A Hull–Engine–Propeller Matching Method for Shaftless Rim-Driven Thrusters

Abstract

As an innovative underwater propulsion technology, the rim-driven thruster (RDT) has garnered increasing attention due to its advantages over conventional diesel or gas turbine propulsion systems, including reduced noise, higher efficiency, and a compact structure. However, traditional hull–engine–propeller matching theories are not directly applicable to RDTs because of their unique shaftless and ducted characteristics. Based on conventional hull–engine–propeller matching theory and propeller design methodology, this study proposes a novel hull–engine–propeller matching approach tailored specifically to RDTs. The method enables rapid matching by using open-water characteristics for hull–engine–propeller matching. In the absence of open-water test data for shaftless propellers, key parameters derived from ducted propeller tests are used for matching based on open-water characteristics to design the shaftless propeller. The propeller is then optimized through computational fluid dynamics (CFD) simulations to achieve the required thrust performance, effectively enabling an equivalent replacement. The proposed method provides a practical framework for selecting and designing RDTs, improves overall propulsion efficiency, and offers specific guidelines for determining optimal motor design parameters.

1. Introduction

The electrification of ship propulsion systems has become a prominent research focus in recent years, driven by the need for more sustainable and environmentally friendly maritime solutions. Among the innovations in electric thruster technologies, the rim-driven thruster (RDT) stands out as a promising alternative to traditional propulsion systems, offering reduced noise, a more compact design, and higher operational efficiency [1].

One of the most notable features of the RDT is its shaftless structure, which significantly reduces the risk of entanglement with fishing nets, seaweed, and other debris—thus ensuring reliable operation in complex and challenging marine environments. This characteristic makes the RDT particularly well-suited for specialized vessels such as marine rescue ships, seaweed collection vessels, and other maritime applications where reliability and performance in adverse conditions are critical.

In the late 1990s, the General Dynamics Electric Boat Company in the United States conducted research and design work on tip-driven pod propulsion. Experimental comparisons revealed that the open-water efficiency of tip-driven pod systems was 5–10% higher than that of conventional shaft-driven pod propulsion systems [2], marking a significant milestone in the commercialization of RDT technology.

Since 2002, Brunvoll, a Norwegian propulsion manufacturer, has been actively involved in the design, development, and production of RDTs. One of its early commercial applications was the installation of an RDT on the Norwegian ferry M/F Eiksund, as shown in Figure 1 [1].

Rolls-Royce also began developing RDT technology as early as 2005, introducing a version with a central support shaft to improve reliability and propulsion efficiency [3]. In 2012, the company delivered the TT-PM1600 RDT model to the Norwegian University of Science and Technology [4]. Two p.m. RDTs were installed on the Gunnerus ocean survey vessel and have since accumulated over 1500 h of trouble-free operation.

Germany’s Voith has likewise been a key player in the research and manufacturing of RDTs. Since 2008, the company has sold more than 55 units, the majority of which remain in active service.

Today, companies such as Rolls-Royce, Schottel, Brunvoll, and Voith have successfully brought RDTs into commercial production, with their product lines continuing to expand in terms of power range.

Table 1. Several commercial rim-driven thrusters [1].

No.	Manufacturer	Location	Typical Model Type	Rated Power (kW)	Propeller Diameter (mm)
1	TSL	England	Diameter-based Nomenclature	0.1–4	50–300
2	HAOYE	China	R21	0.35	93
3	Copenhagen Subsea	Denmark	VM/VL	7–10.5	140–168
4	Vetus	Germany	BOW PRO’S	1–20	100–300
5	Schottel	Germany	SRTxxx	200–800	800–1600
6	GH thru	China	GRT/GRM	5–3000	150–2600
7	Brunvoll	Norway	RDTxxx	200–1600	800–2100
8	Voith	Germany	VIT/VIP	100–1200	400–1000
9	Rolls-Royce	England	AZ-PM	500–2600	560–2850

presents the parameters for several commercially available RDT models.

While the traditional hull–engine–propeller matching in ship propulsion systems is well-established, the development of new types of propellers, such as shaftless RDTs, remains an underexplored area of research. Further investigation into hull–engine–propeller matching for shaftless RDTs is essential to fully realize their potential and advance the state of electric propulsion technology for maritime applications.

