# Development of Enhanced Two-Time-Scale Model for Simulation of Ship Maneuvering in Ocean Waves

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

#### 2.1. Prediction of Wave Drift Force and Moment (Seakeeping Analysis)

_{0}, v

_{0}, and r

_{0}indicate surge, sway, and yaw directional components, respectively.

_{F}.

_{K}.) and hydrodynamic forces (F

_{H}.

_{D}.) are calculated. Finally, wave-induced ship motion responses can be obtained from the 6-DoF equations of motion in Equation (5), Where M

_{ij}and C

_{ij}are the ship inertial and restoring matrix, respectively.

#### 2.2. MMG Model (Maneuvering Analysis)

_{xx}, and I

_{zz}indicate the mass and the moments of inertia of a ship. X, Y, K, and N denote the surge, sway, roll, and yaw directional force and moment, and p

_{0}is roll velocity. In the MMG model, the hydrodynamic forces induced by propeller rotation (F

_{propeller}), rudder deflection (F

_{Rudder}), and ship operation (F

_{Hull}) are calculated separately, whereas the mutual interaction between each component is considered by introducing several interaction coefficients. Moreover, the influence of the wave is taken into account by introducing second-order mean drift force and moment (F

_{wave}). The MMG model adopted in this study is based on the hydrodynamic derivatives and interaction coefficients by [9] (S175) and [30] (KVLCC2). Details of the present MMG model are available in the abovementioned papers.

#### 2.3. Seakeeping-Maneuvering Coupling Analysis

_{0}(=tan

^{−1}(−v

_{0}/u

_{0})) indicates the ship drift angle; χ, ω, and A denote wave heading, frequency, and amplitude, respectively. In the DCM, the seakeeping and maneuvering equations are solved simultaneously, and the physical quantities are exchanged at every time step. Concretely, the ship forward speed, steady drift, and yaw motion calculated by the maneuvering equations are considered in the evaluation of wave drift force and moment. The evaluated drift force is imposed on the maneuvering equations of motion such that the mean drift effect induced by the wave can be reflected in the assessment of ship operation velocities and the maneuvering trajectory. Because the seakeeping-maneuvering coupling procedure is directly performed at every time step, a more realistic analysis can be conducted by continuously updating the wave drift force and moment. On the other hand, based on the assumption that seakeeping and maneuvering equations can be solved independently, the ETM considers the wave effect by imposing a precomputed drift force. Furthermore, the effects of the steady drift and yaw motion are not reflected.

_{0}and r

_{0}values must be selected to calculate the linear derivatives. If extremely small values are selected, the overall tendency of the wave drift force will not be represented well. Linear derivatives calculated by excessively large drift angles and yaw rates can result in overestimation. In this study, the drift angle and yaw rate values were selected based on the converged operation velocities in the calm water turning simulation, i.e., β

_{0}= 0~10 [degree] (S175), β

_{0}= 0~15 [degree] (KVLCC2) and r

_{0}= 0~0.8 [degree/s] (S175, KVLCC2).

## 3. Numerical Result

_{B}= 0.572) and KVLCC2 tanker (C

_{B}= 0.810)—were chosen for maneuvering in wave analysis. Figure 5 shows the solution grids and waterline contours of the two target ship models. As shown in the Figure 5, the KVLCC2 tanker exhibits blunter waterline characteristics than the S175 container ship, particularly near the bow and stern shoulder region. Because the experimental data on the wave drift force and turning trajectories were released for both ship models, the numerical result obtained from the present study can be validated. The principal dimensions of the two ships are listed in Table 1.

#### 3.1. Characteristics of Bilinear Coefficients

_{β}

^{w}always shows a positive value regardless of the operating condition, which indicates that the sway drift force increased in the opposite direction of drift motion for both ship models. On the other hand, the yaw drift moment varied differently with respect to the two target ships. The yaw drift moment of the S175 container ship increased in the opposite direction of the bow drift, whereas that of the KVLCC2 decreased. The variation of the drift force led by the steady yaw motion is shown in Figure 7. The yaw drift moment increased in the opposite direction of the steady yaw motion for both ship models. At the same time, the sway drift force exhibited a different tendency with respect to the ship models.

