## Abstract

For maneuverability in waves, all the current theoretical methods rely on an unverified hypothesis that the hydrodynamic derivatives in waves are consistent with those in calm water. Many scholars attribute the errors between the predicted motions using the theoretical methods and the experimental motions to this irrational hypothesis. To explore the rationality of the hypothesis and improve the theoretical methods, the planar motion mechanism tests of KVLCC2 in calm water and regular head waves were carried out in the towing tank. Different components of the hull forces are obtained by the fast Fourier transform technique. Then, maneuvering hydrodynamic derivatives in calm water and head waves are calculated to assess the wave effect on them quantitatively. The results reveal that the wave effect on $Yv\u2032$, $Nr\u2032$ and the high-order derivatives are obvious, and the influence on $Nr\u2032$ and $Yv\u2032$ is small. This finding sheds light on the error of the earlier hypothesis and is conducive to the improvement of potential flow methods in the future.

## 1 Introduction

The maneuverability of the ship in waves has been a significant research topic with increasing attention nowadays. Compared with the traditional maneuverability in calm water ignoring the external environmental forces and the seakeeping study assuming the heading angle is constant, the maneuverability in waves is closer to the real navigation state with steering. At present, there are mainly three different methods on this topic, which are the experimental method, the theoretical method based on the potential flow theory, and the pure computational fluid dynamics (CFD) method. Compared with the other two methods, the theoretical method is the most widely used method due to its efficiency. The theoretical method can be classified into two-time-scale method and hybrid method. For the two-time-scale method [1,2], the total velocity is the superposition of the velocities calculated by the maneuvering models and seakeeping models. For the hybrid method [3,4], the total velocity is calculated based on a group of rigid-body motion equations in which the total forces are the superposition of the maneuvering forces and the wave forces. Both methods rely on the separated maneuvering model in which the hydrodynamic derivatives are extremely important. Hence, it is critical to obtain the hydrodynamic derivatives in different conditions. However, both methods rely on an assumption that the hydrodynamic derivatives in waves are considered as the same as those in calm water, i.e., the wave effect on the hydrodynamic derivatives is ignored.

Some scholars [5] contribute the error between the predicted trajectories using the potential flow methods and those obtained from the experiments to this unverified hypothesis. In addition, the International Towing Tank Conference (ITTC) specialist conference [6] also recommends further research on the captive model tests in waves. Therefore, it is urgent and important to assess the rationality of the hypothesis by the captive model tests in waves.

Actually, some scholars have conducted relevant experimental studies earlier. Motora et al. [7] conducted planar motion mechanism (PMM) tests in following waves to obtain hydrodynamic derivatives in waves and analyze the reason for the occurrence of the broaching-to phenomenon. Then, Son and Nomoto [8] performed similar tests in following waves and considered the effect of the roll motion for the first time. Araki et al. [9] obtained the hydrodynamic derivatives in waves using the experiments to improve the prediction accuracy of the broaching-to phenomenon. Recently, Bonci et al. [10] carried out PMM tests in following and stern quartering waves and proposed a correction method to predict the broaching-to phenomenon. It is clear that all the aforementioned experiments were performed under the following wave condition. Few scholars have carried out the captive model tests in other wave directions. Yasukawa and Adnan [11] studied the hydrodynamic derivatives in head waves but only performed the oblique towing tests. On one hand, the following wave condition is important because the dangerous broaching-to phenomenon might occur under this condition. On the other hand, the effect of wave parameters on the maneuvering forces can be obtained easily under this condition. For example, under the surf-riding condition in following waves, the relative position between the ship and wave (*ξ*_{G}/*λ*) will be fixed all the time. However, this parameter will change frequently in head waves or beam seas.

Even though there are more difficulties, research on the other wave directions is necessary for predicting the maneuverability in adverse sea conditions. Therefore, a method of separating the high-frequency and low-frequency forces by the fast Fourier transform (FFT) technique is adopted in this paper to deal with this problem in head waves. The effect of the frequently changed parameter *ξ*_{G}/*λ* on the low-frequency maneuvering forces is considered as a time-averaged effect. Then, the low-frequency components of the hydrodynamic forces in waves can be separated and analyzed.

In this study, the PMM tests of KVLCC2 are performed in calm water and regular heading waves to investigate the wave effect on the hydrodynamic derivatives. The maneuvering components of hydrodynamic forces are obtained by excluding the forces at different frequencies using the FFT technique. The hydrodynamic derivatives in waves are calculated by the integral method and the curve-fitting method. Based on the comparison between these hydrodynamic derivatives, the unverified hypothesis is evaluated systematically. Meanwhile, some suggestions that the wave effect on $Yv\u2032$ and $Nr\u2032$ should be considered are concluded to improve the accuracy of the theoretical methods for maneuverability prediction in waves. In addition, the experimental data will be useful materials for future research on the numerical simulations on this topic.

## 2 Experimental Arrangements

The PMM tests of KVLCC2 in calm water and regular waves are carried out in the towing tank of Shanghai Jiao Tong University, China. The main particulars of the ship are listed in Table 1. The ship model is made of fiber-reinforced plastics with a scale of 1:70. The towing tank (see Fig. 1) is 300 m long, 16 m wide, and 7.5 m deep and equipped with a drive carriage, PMM carriage, wavemaker, and wave-dampening beach. The drive carriage is instrumented with several data acquisition computers, speed circuits, and signal conditioning for analog voltage measurements of such as forces and carriage speed. The drive carriage pulls the PMM carriage where the ship model KVLCC2 is attached. The wave is generated by the multi-unit wavemaker system which uses 40 rocker flaps to generate the preset waves. The wave-absorption system includes a wave-dampening beach and an active wave-absorption system.

Symbol | Unit | Full-scale | Model-scale |
---|---|---|---|

Length between perpendiculars, L_{PP} | m | 320 | 4.5714 |

Breadth, B | m | 58 | 0.8286 |

Draft, d | m | 20.8 | 0.2971 |

Displacement, $\u2207$ | m^{3} | 312621.7 | 0.9114 |

Initial metacentric height, GM | m | 5.71 | 0.2393 |

Longitudinal position of the center of buoyancy, LCB (+) | m | 11.04 | 0.1600 |

Block coefficient, C_{b} | — | 0.8098 | 0.8098 |

Yaw radius of gyration, k_{zz}/L_{PP} | — | 0.25 | 0.25 |

Symbol | Unit | Full-scale | Model-scale |
---|---|---|---|

Length between perpendiculars, L_{PP} | m | 320 | 4.5714 |

Breadth, B | m | 58 | 0.8286 |

Draft, d | m | 20.8 | 0.2971 |

Displacement, $\u2207$ | m^{3} | 312621.7 | 0.9114 |

Initial metacentric height, GM | m | 5.71 | 0.2393 |

Longitudinal position of the center of buoyancy, LCB (+) | m | 11.04 | 0.1600 |

Block coefficient, C_{b} | — | 0.8098 | 0.8098 |

Yaw radius of gyration, k_{zz}/L_{PP} | — | 0.25 | 0.25 |

The PMM facility is fixed to the carriage firmly so that their longitudinal velocities are the same. The ship model is attached to the PMM facility by two heave rods and restrained horizontally by springs on the PMM facility. Therefore, the horizontal motion (surge-sway-yaw motion) of the ship model is controlled by the PMM facility, and the ship can oscillate induced by the wave within limits to consider the radiation component on the hydrodynamic forces. The pitch and heave motions are totally free. As for the roll motion, it is restrained for simplification, which is accepted by many earlier studies [12].

