The heelinduced sway force and yaw moment of a highspeed craft in following regular waves
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Abstract
The coupling between heel and the loads in the horizontal plane is usually neglected in manoeuvrability studies. However, the heel–sway and heel–yaw coupling can play an important role in potentially unsafe conditions, such as in a following sea. In these conditions, small fast vessels experience dynamic instabilities which threaten their ability to maintain a straight course. In this study, the coupling between the static heel and the sway force and yaw moment was investigated for a highspeed craft. The objective of this work is to understand the effect of heel on the manoeuvring in following waves, and to predict this effect by means of numerical tools for different combinations of wave characteristics and vessel speeds. A dedicated captive model test campaign was conducted to evaluate the manoeuvring loads in sway and yaw when the craft has a heel angle in following regular waves. The tests were performed in the towing tank of Delft University of Technology. The heelinduced loads depend strongly on the longitudinal position of the vessel in the wave, and they significantly differ from the heelinduced loads in calm water at the respective speed. The data carried out in the model tests were used to describe empirically the heelinduced loads for several combinations of ship speeds and wave characteristics. This empirical description was meant to correct a 3D potential flow boundary element method (BEM), with the objective of being able to predict these loads on a wide range of conditions. The corrected 3D BEM was used to simulate the behaviour of the highspeed craft in following regular waves. This analysis showed that the heelinduced loads have the effect of stabilizing the ship to the inception of dynamic instabilities in the following sea.
Keywords
Highspeed craft Manoeuvrabilityinwaves Heel Captive model tests Boundary element method; broachingtoList of symbols
 β
Drift angle
 δ
Steering angle
 Φ
Heel angle
 θ
Pitch angle
 σ
Ship heave
 μ
Wave incidence angle
 ξ_{G}/λ
Nondimensional location of the ship centre of gravity in the wave
 λ
Wave length
 ω
Wave frequency
 ω_{E}
Wave encounter frequency
 φ
Wave phase
 ς
Wave elevation above calm water line
 A
Wave amplitude
 c
Wave celerity
 Fr
Froude number
 G
Ship centre of gravity
 H/λ
Wave steepness
 m
Wave signal mean value
 r
Yaw rate
 T
Wave period
 T_{E}
Wave encounter period
 u
Ship surge
 U
Ship total speed
 U_{R}
Load uncertainty
 x_{T}
Position of the model along the towing tank
 Y
Sway force
 Z
Heave force
 M
Pitch moment
 N
Yaw moment
 Y_{Φ}, Y_{σ}, Y_{θ}
Sway force derivative in heel, heave, and pitch
 Z_{σ}, Z_{θ}
Heave force derivative in heave and pitch
 M_{σ}, M_{θ}
Pitch moment derivative in heave and pitch
 N_{Φ}, N_{σ}, N_{θ}
Yaw moment derivative in heel, heave, and pitch
1 Introduction
Everybody is aware of the danger of a ship sailing in rough seas. The risk significantly increases for small and fast vessels, since they are more exposed to relatively large motions. The integrity of those ships and the safety of the crew are compromised by the action of the waves on the hull, which can result in lower transverse stability, high accelerations, and course keeping problems. The seakeeping of these ships is widely studied in head and bow quartering sea conditions, since the accelerations in the vertical plane are often a limiting factor for the vessel operability. However, several past studies [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] focused on the problem of a vessel sailing in following and sternquartering waves, recognizing it as a dangerous situation for a ship at sea. In these conditions, the large waveinduced yawing moment turns the ship from its original course despite any steering counteractions. The danger is also recognized by mariners, who are particularly concerned by the difficulty of course keeping and the associated potential hazards such broachingto and capsizing.
The dynamic instability of ships sailing in following seas is extremely complex and the causes governing this type of phenomena are not yet well understood. In most cases, the vessel is captured by the incoming stern wave and accelerated to its speed (surfriding); then, the vessel is suddenly turned beam on to the seaway, phenomenon known as broachingto. In extreme cases, these events are quick and violent, and can lead to a capsize. Surfriding and broachingto are strictly connected phenomena. Usually broachingto is preceded by surfriding. These phenomena can be studied in a quasisteady fashion [1], since the relative encounter frequency between the waves and the ship is very low during a surf. The vessel is “frozen” on a certain wave position, and the hull loads are related to that location on the wave.
