Numerical Study of the Energy-Saving Effect of the Gate Rudder System

Abstract

Energy-saving device (ESD) plays an important role in mitigating the emission of greenhouse gases in ship industry. It is necessary to study a promising ESD, a gate rudder, for its great potential in promoting energy efficiency. In the present study, ship resistance and self-propulsion simulations were conducted to investigate the energy-saving effects of gate rudder using a viscous in-house CFD solver. First, verification and validation studies were performed to estimate the accuracy and reliability of the numerical method and the results are in good agreement with experimental data. Afterward, resistance and self-propulsion simulations of a crude carrier equipped with the conventional rudder and the gate rudder were carried out respectively. Ship resistance and self-propulsion characteristics with different sailing velocities and propeller revolution rates were compared to study the energy-saving ability of the gate rudder as well as its effects on ship hydrodynamic performance. The results indicate that the gate rudder can greatly optimize the energy efficiency of the ship. Meantime, the ship equipped with the gate rudder shows better resistance and propulsion performance in a self-propelled state.

1. Introduction

Recently, the demand for reducing greenhouse gas emissions from the International Maritime Organization (IMO) has been continuously rising [1]. To meet the increasingly stringent requirements of the Energy Efficiency Design Index (EEDI), improving the energy saving of ship navigation has become a topic of great interest in ship engineering nowadays. Currently, the main approaches to ship energy saving include hull form optimization; ship propulsion performance optimization; ship navigation considering matching of ship-machine-propeller-rudder; and installation of Energy Saving Devices (ESD) [2]. The first two are only applicable at the beginning of ship design and construction and can’t be changed once determined. Navigation optimization brings extra operating tasks. It’s quite straightforward and effective to optimize the ship’s energy performance by installing energy-saving devices, which are easy to install and replaceable, and thus a more economical way. After the 1970s energy crisis, energy-saving devices have been the subject of comprehensive investigation within maritime engineering research [3], of which energy-saving devices improving propulsion performance occupy the most important position in the field of ESD nowadays.

Generally, the energy-saving principle of ESD lies in the reduction of flow field losses, which specifically includes equalizing the wake, decreasing propeller energy losses, and generating additional thrust, among other mechanisms [4]. Wake Equalizing Duct (WED) was first proposed by Schneekluth [5], which homogenizes the inflow of the propeller by increasing the flow velocity upstream of the propeller disc and reducing the tangential velocity of the wake field, thereby improving the overall propulsion efficiency. Lu et al. [6] developed a response surface model to optimize the design parameters of the classical wake equalizing duct and achieved a 37.82% increase in the efficiency improvement ratio. Pre-swirl stators consisting of a series of fins are installed upstream of the propeller. The fins generate a pre-swirl flow to straighten the propeller wake, consequently improving propulsive efficiency. Post-swirl stators are positioned downstream of the propeller to directly weaken the rotation of the flow within the propeller slipstream. Considering the scale effect, Koushan et al. [7] conducted numerical analyses and model tests to quantify the energy-saving effect of pre-swirl stators (PSS) for a chemical tanker. The results also indicate that PSS can provide additional power savings when combined with post-swirl devices. Gaggero et al. [8] employed BEM and RANSE calculations to design energy-saving devices incorporating both pre- and post-swirl fins. The optimized combination achieved an overall efficiency improvement of 4.9%, closely aligning with the combined individual contributions of the two devices. The Mewis Duct developed by Mewis [4] is a kind of pre-swirl duct (PSD). It’s designed to homogenize the inflow to the propeller through the duct positioned upstream and to reduce rotational losses in the slipstream through the integrated pre-swirl fin system. The energy saving can reach 7~9%. Trimulyono et al. [9] investigated the impact of the Mewis Duct on propeller efficiency. The optimized design provides approximately 4% additional thrust and torque, thus improving the propeller performance by 5%. In order to extend the Mewis Duct concept for faster vessels characterized by lower block coefficients, the Becker Twisted Fins (BTF) was developed [10]. Similar to the Mewis Duct, the Becker Twisted Fins incorporates a radial array of pre-swirl fins enclosed within a pre-duct. Additionally, it features a set of radial outer pre-swirl fins equipped with winglet-style end plates, with all fins being twisted to optimize pre-swirl generation. Nowruzi et al. [11] performed a comparative investigation on the effects of various pre-swirl ducts on ship propulsion characteristics by employing EFD model tests and CFD numerical simulations. Mewis duct, Becker twisted fins, and unconventional half-circular ducts were investigated against the case without any duct. Kang et al. [12] proposed a new type of PSD called a ring stator and conducted a parametric study. The optimized design is suitable for low-speed full-body ships with a slender stern and can increase propulsion efficiency by 3%. The hub cap fin on the propeller can eliminate the low-pressure zone on the hub cap, weaken the hub vortex, and recover the wake energy. At the same time, the negative torque will be generated on the fin. Based on the developed automatic optimization technology, Mizzi et al. [13] applied the CFD method to carry out the design and optimization procedure of Propeller Boss Cap Fins (PBCF), mainly considering the influence of length, height, and installation angle of the PBCF. The energy-saving efficiency of PBCF in open water conditions is no less than 1.3%. Furthermore, this method is also applicable to the design of other energy-saving devices. In addition to installing energy-saving appendages on the rudder, special energy-saving rudders can be used directly. Zhu et al. [14] designed a twisted rudder based on the panel method to produce the maximum additional thrust and analyzed its influence on the propeller propulsion performance. The results show that it can save about 3% of energy.