### 2.1 Mathematical Modeling for Planar Motion Mechanism Tests.

Two right-hand coordinate systems (see Fig. 2) are utilized in the present mathematical model. One is the earth-fixed coordinate system *O*_{0} − *X*_{0}*Y*_{0}*Z*_{0}. The origin locates at the undisturbed free surface. The other is the ship-fixed coordinate system *G* − *XYZ*. The origin locates at the center of gravity *G*; the *x*-axis points to the bow and the *y*-axis to the starboard. Heading angle *ψ* is the angle between the *GX* axis and the *O*_{0}*X*_{0} axis and the clockwise from the *O*_{0}*X*_{0} axis to the *GX* axis is positive. The drift angle *β* is the angle between the resultant velocity *V*_{R} and the *GX* axis. The positive direction is clockwise from the resultant velocity to the *GX* axis. The velocities with the subscript M, such as *u*_{M} (equals to the carriage velocity *V*) and *v*_{M}, represent the velocities of the hull driven by the PMM facility. *u* and *v* are ship longitudinal and lateral speeds in the ship-fixed coordinate system, *r* is the ship yaw rate around the ship gravity center.

*m*stands for the ship mass,

*m*

_{x}and

*m*

_{y}are the added mass,

*I*

_{z}is the moment of inertia around the

*GZ*axis,

*J*

_{z}is the added moment of inertia around the

*GZ*axis, and the subscripts

*H*and

*E*represent the hull forces and the external forces provided by the PMM facility, respectively. In present study, the longitudinal force is not considered. The interaction between the longitudinal force and the forces in other directions can also be ignored.

*L*

_{PP},

*V*,

*V*/

*L*

_{PP}, and $12\rho LPP2V2$ are the characteristic scales for the length, velocity, angular velocity, and force, respectively. Non-dimensional variables are shown as

*v*and

*r*according to the Maneuvering Modeling Group (MMG) model [12]. Then, Eq. (1) is transferred into the non-dimensional one as follows

These polynomial coefficients in Eq. (3) are the maneuvering hydrodynamic derivatives obtained from the PMM tests. By controlling the horizontal motions of the ship through the PMM facility, only the required terms of hydrodynamic derivatives in the mathematical equations will be retained in each kind of captive model test. Table 2 shows the motions and the computable hydrodynamic derivatives for each kind of captive model test. *ω* (*ω* = 2*πf* = 2*π*/*T*, where *T* is the oscillation period) represents the circular frequency of the dynamic motions.

Test | Motions | Corresponding hydrodynamic derivatives |
---|---|---|

Oblique towing | β = constant$vE=v\u02d9=r=r\u02d9=0$ | $Yv\u2032$, $Yv|v|\u2032$, $Nv\u2032$, $Nv|v|\u2032$ |

Pure sway | v_{E} = −y_{max}ωcos(ωt)$r=r\u02d9=0$ | $Yv\u02d9\u2032$, $Nv\u02d9\u2032$, $Yv\u2032$, $Yv|v|\u2032$, $Nv\u2032$, $Nv|v|\u2032$ |

Pure yaw | ψ = −(y_{max}ω/U_{c})cos(ωt)v _{E} = −y_{max}ωcos(ωt)$v=v\u02d9=\beta =0$ | $Yr\u02d9\u2032$, $Nr\u02d9\u2032$, $Yr\u2032$, $Yr|r|\u2032$, $Nr\u2032$, $Nr|r|\u2032$ |

Combined yaw and drift | ψ = −(y_{max}ω/U_{c})cos(ωt)v _{E} = −y_{max}ωcos(ωt)β = constant | Y_{v|r|}, Y_{|v|r}, N_{v|r|}, N_{|v|r} |

Test | Motions | Corresponding hydrodynamic derivatives |
---|---|---|

Oblique towing | β = constant$vE=v\u02d9=r=r\u02d9=0$ | $Yv\u2032$, $Yv|v|\u2032$, $Nv\u2032$, $Nv|v|\u2032$ |

Pure sway | v_{E} = −y_{max}ωcos(ωt)$r=r\u02d9=0$ | $Yv\u02d9\u2032$, $Nv\u02d9\u2032$, $Yv\u2032$, $Yv|v|\u2032$, $Nv\u2032$, $Nv|v|\u2032$ |

Pure yaw | ψ = −(y_{max}ω/U_{c})cos(ωt)v _{E} = −y_{max}ωcos(ωt)$v=v\u02d9=\beta =0$ | $Yr\u02d9\u2032$, $Nr\u02d9\u2032$, $Yr\u2032$, $Yr|r|\u2032$, $Nr\u2032$, $Nr|r|\u2032$ |

Combined yaw and drift | ψ = −(y_{max}ω/U_{c})cos(ωt)v _{E} = −y_{max}ωcos(ωt)β = constant | Y_{v|r|}, Y_{|v|r}, N_{v|r|}, N_{|v|r} |

For the harmonic experiments, interval integral method or the Fourier integral method is required to separate the components of different phases to obtain the hydrodynamic derivatives [14]. The former method is to multiply the force function by sin(*ωt*) or cos(*ωt*) and integrate it in the interval of (0, 2*π*) to obtain different phase components relying on the integral characteristics of a trigonometric function. The latter is to integrate the force function in different intervals, and then separate the different phase components. In this study, the interval integral method is utilized, and the derived expressions are shown in Table 3. In this table, *Y*_{out} and *N*_{out} represent the parameters of the out-of-phase part with sin(*ωt*) and can be utilized to obtain the hydrodynamic derivatives. For one calculation period *T*, the start time is determined by a special signal by checking the ship’s position.

Test | Mathematic models |
---|---|

Pure sway | ${Yout=14(\u222b0\pi 2YEd\omega t\u2212\u222b\pi 23\pi 2YEd\omega t+\u222b3\pi 22\pi YEd\omega t)=\u2212Yva\omega \u2212Yv|v|(a\omega )2\pi 4Nout=14(\u222b0\pi 2NEd\omega t\u2212\u222b\pi 23\pi 2NEd\omega t+\u222b3\pi 22\pi NEd\omega t)=\u2212Nva\omega \u2212Nv|v|(a\omega )2\pi 4$ |

Pure yaw | ${Yout=14(\u222b0\pi YEd\omega t\u2212\u222b\pi 2\pi YEd\omega t)=(Yr\u2212mu\u2212mxu)a\omega 2/V+Yr|r|(a\omega 2/V)2\pi 4Nout=14(\u222b0\pi NEd\omega t\u2212\u222b\pi 2\pi NEd\omega t)=Nra\omega 2/V+Nr|r|(a\omega 2/V)2\pi 4$ |

