Refinement of Norrbin Model via Correlations between Dimensionless Cross-Flow Coefficient and Hydrodynamic Derivatives

Abstract

To develop a simplified and highly accurate ship motion model, this study thoroughly investigated the relationship between the dimensionless cross-flow coefficient and the four hydrodynamic derivatives of the Norrbin model. Eight different types of ships were simulated to explore the impact of dimensionless cross-flow coefficients and individual hydrodynamic derivatives on the ship’s turning circle. A set of precise formulas is proposed to depict the interplay between these variables. The simulation outcomes indicate that the average deviation in the agreement between the turning circles produced by adjusting the dimensionless cross-flow coefficient and those predicted by modifying the four hydrodynamic derivatives was only 2.70%. Furthermore, the similarities between the two circles and the sea trail were significantly higher at 91.45% and 92.87% compared with the original Norrbin model’s accuracy of 78.12%. Adjusting the dimensionless cross-flow coefficients enabled the rapid identification of a curve that closely mirrored the sea trail. This research aimed to improve the accuracy of the Norrbin model and resolve issues related to determining the magnification of the hydrodynamic derivatives, laying a robust foundation for subsequent studies and applications in relevant domains.

1. Introduction

Ship maneuverability, also referred to as ship motion performance, is a vital factor in maritime operations, serving as a key indicator that impacts ship design and the advancement of the international shipping industry. In 2002, the International Maritime Organization (IMO) embraced the Ship Maneuverability Standard Resolution MSC.137(76) [1], laying out essential criteria that ships are required to adhere to [2]. Without regard to the influence of the navigation environment, that is, the water and the depth of the water are not limited [3], the IMO Ship Maneuverability Standard evaluates a ship’s maneuverability based on six key properties: inherent stability [4], direction keeping, redirection, bow-rocking suppression [5], gyration, and stopping [6]. Maneuverability is fundamental in assessing a ship’s steering performance, impacting various scenarios, such as ship collision avoidance [7], pulling off the pier [8], inshore sailing [9], and ship turning. Yim [10] investigated the effectiveness of executing full-rudder turns to rotate the ship [11]. It is a performance index that measures the size of the minimum water area occupied by the ship’s rotary motion [12] and the degree of rapid rotation [13]. Particularly important in nautical practice, a ship’s turning ability has been examined by many investigators. Gao [14] proposed a more comprehensive evaluation method for ship maneuverability based on cluster analysis of the IMO maneuverability index. Li [15] achieved the accurate measurement of ship model coordinates through self-winding ship model experiments. Kim [16] examined the turning characteristics of a KCS (KRISO container ship) model in regular waves through free model experiments, and proposed the concept of a safety index and investigated its relationship with waves. Chillcce [17] proposed a numerical method to simulate the wave maneuvering of a ship to calculate ship motion in calm water and waves. Suzuki [18] developed a numerical simulation approach to estimate the 6DOF (six degrees of freedom) motion of a ship maneuvering in a regular wave and conduct free model tests. Their obtained results indicate that the proposed methodology can rationally capture the test results. Ren [19] established an effective adaptive Nomoto model to deal with the ship path-tracking problem. Yasukawa [20] employed the proposed hydrodynamic coefficients for KVLCC2 to predict ship motion. Guo [21] developed a new design approach for ship-tracking control based on the Norrbin model with enhanced system stability. Liu [22] considered both random noise and switching control in the Norrbin model to discuss the steering control of a ship and solve the adaptive control problem. Zhou [23] effectively implemented the PID controller based on the Norrbin model. For this purpose, a closed-loop gain-shaping algorithm was adopted to solve the long delay time problem of large ship movement. In summary, the common methods for predicting the turning circle of a ship are the free self-sailing model testing method [24], computer numerical simulation method [25], and semi-empirical estimation method. Along with the development of computer technology, more and more numerical simulation methods have been extensively implemented in ship-turning-ability forecasting. The commonly used methodologies are the MMG model [9,26], the Norrbin model [27], and the Nomoto model. Herein, a modified Norrbin model was employed to examine the problem at hand.

The simulation of ship maneuvering motion is based on an accurate mathematical model [28,29]. As ship design continues to evolve toward larger size, faster speeds, more specialized functions, and modernization, the accuracy of the Norrbin model increasingly falls short, necessitating significant improvements. Li [30] improved the accuracy of the model by scaling the hydrodynamic derivatives of the Norrbin model, resulting in a high degree of agreement with the experimental results of the ship-turning-circle prediction, Zhang [31] confirmed the effectiveness of Norrbin’s four-dimensional nonlinear mathematical model and second-order closed-loop gain-shaping algorithm for design control strategies. The main aim was to ensure the safe navigation of ships in waters with various sea conditions, solving the problem of the adaptive control of ship heading. A close survey of the conducted research on ship maneuvering performance by the aforementioned researchers was reviewed. Li [32] revealed that ship maneuverability experiments have mainly focused on ship heading control and ship maneuverability simulations, and the Norrbin model has been commonly utilized to perform ship control simulation experiments. However, the major problem is that the types of ships involved in the pilot tests are small, most of the ships simulated in the tests are below 100,000 tons, and there are no simulation tests on the maneuvering performance of large ships. Subsequently, to investigate whether the Norrbin model still has a high degree of conformity to the maneuvering performance of large ships, it was improved by incorporating data from large ships. In this regard, Yang [33] enlarged the hydrodynamic derivatives of the Norrbin model and investigated to test the turning circle of a 300,000-ton ballast displacement ship. They improved the Norrbin model based on matching some hydrodynamic derivatives with two nonlinear force moment empirical formulas identified by Norrbin and obtained a turning circle closer to the actual ship’s path with higher accuracy. The dimensionless cross-flow coefficient was of concern to scholars several times in the above articles, and it was briefly passed over and a value was considered for it. More specifically, what factors are associated with its value and how its variation affects the ship’s turning circle have not been thoroughly discussed.