Recently, a new concept of GATE RUDDER (GR) was proposed by Sasaki et al. [15]. The system substitutes the conventional single rudder blade with dual rudder blades distributed on both sides of the propeller. The gate rudder generates additional thrust by accelerating the flow velocity upstream of the propeller plane and reducing the tangential velocity of the wake field, thereby improving the overall propulsive efficiency. It’s like a ducted propeller, which functions as a source of thrust instead of resistance like a conventional rudder. And gate rudder system has more space to install a larger propeller [16]. Moreover, each blade can be controlled separately, which improves the ship’s maneuverability and seakeeping ability [17]. The No More Artistic Hull project (NOAH) of the National Maritime Research Institute (NMRI) is one of the earliest studies on gate rudders. The gate rudder achieved more than 10% power saving and performed better at high speed in the model test [15]. Sasaki and Atlar [18] compared the propulsive efficiency of the gate rudder system and the conventional rudder system. The results demonstrated that the gate rudder blades functioned similarly to a duct, generating additional thrust equivalent to 5–15% of the propeller’s thrust. It can reduce the propeller’s load and improve the overall propulsion efficiency. The gate rudder system was first implemented on the 2400 GT container ship “Shigenobu”, with full-scale sea trials carried out in Japan in November 2017 [19]. Compared with the sister ship “Sakura” equipped with the conventional rudder system, “Shigenobu” showed improved propulsive efficiency, maneuverability, noise, and vibration characteristics. However, the power saving in sea trail was more than that of prediction on the basis of CFD and model test [20]. Considering the scale effect, Hussain et al. [21] investigated the propulsive performance and energy efficiency of a cargo ship under full load and sea trial conditions. Both the gate rudder system and conventional rudder are considered. The findings indicated that the gate rudder system enhanced the ship’s propulsion performance under both conditions, achieving power savings exceeding 12% compared to the conventional rudder. Moreover, the gate rudder of a larger scale produced a larger proportion of additional thrust than the smaller model. Meantime, multiple maneuvering modes provide superior maneuverability performance. Based on the GATERS project, Uyan et al. [22] proposed a comprehensive framework for assessing energy consumption and carbon footprint in retrofitting gate rudder systems on existing ships. Conventional ducted propellers are susceptible to cavitation and vibration issues. Given that the gate rudder system functions as an open-type ducted propeller, further investigation into its hydrodynamic characteristics related to these phenomena is warranted. Through experimental investigation, Turkmen et al. [23] systematically compared cavitation performance and noise characteristics between gate rudder and conventional rudder systems. The study revealed that while both systems exhibited similar cavitation patterns, the gate rudder configuration demonstrated a narrower cavitation range and lower intensity. Additionally, the gate rudder system’s ability to generate additional thrust reduced engine load, consequently leading to a decrease in structural noise levels.

The existing studies have demonstrated the significant energy-saving potential of the gate rudder system, including improving the maneuvering abilities and the seakeeping performance under complex sea conditions and reducing the ship stern vibration and noise. Despite its promising capabilities as an energy-efficient and maneuver-enhancing device, the gate rudder system has a relatively short development history and its related performance is worth being investigated in detail. The present study focuses on the energy-saving capability of the gate rudder system. The geometric configurations of the ship, including the hull, propeller, and both conventional and gate rudders, are detailed in Section 2. Section 3 outlines the numerical methods employed in this study. The computational domain and mesh generation processes are shown in Section 4. To evaluate the reliability and accuracy of the numerical method, comprehensive verification and validation were performed in Section 5. Section 6 conducts a comparative analysis of the hydrodynamic performance of the KVLCC2 model equipped with traditional and gate rudder systems respectively, focusing on resistance and self-propulsion characteristics to assess the energy-saving potential of the gate rudder. Finally, conclusions are drawn in Section 7.

2. Geometry and Simulation Conditions

2.1. The KVLCC2 Model

The KVLCC2 model with a scale ratio of 1:110 is selected in this study. Relevant experimental investigations, including propeller open-water tests and ship resistance tests, have been performed in the Japanese NMRI tank. Therefore, a model of this scale is selected for research in this study to facilitate the verification of numerical simulation results. The principal particulars of the hull are shown in

Table 1. Main parameters of KVLCC2.

Parameters	Symbol	Units	Full-Scale	Model-Scale
Scale ratio	λ	-	1	110
Length of perpendicular	${\mathrm{L}}_{\mathrm{pp}}$	m	320	2.9091
Depth	D	m	30	0.2727
Draft	T	m	20.8	0.1891
Displacement	∆	m3	312,622	0.2349
Wetted surface area	Sw	m2	27194	2.2475
Distance from center of buoyancy to midship	LCB	-	3.48	3.48
Distance from center of gravity to midship	LCG	m	11.1	0.101
Inertia radius about x-axis	ixx/${\mathrm{B}}_{\mathrm{wl}}$	-	0.4	0.4
Inertia radius about y and z-axis	iyy, izz/Lpp	-	0.25	0.25
Rudder total wetted area	SR	m2	273.3	0.0226
Rudder rate	-	deg/s	2.32	24.5

, and the geometric configuration of the hull is presented in Figure 1. The conventional flap rudder without any appendages (as shown in Figure 1) is considered a conventional rudder.

The geometry and parameters of the propeller are as follows (

Table 2. Parameters of the propeller.

Parameters	Symbol	Units	Full-Scale	Model-Scale
Scale ratio	λ	-	1	110
Propeller diameter	DP	m	9.86	0.0896
Pitch ratio (0.7R)	P/D (0.7R)	-	0.721	0.721
Area ratio	Ae/A0	-	0.431	0.431
Rotation	-	-	Right Hand	Right Hand
Hub ratio	-	-	0.155	0.155

and Figure 2):

2.2. The Gate Rudder Model

The gate rudder model is constructed with reference to Turkmen et al. [24] and Yilmaz et al. [25]. The geometry and relative position of the gate rudder blades and the propeller are shown in Figure 3. The variables of the gate rudder are shown in Figure 4a. The angle between the midline of the rudder profile and the propeller axis is defined as the angle of attack $\alpha$, and the distance between the midpoints of the two rudder blades is defined as the distance of the gate rudder $D_G$. As the propeller performance with the gate rudder has been investigated in previous research, a gate rudder model with $\alpha =6deg$ and $D_G=1.20D_P$ selected for resistance research in this study, as shown in Figure 4b.