Currently, many companies, both domestically and internationally, offer commercial shaftless RDTs. However, due to commercial interests, the core technical information of these companies remains confidential. Most of the current research on RDTs has focused on the study of the hydrodynamic characteristics of RDTs. An optimization design method for a RDTs using the adjoint method was proposed by B. Liu et al. [5], which can effectively improve the efficiency of the thruster and has better cavitation performance. Through CFD simulation, C. Chen et al. [6] compared the open-water performance, wake distribution, and sheet cavitation characteristics of shaftless RDTs with different tip rake distributions. P. Li et al. [7] analyzed the effect of domain division types on simulation performance using the Delayed Detached Eddy Simulation (DDES) turbulence model. B. Liu et al. [8] analyzed the vortical flow structures generated in the wake of ducted propellers and RDT in bollard conditions. Multi-objective optimization of the duct for an RDT was conducted by S.Zhai et al. [9], and it was found that duct optimization can improve efficiency by 3.3% or minimum pressure by 5.3%. A numerical design for RDTs was proposed by S.Gaggero [10], which involves a parametric description of the rim blade geometry combined with a multi-objective optimization algorithm to guide the selection of optimal blade shapes. The gap fluid of RDTs was investigated by H.Jiang et al. [11], who found that the gap has a great influence on the thrust coefficient, torque coefficient, and efficiency. Comparative hydrodynamic characterization of hub-type and hubless-type RDTs using CFD was conducted by B.Song et al. [12], revealing that the hubless-type thruster exhibits superior performance. D. Li et al. [13] employed finite element software to optimize the design of the propeller and rim motor for a thruster intended for deep-sea applications. The thruster was subsequently tested to ensure it met the design specifications and fulfilled the power requirements of the underwater vehicle.

L. Amri et al. [14] introduced a multi-objective optimization algorithm for RDTs, which enhances performance by constraining the feasible domain of design variables to determine the optimal parameter combination. Based on the optimization results, the research team designed a pump prototype for verification. Comparisons between finite element analysis and experimental testing confirmed a 95% consistency between the simulation and experimental results. M. F. Hsieh et al. [15] developed an RDT for a cable-operated remotely operated vehicle (ROV). The study revealed that the actual operating speed was lower than the theoretical value due to bearing friction exceeding the expected design limits. As a result, achieving the rated speed required the system to deliver a higher drive current. G. Zhou et al. [16] proposed a mixed-lubrication analysis method for water-lubricated spiral-groove rubber thrust bearings. This approach innovatively incorporated the effects of surface roughness, enabling the optimal design of spiral-groove structural parameters and effectively addressing the mixed-lubrication challenges commonly encountered in spiral surface contact. W. Ouyang et al. [17] proposed an RDT optimization method that combines CAESES for geometric modeling, CFD software for performance simulation, and a genetic algorithm for parameter optimization. Their results showed that the efficiency of the optimized RDT was 5.78% higher than that of the original design.

The optimal matching of hull–engine–propeller systems can reduce energy consumption, enhance economic efficiency, and contribute to environmental protection, among other benefits. These advantages have attracted significant attention from leading researchers in the field. A concept known as the ship operation diagram was proposed by TG. Tran et al. [18], which integrates the characteristics of the ship, propeller, and main engine under various operating conditions into a single framework. This diagram is used to optimize the matching of the ship’s engine and propeller. A methodology was proposed by CH. Marques et al. [19] for the preliminary design phases of an electric vessel, wherein the propeller is optimized to ensure compatibility with the electric motor. This involves iterating the propeller design to achieve the optimal operational speed, thereby maximizing efficiency and performance. CH. Marques et al. [20] investigated the influence of weather conditions on the matching of boat engines and propellers, incorporating these effects into a drag model to minimize fuel consumption during a round trip. This was achieved through the application of an optimization algorithm. Y. Wang et al. [21] employed a predictive control strategy to regulate both fuel consumption and the pitch angle of the adjustable-pitch propeller in a gas turbine system. Through effective control, significant energy savings were achieved. A. Liu et al. [22] approached the hull–engine–propeller matching problem as a multi-objective optimization task, focusing on optimizing both efficiency and cost. Their experimental results demonstrated that the particle swarm optimization algorithm is an effective method for addressing this matching problem. Q. Tan et al. [23] proposed a method for hull–engine–propeller matching based on the incorporation of a propeller boss cap fin (PBCF). This approach plays a significant role in enhancing the performance of PBCFs and optimizing the design of ship propulsion systems.