#### 3.2. Turning Circle Simulation in Regular Wave

_{ref}); RPS = 1.42 for S175 and RPS = 1.75 for KVLCC2. It should be noted that the additional steady force in the longitudinal direction induced by the waves—namely added resistance—leads to the decrement in the speed of a ship. Therefore, maneuvering simulation in the presence of the ocean waves started at the corresponding reduced speed, which is the same as the benchmark free-running experiment presented by [6,9]. In addition, the steering device was deflected until 35 degrees with a constant rotating speed. For the environmental conditions, the regular incident wave with λ/L = 0.7 and A/L = 0.01 was chosen, and two different values of the initial wave heading angles were selected (χ

_{initial}= 180°, 270°). Assuming the small amplitude of seakeeping properties, the linear seakeeping theory was applied for this simulation. The resultant turning trajectories and time histories of the drift force were compared with the DCM, and the improvement from the ETM is discussed.

_{initial}= 180° and 270° and portside turning simulations. Compared with the DCM, the ETM showed vastly different trends of the drift force and moment, resulting in different turning trajectories. Meanwhile, the MTM, which introduces effects of the steady drift and yaw motion, yielded results similar to those of the DCM. By applying the MTM, the sway drift force was estimated much more accurately compared with the ETM, and the overall tendency of the yaw drift moment was similar to that obtained using the DCM. Although discrepancies in the yaw drift moment were observed at the bow quartering wave region, the global turning trajectory agreed well with the results obtained using the DCM.

_{initial}= 270°). By considering the effects of the drift and yaw motion, it can be observed that the sway drift force and yaw drift moment were significantly affected. Especially, the influence of the drift motion was more remarkable than that of the yaw motion. Increment of the drift force and moment induced by the two operation velocities were most prominent in the bow quartering wave region of S175 container ship and the beam wave region of the KVLCC2 tanker. In other words, to more elaborately reflect the seakeeping-maneuvering coupling effects, all directions of operation velocities should be considered in evaluating the drift force and moment.

#### 3.3. Drift Index

## 4. Conclusions

- The effects of operation velocities on the drift force and moment were observed by comparing the bilinear coefficients and interpreted by investigating the relative wave elevation along the ship waterline. From the results, it was confirmed that the characteristics of the bilinear coefficients showed different tendencies based on the hull geometry.
- An MTM was applied for the evaluation of ship maneuverability in regular waves, and its performance was validated by comparing the turning trajectory and the time histories of drift force with those of the DCM. The numerical results of the MTM agreed well with the DCM, whereas a computation speed similar to that of the ETM was obtained.
- The relative contributions of the steady drift and yaw motion were investigated by comparing each component of the drift force and moment. Consequently, the effect of the steady drift motion was more prominent than that of the steady yaw motion in estimating ship maneuvering performance in waves.
- For the purpose of evaluating the quantitative performance of the MTM, several drift indices were defined and compared. Although the variation of the drift indices showed different tendencies based on the ship models, it can be concluded that the overall accuracy improved by applying the MTM.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Definition | S175 Container Ship | KVLCC2 Tanker |
---|---|---|

Length [m] | 175.0 | 320.0 |

Breadth [m] | 25.4 | 58.0 |

Draft [m] | 9.5 | 20.8 |

Block coefficient | 0.572 | 0.810 |

S175 Container Ship | KVLCC2 Tanker | |
---|---|---|

Froude number | 0.036, 0.072, 0.110, 0.150 | 0.030, 0.055, 0.100, 0.142 |

Wave heading | χ = 0~180° (30° interval) |

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**MDPI and ACS Style**

Lee, J.; Nam, B.W.; Lee, J.-H.; Kim, Y.
Development of Enhanced Two-Time-Scale Model for Simulation of Ship Maneuvering in Ocean Waves. *J. Mar. Sci. Eng.* **2021**, *9*, 700.
https://doi.org/10.3390/jmse9070700

**AMA Style**

Lee J, Nam BW, Lee J-H, Kim Y.
Development of Enhanced Two-Time-Scale Model for Simulation of Ship Maneuvering in Ocean Waves. *Journal of Marine Science and Engineering*. 2021; 9(7):700.
https://doi.org/10.3390/jmse9070700

**Chicago/Turabian Style**

Lee, Jaehak, Bo Woo Nam, Jae-Hoon Lee, and Yonghwan Kim.
2021. "Development of Enhanced Two-Time-Scale Model for Simulation of Ship Maneuvering in Ocean Waves" *Journal of Marine Science and Engineering* 9, no. 7: 700.
https://doi.org/10.3390/jmse9070700