Yaw and drift | ${Yc=14\u222b02\pi YEd\omega t=Yv|r|va\omega 2/u+(Yvv+Yv|v|v|v|)\pi 2Nc=14\u222b02\pi NEd\omega t=Nv|r|va\omega 2/u+(Nvv+Nv|v|v|v|)\pi 2Yout=14(\u222b0\pi YEd\omega t\u2212\u222b\pi 2\pi YEd\omega t)=(Yr\u2212mu\u2212mxu)a\omega 2/V+Yr|r|(a\omega 2/V)2\pi 4+Y|v|r|v|a\omega 2/VNout=14(\u222b0\pi NEd\omega t\u2212\u222b\pi 2\pi NEd\omega t)=Nra\omega 2/V+Nr|r|(a\omega 2/V)2\pi 4+N|v|r|v|a\omega 2/V$ |

Test | Mathematic models |
---|---|

Pure sway | ${Yout=14(\u222b0\pi 2YEd\omega t\u2212\u222b\pi 23\pi 2YEd\omega t+\u222b3\pi 22\pi YEd\omega t)=\u2212Yva\omega \u2212Yv|v|(a\omega )2\pi 4Nout=14(\u222b0\pi 2NEd\omega t\u2212\u222b\pi 23\pi 2NEd\omega t+\u222b3\pi 22\pi NEd\omega t)=\u2212Nva\omega \u2212Nv|v|(a\omega )2\pi 4$ |

Pure yaw | ${Yout=14(\u222b0\pi YEd\omega t\u2212\u222b\pi 2\pi YEd\omega t)=(Yr\u2212mu\u2212mxu)a\omega 2/V+Yr|r|(a\omega 2/V)2\pi 4Nout=14(\u222b0\pi NEd\omega t\u2212\u222b\pi 2\pi NEd\omega t)=Nra\omega 2/V+Nr|r|(a\omega 2/V)2\pi 4$ |

Yaw and drift | ${Yc=14\u222b02\pi YEd\omega t=Yv|r|va\omega 2/u+(Yvv+Yv|v|v|v|)\pi 2Nc=14\u222b02\pi NEd\omega t=Nv|r|va\omega 2/u+(Nvv+Nv|v|v|v|)\pi 2Yout=14(\u222b0\pi YEd\omega t\u2212\u222b\pi 2\pi YEd\omega t)=(Yr\u2212mu\u2212mxu)a\omega 2/V+Yr|r|(a\omega 2/V)2\pi 4+Y|v|r|v|a\omega 2/VNout=14(\u222b0\pi NEd\omega t\u2212\u222b\pi 2\pi NEd\omega t)=Nra\omega 2/V+Nr|r|(a\omega 2/V)2\pi 4+N|v|r|v|a\omega 2/V$ |

*w*represents the hydrodynamic derivative in waves, and the subscript

*W*represents the first-order wave forces. The hydrodynamic derivatives here are different from those in calm water. The wave effect, such as the wavelength, wave steepness, and

*ξ*

_{G}/

*λ*, has been considered in these derivatives.

The monitored forces can be preprocessed by the FFT method to leave the PMM motion frequency component. Then, this low-frequency component, which excludes the high-frequency wave force, is utilized to obtain the hydrodynamic derivatives in waves relying on similar methods as those for calm water conditions. Finally, the comparison between these hydrodynamic derivatives in waves and those in calm water can reveal the wave effect on the low-frequency forces.

In fact, the measured low-frequency forces contain not only the maneuvering forces considering the wave effect, but also the wave drift forces and their interactions. In experiments, second-order drift forces are obtained by subtracting the forces in calm water conditions from the time-averaged forces in wave conditions. However, there exist several theoretical limitations. First, the wave effects on the maneuvering forces are totally ignored. Second, the motions of ship maneuvering in waves are significantly different from the working condition of drift force obtained by experiments. The drift force is obtained under the condition of constant speed and fixed drift angle. During the maneuvers in waves, the speed and drift angle change in real-time, and the wave direction also changes, which will cause the accuracy of drift force to be questioned. To avoid these limitations, some scholars proposed a method to add the ship-induced motions in waves to the traditional low-frequency motion equations. Yao et al. [15,16] conducted simulations by considering the drift forces in the low-frequency part and showed a good prediction result. Their calculations are totally focused on the CFD methods which require experiments. In addition, compared with the linear superposition of motion, the author believes that the superposition of force is more reasonable. Therefore, in the present research, the drift forces are considered in the low-frequency part and systematical experiments are carried out.

### 2.2 Experimental System and Conditions.

*F*

_{yf}and

*F*

_{ya}represent the lateral forces monitored on the fore and aft of the ship respectively,

*l*

_{pmm}represents the distance between two strain gauge balances as shown in Fig. 3.

The ship model is free to pitch and heave. The surge-sway-yaw motion is fixed and determined by the PMM facility.

Test conditions can be divided into the calm water condition tests and the wave condition tests, which are summarized in Tables 4 and 5, respectively. In calm water conditions, two carriage velocities are adopted. One (0.953 m/s) is corresponding to the same Froude number as the full-scale vessel. The other is 60% of the first one to evaluate the speed effect on the hydrodynamic derivatives in calm water. Through the parameters listed in Tables 4 and 5, the PMM motions (see Table 2) can be calculated and determined [13]. For the dynamic tests, there are two methods to design the test conditions, one is to change the lateral amplitude and the other is to change the oscillation period. The first method is chosen in the present study according to the ITTC 7.5-02-06-02 [17].

Test | U_{c}(m/s) | Drift angle β(deg) | Oscillation period T(s) | Lateral amplitude y_{max}(m) |
---|---|---|---|---|

Oblique towing | 0.953, 0.572 | 0, ±2, ±4, ±8, ±12, ±16, ±20, ±25, ±30 | / | / |

Pure yaw | 0.953 | / | 17.26, 14.10, 12.21, 10.92, 9.23 | 0.3 |

0.572 | / | 28.76, 23.49, 20.34, 18.19, 15.38 | 0.3 | |

Combined yaw and drift | 0.953 | 4, 8, 12 | 17.26, 14.10, 12.21, 10.92, 9.23 | 0.3 |

0.572 | 4, 8, 12 | 28.76, 23.49, 20.34, 18.19, 15.38 | 0.3 |

Test | U_{c}(m/s) | Drift angle β(deg) | Oscillation period T(s) | Lateral amplitude y_{max}(m) |
---|---|---|---|---|

Oblique towing | 0.953, 0.572 | 0, ±2, ±4, ±8, ±12, ±16, ±20, ±25, ±30 | / | / |

Pure yaw | 0.953 | / | 17.26, 14.10, 12.21, 10.92, 9.23 | 0.3 |

0.572 | / | 28.76, 23.49, 20.34, 18.19, 15.38 | 0.3 | |

Combined yaw and drift | 0.953 | 4, 8, 12 | 17.26, 14.10, 12.21, 10.92, 9.23 | 0.3 |

0.572 | 4, 8, 12 | 28.76, 23.49, 20.34, 18.19, 15.38 | 0.3 |

Test | Drift angle β(deg) | Oscillation period T(s) | λ/L_{PP} | H_{W}/λ |
---|---|---|---|---|