The contributions of this research can be chiefly summarized in the following three items:

(1)

A generalization formula connecting the dimensionless cross-flow coefficients with the four hydrodynamic derivatives was designed.

(2)

This formula possesses a certain universality and aids ships in promptly predicting the turning circle.

(3)

The research methodology presented in this article efficiently simplifies the Norrbin model while enhancing its accuracy.

The other sections of this paper are organized as follows. In Chapter 2, the Norrbin model used in this paper is introduced. Chapter 3 investigates the effects of the dimensionless cross-flow coefficient and hydrodynamic derivatives on the turning circle of ships. The results obtained suggest similarities, prompting a detailed examination of a potential relationship between the two variables. Subsequently, the accurate analysis results of the impact of individual adjusted hydrodynamic derivatives on a ship’s turning circle are given, establishing a correlation through an exchange of communication data between different vessels. In Chapter 4, validation of the generalizability of the formulated equation by applying it to another ship is shown, demonstrating that the average discrepancy fell within the anticipated range.

2. Mathematical Modeling

The ship motion model of the state-space type is the essential basis for ship controller design. To enhance the accuracy of the model, the nonlinear hydrodynamic term ($F_{NON}^{\prime }$), wind term ($F_{WIND}^{\prime }$), and wave force term ($F_{WAVE}^{\prime }$) can be added to the right-hand side of the nonlinear mathematical model of ship movement [30]. Then, the dimensionless mathematical model associated with the nonlinear ship motion can be organized into the following forms:

$$I_{(2)}^{\prime }{\stackrel{.}{X}}_{(2)}=P_{(2)}^{\prime }{X}_{(2)}+Q_{(2)}^{\prime }U+F_{NON}^{\prime }+F_{WIND}^{\prime }+F_{WAVE}^{\prime }$$

(1)

$$\{\begin{array}{l}{X}_{(2)}=[v{r]}^T,U=\delta \\ I_{(2)}^{\prime }=[\begin{array}{l}{m}^{\prime }-Y_{\stackrel{.}{v}}^{\prime }\\ m^{\prime }{x}_C^{\prime }-N_{\stackrel{.}{v}}^{\prime }\end{array}\begin{array}{l}\hspace{1em}L(m^{\prime }{x}_C^{\prime }-Y_{\stackrel{.}{r}}^{\prime })\\ \hspace{1em}L(I_{zz}^{\prime }-N_{\stackrel{.}{r}}^{\prime })\end{array}]\\ P_{(2)}^{\prime }=\left[\begin{array}{l}\frac{V}{L}{Y}_v^{\prime }\hspace{1em}\hspace{1em}V(Y_r^{\prime }-m^{\prime })\\ \frac{V}{L}{N}_v^{\prime }\hspace{1em}\hspace{1em}V(N_r^{\prime }-m^{\prime }{x}_C^{\prime })\end{array}\right]\\ Q_{(2)}^{\prime }=\left[\begin{array}{l}\frac{V^2}{L}{Y}_{\delta }^{\prime }\\ \frac{V^2}{L}{N}_{\delta }^{\prime }\end{array}\right]\end{array}$$

(2)

where the superscript prime symbol ”$\prime$“ stands for the non-dimensionalization, $m^{\prime }$ refers to the ratio of $m$ to $\rho L^3/2$, $x_C^{\prime }$ denotes the ratio of $x_C$ and $L$, $I_{zz}^{\prime }$ represents the ratio of $I_{zz}$ and $\rho L^5/2$, $V$ is the total speed, $v$ denotes the transverse velocity, $r$ is the turning rate, $X_{(2)}$ is the state vector, $U$ is the control input, m denotes the mass, I_zz is the moment of inertia of the ship’s mass around the vertical axis passing through the center of gravity, L is the length between perpendiculars, x_c denotes the longitudinal center of gravity, and ρ is the water density.