2.3. Simulation Conditions

The hydrodynamic performance data for the KVLCC2 model investigated in this research, including resistance and self-propulsion characteristics of the ship-conventional rudder system at V = 0.60, 0.70, 0.76, and 0.80 m/s, are provided by NMRI in Japan (SIMMAN, 2014). The design velocity is V = 0.76 m/s ($F_r=0.142$). The definition of the Froude number $F_r$ is expressed as:

$$F_r={\displaystyle \frac{v}{\sqrt{gL}}}$$

(1)

where v is ship’s velocity, g is the gravitational acceleration and L is the length of the ship.

The simulation conditions of the research are listed in

Table 3. Simulation conditions.

Case	Velocity (m/s)	Rudder	Simulation
1	0.60	Conventional rudder	Resistance
2	0.70
3	0.76
4	0.80
5	0.60	Gate rudder	Resistance
6	0.70
7	0.76
8	0.80
9	0.60	Conventional rudder	Self-propulsion
10	0.70
11	0.76
12	0.80
13	0.60	Gate rudder	Self-propulsion
14	0.70
15	0.76
16	0.80

. These four speeds are selected for resistance and self-propulsion simulation in this study.

3. Mathematical Formulation

3.1. Governing Equation

A viscous in-house CFD solver is used to solve the RANS equation for a three-dimensional, unsteady, and incompressible viscous fluid. The continuity and momentum conservation equations are defined as:

$${\displaystyle \frac{\partial U_i}{\partial x_i}}=0$$

(2)

$${\displaystyle \frac{\partial U_i}{\partial t}}+{\displaystyle \frac{\partial U_i{U}_j}{\partial x_j}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial }{\partial x_j}}\left(\mu {\displaystyle \frac{\partial U_i}{\partial x_j}}-\rho \overline{u_i{u}_j}\right)+f_{b_i}$$

(3)

where $\rho$ is the fluid density, $U_i$ is the time-averaged velocity component, $u_{i,j}$ is the fluctuating velocity component, $P$ is the time-averaged pressure, $\mu$ is the dynamic viscosity, and $\rho \overline{u_i{u}_j}$ is the Reynolds stress term. The $\mathrm{S}\mathrm{S}\mathrm{T}k\text{-}\omega$ turbulence model developed by Menter [26] is employed to solve the URANS equations. The finite difference method is applied to discrete the temporal and spatial terms. The pressure-velocity coupling equation is solved with the projection algorithm established by Howell et al. [27] The free surface in the fluid is calculated with the level-set approach [28].

The body-force source term $f_{b_i}$ is incorporated into the momentum equation for self-propulsion simulations. The body force model applied in this study is an iterative body force model called the Osaka University Method (OUM) [29] and has been validated in earlier works [30,31,32].

3.2. Six-DOF Equations

The ship will be subjected to the force and moment of the surrounding fluid in the flow field, it will make the ship move and change its position and heading. The forces and moments acting on the ship are first solved within the Earth-fixed reference frame. Afterwards, the results are transposed to the ship-fixed reference frame. The 6DOF motion equations of the ship based on the ship-fixed reference frame are defined as:

$$m\left[\dot{u}-vr+wq\right]=X$$

(4)

$$m\left[\dot{v }-wp+ur\right]=Y$$

(5)

$$m\left[\dot{w}-uq+vp\right]=Z$$

(6)

$$I_x\dot{p}+\left[I_z-I_y\right]qr=K$$

(7)

$$I_y\dot{q}+\left[I_x-{ I}_z\right]rp=M$$

(8)

$$I_z\dot{r}+\left[I_y-{ I}_x\right]pq=N$$

(9)

where $m$ is the ship’s mass; u, v, w and p, q, r are the translational velocity components and angular velocity components, respectively; X, Y, Z and K, M, N are the forces and moments acting on the hull; $I_x$, $I_y$, $I_z$ are the moments of inertia relative to the ship‘s center of gravity.

3.3. The P-I Speed Controller

For the purpose of controlling the ship at a certain speed, a proportional-integral (PI) controller is implemented to modulate the propeller rotation rate. When the ship speed is less than the target value, the propeller rotation rate will be increased to generate more thrust. In the opposite case, the rotation rate will be reduced to generate less thrust. The propeller maintains a constant rotation rate upon reaching the target ship speed. The expression equation is defined as:

$$n=Pe+I{\int }_0^{t}edt$$

(10)

where $n$ is the propeller rotation rate; $P$ is the proportional gain; $I$ is the integral gain; $e=U_{target}-U_{ship}$ is the deviation between the ship’s instantaneous velocity $U_{ship}$ and its target velocity $U_{target}$.

4. Numerical Method

4.1. Computational Domain and Boundary Conditions

In both resistance and self-propulsion simulations, identical computational domain and boundary conditions are employed. The range of the domain is $-1<X/L_{pp}<4$, $-1<Y/L_{pp}<1$ and $-1<Z/L_{pp}<0.4$, where $L_{pp}$ is the length of perpendicular. In the initial state, the free surface is positioned on $Z/L_{pp}=0$, and the longitudinal section of KVLCC2 is positioned on $Y/L_{pp}=0$ plane. As shown in Figure 5, the velocity inlet and pressure outlet are applied on the front and end of the domain respectively; the two sides are set as zero-velocity-gradient boundary condition; the top and bottom of the computational domain are set as zero-pressure-gradient and zero-pressure boundary conditions; the surface of the model is set as fixed no-slip boundary condition.

4.2. Grid Generation

For resistance simulation, the grid of the computational domain includes background grid, hull grid, rudder grid, and refined grid. According to the parametric modeling method for a numerical tank of ship resistance and self-propulsion performance [33], local refinement is applied to the background grid so that the flow field can be solved precisely and the free surface can be captured accurately. Meanwhile, the grid around the stern and rudder is refined to improve the solution accuracy. The grid and details of the overset grids are shown in Figure 6 and the grid of the gate rudder is shown in Figure 7. It just needs to replace the conventional rudder grid with gate rudder grid when carrying out simulations with the gate rudder.

The stern grid of self-propulsion simulation with a conventional rudder is shown in Figure 8. Computational domain grids of self-propulsion simulation are generally the same as those of resistance simulation. However, the refinement zone around the stern is expanded to precisely resolve the intricate flow induced by the interactions between body force models and ship-rudder systems.