E.S. Koenhardono et al. [24] developed a neural network model for engine performance prediction aimed at optimizing propulsion system performance and minimizing fuel consumption. Data collected from patrol vessels demonstrated that the model’s prediction error for fuel consumption was less than 1%. O.B. Ogar [25] proposed a matching method for variable-pitch propeller propulsion systems that comprehensively accounts for both the thrust–torque characteristics of the propeller and the dynamic response of the turbocharger. This method was successfully applied to the hull–engine–propeller matching design of an F90 frigate by calculating the mass flow rate under various speeds and boost pressures. The simulation results showed that the matched values of propeller speed, engine power, and pitch ratio closely aligned with the design parameters of the frigate, while key performance metrics such as the thrust coefficient, torque coefficient, and open-water efficiency were consistent with actual ship data.

Extensive research has been carried out on the structural design and hydrodynamic characteristics of RDTs, and the hull–engine–propeller matching theory has become relatively well-established. However, most studies focus on conventional propulsion systems, with limited investigations addressing motor–propeller matching specifically for RDTs. In this study, a hull–engine–propeller matching method is proposed for RDTs. The propeller is analyzed using blade element theory, and a matching approach is developed based on its open-water characteristics. The method is applicable to both conventional propellers and shaftless RDTs, enabling the rapid determination of matching parameters and streamlining the overall design process of hull–engine–propeller integration. However, the variety of available RDTs is limited, and the systematic open-water performance characterization of these systems remains insufficient.

We focus on an RDT concept similar to the duct propeller, which serves as the baseline for comparison. The matching results obtained through the proposed method are then applied to the design of a shaftless propeller, enabling the determination of key parameters for both the propeller and the motor. This approach not only provides a rapid means of generating matching results, but also offers a practical tool for designing more efficient, environmentally friendly propulsion systems for specialized maritime applications.

2. Blade Element Theory

In blade element theory, the forces acting on a propeller are calculated by treating each small radial segment of the blade, referred to as a blade element, as a two-dimensional airfoil. Given the propeller’s advance speed $V_{\mathrm{A}}$ and rotational speed n, the induced velocities $u_{\mathrm{a}}$ and $u_{\mathrm{t}}$ can be determined. Using blade element theory, the force on the blade element at any radius can then be calculated, allowing for the determination of the total force on the entire propeller.

Consider a blade element $dr$ located at a radial position r, as illustrated in Figure 2. When water approaches the blade element with a resultant velocity $V_{\mathrm{R}}$ and an angle of attack ${\alpha }_k$, it generates a lift force $dL$ and a drag force $dD$. The lift $dL$ can be decomposed into two components: $d{L}_a$, which acts along the propeller axis, and $d{L}_t$, which acts in the tangential (rotational) direction. Similarly, the drag dD is also decomposed into two components: $d{D}_a$, along the axis, and $d{D}_t$, in the tangential direction. Here, ${\beta }_i$ denotes the angle between the resultant velocity and the rotational velocity. The resulting thrust- $dT$ and torque-induced resistance $dF$ acting on the blade element can thus be expressed as follows:

$$\begin{array}{cc}\hfill dT& =d{L}_a-d{D}_a=dLcos\left({\beta }_i\right)-dDsin\left({\beta }_i\right)\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill dF& =d{L}_t+d{D}_t=dLsin\left({\beta }_i\right)+dDcos\left({\beta }_i\right)\hfill \end{array}$$

(2)

According to the Kutta–Joukowski lift theorem, assuming the circulation at radius r is $\mathrm{\Gamma }\left(r\right)$, the lift generated by the blade element located in position r is given by

$$dL=\rho \mathrm{\Gamma }\left(r\right)V_{\mathrm{R}}dr$$

(3)