Additionally, $I_{(2)}^{\prime },P_{(2)}^{\prime },$ and $Q_{(2)}^{\prime }$ are the inertial force matrix, viscous force matrix, and rudder force matrix, respectively. Moreover, the required factors in Equation (2) can be obtained from the 10 linear hydrodynamic derivative regression formulas collated by Clark [34], as illustrated in Equation (3):

$$\{\begin{array}{l}{Y}_{\delta }^{\prime }=\frac{3.0A_{\delta }}{L^2}\\ Y_v^{\prime }=-\left[1+\frac{0.40C_{b}B}{T}\right]\cdot \pi (T/L{)}^2-0.30Y_{\delta }^{\prime }\\ Y_{\dot{v}}^{\prime }=-\left[1+0.16C_{b}B/T-5.1{(B/L)}^2\right]\cdot \pi (T/L{)}^2\\ Y_r^{\prime }=-\left[-1/2+2.2B/L-0.080B/T\right]\cdot \pi (T/L{)}^2+0.15Y_{\delta }^{\prime }\\ Y_{\dot{r}}^{\prime }=-\left[0.67B/L-0.0033{(B/T)}^2\right]\cdot \pi (T/L{)}^2\\ N_{\delta }^{\prime }=-(\frac{1}{2})Y_{\delta }^{\prime }\\ N_v^{\prime }=-\left[1/2+2.4T/L\right]\cdot \pi (T/L{)}^2+0.15Y_{\delta }^{\prime }\\ N_{\dot{v}}^{\prime }=-\left[1.1B/L-0.041B/T\right]\cdot \pi (T/L{)}^2\\ N_r^{\prime }=-\left[1/4+0.039B/T-0.56B/L\right]\cdot \pi (T/L{)}^2-0.075Y_{\delta }^{\prime }\\ N_{\dot{r}}^{\prime }=-\left[\frac{1}{12}+\frac{0.017C_{b}B}{T}-\frac{0.33B}{L}\right]\cdot \pi (T/L{)}^2\end{array}$$

(3)

where $B,T,C_b,$ and $A_{\delta }$ are the width of the ship, the draft, block coefficient, and the rudder’s area, respectively, and $Y_{\dot{r}}^{\prime }$, $N_{\dot{r}}^{\prime }$, $Y_r^{\prime }$, and $N_r^{\prime }$ represent the hydrodynamic derivatives of the ship itself.

Norrbin presented a concise expression for the nonlinear fluid force in the study of the ship parameter identification problem, which is presented as follows:

$$F_{NON}^{\prime }=\left[\begin{array}{l}{Y}_{NON}^{\prime }\\ N_{NON}^{\prime }\end{array}\right]=\left[\begin{array}{l}{C}_{\mathrm{r}}{f}_Y(v,r)\\ C_{\mathrm{r}}{f}_N(v,r)\end{array}\right]$$

(4)

The calculations of $f_Y(v,r)$ and $f_N(v,r)$ are given in Equations (5) and (6):

$$f_Y(v,r)=\{\begin{array}{l}T\cdot r\left|r\right|\left[-\frac{1}{12}-\frac{1}{L^2}{(\frac{v}{r})}^2\right]\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\infty <-\frac{1}{L}\frac{v}{r}<-\frac{1}{2}\\ T\cdot r\left|r\right|\left[-\frac{1}{2}\frac{1}{L}\frac{v}{r}-\frac{2}{3}\frac{1}{L^3}{(\frac{v}{r})}^3\right]\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}-\frac{1}{2}\le -\frac{1}{L}\frac{v}{r}\le \frac{1}{2}\\ T\cdot r\left|r\right|\left[\frac{1}{12}+\frac{1}{L^2}{(\frac{v}{r})}^2\right]\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \frac{1}{2}<-\frac{1}{L}\frac{v}{r}<\infty \end{array}$$

(5)

$$f_N(v,r)=\{\begin{array}{l}T\cdot r\left|r\right|\left[-\frac{1}{6}\frac{1}{L}(\frac{v}{r})\right]\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\infty <-\frac{1}{L}\frac{v}{r}<-\frac{1}{2}\\ T\cdot r\left|r\right|\left[-\frac{1}{32}-\frac{1}{4}\frac{1}{L^2}{(\frac{v}{r})}^2+\frac{1}{6}\frac{1}{L^4}{(\frac{v}{r})}^4\right]\hspace{1em}-\frac{1}{2}\le -\frac{1}{L}\frac{v}{r}\le \frac{1}{2}\\ T\cdot r\left|r\right|\left[\frac{1}{6}\frac{1}{L}(\frac{v}{r})\right]\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{1}{2}<-\frac{1}{L}\frac{v}{r}<\infty \end{array}$$

(6)

where $C_r$ denotes the dimensionless cross-flow coefficient, whose value is usually taken in the range of 0.3 to 0.8. Since the sea trail is commonly disturbed by wind, current, and other effects, the wind-generated wave model used in [30] can be employed to deal with the disturbance of wind, current, and waves.

3. Results

3.1. Effect of the Dimensionless Cross-Flow Coefficient on the Ship’s Turning Circle Performance

In the Norrbin model, the range of the dimensionless cross-flow coefficient is 0.3–0.8. To investigate the effect of this coefficient on the turning performance of the ship, the authors employed the ship “Yuan Kun Yang” for the simulations, where its ship parameters are provided in

Table 1. Particulars of “Yuan Kun Yang”.