5. Verification and Validation

In order to evaluate the reliability of the numerical methods, verification, and validation, procedures have been conducted according to the guidance of ITTC [34].

5.1. Verification

To assess the uncertainty of the numerical method, systematic convergence analyses were performed for both grid and time step, following the established protocol proposed by Xing et al. [35] and validated against existing test data [36].

The solver has been verified by a large number of calculations, and its iterative uncertainty and other parameter uncertainties can be ignored. Therefore, the numerical uncertainty $U_{SN}$ can be defined as:

$$U_{SN}=\sqrt{U_G^2+U_T^2}$$

(11)

where $U_G$ is grid uncertainty and $U_T$ is time step uncertainty.

According to the guidelines of ITTC, the convergence coefficient $R_i$($i=G$ represents grid, $i=T$ represents time step) is applied to assess the convergence characteristics. At least three different grid and time step sizes (fine, medium, and coarse) are required for the definition of $R_i$. The refinement ratio $r_i$ is expressed as:

$$r_i={\displaystyle \frac{∆x_{i,2}}{∆x_{i,1}}}={\displaystyle \frac{∆x_{i,3}}{∆x_{i,2}}}$$

(12)

where $∆x_{i,1}$, $∆x_{i,2}$ and $∆x_{i,3}$ represents the fine, medium and coarse sizes respectively. $r_G=\sqrt{2}$, $r_T=2$. The convergence ratio is defined as:

$$R_i={\displaystyle \frac{{\epsilon }_{21}}{{\epsilon }_{32}}}$$

(13)

where ${\epsilon }_{21}=S_2-S_1$, ${\epsilon }_{32}=S_3-S_2$, $S_1$, $S_2$ and $S_3$ are the results using fine, medium, and coarse grid or time step respectively. For different values of $R_i$, the convergence conditions are separated as:

(1)

$0<R_i<1$, Monotonic convergence

(2)

$\mathrm{-1}<R_i<0$, Oscillatory convergence

(3)

$\left|R_i\right|>1$, Divergence

When it achieves monotonic convergence, the error ${\delta }_{RE}$ and order of accuracy $P_{RE}$ are obtained with the Richardson extrapolation method:

$${\delta }_{RE}={\displaystyle \frac{{\epsilon }_{21}}{r^{P_{RE}}-1}}$$

(14)

$$P_{RE}={\displaystyle \frac{\ln \left(\frac{{\epsilon }_{32}}{{\epsilon }_{21}}\right)}{\ln r_i}}$$

(15)

The factor of safety method uncertainty is defined as [37]:

$$U_{FS}=\left\{\begin{array}{c}\left(2.45-0.85P\right)\left|{\delta }_{RE}\right|, 0<P\le 1\\ \left(16.4P-14.8\right)\left|{\delta }_{RE}\right|, P>1\end{array}\right.$$

(16)

where the distance metric $P$ is defined as the ratio of $P_{RE}$ to $P_{th}$, $P_{th}$ is set as 2 according to the theoretical order of accuracy. For oscillatory convergence, the numerical uncertainty is defined as [38]:

$$U_i={\displaystyle \frac{1}{2}}\left(S_U-S_L\right)$$

(17)

where $S_U$ is the maximum of the solutions and $S_L$ is the minimum of the solutions.

Convergence studies were conducted by analyzing the total resistance $R_T$ of KVLCC2 at a velocity of $\mathrm{V}=0.76 \mathrm{m}/\mathrm{s}$ ($F_r=0.142$) in resistance simulations. Three different grid sizes are generated with a refinement ratio of $\sqrt{2}$ and three different time-step sizes were generated with a refinement ratio of 2. The results of the convergence study for grid and time-step are shown in

Table 4. Results of grid convergence study.

Grid	Coarse	Medium	Fine	EFD
Size (M)	1.55	4.18	11.70	-
$R_T$ (N)	3.239	3.345	3.358	3.389
Diff. (%)	−4.43	−1.29	−0.92	-
$R_G$	0.116
Convergence	Monotonic
P	3.104
$U_G$$(\%\mathrm{D}$)	1.740

and

Table 5. Results of time-step convergence study.

Time-Step	Coarse	Medium	Fine	EFD
Size (s)	0.01913	0.00957	0.00478	-
$R_T$ (N)	3.260	3.345	3.355	3.389
Diff. (%)	−3.81	−1.29	−1.02	-
$R_T$	0.116
Convergence	Monotonic
P	1.621
$U_T$$(\%\mathrm{D}$)	0.371

. The uncertainty results are calculated based on the Factor of Safety approach.

The monotonic convergence is attained with respect to both grid and time-step, and uncertainties of grid and time-step are $1.740\%\mathrm{D}$ and $0.371\%\mathrm{D}$ respectively. The results demonstrate that the numerical method for ship resistance simulation employed in this study is reliable, satisfying both convergence and accuracy requirements. Considering the calculation accuracy of the results and the consumption of computing resources, the grid and time-step of medium size were used for subsequent ship resistance and self-propulsion simulation research.

5.2. Validation

To assess the accuracy of the numerical simulations, the discrepancies between the results of CFD and EFD are evaluated. The error $E$ is defined to quantify the deviations.

$$E=D-S$$

(18)

where $D$ is the experimental data and $S$ is the simulation results with the grid and time-step of medium size.

The validation uncertainty $U_V$ is expressed as

$$U_V^2=U_D^2+U_{SN}^2$$

(19)

The uncertainty of the experimental data $U_D$ is assumed as 1.00%D when it is unknown [39]. If $\left|E\right|<U_V$, the simulation results are validated under the $U_V$ level.

Table 6. Results of the validation study.

${\mathit{U}}_{\mathit{S}\mathit{N}}\left(\mathit{\%}\mathit{D}\right)$	${\mathit{U}}_{\mathit{D}}\left(\mathit{\%}\mathit{D}\right)$	${\mathit{U}}_{\mathit{V}}\left(\mathit{\%}\mathit{D}\right)$	$\mathit{E}\left(\mathit{\%}\mathit{D}\right)$
1.779	1.000	2.041	−1.29

presents the results of the validation analysis, demonstrating the stability and reliability of the numerical method for further investigations.