Substituting Equation (3) into Equations (1) and (2), and setting $dD=\epsilon$$dL$ (where $\epsilon$ represents the drag-to-lift ratio of the blade element), the torque on the blade element, $dQ=rdF$, can be derived as follows:

$$\begin{array}{cc}\hfill dT& =\rho \mathrm{\Gamma }\left(r\right)V_{\mathrm{R}}cos\left({\beta }_i\right)(1-\epsilon tan\left({\beta }_i\right))dr\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill dQ& =\rho \mathrm{\Gamma }\left(r\right)V_{\mathrm{R}}sin\left({\beta }_i\right)(1+\epsilon cot\left({\beta }_i\right))rdr\hfill \end{array}$$

(5)

From Figure 2, the following relationship can be derived:

$$\begin{array}{cc}\hfill V_{\mathrm{R}}cos\left({\beta }_i\right)& =2\pi nr-{\displaystyle \frac{1}{2}}{u}_{\mathrm{t}}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill V_{\mathrm{R}}sin\left({\beta }_i\right)& =V_{\mathrm{A}}+{\displaystyle \frac{1}{2}}{u}_{\mathrm{a}}\hfill \end{array}$$

(7)

Substituting these relationships into Equations (4) and (5), the following expression is obtained:

$$\begin{array}{cc}\hfill dT& =\rho \mathrm{\Gamma }\left(r\right)\left(2\pi nr-{\displaystyle \frac{1}{2}}{u}_{\mathrm{t}}\right)(1-\epsilon tan\left({\beta }_i\right))dr\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill dQ& =\rho \mathrm{\Gamma }\left(r\right)\left(V_{\mathrm{A}}+{\displaystyle \frac{1}{2}}{u}_{\mathrm{a}}\right)(1+\epsilon cot\left({\beta }_i\right))rdr\hfill \end{array}$$

(9)

By integrating Equations (8) and (9) along the radial direction from the hub to the blade tip, and multiplying the result by the number of blades Z, the total thrust and torque generated by the entire propeller can be determined as follows:

$$\begin{array}{cc}\hfill T& =Z\rho {\int }_{r_{\mathrm{h}}}^R\mathrm{\Gamma }\left(r\right)\left(2\pi nr-{\displaystyle \frac{1}{2}}{u}_{\mathrm{t}}\right)(1-\epsilon tan\left({\beta }_i\right))dr\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill Q& =Z\rho {\int }_{r_{\mathrm{h}}}^R\mathrm{\Gamma }\left(r\right)\left(V_{\mathrm{A}}+{\displaystyle \frac{1}{2}}{u}_{\mathrm{a}}\right)\left(1+{\displaystyle \frac{\epsilon }{tan\left({\beta }_i\right)}}\right)rdr\hfill \end{array}$$

(11)

where $R_{\mathrm{h}}$ is the hub radius, and R is the propeller radius.

3. Matching Strategy

3.1. Propeller Open-Water Characteristics

The hydrodynamic performance of a propeller is characterized by the relationships among the thrust T, torque Q, and efficiency ${\eta }_0$, as influenced by the propeller’s advance speed $V_{\mathrm{A}}$ and rotational speed n when operating in water.

The propeller advance refers to the linear distance it travels along its shaft during one complete revolution. This distance can be calculated using the following formula:

$$h_{\mathrm{p}}={\displaystyle \frac{V_{\mathrm{A}}}{n}}$$

(12)

where $h_{\mathrm{p}}$ is the propeller advance per revolution, $V_{\mathrm{A}}$ is the propeller advance speed, and n is the propeller rotational speed.

The propeller advance coefficient, denoted as J, is defined as the ratio of the propeller’s advance $h_{\mathrm{p}}$ to the propeller diameter D. It is expressed as

$$J={\displaystyle \frac{h_{\mathrm{p}}}{D}}={\displaystyle \frac{V_{\mathrm{A}}}{nD}}$$

(13)

According to dimensional analysis, the thrust and torque of the propeller can be calculated using the following formulas:

$$\begin{array}{c}\hfill T=K_{\mathrm{T}}\rho n^2{D}^4\end{array}$$

(14)

$$\begin{array}{c}\hfill Q=K_{\mathrm{Q}}\rho n^2{D}^5\end{array}$$

(15)

where

$K_{\mathrm{T}}$ is the thrust coefficient;

$K_{\mathrm{Q}}$ is the torque coefficient;

$\rho$ is the water density;

D is the propeller diameter.