Name	Symbol	Unit	Yuan Kun Yang
Length	$L_{pp}$	m	324.000
Width	$B$	m	60.000
Design draft	$d$	m	20.500
Displacement	∆	t	310,297
Square coefficient	$C_b$	/	0.8215
Design speed	$v_0$	kn	18.7
Longitudinal center of gravity	Xc	m	9.91
Area of rudder	Ad	m2	139.217
Molded depth	$T_{\mathrm{d}}$	m	30.0

. In the actual turning test of the ship, the sea state had 2–3 levels, the wind had three levels, the actual wind direction was 175.5°, the tide in the test area exhibited back-and-forth flow, the maximum speed of the rising tide was 2.5 kn, the associated flow direction was S, the maximum speed of the falling tide was 1.5 kn, the corresponding flow direction was N, and the initial heading was 190.2°. In the simulation, we chose various values of the dimensionless cross-flow coefficient in the range of 0.3–0.8 to observe its effect on the ship’s turning performance, and the experimental results of the simulation are presented in Figure 1 and

Table 2. Parameter table of the influence of the dimensionless cross-flow coefficient on the turning circle.

Category	${\mathit{C}}_{\mathit{r}}=0.3$	${\mathit{C}}_{\mathit{r}}=0.4$	${\mathit{C}}_{\mathit{r}}=0.5$	${\mathit{C}}_{\mathit{r}}=0.6$	${\mathit{C}}_{\mathit{r}}=0.7$	${\mathit{C}}_{\mathit{r}}=0.8$
$A_{\mathrm{dm}ax}/L$	2.62	2.65	2.69	2.73	2.78	2.82
$A_{\mathrm{dmin}}/L$	0.37	0.10	−0.2	−0.5	−0.7	−1.0
$T_{\mathrm{r}\max }/L$	2.53	2.74	2.93	3.15	3.33	3.53
$(A_{\mathrm{dmax}}-A_{\mathrm{dmin}})/L$	2.25	2.55	2.87	3.19	3.52	3.83

.

Figure 1 and Table 2 show the significant effect of the variation of the dimensionless cross-flow coefficient on the turning circle of the ship. The simulation results indicate that the turning circle of the ship grew with the growth of the dimensionless cross-flow coefficient, and generally, the longitudinal growth was substantially lower than the transverse growth.

3.2. Influence of the Hydrodynamic Conductivity on the Rotational Performance

To investigate the effect of the hydrodynamic derivatives on the ship’s turning circle performance, “Yuan Kun Yang” was again utilized for the simulations. To explore the effect of the magnification of hydrodynamic derivatives, its value was considered in the range of 1.1 to 1.6. The hydrodynamic derivatives on the turning circle of the ship and the obtained simulation results are presented in Figure 2 and

Table 3. Parameter table of influence of the hydrodynamic derivative magnification on the ship’s turning circle.

Category	1.1 Times	1.2 Times	1.3 Times	1.4 Times	1.5 Times	1.6 Times
$A_{\mathrm{dm}ax}/L$	2.61	2.72	2.83	2.94	3.05	3.17
$A_{\mathrm{dmin}}/L$	0.39	0.30	0.16	0.00	−0.2	−0.5
$T_{\mathrm{r}\max }/L$	2.57	2.73	2.92	3.13	3.35	3.60
$(A_{\mathrm{dmax}}-A_{\mathrm{dmin}})/L$	2.22	2.42	2.67	2.94	3.26	3.64

.

From Figure 2 and Table 3, it is clear that changing the hydrodynamic derivatives had the same significant effect on the turning circle of the ship. The simulation results show that the turning circle of the ship increased with the growth of the hydrodynamic derivatives, and the longitudinal growth was slightly smaller than the transverse growth.

By comparing Table 2 and Table 3, it was found that both the enlargement of the dimensionless cross-flow coefficient and the enlargement of the hydrodynamic derivative could combine to significantly increase the turning circle of the ship. In addition, the increasing effect of the hydrodynamic derivative in the longitudinal distance of the turning circle was greater than the dimensionless cross-flow coefficient, and the discrepancies between these two were not significant in increasing the transverse distance of the ship’s turning circle.

3.3. Influence of the Single Amplified Hydrodynamic Conductivity on the Ship’s Turning Circle Performance

Based on navigational practices, it was discovered that after the enlargement of ships, the forces and moments related to the yaw rate and yaw acceleration increased. To examine the effect of a single magnification of hydrodynamic derivatives on the ship-turning circle, the magnification was chosen to select various values in the range of 1.1–1.6, while the magnification values of other hydrodynamic derivatives were kept fixed. The obtained results reveal that when the hydrodynamic derivatives were all amplified by 1.1 times, the formed turning circle was the most consistent with the actual ship trajectory. Therefore, we set the hydrodynamic derivatives to be enlarged by 1.1 times, adjusted the value of a specific hydrodynamic derivative to ensure that there was only one variable in the simulation, and adopted the “Yuan Kun Yang” simulation. For the sake of convenience in plotting the results, the simulation results are illustrated in Figure 3, which are represented in the graph respectively.

According to Figure 3, it can be seen that the effect of the single amplification on the turning circle changed after different degrees of single amplification of the hydrodynamic derivative, and the influence law of single amplification on the turning circle was also presented in

Table 4. Table of the effect of single magnification of $Y_{\dot{r}}^{\prime }$, $N_{\dot{r}}^{\prime }$, $Y_r^{\prime }$, and $N_r^{\prime }$ on ship-turning circle.