A validation study was conducted to assess the accuracy of the numerical method, and simulations of ship resistance of the KVLCC2 were carried out. The simulation results were subsequently subjected to comparative analysis against the experimental data. Resistance simulations were conducted with the ship free to heave and pitch. The comparison is shown in Figure 9. The difference between the results of the simulation with medium size and EFD is −1.29%, which is less than the total numerical uncertainty $U_{SN}=1.779\%\mathrm{D}$. The results demonstrate that the ship resistance values obtained from CFD simulations are highly consistent with the experimental data, confirming that the numerical method applied in this study has high prediction accuracy for the simulation and prediction of resistance at different speeds.

6. Results and Discussion

To evaluate the impact of the gate rudder configuration on ship energy efficiency, the ship resistance and self-propulsion simulations with different velocities are carried out. In this section, the results of the ship-gate rudder (SGR) system and the ship-conventional rudder (SCR) system are compared.

6.1. Results of Ship Resistance Simulation

6.1.1. Resistance Characteristics

Numerical simulations of resistance of the SCR system and SGR system are carried out respectively. The ship resistance obtained from CFD simulations is presented in

Table 7. Resistance of SCR system and SGR system.

Velocity (m/s)	${\mathbf{R}}_{\mathbf{T}}$ (N)			${\mathbf{R}}_{\mathit{f}}$ (N)			${\mathbf{R}}_{\mathbf{r}}$ (N)
	SCR	SGR	Δ(%)	SCR	SGR	Δ(%)	SCR	SGR	Δ(%)
0.60	2.144	2.200	2.612	1.625	1.652	1.662	0.520	0.548	5.385
0.70	2.870	2.946	2.648	2.161	2.196	1.620	0.709	0.750	5.783
0.76	3.345	3.434	2.661	2.517	2.558	1.629	0.828	0.876	5.797
0.80	3.689	3.787	2.657	2.766	2.809	1.555	0.923	0.977	5.850

Where $R_T=R_f+R_r$, $R_T$, $R_f$ and $R_r$ are the total resistance, frictional resistance, and residual resistance, respectively; Δ = (R_SGR − R_SCR)/R_SCR, represents the differences between SGR system relative to SCR system.

and the comparison of resistance components is shown in Figure 10. As the velocity increases, a consistent upward trend is observed in the resistance components of both ship-rudder system. The frictional resistance is about 75% of the total resistance, which accounts for the majority of the total resistance. At the same speed, the resistance components of the SGR system are all larger than those of the SCR system. The total resistance of the SGR system is about 2.6% larger than that of the SCR system, and the higher the speed, the greater the difference.

The rudder resistance and hull resistance of the SCR system and the SGR system are shown in

Table 8. Rudder resistance and hull resistance of SCR system.

Velocity (m/s)	$\mathit{S}$-CR (m2)	${\mathit{R}}_{\mathit{T}}$-CR (N)	${\mathbf{R}}_{\mathit{f}}$-CR (N)	${\mathbf{R}}_{\mathit{r}}$-CR (N)	${\mathit{R}}_{\mathit{T}}$-S (N)	${\mathbf{R}}_{\mathit{f}}$-S (N)	${\mathbf{R}}_{\mathit{r}}$-S (N)	$({\mathbf{R}}_{\mathbf{T}}$$-\mathbf{CR})/{\mathbf{R}}_{\mathbf{T}}$ (%)
0.60	0.0265	0.0531	0.0132	0.0399	2.0912	1.6115	0.4797	2.48
0.70		0.0727	0.0178	0.0549	2.7974	2.1435	0.6539	2.53
0.76		0.0861	0.0207	0.0654	3.2594	2.4967	0.7627	2.57
0.80		0.0952	0.0228	0.0724	3.5942	2.7432	0.8510	2.58

and

Table 9. Rudder resistance and hull resistance of SGR system.

Velocity (m/s)	$\mathit{S}$-GR (m2)	${\mathit{R}}_{\mathit{T}}$-GR (N)	${\mathbf{R}}_{\mathit{f}}$-GR (N)	${\mathbf{R}}_{\mathit{r}}$-GR (N)	${\mathit{R}}_{\mathit{T}}$-S (N)	${\mathbf{R}}_{\mathit{f}}$-S (N)	${\mathbf{R}}_{\mathit{r}}$-S (N)	$({\mathbf{R}}_{\mathbf{T}}$$-\mathbf{GR})/{\mathbf{R}}_{\mathbf{T}}$  (%)
0.60	0.0354	0.1084	0.0394	0.0690	2.0916	1.6126	0.4790	4.93
0.70		0.1457	0.0515	0.0942	2.7999	2.1446	0.6553	4.95
0.76		0.1691	0.0596	0.1095	3.2645	2.4980	0.7665	4.92
0.80		0.1862	0.0651	0.1212	3.6005	2.7442	0.8563	4.92

respectively, where CR refers to the conventional rudder, GR refers to the gate rudder, and S refers to the hull. As presented in Figure 10, the resistance components of the two rudders are compared. As the speed increases, both the rudder and hull resistance components of the two ship-rudder systems show proportional growth, leading to a corresponding increase in the total resistance of the systems. The wet surface area of the gate rudder is about 1.3 times that of the conventional rudder, contributing to larger rudder resistance. The gate rudder exhibits twice the total resistance and 3 times the frictional resistance of the conventional rudder. Compared with the SCR system, the relative contribution of rudder resistance to the total resistance of the SGR system increases from about 2.5% to about 4.9%. The frictional component constitutes a significantly higher percentage in the gate rudder’s total resistance when contrasted with the conventional rudder (the former 35% and the latter 24%), demonstrating that the gate rudder exhibits a significant impact on its resistance components. The increase of rudder resistance is the main reason why the resistance of the SGR system is greater than that of the SCR system.