The propeller efficiency ${\eta }_0$ can also be expressed using the dimensionless coefficients $K_{\mathrm{T}}$ and $K_{\mathrm{Q}}$ and the advance coefficient J as follows:

$${\eta }_0={\displaystyle \frac{T{V}_{\mathrm{A}}}{2\pi nQ}}={\displaystyle \frac{K_{\mathrm{T}}\rho n^2{D}^4{V}_{\mathrm{A}}}{2\pi n{K}_{\mathrm{Q}}\rho n^2{D}^5}}={\displaystyle \frac{K_{\mathrm{T}}}{K_{\mathrm{Q}}}}·{\displaystyle \frac{J}{2\pi }}$$

(16)

For a given propeller, the thrust coefficient $K_{\mathrm{T}}$, torque coefficient $K_{\mathrm{Q}}$, and efficiency ${\eta }_0$ depend solely on the advance coefficient J. The curves of $K_{\mathrm{T}}$, $K_{\mathrm{Q}}$, and ${\eta }_0$ as functions of J represent the open-water characteristic curves of the propeller. These characteristic curves can be determined experimentally, as illustrated in Figure 3. Because the value of $K_{\mathrm{Q}}$ is relatively small, it is often scaled up by a factor of 10 and plotted on the same coordinate system as $K_{\mathrm{T}}$.

For a given marine propeller, the ship’s sailing conditions determine the propeller advance coefficient J. That is, when the ship is cruising steadily under a specific condition, the propeller operates at a fixed J value, which, in turn, corresponds to specific values of the thrust coefficient $K_{\mathrm{T}}$ and torque coefficient $K_{\mathrm{Q}}$.

Based on the duct thrust coefficient $K_{\mathrm{TN}}$, propeller thrust coefficient $K_{\mathrm{TP}}$, and torque coefficient $K_{\mathrm{Q}}$ obtained from open-water experiments, the relationship between these variables and the pitch ratio $P/D$ as well as the advance coefficient J is described using a bivariate polynomial empirical regression surface. The form of the regression polynomial is as follows:

$$\begin{array}{c}\hfill K_{\mathrm{TP}}=\sum \sum A_{ij}{\left({\displaystyle \frac{P}{D}}\right)}^i{\left(J\right)}^j\end{array}$$

(17)

$$\begin{array}{c}\hfill K_{TN}=\sum \sum C_{ij}{\left({\displaystyle \frac{P}{D}}\right)}^i{\left(J\right)}^j\end{array}$$

(18)

$$\begin{array}{c}\hfill K_{\mathrm{Q}}=\sum \sum B_{ij}{\left({\displaystyle \frac{P}{D}}\right)}^i{\left(J\right)}^j\end{array}$$

(19)

3.2. Hull–Engine–Propeller Matching Based on Open-Water Characteristics

The preliminary matching design of the hull–engine–propeller system involves selecting the most suitable propeller based on the requirements outlined in the ship’s design specification. Subsequently, the main engine power $P_{\mathrm{S}}$ is determined based on the propeller’s rotational speed n and efficiency ${\eta }_0$.

Most traditional methods for hull–engine–propeller matching utilize the $B_P$-$\delta$ graph. While the $B_P$-$\delta$ method is simple, easy to understand, and convenient for calculations, it is challenging to organize the open-water characteristic curves into the $B_P$-$\delta$ graph. The conversion process is cumbersome, difficult to implement in computer models, and prone to errors during the conversion. Therefore, after examining the principles of the $B_P$-$\delta$ method and the ship energy balance relationship, this paper proposes a new calculation approach. This method eliminates the need to convert the open-water characteristic curves into the $B_P$-$\delta$ graph and instead directly utilizes the open-water characteristic curves for preliminary matching.

In the open-water characteristic curve, the horizontal axis represents the advance coefficient, while the vertical axis includes the thrust coefficient, ten times the torque coefficient, and efficiency. Research has shown that, for a given advance coefficient J, the open-water efficiency varies with different pitch-to-diameter ratios $P/D$. There exists a maximum open-water efficiency for each J, and the corresponding $P/D$ at this point is considered the optimal pitch ratio for that advance coefficient. Thus, for each advance coefficient J, there is a corresponding optimal open-water efficiency and an optimal pitch ratio.