The ship dynamics in following waves results from a combination of the seakeeping and manoeuvrability characteristics of the vessel. Hull hydrodynamic and hydrostatic loads, steering forces, and wave destabilizing effect contribute to different extents to the onset of loss of control. Manoeuvring loads on the hull are largely dependent on the wave characteristics and in the relative position that the vessel assumes on a wave. Variations of the hull submerged geometry result in great differences in the loads acting on the ship. These variations are particularly significant on small craft: the waves are relatively large with respect to the hull size. These loads must be predicted as a function of the longitudinal position in the wave and considering the correct hull submerged geometry at that position.
Ship manoeuvrability studies have usually been confined to the three degrees of freedom in the horizontal plane: surge, sway, and yaw. Roll is often neglected in manoeuvrability research [14]. In other works [15, 16, 17, 18, 19, 20, 21, 22], the effect of roll on manoeuvrability was considered, mainly in calm water, both for displacement and semidisplacement vessels. Roll can affect the manoeuvrability of small craft in waves, since the heel angle significantly changes the submerged hull characteristics. When heeled, the vessel experiences a side force and a yawing moment due to the nonsymmetric submerged geometry. In the literature, these effects are denominated as heel–yaw and heel–sway coupling [18] or as heelinduced loads [23]. Hereby, both the expression will be used in an equivalent way. Because of the already great complexity of the problem, the effect of the heel–sway and heel–yaw hydrodynamic coupling in following waves is not commonly investigated, either experimentally or numerically. This paper presents an investigation of the coupling between heel and the manoeuvring loads in following waves by means of experimental captive model tests. The experiments are a starting point in the understanding of this physical problem and its characterization by numerical tools.
In an earlier work by Hashimoto et al. [23], captive model tests were carried out in following waves to determine the heelinduced loads on a fishing vessel, for one wave condition and one speed. That study considered high heel angles to realize quantitative prediction of the capsize dynamics due to broachingto. In the current work, the heelinduced loads on a highspeed craft are investigated experimentally, but at a lower range of heel angles. The interest is directed to the linear characterization of the manoeuvring loads and the inception of dynamic instabilities. Furthermore, hardchine craft in these sea conditions usually experiences moderate roll angles which are often between 10° and 15°.
In the experimental setup in [23], the vertical position of the ship model was fixed during each run, and then, the coefficients were corrected to account for the error in the ship heave and pitch position on the wave. In the current study, the experimental setup was different. The ship model was fully constrained by a 6DOF oscillator (Hexapod), which moved in heave and pitch to achieve the vertical equilibrium of forces and moments of the model in the wave. This was meant to reproduce the most natural vertical attitude of the craft in waves, since the manoeuvrability of the vessel is affected by the ship position in the water. The oscillations were calculated beforehand and used as the main input to the Hexapod.
Fully constraining the model to an oscillator ensures a better stiffness of the experimental setup: a setup moving in heave and pitch and fixed in the other motions might interfere with the measurement of the loads in an unpredictable manner. Furthermore, the use of an oscillator allows the model to be orientated in every desired direction making it possible to measure the sensitivity of the loads with respect to welldefined positions of the model. This is very meaningful for manoeuvring and seakeeping applications: it is possible to measure all the forces and moments acting on the model in waves in static conditions or with forced oscillations [24]. In this work, the model was forced to assume its equilibrium position on the wave; more in general, the model can be forced to assume every prescribed position or to perform every oscillation on the waves.
Given the high complexity of the problem, the numerical determination of the manoeuvring loads on a heeled ship is difficult. Highspeed craft are commonly assumed to be symmetrical for the purpose of validation of the calculation of the hydrodynamic pressure distribution, devoting attention to the dynamics in the vertical plane [25, 26]. Nonsymmetric hardchine heeled hulls have been studied numerically by several past researchers [27, 28, 29], mainly in calm water or head waves, by means of strip theory codes or simplified 2D BEM. The production of lift in heeled conditions is governed by very complex viscous phenomena (such as the separation of the flow at the chine), which are difficult to predict with slender body theories or potential flow models. Often, the results given by these numerical tools are not satisfactory, and do not focus on the effect of heel on the manoeuvring characteristics of the vessel. A validation of more sophisticated numerical fluid solvers for these conditions is still lacking.
In this work, the experimental data collected were used to define an empirical description of the loads in the domain of ship speeds and wavelengths. The aim is to be able to predict these loads over the widest interesting range of ship speeds and wavelengths. This empirical description of heelinduced loads was used to correct a 3D blended potential flow BEM with the final objective of understanding the effect of the heelinduced loads in the inception of dynamic instabilities.