Oblique towing | −4 | / | 0.5, 0.6, 0.8, 1.0 | 0.03 |

0.5 | 0.04 | |||

−8, −2 | / | 0.5 | 0.03 | |

Pure yaw | / | 17.26, 14.10, 12.21, 10.92 | 0.5 | 0.03 |

12.21 | 0.8 | 0.03 | ||

Combined yaw and drift | 8 | 17.26, 14.10, 12.21, 10.92 | 0.5 | 0.03 |

12.21 | 0.8 | 0.03 |

Test | Drift angle β(deg) | Oscillation period T(s) | λ/L_{PP} | H_{W}/λ |
---|---|---|---|---|

Oblique towing | −4 | / | 0.5, 0.6, 0.8, 1.0 | 0.03 |

0.5 | 0.04 | |||

−8, −2 | / | 0.5 | 0.03 | |

Pure yaw | / | 17.26, 14.10, 12.21, 10.92 | 0.5 | 0.03 |

12.21 | 0.8 | 0.03 | ||

Combined yaw and drift | 8 | 17.26, 14.10, 12.21, 10.92 | 0.5 | 0.03 |

12.21 | 0.8 | 0.03 |

## 3 Results and Discussion

### 3.1 Uncertainty Analysis.

*U*can be classified into two types: type A uncertainty and type B uncertainty. For the type A uncertainty, it is determined from repeated measurement and the estimation is given by

*N*is the number of the repeat experiments, and

*S*

_{R}is the standard deviation which is defined as

*R*_{i} is the measurement variable for the *i*th time, and $R\xaf$ denotes the average of the measured values for *N* times.

For the type B uncertainty, it is determined from the earlier test data, the experience and knowledge of the experimenters, and the indicated specifications of the instruments.

*U*

_{cR}can be obtained using the standard uncertainty

*U*and the sensitivity coefficient

*θ*. For the PMM tests in the towing tank, the combined uncertainty for the forces can be defined as

*R*represents the non-dimensional force and moment,

*U*

_{i}is the standard uncertainty of each measured physical parameter

*x*

_{i}, and

*θ*

_{i}is the corresponding sensitivity coefficient reflecting the variation degree of the total deviation with the subject parameter

*x*

_{i}.

*θ*

_{i}can be expressed as Eq. (11) for forces and Eq. (12) for moments.

The uncertainty of the water density $U\rho $ comes from the water temperature probe accuracy. The density-temperature relationship for *g* = 9.81 m/s can be expressed as: *ρ* = 1000.1 + 0.0552*t*^{o} − 0.0077(*t*^{o})^{2} + 0.00004(*t*^{o})^{3} [19]. Hence, with $to=11.9\u2218C$ and the temperature probe accuracy $Uto=0.2\u2218C$ in present experiments, $U\rho =|\u2202\rho /\u2202to|Uto=0.022kg/m3$.

$ULpp$ is based on the model manufacturing precision. According to the manufacturing tolerances [20], the tolerance for model length should be within $0.05%Lpp$ or 1.0 mm. Therefore, we can obtain the uncertainty of the ship length $ULpp=2.286mm$.

*U*_{V} is obtained by repeated measurement. Based on the estimate formulas of Eqs. (7) and (8), *U*_{V} = 0.003 m/s which is 0.31% of the set velocity.

For the uncertainty of the measured forces $UYmeasured$ and $UNmeasured$, they are calculated based on the 11 repeated runs for the oblique towing tests and 12 repeated runs for the harmonic dynamic tests.

Finally, the estimation of uncertainty for the non-dimensional forces can be obtained and listed in Table 6. For the oblique towing tests, the largest uncertainty resource is the measured force, and the carriage velocity is the second-largest uncertainty resource. The measured force uncertainty contributes over 80% to the combined uncertainty for the forces. Water density and the ship length have negligible proportions in the combined uncertainty. The combined uncertainties are reasonably small, 2.44% and 1.59% for *Y*′ and *N*′, respectively.

Source of uncertainty | Density (kg/m ^{3}) | Ship length (m) | Velocity (m/s) | Y(N) | N(N · m) |
---|---|---|---|---|---|

Nominal value | 9.995 × 10^{2} | 4.574 × 10^{0} | 9.560 × 10^{−1} | −1.666 × 10^{1} | −5.023 × 10^{1} |

Standard uncertainty | 2.201 × 10^{−5} | 2.286 × 10^{−3} | 3.148 × 10^{−3} | 3.948 × 10^{−1} | 7.279 × 10^{−1} |

Sensitivity coefficient for Y | −1.744 × 10^{−6} | −7.622 × 10^{−4} | −3.647 × 10^{−3} | 1.046 × 10^{−4} | |

Sensitivity coefficient for N | −1.150 × 10^{−6} | −7.538 × 10^{−4} | −2.404 × 10^{−3} | 2.288 × 10^{−5} | |

$\theta x2Ux2UcR2$ for Y (%) | 0.0 | 0.2 | 7.2 | 92.7 | |

$\theta x2Ux2UcR2$ for N (%) | 0.0 | 0.9 | 17.0 | 82.2 | |

$UcY\u2032/Y\u2032=2.44%$$UcN\u2032/N\u2032=1.59%$ |

Source of uncertainty | Density (kg/m ^{3}) | Ship length (m) | Velocity (m/s) | Y(N) | N(N · m) |
---|---|---|---|---|---|

Nominal value | 9.995 × 10^{2} | 4.574 × 10^{0} | 9.560 × 10^{−1} | −1.666 × 10^{1} | −5.023 × 10^{1} |

Standard uncertainty | 2.201 × 10^{−5} | 2.286 × 10^{−3} | 3.148 × 10^{−3} | 3.948 × 10^{−1} | 7.279 × 10^{−1} |

Sensitivity coefficient for Y | −1.744 × 10^{−6} | −7.622 × 10^{−4} | −3.647 × 10^{−3} | 1.046 × 10^{−4} | |

Sensitivity coefficient for N | −1.150 × 10^{−6} | −7.538 × 10^{−4} | −2.404 × 10^{−3} | 2.288 × 10^{−5} | |

$\theta x2Ux2UcR2$ for Y (%) | 0.0 | 0.2 | 7.2 | 92.7 | |

$\theta x2Ux2UcR2$ for N (%) | 0.0 | 0.9 | 17.0 | 82.2 | |

$UcY\u2032/Y\u2032=2.44%$$UcN\u2032/N\u2032=1.59%$ |

Note: The source of uncertainty greater than 5% is marked in bold.

For the harmonic dynamic tests, the forces are in a sine form rather than a constant. Therefore, we utilized the parameters processed by the interval integral methods to conduct the uncertainty analysis. The result of the pure yaw tests is shown in Table 7 as an example. Obviously, the proportion of *U*_{V} for *Y*′ and *N*′ is large due to the large sensitive coefficients resulting from the huge lateral forces and yaw moments. Meanwhile, the proportion of the uncertainty of velocity on the combined uncertainty increases clearly. The combined uncertainties for *Y*′ and *N*′ are under 2% of the dimensionless forces.