Based on these findings, it is possible to plot two key curves: one representing the relationship between the advance coefficient J and the maximum open-water efficiency, and the other showing the optimal pitch ratio as a function of J. When the advance coefficient J and $P/D$ ratio are known, the corresponding torque coefficient $K_{\mathrm{Q}}$ can be determined. This $K_{\mathrm{Q}}$ value can then be plotted as a curve in relation to J.

Building on these relationships, a J–N diagram is proposed in this study, where the horizontal axis denotes the advance coefficient J, and the vertical axis, denoted as N, represents three dimensionless parameters, the open-water efficiency ${\eta }_0$, torque coefficient $K_{\mathrm{Q}}$, and pitch ratio $P/D$, as illustrated in Figure 4. The diagram consists of three curves: (1) the relationship between the advance coefficient J and the maximum open-water efficiency; (2) the relationship between the advance coefficient J and the optimal pitch ratio corresponding to the maximum efficiency; and (3) the relationship between the advance coefficient J and the torque coefficient at the optimal pitch ratio.

When the propeller diameter, thrust deduction fraction, wake fraction, and effective power at the target speed are known, the power absorbed by the propeller and the corresponding propeller speed can be determined. To achieve this, a set of propeller speeds must be assumed for the calculation, using the J–N curve. The specific steps for the initial propeller matching are outlined in

Table 2. Confirmation of the calculation of the optimal speed.

No.	Name	Unit	Data
1	Propeller diameter D	m
2	${\eta }_{\mathrm{H}}={\displaystyle \frac{1-t}{1-\omega }}$
3	$V_{\mathrm{A}}=V(1-\omega )$	m/s
4	$P_{\mathrm{E}}$ (given)	kW
5	Assume a given set of rotational speeds n	r/min	$n_1$    $n_2$    $n_3$    $n_4$
6	Advance coefficient J		$J_1$   $J_2$   $J_3$   $J_4$
7	Determine the maximum efficiency as a function of the advance coefficient J,
	along with the corresponding pitch ratio and torque coefficient $K_{\mathrm{Q}}$.
8	Propeller absorbed power: $P_{\mathrm{D}}=2\pi n{K}_{\mathrm{Q}}\rho n^2{D}^5$	kW	$P_{D1}$    $P_{D2}$    $P_{D3}$    $P_{D4}$
9	Effective thrust power: $P_{\mathrm{TE}}=P_{\mathrm{D}}{\eta }_0{\eta }_{\mathrm{H}}$	kW	$P_{TE1}$    $P_{TE2}$    $P_{TE3}$    $P_{TE4}$

.

The calculation results presented in Table 2 are plotted in Figure 5. The rotational speed of the propeller n, is shown on the horizontal axis, while the vertical axis displays four key quantities: the absorbed power of the propeller $P_{\mathrm{D}}$, the effective thrust power of the propeller $P_{\mathrm{TE}}$, the pitch ratio of the propeller $P/D$ and the efficiency of the open-water ${\eta }_0$. To determine the required propeller specifications, a horizontal line representing the effective power $P_E$ at the target ship speed V is plotted to intersect the $P_{TE}$ curve. The point of intersection represents the required propeller operating point, from which the propeller rotational speed n, propeller absorbed power $P_{\mathrm{D}}$, the propeller pitch ratio $P/D$, and the efficiency ${\eta }_0$ can be directly obtained.

3.3. Development of a Software Flowchart for Hull–Engine–Propeller Matching

The open-water characteristic curve for the ducted propeller is derived by discretizing the advance coefficient J and the pitch ratio $P/D$ using Formulas (17)–(19), as shown in Figure 6. Figure 6 presents the thrust coefficient $K_{\mathrm{T}}$, ten times the torque coefficient 10 $K_{\mathrm{Q}}$, and the open-water efficiency ${\eta }_0$ under different pitch ratios $P/D$. The maximum efficiency $\eta$, optimal pitch ratio $P/D$, and corresponding optimal torque $K_{\mathrm{Q}}$ for different advance coefficients J are determined using the maximum and minimum value functions. Subsequently, linear interpolation is employed to connect adjacent data points with straight lines, allowing for the fitting of the relevant parameters and the generation of the J–N curve, as illustrated in Figure 4. Considering the discrete nature of the available data, the curve is exported in .txt format to facilitate direct import into the subsequent hull–engine–propeller matching analysis.