Category	Max. Longitudinal Front Distance	Max. Longitudinal Aft Distance	Maximum Horizontal Distance	Integral Turning Circle
Single enlarged $Y_{\dot{r}}^{\prime }$	Slightly smaller	Slightly larger	Slightly smaller	Slightly down
Single enlarged $N_{\dot{r}}^{\prime }$	Larger	Smaller	Virtually unchanged	Upward
Single enlarged $Y_r^{\prime }$	Slightly larger	Larger	Larger	Downward
Single enlarged $N_r^{\prime }$	Significantly larger	Significantly larger	Significantly larger	Significantly larger

.

From Table 4, it can be seen that after the increase in the single magnification, the first longitudinal distance of the ship circle was slightly reduced and the trailing longitudinal distance was slightly increased. Furthermore, the change in the transverse distance was not clear, and the overall circle was slightly shifted downward. After the single magnification, the first longitudinal distance of the ship circle grew, the trailing longitudinal distance lessened, the transverse distance almost remained unchanged, and the overall circle was shifted upward. After the single magnification, the first longitudinal distance of the ship circle was slightly increased, the trailing longitudinal distance grew, the transverse distance also enlarged, and the overall circle was shifted downward. Finally, after the single magnification, the first longitudinal distance of the ship circle substantially grew, the trailing longitudinal distance noticeably grew, the transverse distance also increased significantly, and the overall circle increased significantly.

The main conclusions obtained from Table 4 and Figure 3 can be summarized as follows: $Y_{\dot{r}}^{\prime }$ had a minimal effect on the turning circle, $N_{\dot{r}}^{\prime }$ had a greater effect on the longitudinal distance, $Y_r^{\prime }$ had a greater effect on the transverse distance, and $N_r^{\prime }$ was the main influential factor of the turning circle.

3.4. Investigation of the Relationship between the Dimensionless Cross-Flow Coefficient and the Hydrodynamic Derivatives

In conclusion, the change in the dimensionless cross-flow coefficient and the hydrodynamic derivatives both had remarkable and similar impacts on the turning circle of the ship, and thereby, the effect of a single change on the turning circle of the ship was methodically planned. In theory, by unequally scaling the hydrodynamic derivatives, a turning circle could be obtained that was approximately equivalent to the variation in the dimensionless cross-flow coefficient. To investigate the relationship between the dimensionless cross-flow coefficient and the hydrodynamic derivative, the following simulation was performed: after first matching with $C_r$ (i.e., in the cases of 1.6 and 0.8, the turning circles were similar; therefore, in the case of 0.8, $N_r^{\prime }$ was 1.6, and the others could be sequentially matched), $N_{\dot{r}}^{\prime }$ was reduced to adjust the longitudinal distance, $Y_r^{\prime }$ was altered to adjust the transverse distance, and finally $Y_{\dot{r}}^{\prime }$ was changed to achieve a fine adjustment of the turning circle.

The simulation results are presented In

Table 5. Table of the corresponding relation between unequal scaling of the hydrodynamic derivatives and the dimensionless cross-flow coefficient.

${\mathit{C}}_{\mathit{r}}$	${\mathit{N}}_{\mathit{r}}^{\prime }$	${\mathit{N}}_{\dot{\mathit{r}}}^{\prime }$	${\mathit{Y}}_{\mathit{r}}^{\prime }$	${\mathit{Y}}_{\dot{\mathit{r}}}^{\prime }$
0.4	1.2	0.80	1.400	1.5
0.5	1.3	0.68	1.635	3.0
0.6	1.4	0.50	1.748	4.6
0.7	1.5	0.36	1.780	6.6

and Figure 4, and the curve-fitting results were found, as illustrated in Figure 5. These are collated in Equation (7):

$$\{\begin{array}{l}fd{Y}_{\dot{r}}^{\prime }=16.9C_r-5.37\\ fd{N}_{\dot{r}}^{\prime }=-1.5C_r+1.41\\ fd{Y}_r^{\prime }=1.782\sin (2.454C_r+6.208)\\ fd{N}_r^{\prime }=C_r+0.8\end{array}$$

(7)

where $C_r$ represents the dimensionless cross-flow coefficient, whose value in the simulation was taken in the range of 0.4 to 0.7. The factors $fd{N}_{\dot{r}}^{\prime }$, $fd{N}_{\dot{r}}^{\prime }$, $fd{Y}_r^{\prime }$, and $fd{N}_r^{\prime }$ denote the magnifications of $Y_{\dot{r}}^{\prime }$, $N_{\dot{r}}^{\prime }$, $Y_r^{\prime }$, and $N_r^{\prime }$, respectively. It is noteworthy that the corresponding four formulations could be adopted to fit the dimensionless cross-flow coefficient changes by the curve fitting toolbox in MATLAB, and unequal scaling of $Y_{\dot{r}}^{\prime }$, $N_{\dot{r}}^{\prime }$, $Y_r^{\prime }$, and $N_r^{\prime }$ was utilized to form the turning circle. Hence, the four magnification values must be determined based on the value of one given $C_r$ and only one so that the resulting correspondences will be meaningful.

To check whether Equation (7) is still applicable for other ships, seven other different ships with tonnages ranging from 50,000 tons to 300,000 tons and different types of bulk carriers, container ships, and tankers were selected, where the data of the ship type parameters are presented in

Table 6. Ship type data parameters.

