in Figure 3 where also the response provided by (20) is shown as a dashed line. It is evident that this simple model provides a fair averaged estimation for the set of empiric data but in many cases, the magnitude of the surge force is substantially underestimated. To provide more conservative estimates, the following two-term equation can be proposed:

$$\mathcal{C}\_{AX} = -\mathcal{C}\_{AX0}^{M}\cos\beta\_A + \mathcal{C}\_{AX0}^{5}\cos 5\beta\_A \tag{22}$$

with *C<sup>M</sup> AX*<sup>0</sup> = 1.35 and *<sup>C</sup>*<sup>5</sup> *AX*<sup>0</sup> = 0.2, and the corresponding response is shown in Figure 3 with a solid line. It is obvious that the formula (22) does really provide an upper limit for the absolute value of the surge force for most configurations, The experimental points more or less significantly dropping outside the envelope at air drift angles around 30 and 150 degrees correspond not exactly to a ship but to an empty floating dock with rather high walls but with a relatively small transverse section area. So, Equation (22) with the indicated values of the constant parameters can be recommended if really conservative estimates are desirable. A similar analysis was also performed for the sway force coefficient in Figure 4 where the dashed line represents the response provided by the Equation (20) and the solid line—by the following model:

$$\mathbb{C}\_{AY} = \mathbb{C}\_{AY0}^{M} \sin \beta\_A + \mathbb{C}\_{AY}^{3} \sin \mathfrak{B} \beta\_A \tag{23}$$

with *C<sup>M</sup> AY*<sup>0</sup> = 1.17 and *<sup>C</sup>*<sup>3</sup> *AY* = 0.15.

The latter formula produces a rather clear envelope for almost all experimental data except those corresponding to the same floating dock and, at *β<sup>A</sup>* = 60◦, to a car carrier with an exceptionally high freeboard.

Analysis of data for the yaw moment coefficient turns out somewhat more complicated as an additional parameter comes into play. The generic model for the yaw moment coefficient is [57]:

$$\mathbb{C}\_{AN}(\beta\_A) = \mathbb{C}\_{AY}(\beta\_A) \cdot \mathbf{x}\_A'(\beta\_A),\tag{24}$$

where *x <sup>A</sup>* is the relative lever of the sway force which, in its turn, is represented as

$$
\mathfrak{x}'\_A(\beta\_A) = \mathfrak{x}'\_w + \Delta \mathfrak{x}'\_A(|\beta\_A|),
\tag{25}
$$

where *x <sup>w</sup>* = *xw*/*LOA* is the relative abscissa of the centroid of the lateral windage area and the function Δ*x <sup>A</sup>*() accounts for the displacement of the sway force application point and can be approximated as

$$
\Delta \mathbf{x}'\_A = k\_\mathbf{x} \left( \frac{1}{4} - \frac{|\mathcal{B}\_A|}{2\pi} \right),
\tag{26}
$$

which is generalization of the equation recommended in [56,57] where it is assumed that *kx* ≡ 1.0.

Blendermann's data on the sway force and yaw moment coefficients were used to generate experimental values of Δ*x <sup>A</sup>* using the evident relation:

$$
\Delta \mathbf{x}'\_A = \mathbb{C}\_{AN}(\beta\_A) / \mathbb{C}\_{AY}(\beta\_A) - \mathbf{x}'\_w \tag{27}
$$

and the resulting values are plotted as symbols in Figure 5 except for the values corresponding to the drift angles close to 0 or 180 degrees as the absolute values of *CAY* and *CAN* are very small which may lead to unnaturally high relative errors in Δ*x A*.

**Figure 3.** Experimental (circles) and analytical (lines) data for surge force coefficient.

**Figure 4.** Experimental (circles) and analytical (lines) data for sway force coefficient.

The responses provided by (24) at *kx* = 1.0 (dashed line) and *kx* = 0.8 (solid line) are also plotted there. It is obvious that the scatter of empiric data is considerable, and in any particular case, the error caused by using (24) can be very large. However, in general, the linear approximation looks reasonably adequate and the value *kx* = 1 leads to somewhat more conservative estimates. At the same time, the value *kx* = 0.8 worked somewhat better delimiting the stripe of the *x <sup>A</sup>* values for various *x <sup>w</sup>*. The lower line in Figure 6 was obtained from (24) at *kx* = 0.8 and *x <sup>w</sup>* = −0.11 which is the lowest value of the centroid's abscissa for all configurations presented in [78] while the upper line corresponds to *x <sup>w</sup>* = +0.13 which is the highest value.