The ship attitude of the two systems is shown in

Table 10. Ship attitude of SCR system and SGR system.

Velocity (m/s)	SCR		SGR
	Heave (×10−3 m)	Trim (deg)	Heave (×10−3 m)	Trim (deg)
0.60	−1.800	−0.0747	−1.799	−0.0748
0.70	−2.487	−0.1037	−2.485	−0.1040
0.76	−2.951	−0.1242	−2.951	−0.1243
0.80	−3.287	−0.1387	−3.279	−0.1388

. The ship is in a sinking and bow-down state. As the ship’s speed increases, both the sinking depth and bowing angle increase. The differences in ship heave and trim motions between SCR and SGR systems are less than 0.32%, indicating the limited effect of rudder structures on the attitude of the forwarding ship in calm water.

6.1.2. Flow Field Characteristics

Dynamic pressure distribution patterns on both rudder and hull stern surfaces of the two systems at different speeds are shown in Figure 11, where $\mathrm{P}\mathrm{r}=P/0.5\rho V^2$ is the dimensionless dynamic pressure, $P$ is the dynamic pressure, $\rho$ is the fluid density, and $V$ is the ship speed.

As shown in Figure 11, the hydrodynamic pressure distribution exhibits symmetrical patterns across the rudder and stern surfaces of both systems. The stagnation region induced by incoming flow is predominantly located at the leading edge of the conventional rudder, where the pressure is high. Because the thickness of the conventional rudder increases first and then decreases along the chord length, different positions of the rudder have different crowding effects on the flow, making the flow velocity across the rudder surface increase first and then decrease. Therefore, the surface pressure shows a high-low-high distribution along the chord length. The surface pressure distribution of the gate rudder blades along the chord length shows comparable characteristics to the conventional rudder. However, the presence of a gate rudder attack angle ($\alpha =6deg$) induces substantial pressure differentials between its inner and outer surfaces, consequently increasing the residual resistance.

Figure 12 illustrates the longitudinal flow velocities across the propeller disc of the two systems, which are generally symmetrically distributed. The wake effect generates a pronounced velocity gradient differential between the propeller disc region and the undisturbed far field, particularly evident in the axial flow characteristics. Because the gate rudder blades are positioned symmetrically about the propeller axis, they exert more pronounced hydrodynamic effects on the propeller disc flow field compared to the conventional rudder. The SGR system exhibits a reduced low-speed region at the propeller disc center relative to the SCR system.

6.2. Results of Ship Self-Propulsion Performance

6.2.1. Self-Propulsion Characteristics

The ship self-propulsion simulations were carried out with surge, heave, and pitch motion released. The OUM body force model is applied to replace physical propellers for propulsion. The PI control method is used to control and adjust the propeller rotation rate. Upon achieving the target ship speed, the propeller maintains a constant rotation rate. Afterwards, the performance parameters of both the hull and propeller will be obtained, as shown in

Table 11. Self-propulsion characteristics of SCR system and SGR system.

Velocity (m/s)	n(rps)			T(N)			Q(N × m)
	SCR	SGR	Δ(%)	SCR	SGR	Δ(%)	SCR	SGR	Δ(%)
0.60	14.54	13.59	−6.56	2.690	2.483	−7.71	0.0242	0.0224	−7.49
0.70	16.93	15.74	−7.07	3.605	3.306	−8.30	0.0326	0.0299	−8.24
0.76	18.34	17.05	−7.02	4.203	3.856	−8.26	0.0381	0.0350	−8.20
0.80	19.31	17.95	−7.01	4.642	4.243	−8.58	0.0421	0.0386	−8.37

Where n is the rotational speed, T is the propeller thrust and Q is the propeller torque, Δ = (R_SGR − R_SCR)/R_SCR, represents the differences between SGR system relative to SCR system.

.

With increasing target ship speed, the PI controller proportionally adjusts the propeller’s rotational velocity to deliver the necessary thrust and torque output. Compared with the SCR system, the propeller rotation rate required for the SGR system to achieve the same velocity is reduced by 6.56~7.07%, the propeller thrust is reduced by 7.71~8.58%, and the propeller torque is reduced by 7.49~8.37%. The propeller thrust equals the ship’s total resistance under self-propulsion conditions. The results show that the resistance of the SGR system is smaller than that of the SCR system when self-propelled at the same speed, which means the effective power required for the SGR system to reach the same velocity as the SCR system is smaller. In other words, the SGR system can reach a higher speed than the SCR system with the same propeller power.

The thrust deduction and thrust deduction factor of the two systems are shown in

Table 12. Thrust deduction factor of SCR system and SGR system.

Velocity (m/s)	$∆\mathit{T}$(N)		$\mathit{t}$
	SCR	SGR	SCR	SGR	Δ(%)
0.60	0.546	0.283	0.203	0.114	−43.86
0.70	0.735	0.360	0.204	0.109	−46.53
0.76	0.858	0.422	0.204	0.109	−46.34
0.80	0.952	0.457	0.205	0.108	−47.56

Where $∆T=T-R$, $\mathrm{t}=∆T/T$, $R$ is the total resistance of the ship in resistance simulation, $∆T$ is the thrust deduction of the propeller and $t$ is the thrust deduction factor of the propeller.

. The thrust deduction and thrust deduction factor of the SGR system are much smaller than those of the SCR system. Specifically, the thrust deduction factor is reduced by 43.86~47.56%. Based on the simulation results, the required power and propulsive efficiency of the SCR system and the SGR system are obtained and compared, as shown in

Table 13. Power and propulsive efficiency of SCR system and SGR system.