Name	Symbol	Unit	Monte Granada	Pacific Sapphire	Yuan Kun Yang	KVLCC2	Hanover Square	Emerald Splendor	Long Hu San	Saturn Moon
Ship type	/	/	Bulk carrier	Oil tanker	Oil tanker	Oil tanker	Bulk carrier	Container vessel	Oil tanker	Oil tanker
Length	$L_{pp}$	m	274.20	179.99	324.00	320.00	249.90	228.00	333.00	274.00
Width	$B$	m	48.00	32.20	60.00	58.00	44.00	32.24	60.00	48.00
Draft	$d$	m	15.30	12.62	20.50	20.80	21.50	12.22	21.00	17.15
Displacement	∆	t	173,054	58,025	310,290	312,600	64,200	74,789	366,534	168,729
Block coefficient	$C_b$	/	0.781	0.8095	0.8215	0.81	0.8085	0.8430	0.821	0.8026
Speed	$v_0$	kn	15.73	15.30	18.7	15.8	14.45	15.60	12.20	13.1
Molded depth	$T_{\mathrm{d}}$	m	22.40	19.05	30.00	30.00	30.00	20.65	30.50	23.40

.

Then, using the simulation of other ship data according to the above steps, the matching between $Y_{\dot{r}}^{\prime }, N_{\dot{r}}^{\prime }, Y_r^{\prime }, N_r^{\prime }$, and $C_r$ is organized in

Table 7. Ship’s $C_r$ and $Y_{\dot{r}}^{\prime },N_{\dot{r}}^{\prime },Y_r^{\prime },$ and $N_r^{\prime }$ correspondence.

Yuan Kun Yang					Monte Granada
$C_r$	${\mathit{N}}_{\mathit{r}}^{\prime }$	${\mathit{N}}_{\dot{\mathit{r}}}^{\prime }$	${\mathit{Y}}_{\mathit{r}}^{\prime }$	${\mathit{Y}}_{\dot{\mathit{r}}}^{\prime }$	$C_r$	${\mathit{N}}_{\mathit{r}}^{\prime }$	${\mathit{N}}_{\dot{\mathit{r}}}^{\prime }$	${\mathit{Y}}_{\mathit{r}}^{\prime }$	${\mathit{Y}}_{\dot{\mathit{r}}}^{\prime }$
0.4	1.2	0.80	1.400	1.5	0.4	1.2	0.78	1.4	1.39
0.5	1.3	0.68	1.635	3.0	0.5	1.3	0.6	1.3	3.08
0.6	1.4	0.50	1.748	4.6	0.6	1.42	0.5	1.25	6.4
0.7	1.5	0.36	1.780	6.6	0.7	1.55	0.36	1.1	8.0
Hanover Square					KVLCC2
$C_r$	${\mathit{N}}_{\mathit{r}}^{\prime }$	${\mathit{N}}_{\dot{\mathit{r}}}^{\prime }$	${\mathit{Y}}_{\mathit{r}}^{\prime }$	${\mathit{Y}}_{\dot{\mathit{r}}}^{\prime }$	$C_r$	${\mathit{N}}_{\mathit{r}}^{\prime }$	${\mathit{N}}_{\dot{\mathit{r}}}^{\prime }$	${\mathit{Y}}_{\mathit{r}}^{\prime }$	${\mathit{Y}}_{\dot{\mathit{r}}}^{\prime }$
0.4	1.2	0.81	1.403	1.39	0.4	1.15	0.81	1.403	1.39
0.5	1.3	0.66	1.298	3.08	0.5	1.21	0.66	1.628	3.08
0.6	1.4	0.51	1.2	4.77	0.6	1.29	0.51	1.755	4.77
0.7	1.5	0.36	1.1	6.46	0.7	1.37	0.36	1.777	6.46
Pacific Sapphire					Long Hu San
$C_r$	${\mathit{N}}_{\mathit{r}}^{\prime }$	${\mathit{N}}_{\dot{\mathit{r}}}^{\prime }$	${\mathit{Y}}_{\mathit{r}}^{\prime }$	${\mathit{Y}}_{\dot{\mathit{r}}}^{\prime }$	$C_r$	${\mathit{N}}_{\mathit{r}}^{\prime }$	${\mathit{N}}_{\dot{\mathit{r}}}^{\prime }$	${\mathit{Y}}_{\mathit{r}}^{\prime }$	${\mathit{Y}}_{\dot{\mathit{r}}}^{\prime }$
0.4	1.2	0.78	1.4	1.39	0.4	1.2	0.7	1.3	1.5
0.5	1.3	0.6	1.3	3.08	0.5	1.3	0.6	1.25	2.5
0.6	1.4	0.5	1.0	4.0	0.6	1.4	0.5	1.22	5.0
0.7	1.5	0.4	0.9	5.2	0.7	1.5	0.4	1.18	6.4
Emerald Splendor					Saturn Moon
$C_r$	${\mathit{N}}_{\mathit{r}}^{\prime }$	${\mathit{N}}_{\dot{\mathit{r}}}^{\prime }$	${\mathit{Y}}_{\mathit{r}}^{\prime }$	${\mathit{Y}}_{\dot{\mathit{r}}}^{\prime }$	$C_r$	${\mathit{N}}_{\mathit{r}}^{\prime }$	${\mathit{N}}_{\dot{\mathit{r}}}^{\prime }$	${\mathit{Y}}_{\mathit{r}}^{\prime }$	${\mathit{Y}}_{\dot{\mathit{r}}}^{\prime }$
0.4	1.2	0.8	1.20	1.5	0.4	1.2	0.8	1.20	1.5
0.5	1.3	0.7	1.15	3.5	0.5	1.3	0.7	1.15	3.0
0.6	1.4	0.6	1.10	6.0	0.6	1.4	0.6	1.10	5.5
0.7	1.5	0.5	1.05	9.0	0.7	1.5	0.55	1.05	7.0