Source of uncertainty | Density (kg/m ^{3}) | Ship length (m) | Velocity (m/s) | Y(N) | N(N · m) |
---|---|---|---|---|---|

Nominal value | 1.00 × 10^{3} | 4.57 × 10^{0} | 9.56 × 10^{−1} | −5.77 × 10^{1} | −4.56 × 10^{1} |

Standard uncertainty | 2.20 × 10^{−5} | 2.29 × 10^{−3} | 3.15 × 10^{−3} | 4.66 × 10^{−1} | 4.25 × 10^{−1} |

Sensitivity coefficient for Y | −6.04 × 10^{−6} | −2.64 × 10^{−3} | −1.26 × 10^{−2} | 1.05 × 10^{−4} | |

Sensitivity coefficient for N | −1.04 × 10^{−6} | −6.84 × 10^{−4} | −2.18 × 10^{−3} | 2.29 × 10^{−5} | |

$\theta x2Ux2UcR2$ for Y (%) | 0 | 0.9 | 39.5 | 59.6 | |

$\theta x2Ux2UcR2$ for N (%) | 0 | 1.7 | 32.7 | 65.6 | |

$UcY\u2032/Y\u2032=1.04%$$UcN\u2032/N\u2032=1.14%$ |

Source of uncertainty | Density (kg/m ^{3}) | Ship length (m) | Velocity (m/s) | Y(N) | N(N · m) |
---|---|---|---|---|---|

Nominal value | 1.00 × 10^{3} | 4.57 × 10^{0} | 9.56 × 10^{−1} | −5.77 × 10^{1} | −4.56 × 10^{1} |

Standard uncertainty | 2.20 × 10^{−5} | 2.29 × 10^{−3} | 3.15 × 10^{−3} | 4.66 × 10^{−1} | 4.25 × 10^{−1} |

Sensitivity coefficient for Y | −6.04 × 10^{−6} | −2.64 × 10^{−3} | −1.26 × 10^{−2} | 1.05 × 10^{−4} | |

Sensitivity coefficient for N | −1.04 × 10^{−6} | −6.84 × 10^{−4} | −2.18 × 10^{−3} | 2.29 × 10^{−5} | |

$\theta x2Ux2UcR2$ for Y (%) | 0 | 0.9 | 39.5 | 59.6 | |

$\theta x2Ux2UcR2$ for N (%) | 0 | 1.7 | 32.7 | 65.6 | |

$UcY\u2032/Y\u2032=1.04%$$UcN\u2032/N\u2032=1.14%$ |

Note: The source of uncertainty greater than 5% is marked in bold.

### 3.2 Results of Oblique Towing Tests in Calm Water and Regular Waves.

For the steady experiment, i.e., the oblique towing test, an example of the measured forces in waves and calm water is drawn in Fig. 4. The abscissa is time, and the ordinates are the force and moment under still water and waves. The measured forces in calm water fluctuate slightly around the average values. The measured forces in waves oscillate violently.

Then, the FFT method is utilized to transform the data from the time domain to the frequency domain. The frequency spectrums of the forces in calm water are shown in Fig. 5. The peak frequencies except the zero frequency are the carriage wheel frequency, higher-order harmonics of the wheel frequency, natural frequencies, and the frequencies of the mechanical vibrations. All of them are tiny compared with the value at the zero frequency, which means that the measurement is acceptable for the minor effect of the environment noise.

The FFT results of the forces measured in regular waves with the same wavelength (*λ*/*L*_{pp} = 0.5) and different wave steepness (*H*_{W}/*λ*) are drawn in Fig. 6. The abscissa is the frequency, and the ordinates are the amplitudes of FFT results for the lateral force *Y* and yaw moment *N*. Obviously, the wave frequency components are larger than those at zero frequency. The peak frequencies are the wave encounter frequency (*f*_{e} = 1.24 Hz) and its multiples. For the forces at the wave encounter frequency and its multiples, they increase significantly with the wave steepness increasing. The increase in zero frequency components with an increase in wave height is less as compared to the increase in force components at wave frequency for a change in wave steepness. In Fig. 7, the frequency spectrums of the forces in regular waves with the same wave steepness and different wavelengths are drawn. Similarly, the peak frequencies are the encounter frequencies and their multiples. The wave encounter frequency is the dominant frequency. With the increase of the wavelength to ship length ratio (*λ*/*L*_{PP}), the yaw moments at the encounter frequency raise obviously. In Fig. 8, the frequency spectrum of the lateral forces measured by the strain gauge balance on the fore is drawn. It reveals that the wave-frequency lateral forces also rise with the increase of the wavelength to ship length ratio.

To assess the wave effect on the hydrodynamic forces quantitatively, the maneuvering forces in calm water and waves are listed in Table 8, where the parameters *D*_{Y} and *D*_{N} represent the differences between the forces in calm water and those in waves. It indicates that the lateral force *Y* and yaw moment *N* in waves are larger than those in calm water. With the increase of the wavelength (*λ*/*L*_{pp} increases from 0.5 to 1.0), the maneuvering components of the lateral force increase significantly (33.1% to 66.2%). So is the yaw moment (7.0% to 24.7%). With the increase of the wave steepness, the maneuvering components of the lateral force and yaw moment also increase obviously. In addition, the comparison between the forces in different drift angles in the same wave conditions reveals that the wave effect on the maneuvering forces is larger with the increase of the drift angle. All the cases also show that the wave effect on the lateral force *Y* is obviously larger than that on the yaw moment *N* for the oblique towing tests. For the oblique towing tests in waves, the lateral forces on the aft and fore of the ship are higher than those in calm water conditions. Hence, the total lateral force on the hull in waves will be much larger than that in calm water conditions. However, due to the opposite directions of the increased yaw moment generated by the wave on the fore and aft, the increased proportion of yaw moment in regular waves will be less than that of transverse force in waves, which makes $Yv\u2032$ greater than $Nv\u2032$.