In accordance with the ship design specifications, the following parameters are input: the propeller diameter, the thrust deduction fraction, the wake fraction, the effective power, and other relevant variables. Subsequently, the maximum efficiency $\eta$, the optimal pitch ratio $P/D$, and the torque $K_{\mathrm{Q}}$ curves corresponding to different advance coefficients J are imported. Multiple rotational speeds are selected, and the corresponding advance coefficients are calculated. The values of each curve at the respective advance coefficients are then extracted and processed to fit the curves, as shown in Figure 5. The horizontal axis represents the rotational speed, while the vertical axis corresponds to the power absorbed by the propeller, pitch ratio, and other related parameters. By performing a secondary curve fitting, the rotational speed $n_0$ at which the effective power curve of the propeller intersects with that of the ship is determined. The power absorbed by the propeller corresponding to $n_0$ represents the matching power, and the pitch ratio at $n_0$ corresponds to the required pitch ratio. Finally, the matching absorbed power, rotational speed, and pitch ratio are output, as illustrated in the propeller matching flowchart in Figure 7.

3.4. Equivalent Substitution

When other propeller parameters are held constant, the primary differences between shaftless and shafted propellers lie in the blade thickness distribution, the blade chord length distribution, and the presence of a drive shaft—factors that exert only a limited influence on the overall hydrodynamic performance. In light of the current absence of systematic open-water characteristic curves for shaftless propellers, this study proposes a matching design method based on parameter equivalence. This approach utilizes the geometric similarities between shaftless and ducted propellers, as illustrated in Figure 8. The method begins by applying the J–N curve to match the key parameters of a ducted propeller, after which critical geometric attributes—such as diameter, pitch ratio, and disk area ratio—are inherited and refined to construct an initial shaftless propeller model. The pitch ratio is then iteratively optimized through CFD simulations to achieve thrust–resistance (T–R) matching. This process ultimately determines the optimal geometric configuration and operating parameters for the shaftless propeller.

This study achieves equivalent propeller replacement by adjusting the design of the shaftless propeller. The overall process is illustrated in the propeller equivalence flow chart shown in Figure 9. The optimization of the pitch ratio adopts a closed-loop adjustment strategy: if the simulated thrust is insufficient (T < R), the pitch ratio is gradually increased and re-evaluated until the thrust exceeds the resistance; conversely, if the thrust is excessive (T > R), the pitch ratio is progressively reduced until the thrust falls below the resistance. This iterative process defines the feasible range of the pitch ratio.

Within this range, a bisection method is applied to accurately search for the optimal pitch ratio, ensuring that the relative error between thrust and resistance is within 1%. This approach balances computational efficiency and accuracy, ensuring that the propulsion performance of the shaftless propeller is precisely matched to the ship’s operational requirements.

3.5. Case Study

To verify the effectiveness of the proposed matching method, a shaftless propeller based on the NACA 66-mod airfoil with a = 0.8 was selected as the research subject. Using the hull parameters of a specific type of fishing vessel (listed in

Table 3. Input parameters for matching.

No.	Name	Unit	Data
1	D	m	1.4
2	${\eta }_{\mathrm{H}}={\displaystyle \frac{1-t}{1-\omega }}$		1.1146
3	$V_{\mathrm{A}}=V(1-\omega )$	m/s	3.8111
4	$P_{\mathrm{E}}$	kW	55.2
5	n	r/min	200 210 220 230

), a matching analysis was conducted.

Due to the absence of open-water characteristic curves for shaftless propellers, the J–N curve of the No. 19 + Ka4-70 ducted propeller—which exhibits characteristics similar to the RDT design—was selected as a reference. This reference curve was fitted and analyzed using professional software tools, and the key performance parameters of the No. 19 + Ka4-70 propeller were extracted, as shown in

Table 4. Matching results of the No. 19 + Ka4-70.

Name	Value
Rotational speed	208.6 rpm
Pitch ratio	1.32
Open-water efficiency	0.605
Power absorbed by the propeller	81.8 kW

. These results provided essential data support for the subsequent equivalent matching design.