, which was found by fitting the respective formulas. Finally, a set of formulas for the dimensionless cross-flow coefficients was obtained by averaging the coefficients in each formula, as provided in the following:

$$\{\begin{array}{l}fd{Y}_{\dot{r}}^{\prime }=17.781C_r-6.044\\ fd{N}_{\dot{r}}^{\prime }=-1.182C_r+1.286\\ fd{Y}_r^{\prime }=1.877\sin (0.971C_r+7.040)\\ fd{N}_r^{\prime }=0.992C_r+0.779\end{array}$$

(8)

The hydrodynamic derivatives for various values (i.e., 0.4, 0.5, 0.6, and 0.7) were then appropriately evaluated using Equation (8), and the corresponding values are summarized in

Table 8. The corresponding relationships between $C_{\mathrm{r}}$ and $Y_{\dot{r}}^{\prime }, N_{\dot{r}}^{\prime }, Y_r^{\prime },$ and $N_r^{\prime }$ in Formula (9).

${\mathit{C}}_{\mathit{r}}$	$\mathit{f}\mathit{d}{\mathit{Y}}_{\dot{\mathit{r}}}^{\prime }$	$\mathit{f}\mathit{d}{\mathit{N}}_{\dot{\mathit{r}}}^{\prime }$	$\mathit{f}\mathit{d}{\mathit{Y}}_{\mathit{r}}^{\prime }$	$\mathit{f}\mathit{d}{\mathit{N}}_{\mathit{r}}^{\prime }$
0.4	1.069	0.813	1.709	1.176
0.5	2.847	0.695	1.776	1.275
0.6	4.625	0.576	1.827	1.375
0.7	6.403	0.458	1.860	1.474

.

4. Simulation Tests

Let us introduce a ship called “Cesi Gladstone” to validate Equation (8), where the parameters of the ship are given in

Table 9. “Cesi Gladstone” main parameters.

Name	Symbol	Unit	Cesi Gladstone
Length	$L_{pp}$	m	290.000
Width	$B$	m	45.600
Design draft	$d$	m	11.70
Displacement	∆	t	120,686.5
Block coefficient	$C_b$	/	0.8022
Design speed	$v_0$	kn	14.0
Molded depth	$T_{\mathrm{d}}$	m	26.50

. In addition, the sea state of the sea area was assumed to have 1–2 levels, and the wind force was considered at four levels, and the wind direction was 47.6°. The tides in the offshore area had an alternating current, the maximum speed of the high tide was 1.5 kn, the flow direction was SW, the maximum speed of the falling tide was 1.0 kn, the current direction was NE, and the initial heading was set to 107°. The dimensionless cross-flow coefficients were considered as 0.4, 0.5, 0.6, and 0.7, the corresponding hydrodynamic derivatives are presented in Table 8, and the simulation results are demonstrated in Figure 6.

To quantify the research results, this study referred to the compliance algorithm in [29] and selected the maximum longitudinal distance in the forward direction, the maximum longitudinal distance in the backward direction, and the maximum distance in the transverse direction. The direction and the maximum distance difference in the longitudinal direction were used as evaluation indices of the ship’s turning performance. Then, the average compliance of the ship’s turning performance was appropriately calculated, which is presented in Equation (9). In this relation, $A_{d\max }$ denotes the maximum longitudinal distance in the first direction, $A_{d\min }$ represents the maximum longitudinal distance in the first direction, $T_{r\max }$ stands for the maximum longitudinal distance in the transverse direction, $A_{d\max -d\min }$ signifies the maximum longitudinal distance difference, and ${\overline{C}}_M$ denotes the average compliance.

$${\overline{C}}_M=\frac{1}{4}\left(\begin{array}{l}\frac{\min (A_{d\max \_true},A_{d\max \_\mathrm{mod}el})}{\max (A_{d\max \_true},A_{d\max \_\mathrm{mod}el})}+\frac{\min (A_{d\min \_true},A_{d\min \_\mathrm{mod}el})}{\max (A_{d\min \_true},A_{d\min \_\mathrm{mod}el})}+\\ \frac{\min (T_{r\max \_true},T_{r\max \_\mathrm{mod}el})}{\max (T_{r\max \_true},T_{r\max \_\mathrm{mod}el})}+\frac{\min (A_{d\max \_true-d\min \_true},A_{d\max \_\mathrm{mod}el-d\min \_\mathrm{mod}el})}{\max (A_{d\max \_true-d\min \_true},A_{d\max \_\mathrm{mod}el-d\min \_\mathrm{mod}el})}\end{array}\right)\times 100\%$$

(9)

Given the space limitation, only some compliance calculations are given, and note the distinction in

Table 10. “Cesi Gladstone” wheel ship-turning test parameter table.