In [18], Renilson and Manwarring used the heel–yaw and heel–sway coupling loads obtained experimentally in calm water to assess their effect on the inception of broachingto. The outcomes showed that the heelinduced loads increase the likelihood of broachingto. After the experimental investigation in following waves already mentioned in this introduction, Hashimoto et al. [23] proved that the heelinduced loads have the consequence of stabilizing the ship contrarily to what found in [18]. In the present study, this phenomenon was investigated extending the investigation in [23] to a higher number of wave and vessel speed conditions.
2 The captive model tests
2.1 Test case definition
Particulars of the six conditions tested during the experiments in terms of model speed, wavelength, and wave encounter frequency
Conditions  Fr = U/sqrt(gL)  λ/L  ω′ _{ E} 

1  0.38  1  0.261 
2  0.38  1.5  1.076 
3  0.48  1.5  0.071 
4  0.48  2  0.546 
5  0.57  2.25  1.715 
6  0.57  2.5  0.228 
Main characteristic of the rescue boat NH1816
NH1816 parameter  Values 

Length between perpendicular L  18.37 m 
Overall breadth  5.60 m 
Draft at zero speed  1.10 m 
Longitudinal centre of gravity  6.00 m 
Weight  34 t 
2.2 Experimental procedure
The experiments were conducted in a rectilinear towing tank, the wave incidence angle, μ, and the drift angle, β, being zero during the entire experimental campaign. Thus, sway force, Y′, and yaw moment, N′, acting on the horizontal plane were caused exclusively by the ship heel angle. The loads depend on the longitudinal location of the ship centre of gravity on the wave, ξ_{G}, given nondimensionally as ξ_{G}/λ: ξ_{G}/λ = 0 wave crest, 0 < ξ_{G}/λ < 0.5 wave front, ξ_{G}/λ = 0.5 wave trough, ξ_{G}/λ > 0.5 wave back.
The forces and moments in sway and yaw are expressed by the heel angle Φ multiplied by the hydrodynamic coefficients \(Y^{\prime}_{\varPhi }\) and \(N^{\prime}_{\varPhi }\). These coefficients represent the effect of the ship’s static heel angle on the manoeuvring loads.
The experiments were performed in the Delft University of Technology towing tank. The vessel model was attached to the carriage using an oscillator capable of moving in six degrees of freedom. The forces and moments were measured on a frame installed between the model and the oscillator. The wave elevation was measured by a wave probe rigidly attached to the carriage at the same longitudinal location as the centre of gravity of the model, G.
2.3 Numerical correction of the model vertical equilibrium
2.4 Uncertainty analysis
3 The numerical simulations
The dynamic pressures due to the incoming flow are evaluated only on the fixed below water geometry. Hydrostatic loads (buoyancy), Froude–Krylov wave loads, crossflow drag, and frictional resistance are specified on the actual submerged geometry, i.e., on the instantaneous wavy free surface. The linearization of the boundary conditions and the Green function around the flat waterline results in a less accurate prediction than their computation on the actual wavy free surface. However, the linearization drastically reduces the computational effort allowing this to be applied to a wide range of variables.
4 Results

nondimensional heel–sway hydrodynamic coupling coefficient, Y′_{Φ};

nondimensional heel–yaw hydrodynamic coupling coefficient, N′_{Φ}.
Values of the heel–sway and heel–yaw hydrodynamic coupling obtained in calm water
(Taken from [19])
Fr  Y′ _{ Φ}  N′ _{ Φ} 

0.38  5.282E−03  3.759E−05 
0.48  1.980E−03  − 1.043E−03 
0.58  − 8.009E−04  − 1.215E−03 
The amplitude and the mean value of the sinusoidal signals are greater for greater values of Froude number. Between the lowest speed Fr = 0.38 and the other two higher speed cases Fr = 0.48 and Fr = 0.58, a phase shift of about half wave length was observed. The variations with respect to the calm water values are quite significant. The value of Y′_{Φ} is mostly negative at Fr = 0.38 for all the locations ξ_{G}/λ, meaning that, when heeled at starboard, the sway force is directed to port. For the higher speeds, the sign of Y′_{Φ} changes from negative on the wave front to positive when the ship is located on the wave back.
Unlike the values of N′_{Φ} in calm water which are negative for the speed range investigated, in following waves, the heel–yaw coefficient changes sign. Whereas at Fr = 0.38 N′_{Φ} is always positive for all the longitudinal locations in the wave, ξ_{G}/λ, for the other two cases, the coefficient is negative on the front of the wave, and positive on the back of the wave. Thus, on the front of the wave, a heel angle to starboard means that the coupling effect makes the vessel to turn to port. The opposite occurs on the back of the wave.