Velocity (m/s)	${\mathit{P}}_{\mathit{E}}$(W)			${\mathit{P}}_{\mathit{D}\mathit{B}}$(W)			${\mathit{\eta }}_{\mathit{D}}$
	SCR	SGR	Δ(%)	SCR	SGR	Δ(%)	SCR	SGR	Δ(%)
0.60	1.287	1.320	2.60	2.210	1.910	−13.56	0.5823	0.6911	18.69
0.70	2.009	2.062	2.63	3.465	2.955	−14.73	0.5798	0.6978	20.35
0.76	2.543	2.610	2.63	4.388	3.745	−14.65	0.5796	0.6969	20.25
0.80	2.952	3.029	2.64	5.110	4.354	−14.80	0.5776	0.6958	20.46

Where $P_E=RV$, $P_{DB}=2\pi nQ$, ${\eta }_D=P_E/P_{DB}$, $P_E$ is the effective power, $P_{DB}$ is the delivered power, ${\eta }_D$ is propulsive efficiency and Δ = (R_SGR − R_SCR)/R_SCR, represents the differences of SGR system relative to SCR system.

and Figure 13. Compared with the SCR system, the effective power of the SGR system increased by 2.60%, the delivered power was reduced by 13.56~14.80%, and the propulsive efficiency increased by 18.69~20.46%. Since the hull and propeller of the two systems are identical, the engine power required for the SGR system to reach the same speed as the SCR system is reduced by 13.56~14.80%. The gate rudder system shows excellent power-saving effect.

6.2.2. Resistance Components

In order to explore the reason why the resistance of SGR system is smaller, the resistance components of SCR system and SGR system in self-propulsion simulations are analyzed, as shown in

Table 14. Self-propulsion resistance components of SCR system.

Velocity (m/s)	${\mathbf{R}}_{\mathbf{T}}$ (N)	${\mathbf{R}}_{\mathit{T}}$-S (N)	${\mathbf{R}}_{\mathit{T}}$-CR (N)	${\mathbf{R}}_{\mathit{f}}$ (N)	${\mathbf{R}}_{\mathit{f}}$-S (N)	${\mathbf{R}}_{\mathit{f}}$-CR (N)	${\mathbf{R}}_{\mathit{r}}$ (N)	${\mathbf{R}}_{\mathbf{r}}$-S (N)	${\mathbf{R}}_{\mathit{r}}$-CR (N)	$({\mathbf{R}}_{\mathit{T}}$$-\mathbf{CR})/{\mathbf{R}}_{\mathit{T}}$ (%)
0.60	2.690	2.482	0.2080	1.687	1.624	0.0622	1.004	0.858	0.1457	7.73
0.70	3.605	3.325	0.2797	2.240	2.159	0.0806	1.365	1.166	0.1991	7.76
0.76	4.203	3.874	0.3287	2.608	2.515	0.0924	1.595	1.359	0.2363	7.82
0.80	4.642	4.278	0.3640	2.865	2.764	0.1009	1.777	1.514	0.2631	7.84

and

Table 15. Self-propulsion resistance components of SG system.

Velocity (m/s)	${\mathbf{R}}_{\mathbf{T}}$ (N)	${\mathbf{R}}_{\mathit{T}}$-S (N)	${\mathbf{R}}_{\mathit{T}}$-GR (N)	${\mathbf{R}}_{\mathit{f}}$ (N)	${\mathbf{R}}_{\mathit{f}}$-S (N)	${\mathbf{R}}_{\mathit{f}}$-GR (N)	${\mathbf{R}}_{\mathit{r}}$ (N)	${\mathbf{R}}_{\mathbf{r}}$-S (N)	${\mathbf{R}}_{\mathit{r}}$-GR (N)	$({\mathbf{R}}_{\mathit{T}}$$-\mathbf{GR})/{\mathbf{R}}_{\mathit{T}}$ (%)
0.60	2.483	2.458	0.0251	1.656	1.625	0.0311	0.827	0.833	−0.0060	1.01
0.70	3.306	3.268	0.0376	2.201	2.160	0.0404	1.105	1.108	−0.0029	1.14
0.76	3.856	3.815	0.0411	2.563	2.516	0.0469	1.293	1.298	−0.0059	1.07
0.80	4.243	4.214	0.0291	2.816	2.764	0.0516	1.427	1.450	−0.0224	0.69

, where S represents the hull, CR represents the conventional rudder, and GR represents the gate rudder. Figure 14 shows the comparisons of the resistance components.

As shown in Figure 14a, compared with the SCR system, the decrease in the residual resistance is the main reason for the decrease in the total resistance of the SGR system. Figure 14b,c demonstrates that the gate rudder configuration significantly affects rudder resistance while exhibiting minimal impact on hull resistance. Specifically, the gate rudder resistance is only 8.01~13.43% of the conventional rudder resistance. Both the frictional and residual resistance of the gate rudder are much smaller. Especially, the residual resistance of the gate rudder performs negative values, providing extra thrust in self-propulsion simulations. Moreover, the conventional rudder resistance accounts for about 7.8% of the total resistance of the SCR system, while the gate rudder resistance accounts for only about 1.0% of the total resistance of the SGR system, the proportion of rudder resistance is greatly reduced.

A comprehensive comparative analysis was conducted to evaluate the propeller’s hydrodynamic interaction with both rudder configurations, through systematic examination of resistance and self-propulsion simulation results, as illustrated in Figure 15, Figure 16 and Figure 17. The relative contribution of rudder resistance to the overall system’s total resistance is quantitatively presented in Figure 18. All resistance components of the two systems in the self-propulsion simulation are larger than those in the resistance simulation. The total, frictional, and residual resistance of the SCR system increased by 25.46~25.81%, 3.57~3.80%, and 92.44~93.18% respectively. The total, frictional and residual resistance of the SGR system increased by 12.06~12.86%, 0.21~0.26% and 46.02~50.82% respectively. The increases of all resistance components of the SGR system are less than those of the SCR system. It can be observed that the frictional resistance of both systems changed slightly and the residual resistance dramatically increased. The residual resistance of the hull obviously increased while the frictional resistance increased marginally. Rudder resistance components of both ship-rudder systems are significantly affected by the propeller. The conventional rudder resistance increases by 381.88~391.64% and the proportion of the rudder resistance in SCR system resistance increases from about 2.5% to about 7.8%. On the contrary, the gate rudder resistance is reduced by 74.22~84.35% and the proportion of the rudder resistance in the SGR system decreases from about 4.9% to about 1.0%.