**Figure 5.** Increment of sway force lever as a function of the air drift angle.

**Figure 6.** Sway force lever as a function of the air drift angle.

Finally, the measured values of the yaw moment coefficients *CAN* are plotted in Figure 7 together with the predictions obtained with the formula (24) at *x <sup>w</sup>* = −0.11 (lower line), *x <sup>w</sup>* = 0 (middle line) and *x <sup>w</sup>* = 0.13 (upper line). The outlying empiric values correspond to a ro-ro/lo-lo ship, to a tanker with the aft superstructures and to the floating dock mentioned above.

Regarding some possible future developments, it makes sense to note that Dand [26] proposed two additional parameters describing the windage area geometry: the lateral aspect ratio *kAL* = 2*AL*/*L*<sup>2</sup> *OA* and the transverse aspect ratio *kAT* = <sup>2</sup>*AT*/*L*<sup>2</sup> *OA*. Unfortunately, the authors are not aware of any attempts to use these parameters for the classification and clusterization of ship aerodynamic data.

**Figure 7.** Experimental (circles) and analytical (lines) data for yaw moment coefficient.

#### **6. Discussion and Conclusions**

An attempt to work out rational approaches to the development of manoeuvrability criteria and standards was undertaken and is described in the present article. A definite emphasis was put on additional "environmental" criteria, which aim at complementing the existing IMO set of standards. Even though primarily it was found more rational to avoid criteria related to the resisting capacity of a ship to exogenous factors leaving this issue to performance specifications supposed to be formulated by the customer, the recently emerged danger of the appearance of underpowered and more prone to external factors ships stimulated revision of that attitude. As long as the introduction of the Energy Efficiency Design Indices stimulating the under-powering was performed on the level of officially introduced IMO regulations, it is quite natural to balance those indices with some counter-weighting requirements in order to prevent unacceptable reduction of safety.

A natural solution could be to accompany the reduction of the engine power and the design speed with appropriate augmentation of the effectiveness of the main control device in such a way that the resistance capacity of the ship to external factors remains unchanged. In some respects, such formulation of the manoeuvrability standardization problem is not typical and requires a special analysis, which is presented in this article. The main result of this analysis is an outline of a procedure for modification and extension of the IMO set of manoeuvring standards which seems to be relatively simple and promising. The procedure presumes execution of the following steps:


It can be concluded from the present study that future progress in developing consistent and practical manoeuvring standards depends on development of a fast, reliable and validated method for simulating manoeuvring motion in waves. Systematic simulations with various hull forms are expected to provide sufficient information for confirming or disproving the hypothesis of concurrency of manoeuvring qualities in still water and of behaviour of a ship in waves. If the hypothesis is confirmed, modification of IMO standards can be carried out. Of course, such modification, as well as development of additional standards for controllability in wind, will require adjustment to statistical data on the existing fleet.

**Author Contributions:** Conceptualization, S.S. and C.G.S.; methodology, S.S.; validation, S.S.; formal analysis, S.S.; investigation, S.S.; resources, C.G.S.; data curation, S.S.; writing—original draft preparation, S.S.; writing—review and editing, S.S. and C.G.S.; visualization, S.S.; supervision, C.G.S.; project administration, C.G.S.; funding acquisition, C.G.S. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the project SHOPERA—Energy Efficient Safe SHip OPERAtion, which was partially funded by the EU under contract 605221 and by the project IC&DT–AAC n.º02/SAICT/2017 "Simulation of manoeuvrability of ships in adverse weather conditions" funded by the Portuguese Foundation for Science and Technology (FCT). This work contributes to the Strategic Research Plan of the Centre for Marine Technology and Ocean Engineering (CENTEC), which is financed by the Portuguese Foundation for Science and Technology (Fundação para a Ciência e Tecnologia—FCT) under contract UIDB/UIDP/00134/2020.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors appreciate support obtained from Blendermann and Soeding in relation to the wind tunnel data resulted from studies carried out at the Institute of Shipbuilding of the University of Hamburg.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript or in the decision to publish the results.