Case | Y | N | D_{Y} | D_{N} |
---|---|---|---|---|

β = −4 deg, calm water | 7.16 | 27.48 | — | — |

β = −4 deg, λ/L = 0.5, H/λ = 0.03 | 9.53 | 29.40 | 33.1% | 7.0% |

β = −4 deg, λ/L = 0.5, H/λ = 0.04 | 10.08 | 30.17 | 40.8% | 9.8% |

β = −4 deg, λ/L = 0.5, H/λ = 0.05 | 10.27 | 30.35 | 43.4% | 10.4% |

β = −4 deg, λ/L = 0.6, H/λ = 0.03 | 9.91 | 29.67 | 38.4% | 8.0% |

β = −4 deg, λ/L = 0.8, H/λ = 0.03 | 10.17 | 30.18 | 42.0% | 9.8% |

β = −4 deg, λ/L = 1.0, H/λ = 0.03 | 11.90 | 34.27 | 66.2% | 24.7% |

β = −8 deg, calm water | 16.49 | 50.20 | — | — |

β = −8 deg, λ/L = 0.5, H/λ = 0.03 | 22.94 | 54.92 | 39.1% | 9.4% |

β = −2 deg, calm water | 2.94 | 12.08 | — | — |

β = −2 deg, λ/L = 0.5, H/λ = 0.03 | 3.78 | 12.53 | 28.6% | 3.7% |

Case | Y | N | D_{Y} | D_{N} |
---|---|---|---|---|

β = −4 deg, calm water | 7.16 | 27.48 | — | — |

β = −4 deg, λ/L = 0.5, H/λ = 0.03 | 9.53 | 29.40 | 33.1% | 7.0% |

β = −4 deg, λ/L = 0.5, H/λ = 0.04 | 10.08 | 30.17 | 40.8% | 9.8% |

β = −4 deg, λ/L = 0.5, H/λ = 0.05 | 10.27 | 30.35 | 43.4% | 10.4% |

β = −4 deg, λ/L = 0.6, H/λ = 0.03 | 9.91 | 29.67 | 38.4% | 8.0% |

β = −4 deg, λ/L = 0.8, H/λ = 0.03 | 10.17 | 30.18 | 42.0% | 9.8% |

β = −4 deg, λ/L = 1.0, H/λ = 0.03 | 11.90 | 34.27 | 66.2% | 24.7% |

β = −8 deg, calm water | 16.49 | 50.20 | — | — |

β = −8 deg, λ/L = 0.5, H/λ = 0.03 | 22.94 | 54.92 | 39.1% | 9.4% |

β = −2 deg, calm water | 2.94 | 12.08 | — | — |

β = −2 deg, λ/L = 0.5, H/λ = 0.03 | 3.78 | 12.53 | 28.6% | 3.7% |

The hydrodynamic derivatives in calm water and head waves can be obtained using the experimental data. The results are drawn in Fig. 9 and listed in Table 9. To ensure the accuracy of the experimental results, the benchmark data published on Simman2008 web^{2} are utilized to compare with the present experimental ones. The dotted line “Simman_HMRI (Calm_100%V)” represents the results of the oblique towing tests performed by Hyundai Maritime Research Institute with the 100%V in calm water condition, which reveals the accuracy of the present experiments. The differences between the linear hydrodynamic derivatives in calm water and those in head waves are 24% and 10% for $Yv\u2032$ and $Nv\u2032$ respectively. The velocity effect on the derivatives seems smaller. It should be noted that the hydrodynamic derivatives in waves obtained here are under a relatively lower steepness condition (*H*_{W}/*λ* = 0.03) rather than a higher steepness one. When a steeper wave condition is chosen, the derivatives will change more significantly. Therefore, it reveals that the wave effect on $Yv\u2032$ and $Nv\u2032$ is huge enough to be considered when using these parameters to predict the ship trajectories or motions in waves.

Hydrodynamic derivatives | Calm water V = 0.953 m/s | Calm water V = 0.572 m/s | Head wave λ/L_{PP} = 0.5, H_{W}/λ = 0.03 |
---|---|---|---|

$Yv\u2032\xd7102$ | −1.33 | −1.49 | −1.65 |

$Nv\u2032\xd7102$ | −0.84 | −0.83 | −0.92 |

Hydrodynamic derivatives | Calm water V = 0.953 m/s | Calm water V = 0.572 m/s | Head wave λ/L_{PP} = 0.5, H_{W}/λ = 0.03 |
---|---|---|---|

$Yv\u2032\xd7102$ | −1.33 | −1.49 | −1.65 |

$Nv\u2032\xd7102$ | −0.84 | −0.83 | −0.92 |

As mentioned in Sec. 2.1, the difference between the forces in calm water and waves contains not only the low-frequency maneuvering components but also the second-order drift forces and their interactions. Hence, it is necessary to distinguish the components and their proportions in the differences between the forces in calm water and waves. The commercial potential flow theory software sesamhydrod is adopted to obtain the second-order drift forces. In Fig. 10, the comparison between the experimental and simulated lateral second-order drift force coefficient *C*_{DFY} shows reasonable results. *C*_{DFY} denotes the second-order wave drift force in lateral direction divided by *ρgA*^{2}(*B*^{2}/*L*_{PP}). Then, the drift forces in the experimental conditions can be calculated and summarized in Table 10. In this table, *DFY* and *DN* represent the second-order drift forces in the lateral direction and yaw direction, respectively. *P*_{DFY} indicates the proportion of the lateral second-order drift forces in the difference between the lateral forces in calm water and waves, which can be shown as $PDFY=DFY/(Ywave\u2212Ycalm)\xd7100%$. Similarly, *P*_{DN} is defined as $DN/(Nwave\u2212Ncalm)\xd7100%$. For all the calculated cases, the drift force percentage for the lateral force is smaller than 40% and that for the yaw moment is smaller than 60%. This shows that there might be some error in adding the second-order force to the calm water maneuverability equation as the whole influence of waves on the maneuverability force. In addition, with the increase of the wave height, *P*_{DFY} and *P*_{DN} increase slightly, which show that the proportion of the second-order drift forces in the differences between the forces in calm water and waves will be larger when the wave steepness is larger.