The pitch ratio calculation process is summarized in

Table 5. Calculation of the equivalent pitch.

Iterations	$\mathit{P}/\mathit{D}$	T (kN)	R (kN)	|T−R|/R	${\mathit{P}}_{\mathbf{D}}$ (kW)
1	1.3234	10.13	10.7	0.053	74.443
2	1.4557	13.68		0.278	99.134
3	1.3895	11.897		0.110	86.2
4	1.3564	11.021		0.03	80.381
5	1.3399	10.576		0.0115	77.411
6	1.3481	10.782		0.007	78.712

. During this process, the pitch ratio is adjusted while all other parameters of the shaftless propeller remain unchanged. As shown in Figure 10, the propeller thrust gradually converges toward the ship resistance at the corresponding speed. By the sixth iteration, the pitch ratio reaches 1.3481, generating a thrust of 10.782 kN at the same operating speed. At this point, the absolute difference between the thrust and the resistance, normalized by the resistance, falls below 0.01, indicating that the matching process has been successfully completed.

Neglecting friction losses, the power absorbed by the propeller is assumed to be equal to the output power of the RDT. Assuming a motor efficiency of 0.9, the rated power of the motor is calculated as follows:

$$P_{\mathrm{S}}=P_{\mathrm{out}}/{\eta }_{\mathrm{s}}=78.712/0.9=87.46\mathrm{kW}$$

(20)

Therefore, a shaftless RDT with a rotational speed of 208.6 rpm, a rated power of 87.46 kW, and a pitch ratio of 1.3481 is selected to match the fishing vessel.

4. Conclusions

This study presents a propeller matching optimization method specifically developed for RDTs. To address the difficulty of integrating traditional propeller maps in hull–engine–propeller matching, the proposed approach utilizes open-water characteristic curves as the basis for system matching. Compared with conventional methods, this strategy offers two key advantages: (1) open-water data are more readily digitized, enabling rapid and efficient computation, and (2) the method is highly adaptable and applicable to both conventional shaft-driven and rim-driven propulsion systems through appropriate curve selection.

During the research process, due to the absence of complete open-water characteristic data for RDTs, this study initially utilized the open-water characteristics of geometrically similar ducted propellers to conduct preliminary matching calculations. Based on this, key geometric parameters of the ducted propeller—such as the diameter, pitch ratio, and disk ratio—were inherited and suitably adjusted to establish the initial model of the shaftless propeller. The pitch ratio was subsequently optimized, enabling the determination of essential parameters of the shaftless propulsion system, including the propeller geometry, rated motor speed, and power requirements.

This paper includes a case study involving the matching of a propeller to a fishing vessel. The study demonstrates the rapid matching of a ducted propeller, followed by an adjustment of the pitch ratio to achieve equivalence with a shaftless propeller based on the NACA 66 mod profile. This ensures that the thrust produced by the RDT at the vessel’s target speed closely aligns with the fishing vessel’s resistance, thereby validating the feasibility of the proposed method. Although the current variety of commercially available RDTs remains limited, this constraint is expected to diminish as RDT technology evolves and more systematic open-water data become available. The proposed method is thus anticipated to become increasingly practical and effective for the rapid matching and selection of appropriate RDTs across a broader range of ship types and operational scenarios.

This method holds significant promise for both shipowners and motor engineers. For shipowners, it offers a practical decision-making tool that facilitates the selection of efficient shaftless RDTs aligned with specific vessel performance requirements—particularly in terms of thrust demand and fuel efficiency. For motor engineers, the method provides critical boundary conditions—such as the required torque, rotational speed, and power—that directly inform the motor design phase.

Further research could focus on expanding the database of RDT open-water characteristics, either through experimental testing or high-fidelity CFD simulations. In addition, coupling this matching method with real-time onboard monitoring systems could enable adaptive control strategies that continuously optimize RDT performance under varying operational conditions.Moreover, future work might extend the method to incorporate additional constraints such as cavitation risk, noise emissions, or structural vibration, which are particularly important in specialized vessels like research ships or naval platforms.

In summary, this study presents a practical and forward-looking matching method for RDTs that bridges the gap between conventional propeller theory and emerging propulsion technologies, paving the way for more efficient and integrated RDT design practices.