Category	${\mathit{C}}_{\mathit{r}}=0.4$	Derivatives	${\mathit{C}}_{\mathit{r}}=0.5$	Derivatives	Sea Trajectory
$A_{\mathrm{dmax}}/L$	3.10	3.05	3.12	3.11	3.17
$A_{\mathrm{dmin}}/L$	0.31	0.25	0.18	0.21	0.20
$T_{\mathrm{rmax}}/L$	3.68	3.72	3.89	3.96	3.58
$(A_{\mathrm{dmax}}-A_{\mathrm{dmin}})/L$	2.89	2.80	2.98	2.93	2.97
${\overline{C}}_{M1}$	89.22%	91.68%	85.19%	87.23%	/
${\overline{C}}_{M2}$	93.10%		92.37%		/

, where ${\overline{C}}_{M1}$ represents the calculated compliance of $C_r$ and its derivatives show the turning circle with the real ship trajectory, and ${\overline{C}}_{M2}$ indicates the calculated compliance of $C_r$ with changing hydrodynamic derivatives to form two turning circles. Subsequently, the average compliance difference between $C_r$ and the corresponding hydrodynamic derivatives that formed the turning circle and the real ship trail was calculated, as the results are provided in

Table 11. “Cesi Gladstone” round coincidence difference table.

${\mathit{C}}_{\mathit{r}}$	${\mathit{C}}_{\mathit{r}}=0.4$	${\mathit{C}}_{\mathit{r}}=0.5$	${\mathit{C}}_{\mathit{r}}=0.6$	${\mathit{C}}_{\mathit{r}}=0.7$	Average
Conformity difference	2.24%	2.47%	2.91%	3.18%	2.70%
Contrast correspondence	93.10%	92.37%	91.91%	91.39%	92.19%

.

Table 11 presents the average difference between the turning circle formed by the dimensionless cross-flow coefficient and its corresponding changing hydrodynamic derivative. The conformity difference is the difference in ${\overline{C}}_{M1}$ between different dimensionless cross-flow coefficients and their corresponding hydrodynamic derivatives. The contrast correspondence is the average compliance rate between different dimensionless cross-flow coefficients and their corresponding hydrodynamic derivatives. The obtained results reveal that the conformity difference for the turning circle of the real ship trajectory was predicted to be 2.70% and the average contrast between the two circles was obtained as 92.19%, which fulfilled the expected purpose and showed that Equation (8) had a certain generalization to different ships. The next step was to check whether the accuracy of the simplified model was improved. To this end, the simulation of the “Cesi Gladstone” ship was combined with the Norrbin model and the turning circle, which was formed by changing the dimensionless cross-flow coefficient and the corresponding hydrodynamic derivatives, and the calculated results are presented in Figure 7 and

Table 12. Table of turning test parameters of the ship “Cesi Gladstone”.

Category	Norrbin Model	${\mathit{C}}_{\mathit{r}}=0.43$	Derivative	Sea Trajectory
$A_{\mathrm{dmax}}/L$	3.17	3.10	3.08	3.17
$A_{\mathrm{dmin}}/L$	0.69	0.26	0.23	0.20
$T_{\mathrm{rmax}}/L$	3.58	3.75	3.79	3.58
$(A_{\mathrm{dmax}}-A_{\mathrm{dmin}})/L$	2.48	2.84	2.85	2.97
${\overline{C}}_M$	78.12%	91.45%	92.87%	/

.

The simulation results reveal that the correspondence of the turning circle with the real ship trajectory obtained using the original Norrbin model was obtained as 78.12%, while after adjusting $C_r$ and related hydrodynamic derivatives, the correspondence of the final turning circles with the real ship trajectory were obtained as 91.45% and 92.87%. These values are remarkably higher than that predicted by the original Norrbin model, and the correspondence of the two was very close to the real ship trajectory. The simulation results indicate that the accuracy of the simplified model was also improved.

5. Conclusions

In this study, the impact of the dimensionless cross-flow coefficient and the variations in four hydrodynamic derivative factors on the ship’s turning circle were systematically analyzed. Correlation equations linking the dimensionless cross-flow coefficient to the four hydrodynamic derivative factors were successfully developed. Simulation results demonstrated that our formulated model achieved an average consistency of 90.24% across diverse vessel types. Through adjustments to the dimensionless cross-flow coefficient, a highly accurate representation of the actual ship trajectory could be rapidly achieved. By considering the interplay between the dimensionless cross-flow coefficient and the four hydrodynamic derivative factors, and the individual influence of each hydrodynamic derivative on the ship’s turning circle, the four factors were proportionally scaled and appropriately adjusted to swiftly and effectively enhance the accuracy in aligning with the real ship trajectory. This approach is crucial for refining the prediction accuracy of the ship’s turning circle post changes in wind flow, thereby ensuring navigational safety. However, there is room for further refinement in current research, particularly in examining slight variations in the square coefficients and the effects of increased speed. Given the limited availability of ship data, future studies will focus on expanding data collection efforts to refine Equation (8), bolster the accuracy of the Norrbin model, optimize prediction outcomes, and ultimately fortify navigational security.