The analysis above demonstrates that propeller interactions significantly impact the hydrodynamic performance of both rudder configurations. In resistance simulations, the resistance of the gate rudder is about twice that of the conventional rudder, resulting in a larger total resistance relative to the SCR system. However, in self-propulsion simulations, a significant increase in resistance is observed for the conventional rudder, whereas the gate rudder demonstrates a substantial reduction in resistance. Consequently, the total resistance of the SGR system is smaller than that of the SCR system in self-propulsion simulations.

Table 16. Hull attitude of SCR system and SGR system in self-propulsion.

Velocity (m/s)	SCR		SGR
	Heave (×10−3 m)	Trim (deg)	Heave (×10−3 m)	Trim (deg)
0.60	−1.843	−0.0682	−1.841	−0.0678
0.70	−2.538	−0.0952	−2.533	−0.0951
0.76	−3.016	−0.1140	−3.009	−0.1140
0.80	−3.356	−0.1273	−3.352	−0.1272

presents the ship attitude of the SCR system and the SGR system in the self-propulsion simulation. The ships from both systems are in a sinking and bow-down state, and the sinking depth and bowing angle increase as the speed increases. In line with the resistance simulation results, the ship attitude variations between the two systems are negligible, indicating that the impact of the gate rudder on the motion attitude of the hull in the self-propelled state is insignificant.

6.2.3. Flow Field Characteristics

Figure 19 presents a comparative analysis of the pressure distribution on the rudder and hull stern surfaces for both systems in self-propulsion simulation.

In contrast to resistance simulation results, the SCR system exhibits asymmetric pressure distribution patterns on both the stern and rudder surfaces. Because the propeller is right-handed, the rotation of flow leads to asymmetric pressure distribution across the conventional rudder surfaces. The accelerated flow downstream of the propeller significantly modifies the pressure distribution characteristics, resulting in an expansion of both the high-pressure region at the rudder’s leading edge and the low-pressure region along the lateral surface. And the rudder resistance is larger than that in the resistance simulation. Moreover, the asymmetric flow obstructed by the conventional rudder directly impacts the hull stern, resulting in a distinct pressure asymmetry across the stern region. As to the SGR system, the pressure on the stern and rudder is still roughly symmetrically distributed. A large low-pressure region develops on the inner sides of rudder blades, while the outer sides exhibit an expanded high-pressure region and a corresponding reduced low-pressure region. The differential pressure distribution across the gate rudder’s internal and external surfaces and the existence of the attack angle $\alpha$ (as shown in Figure 4b) make the gate rudder provide thrust, namely, the value of residual resistance is negative.

In self-propulsion simulations, the velocity at the propeller disc of the SCR system and the SGR system is asymmetrically distributed, as shown in Figure 20. The high-speed region within the propeller disc exhibits increased velocity with rising self-propulsion speed. Based on prior findings, the SGR system requires a lower propeller rotation rate than the SCR system to achieve equivalent speeds. Therefore, the longitudinal velocity at the propeller disc of the SGR system is smaller.

Figure 21 illustrates the stern flow field characteristics for both the SCR and SGR systems under self-propulsion simulation conditions. The inflow velocity at the propeller disc, when equipped with a gate rudder, is higher than that when equipped with a conventional rudder at the same ship speed. It can be seen that the propeller wake of the SCR system is more dispersed in height, which is caused by the obstruction of the conventional rudder to the rotating wake, while the gate rudder of the SGR system limits the wake to a certain area, which makes the wake more concentrated and has less influence on the flow field around the hull. In the SCR system, the conventional rudder is positioned within the propeller wake, where it is directly influenced by the flow. The gate rudder in the SGR system avoids the direct impact of the wake and causes a difference in the flow velocity inside and outside the rudder blades, resulting in a smaller resistance of the gate rudder.

7. Conclusions

This study evaluates the energy-saving performance of gate rudder systems through numerical analysis. The ship resistance and self-propulsion simulations were performed for both the ship-conventional rudder (SCR) and ship-gate rudder (SGR) systems to evaluate the energy-saving effect of the gate rudder system. First of all, the resistance simulation of KVLCC2 with a conventional rudder model is carried out. The results of total resistance at different speeds show good agreement with experimental data, confirming the accuracy and reliability of the numerical method applied in this study.

For the ship resistance simulation, the SGR system exhibits higher total, frictional, and residual resistance than the SCR system, with the total resistance increasing by 2.6%. The hull resistance remains relatively unchanged, whereas the gate rudder exhibits higher frictional and residual resistance compared to the conventional rudder. The total resistance of the gate rudder is about twice that of the conventional rudder, which is mainly caused by the larger wet surface area and the extra pressure differential force induced by the attack angle of the rudder blades.

For the self-propulsion simulation, the SGR system exhibits a total resistance reduction of 7.71~8.58% relative to the SCR system, and both the hull resistance and gate rudder resistance are smaller. Especially, the resistance of the gate rudder is only 8.01~13.43% of that of the conventional rudder. The primary contributing factors are twofold: firstly, the gate rudder avoids the direct impact of the propeller wake; secondly, the differential velocity distribution across the rudder blade surfaces induces thrust generation. The difference in hydrodynamic performance between the two systems leads to the reduction of propeller speed by 6.56~7.07%, propeller torque by 7.49~8.37%, thrust deduction factor by 43.86~47.56%, required main engine power by 13.56~14.80%, and propulsion efficiency by 18.69~20.46% when the ship-gate rudder system reaches the same speed as the ship-conventional rudder system in self-propelled state. The gate rudder shows a good energy-saving effect.

Compared with the resistance simulation, the total resistance of the SCR system in the self-propulsion simulation increased by 25.46~25.81% and the resistance of the conventional rudder increased by 381.88~391.64%. It’s primarily attributed to the direct impact of the propeller wake on the conventional rudder. The total resistance of the SGR system only increased by 12.06~12.86%. Due to the avoidance of wake impact, the resistance of the gate rudder decreased by 74.22~84.35%. The gate rudder demonstrates superior energy-saving performance due to its distinct propeller-rudder interaction characteristics compared to conventional rudder.