Case | Y | N | DFY | DN | P_{DFY} | P_{DN} |
---|---|---|---|---|---|---|

β = −4 deg, calm water | 7.16 | 27.48 | – | – | – | – |

β = −4 deg, λ/L = 0.5, H/λ = 0.03 | 9.53 | 29.40 | 0.41 | 2.37 | 17% | 28% |

β = −4 deg, λ/L = 0.5, H/λ = 0.04 | 10.08 | 30.17 | 0.74 | 2.92 | 25% | 35% |

β = −4 deg, λ/L = 0.5, H/λ = 0.05 | 10.27 | 30.35 | 1.15 | 3.11 | 37% | 51% |

β = −4 deg, λ/L = 0.6, H/λ = 0.03 | 9.91 | 29.67 | 0.49 | 2.75 | 18% | 22% |

β = −4 deg, λ/L = 0.8, H/λ = 0.03 | 10.17 | 30.18 | 0.31 | 3.01 | 10% | 23% |

β = −4 deg, λ/L = 1.0, H/λ = 0.03 | 11.90 | 34.27 | 0.14 | 4.74 | 3% | 9% |

β = −8 deg, calm water | 16.49 | 50.20 | – | – | – | – |

β = −8 deg, λ/L = 0.5, H/λ = 0.03 | 22.94 | 54.92 | 0.86 | 6.45 | 13% | 22% |

β = −2 deg, calm water | 2.94 | 12.08 | – | – | – | – |

β = −2 deg, λ/L = 0.5, H/λ = 0.03 | 3.78 | 12.53 | 0.20 | 0.84 | 24% | 59% |

Case | Y | N | DFY | DN | P_{DFY} | P_{DN} |
---|---|---|---|---|---|---|

β = −4 deg, calm water | 7.16 | 27.48 | – | – | – | – |

β = −4 deg, λ/L = 0.5, H/λ = 0.03 | 9.53 | 29.40 | 0.41 | 2.37 | 17% | 28% |

β = −4 deg, λ/L = 0.5, H/λ = 0.04 | 10.08 | 30.17 | 0.74 | 2.92 | 25% | 35% |

β = −4 deg, λ/L = 0.5, H/λ = 0.05 | 10.27 | 30.35 | 1.15 | 3.11 | 37% | 51% |

β = −4 deg, λ/L = 0.6, H/λ = 0.03 | 9.91 | 29.67 | 0.49 | 2.75 | 18% | 22% |

β = −4 deg, λ/L = 0.8, H/λ = 0.03 | 10.17 | 30.18 | 0.31 | 3.01 | 10% | 23% |

β = −4 deg, λ/L = 1.0, H/λ = 0.03 | 11.90 | 34.27 | 0.14 | 4.74 | 3% | 9% |

β = −8 deg, calm water | 16.49 | 50.20 | – | – | – | – |

β = −8 deg, λ/L = 0.5, H/λ = 0.03 | 22.94 | 54.92 | 0.86 | 6.45 | 13% | 22% |

β = −2 deg, calm water | 2.94 | 12.08 | – | – | – | – |

β = −2 deg, λ/L = 0.5, H/λ = 0.03 | 3.78 | 12.53 | 0.20 | 0.84 | 24% | 59% |

### 3.3 Results of the Harmonic Dynamic Tests in Calm Water and Regular Waves.

For the harmonic dynamic tests, the time histories of the measured forces for the pure yaw tests in calm water and regular waves are drawn in Fig. 11. The abscissa is the time, and the ordinates are the the lateral force *Y* and yaw moment *N* in waves. Compared with the forces in calm water conditions, the forces in waves are the superposition of the high-frequency components and the low-frequency components.

The FFT results of the harmonic dynamic tests in calm water are shown in Fig. 12. The forces at noise frequencies are tiny compared with those at the PMM frequency *ω*, which reveals the effectiveness of the present experiments. For the pure yaw tests with different oscillation frequencies (*f*) in the same wave condition (*λ*/*L*_{PP} = 0.5, *H*_{W}/*λ* = 0.03), the FFT results are drawn in Fig. 13. The abscissa is the frequency, and the ordinates are the amplitudes of FFT results for the lateral force *Y* and yaw moment *N*. The dominant frequency of the spectrum is the PMM motion frequency which is different in various runs. The other peak frequencies are around the wave encounter frequency and its multiples. These frequencies are determined by the wave condition and are the same in these runs. Different from the FFT results of the oblique towing tests, the FFT results of the pure yaw tests have two peaks around the wave encounter frequency rather than one at the wave encounter frequency, which can be explained by the motion analysis. The lateral forces are measured at the fore and aft. For the fore measurement point, in step 1 shown in Fig. 14, its longitudinal velocity component is smaller than that of the carriage connection point (i.e., the gravity center) due to the yaw motion. Hence, the wave encounter frequency for the fore point will be smaller than that for the gravity center in step 1. Similarly, in step 2, the longitudinal velocity component of the fore measurement point is larger than that of the gravity center due to the yaw motion, which makes the wave encounter frequency for the fore point larger than that for the gravity center. For the aft measurement point, the wave encounter frequency is larger than that for the gravity center in step 1 and smaller in step 2, which is opposite to the fore measurement point. Therefore, there will be two symmetrical peak frequencies around the wave encounter frequency calculated for the ship gravity center. In fact, the aft measurement point will be affected by the ship wave, which causes multiple peaks.

According to the results obtained from the four pure yaw tests in the same wave condition, the hydrodynamic derivatives related to yaw angular velocity can be calculated. To observe the effect of different wavelengths, another wave condition is chosen to conduct the pure yaw test and the FFT result for that condition is drawn in Fig. 15. The comparison between Figs. 13(c) and 15 shows that with the increase of the wavelength to ship length ratio, the high-frequency forces around the wave encounter frequency increase as well as the low-frequency maneuvering forces.

The parameters *Y*_{out} and *N*_{out},which are obtained using the interval integral method shown in Table 3, are utilized for the comparison of the forces at the PMM motion frequency. The results are summarized in Table 11. Contrary to the conclusion in the oblique towing tests, the wave effect on the yaw moment is more obvious than that on the lateral force. For the yaw moment in waves, the lateral forces increased by waves on the fore and aft will be opposite. In contrast, the direction of the yaw moments increased by waves on the fore and aft will be the same. Hence, the increased proportion of yaw moment in regular waves will be larger than that of transverse force in waves, which makes $Nr\u2032$ greater than $Yr\u2032$.

Case | Y_{out} | N_{out} | D_{Y} | D_{N} |
---|---|---|---|---|

f = 0.058 Hz, calm water | −28.8 | −23.8 | — | — |

f = 0.058 Hz, λ/L = 0.5, H/λ = 0.03 | −29.0 | −23.6 | 0.7% | −0.9% |

f = 0.071 Hz, calm water | −43.9 | −35.2 | — | — |

f = 0.071 Hz, λ/L = 0.5, H/λ = 0.03 | −44.2 | −36.5 | 0.7% | 3.9% |

f = 0.082 Hz, calm water | −57.7 | −49.1 | — | — |

f = 0.082 Hz, λ/L = 0.5, H/λ = 0.03 | −56.0 | −51.3 | −2.9% | 4.3% |

f = 0.082 Hz, λ/L = 0.8, H/λ = 0.03 | −58.8 | −55.7 | 2.0% | 13% |

f = 0.092 Hz, calm water | −71.4 | −63.9 | — | — |

f = 0.092 Hz, λ/L = 0.5, H/λ = 0.03 | −71.5 | −65.3 | 0.2% | 2.3% |

Case | Y_{out} | N_{out} | D_{Y} | D_{N} |
---|---|---|---|---|

f = 0.058 Hz, calm water | −28.8 | −23.8 | — | — |

f = 0.058 Hz, λ/L = 0.5, H/λ = 0.03 | −29.0 | −23.6 | 0.7% | −0.9% |

f = 0.071 Hz, calm water | −43.9 | −35.2 | — | — |

f = 0.071 Hz, λ/L = 0.5, H/λ = 0.03 | −44.2 | −36.5 | 0.7% | 3.9% |

f = 0.082 Hz, calm water | −57.7 | −49.1 | — | — |

f = 0.082 Hz, λ/L = 0.5, H/λ = 0.03 | −56.0 | −51.3 | −2.9% | 4.3% |

f = 0.082 Hz, λ/L = 0.8, H/λ = 0.03 | −58.8 | −55.7 | 2.0% | 13% |

f = 0.092 Hz, calm water | −71.4 | −63.9 | — | — |

f = 0.092 Hz, λ/L = 0.5, H/λ = 0.03 | −71.5 | −65.3 | 0.2% | 2.3% |

Then, the dimensionless coefficients are drawn in Fig. 16 and the hydrodynamic derivatives are listed in Table 12. The abscissa is the dimensionless yaw rate, and the ordinates are the hydrodynamic derivatives in waves. Obviously, the wave and the velocity have small effect on the lateral forces. But for the yaw moment, the wave effect on the hydrodynamic derivatives $Nr\u2032$ and $Nr|r|\u2032$ are obvious. The difference between $Nr\u2032$ in calm water and regular head wave (*λ*/*L*_{pp} = 0.5, *H*_{W}/*λ* = 0.03) is 15.3%. It should be noted that these hydrodynamic derivatives in waves are obtained under the lower steepness wave condition. When the wave is longer or steeper, the wave effect will be larger.