The heelinduced loads originate from the nonsymmetrical hull submerged geometry and by the vertical position of the vessel in the wave. They are caused by the lift developed on the hull bottom, the Froude–Krylov, radiation and diffraction effects, and by more complex viscous phenomena such as the flow separation at the chine. Since only the total forces and moments are measured during a captive model experimental test, it is very hard to examine the characteristics of these loads in detail. Advanced numerical simulations (RANSE solvers) might provide a better insight in the study of these loads.
5 Utilization of the results
6 The effect of the heelinduced loads in the following sea
The investigation covers 7 vessel speeds equally spaced between Fr = 0.3–0.6; for each speed 8 wavelengths are considered, between a minimum of λ/L =0.6 and a maximum of λ/L = 3.0, resulting in 56 total conditions. The wave steepness coincides with the value used in the experiment, H/λ = 0.06, and the initial wave heading angle μ is equal to − 20° in all the conditions simulated.
The effect of the heelinduced loads is analysed by comparing the broaching zones within the range of the conditions examined with and without the utilization of the heelcoupling terms. In the latter case, the coupling terms were set to zero by correcting the 3D BEM outcomes in a similar way to what done in Sect. 5. The results are presented in Fig. 15.
7 Conclusions
This work investigated the hydrodynamic coupling between heel and the manoeuvring loads in sway and yaw in following waves for a highspeed craft. The heelinduced sway force and yaw moment were investigated experimentally by means of captive model tests in six conditions of model speeds and wave lengths. The measured data were utilized for an empirical formulation of these loads on a wider range of speeds and wavelengths. This formulation was implemented into a time domain 3D BEM, and the comparison between experimental and numerical results was presented.
The experimental tests carried out in this study and presented in Sect. 2.4 are based on a not common technique of synchronization of waves and model motions. This work proved that wavemaker, carriage, and model oscillator can be synchronized with a good level of accuracy at low–medium encounter frequency. Although the measured data showed a good reliability, a more rigorous procedure to estimate the uncertainty of the experimental technique is missing. This issue must be addressed in future studies.
The results of the experiments, presented in terms of nondimensional hydrodynamic coefficients Y′_{Φ} and N′_{Φ}, showed that the coupling between heel and the manoeuvring loads in sway and yaw depends strongly on the location of the vessel in the wave. Furthermore, Y′_{Φ} and N′_{Φ} differ significantly from their respective values in calm water. The single contribution of the hydrodynamic lift, the wavemaking effect, and incoming wave to the manoeuvring loads due to heel must be still clarified by deeper hydrodynamic insights. More accurate numerical methods can be used to investigate in detail this important phenomenon which affects significantly the dynamics of the ship in the seaway.
a _{1}  a _{2}  a _{3}  

Y′ _{ Φ}  \(A^{\prime}_{{Y^{\prime}\varPhi }}\)  − 0.0407  0.0118  0.0048 
\(\varphi ^{\prime}_{{Y^{\prime} \varPhi }}\)  2.1898  − 0.6934  − 0.2319  
\(m^{\prime}_{{Y^{\prime}\varPhi }}\)  − 0.0038  0.0131  − 0.0250  
N′ _{ Φ}  \(A^{\prime}_{{N^{\prime}\varPhi }}\)  0.0119  − 0.0013  − 0.0008 
\(\varphi ^{\prime}_{N^{\prime}\varPhi }\)  − 1.7460  0.2292  − 0.1607  
\(m^{\prime}_{N^{\prime} \varPhi }\)  − 0.0252  0.0012  0.0121 
The corrected 3D BEM was used to assess the effect of the heel–yaw and heel–sway hydrodynamic coupling into the inception of broachingto in the following sea. The heelinduced loads cause the ship to be more stable and then reducing the likelihood of broaching inception. The results agree with the investigation of Hashimoto et al. [23], confirming the validity of these considerations to several wave and shipspeed conditions. The study of Renilson and Manwarring [18] shows instead an opposite result of the effect of the heelinduced loads into the inception of broachingto. It must be said that this work was based on a containership, which is naturally very different from a semidisplacement vessel or a highspeed craft. However, most importantly, it considered only the heelinduced loads measured in calm water as first approximation for the more complex dynamics in a seaway. This confirms another fact shown by this work: the nature of heel–sway and heel–yaw coupling in the following sea is very different, even opposite, from the case in calm water. The effects originating from the incoming waves are more significant that the dynamics of the lift and wavemaking in calm water, especially for the heel–yaw hydrodynamic coupling N′_{Φ}.
Notes
Acknowledgements
The authors would like to acknowledge KNRM and DAMEN Shipyards as owner and designer of the vessel used as model in this study, to having made the information of the NH1816 rescue boat available.
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