Hydrodynamic derivatives | Calm water V = 0.953 m/s | Calm water V = 0.572 m/s | Head wave λ/L_{PP} = 0.5, H_{W}/λ = 0.03 |
---|---|---|---|

$Yr\u2032\xd7103$ | 2.967 | 2.935 | 3.313 |

$Nr\u2032\xd7103$ | −2.198 | −1.968 | −2.535 |

$Yr|r|\u2032\xd7103$ | 2.340 | 2.432 | 1.474 |

$Nr|r|\u2032\xd7103$ | −1.404 | −1.340 | −1.242 |

Hydrodynamic derivatives | Calm water V = 0.953 m/s | Calm water V = 0.572 m/s | Head wave λ/L_{PP} = 0.5, H_{W}/λ = 0.03 |
---|---|---|---|

$Yr\u2032\xd7103$ | 2.967 | 2.935 | 3.313 |

$Nr\u2032\xd7103$ | −2.198 | −1.968 | −2.535 |

$Yr|r|\u2032\xd7103$ | 2.340 | 2.432 | 1.474 |

$Nr|r|\u2032\xd7103$ | −1.404 | −1.340 | −1.242 |

Similarly, the wave effect on the high-order hydrodynamic derivatives can be obtained from the combined yaw and drift tests. The comparison of the force parameters *Y*_{out} and *N*_{out} is obtained using the interval integral method, shown in Table 3 and summarized in Table 13. Then, the dimensionless coefficients can be drawn in Fig. 17 and listed in Table 14. It is clear that for the wave condition, even the lower steepness one, the high-order derivatives will change a lot, especially $N|v|r\u2032$.

Case | Y_{out} | N_{out} | D_{Y} | D_{N} |
---|---|---|---|---|

f = 0.058 Hz, calm water | −25.6 | −28.9 | — | — |

f = 0.058 Hz, λ/L = 0.5, H/λ = 0.03 | −27.3 | −31.6 | 6.9% | 9.5% |

f = 0.071 Hz, calm water | −39.8 | −43.9 | — | — |

f = 0.071 Hz, λ/L = 0.5, H/λ = 0.03 | −40.8 | −46.8 | 2.3% | 6.6% |

f = 0.082 Hz, calm water | −52.5 | −62.4 | — | — |

f = 0.082 Hz, λ/L = 0.5, H/λ = 0.03 | −53.8 | −65.9 | 2.5% | 5.6% |

f = 0.082 Hz, λ/L = 0.8, H/λ = 0.03 | −53.8 | −71.1 | 2.5% | 14% |

f = 0.092 Hz, calm water | −65.9 | −80.6 | — | — |

f = 0.092 Hz, λ/L = 0.5, H/λ = 0.03 | −65.0 | −79.4 | −1.4% | −1.6% |

Case | Y_{out} | N_{out} | D_{Y} | D_{N} |
---|---|---|---|---|

f = 0.058 Hz, calm water | −25.6 | −28.9 | — | — |

f = 0.058 Hz, λ/L = 0.5, H/λ = 0.03 | −27.3 | −31.6 | 6.9% | 9.5% |

f = 0.071 Hz, calm water | −39.8 | −43.9 | — | — |

f = 0.071 Hz, λ/L = 0.5, H/λ = 0.03 | −40.8 | −46.8 | 2.3% | 6.6% |

f = 0.082 Hz, calm water | −52.5 | −62.4 | — | — |

f = 0.082 Hz, λ/L = 0.5, H/λ = 0.03 | −53.8 | −65.9 | 2.5% | 5.6% |

f = 0.082 Hz, λ/L = 0.8, H/λ = 0.03 | −53.8 | −71.1 | 2.5% | 14% |

f = 0.092 Hz, calm water | −65.9 | −80.6 | — | — |

f = 0.092 Hz, λ/L = 0.5, H/λ = 0.03 | −65.0 | −79.4 | −1.4% | −1.6% |

Hydrodynamic derivatives | Calm water V = 0.953 m/s | Calm water V = 0.572 m/s | Head wave λ/L_{PP} = 0.5, H_{W}/λ = 0.03 |
---|---|---|---|

$Y|v|r\u2032\xd7103$ | 8.832 | 7.937 | 7.734 |

$N|v|r\u2032\xd7103$ | −3.375 | −3.485 | −5.394 |

Hydrodynamic derivatives | Calm water V = 0.953 m/s | Calm water V = 0.572 m/s | Head wave λ/L_{PP} = 0.5, H_{W}/λ = 0.03 |
---|---|---|---|

$Y|v|r\u2032\xd7103$ | 8.832 | 7.937 | 7.734 |

$N|v|r\u2032\xd7103$ | −3.375 | −3.485 | −5.394 |

## 4 Conclusion

The benchmark PMM tests of a standard ship model KVLCC2 in calm water and regular head waves with various wavelengths and wave steepness are carried out in this study. By the data acquisition and calculation, the hydrodynamic derivatives in calm water and waves are obtained. The unverified hypothesis in the traditional potential flow methods for maneuverability study in waves is evaluated by comparing the hydrodynamic derivatives in calm water and those in regular waves. The main contributions can be listed as follows:

The detailed method of PMM tests in regular head waves is introduced and the benchmark captive model tests in calm water and regular waves are carried out.

The time history and frequency spectrum of the hydrodynamic forces acting on the hull are provided. The distribution of the frequency peaks and the amplitudes at different frequencies are provided, which can be utilized for the validation of the numerical simulations. In addition, the wave effect on lateral force for the oblique towing tests and yaw moment for the pure yaw tests are obvious. With the increase of the wavelength and wave steepness, the wave effect on the maneuvering forces will be larger.

The maneuverability hydrodynamic derivatives of ships in waves are given quantitatively. The comparison between the hydrodynamic derivative in calm water and regular head waves reveals that there is some error in the earlier hypothesis that the hydrodynamic derivatives in waves are the same as those in calm water. The differences between the linear hydrodynamic derivatives in calm water and those in head waves are larger than 10% at least. For the high-order derivatives, the differences will be much larger which should not be ignored. This error in the hypothesis might be the reason for the limited accuracy of the present potential flow methods on maneuverability in waves.

More extensive experiments under different waves have not been carried out at present. However, the provided experimental data can be used for verification of the numerical simulations to conduct a more comprehensive study, to deduce the relevant correction formula to improve the accuracy of maneuverability prediction in waves. Actually, the authors [22] have carried out the numerical simulations for the pure yaw tests, which are validated by the present experimental data, to obtain the deeper physical mechanism and wave effect on the hydrodynamic derivatives.

## Footnote

## Acknowledgment

This study is financially supported by the national project of the Knowledge-based Ship-design Hyper-Integrated Platform II (KSHIP-II, No. GKZY010004), and the project of State Key Laboratory of Ocean Engineering (No. GKZD010077). The authors are grateful to Mr. Tianwei Liu, Dr. Jun Lu, and Mr. Fangyi Wei for their help in the experiments.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## References

*Mathematical Model of Ship Motion—Mechanism Modeling and Identification Modeling*