Performance Assessment of B-Series Marine Propellers with Cupping and Face Camber Ratio Using Machine Learning Techniques

Abstract

This study investigates the performance of B-series marine propellers enhanced through geometric modifications, namely face camber ratio (FCR) and cupping percentage modifications, using a machine learning (ML)-driven optimization framework. A large dataset of over 7000 open-water propeller configurations is curated, incorporating variations in blade number, expanded area ratio (EAR), pitch-to-diameter ratio (P/D), FCR, and cupping percentage. A multi-layer artificial neural network (ANN) is trained to predict thrust, torque, and open-water efficiency (η_o) with a high coefficient of determination (R²), greater than 0.9999. The ANN is integrated into an optimization algorithm to identify optimal propeller designs for the KRISO Container Ship (KCS) using empirical constraints for cavitation and tip speed. Unlike prior studies that rely on boundary element method (BEM)-ML hybrids or multi-fidelity simulations, this study introduces a geometry-coupled analysis of FCR and cupping—parameters often treated independently—and applies empirical cavitation and acoustic (tip speed) limits to guide the design process. The results indicate that incorporating 1.0–1.5% cupping leads to a significant improvement in efficiency, up to 9.3% above the reference propeller, while maintaining cavitation safety margins and acoustic limits. Conversely, designs with non-zero FCR values (0.5–1.5%) show a modest efficiency penalty (up to 4.3%), although some configurations remain competitive when compensated by higher EAR, P/D, or blade count. The study confirms that the combination of cupping with optimized geometric parameters yields high-efficiency, cavitation-safe propellers. Furthermore, the ML-based framework demonstrates excellent potential for rapid, accurate, and scalable propeller design optimization that meets both performance and regulatory constraints.

1. Introduction

In response to the increasingly stringent regulations established by the International Maritime Organization (IMO) and the International Convention for the Prevention of Pollution from Ships (MARPOL), the maritime industry has prioritized the development of innovative solutions to reduce greenhouse gas emissions and enhance energy efficiency across vessel operations [1]. Improving a ship’s performance involves many aspects, from the shape of the hull [2,3,4,5] and the type of machinery used [6,7,8] to the materials on board [9], the routes it takes [10,11], and even the energy-saving devices installed [12]. While each of these contributes in its own way, the propeller stands out as one of the most important components. It directly affects how much thrust the ship produces, how much fuel it burns, and how much it pollutes. That is why designing and optimizing the propeller is so critical for improving efficiency and meeting today’s environmental standards [13].

Traditionally, propeller design has been guided by empirical methods such as the Bp-δ (Blade Loading Coefficient vs. Diameter Ratio) method and open-water diagrams [14]. These tools provide foundational relationships among key parameters such as pitch, diameter, and expanded blade area ratio (EAR), allowing designers to select geometries that meet specified performance targets. However, these diagram-based methods are inherently optimized within constrained conditions and typically fail to account for the full range of propeller geometries or hydrodynamic complexities involved in real-world operations. As a result, identifying a globally optimal propeller across various series and operating profiles becomes both time-consuming and suboptimal [15].

To overcome the limitations of traditional methods, more advanced analytical and numerical techniques, like the lifting line and lifting surface methods, have been developed. These approaches give a better understanding of how water flows around the propeller blades and how the blades generate thrust. By simulating the circulation and pressure distribution along the blade surface, they help designers make more accurate and effective design choices. However, their reliance on simplifying assumptions and the limited scope of validation data means they often struggle to predict performance under off-design or highly dynamic conditions accurately [16,17,18].

Recent advancements in computational fluid dynamics (CFD) have greatly enhanced propeller design accuracy by enabling high-fidelity, three-dimensional simulations of complex flow interactions, including cavitation, wake effects, and propeller–rudder–hull interactions. CFD methods offer high-fidelity insights into key hydrodynamic phenomena, including pressure distribution, flow separation, and blade loading, factors that are critical for the accurate assessment and optimization of propeller performance. These detailed simulations enable designers to address complex challenges such as noise generation, structural vibration, and cavitation-induced erosion. Despite their accuracy, CFD-based analyses are computationally intensive and time-consuming, posing significant limitations when employed within iterative optimization loops or large-scale parametric studies [19,20,21,22].

To mitigate these challenges, surrogate modeling techniques based on machine learning (ML) have emerged as powerful tools in propeller design and selection. These data-driven models are trained on extensive datasets, derived from CFD simulations, experimental testing, or historical performance records, and significantly reduce computational time while maintaining acceptable levels of accuracy [23]. ML models have been used to predict propeller performance, cavitation onset, and underwater radiated noise, often in combination with optimization techniques such as genetic algorithms (GAs) and particle swarm optimization (PSO) [2,24]. These approaches allow for the exploration of broader design spaces and the identification of non-intuitive yet highly efficient propeller configurations [25].

Several studies have demonstrated the success of ML-CFD hybrid models in various contexts. For instance, Doijode et al. [26] and Doijode et al. [27] integrated boundary element methods (BEMs) with ML to optimize B-series propellers for reduced cavitation and enhanced efficiency. Gaggero et al. [28] applied a multi-fidelity ML framework to predict controllable pitch propeller (CPP) performance, while Li et al. [29] validated CFD simulations of high-skew propellers and trained ML models to accelerate performance prediction and optimization. Deep learning approaches have also shown promise; Choi et al. [30] used a neural network trained on CFD and finite element method (FEM) data to predict and minimize cavitation in composite propellers.

In addition to these advanced modeling techniques, modifications to traditional propeller geometries have also been explored to enhance performance. One such modification is the application of propeller cupping, a deformation of the trailing edge that increases the effective pitch of the propeller, enabling greater thrust at lower engine loads and reducing cavitation risk. Light to heavy cupping, typically between 0.5% and 1.5% of the propeller diameter, has been validated through experimental analyses to improve cavitation resistance and reduce hydrodynamic noise [31]. Adjustments in rake, skew, camber, and other blade characteristics have also demonstrated performance gains in optimized designs [32,33,34]. For example, Tadros et al. [35] developed models combining OpenProp with nonlinear optimizers to minimize brake-specific fuel consumption (BSFC) at given operating points. Similarly, Bacciaglia et al. [36] applied PSO to refine 3D propeller geometries at various blade sections for more efficient electric motor operation.

While polynomial equations and standard propeller series, such as those employed in software like HydroComp v2021 [37], are common, machine learning models remain widely used during the initial and detailed design phase due to their simplicity and speed. Machine learning models are being increasingly developed to operate in parallel with these methods, offering enhanced flexibility and accuracy in handling complex, non-linear design spaces, particularly in projects where iterative optimization is required.

Nevertheless, a persistent challenge in machine learning-based approaches lies in the limited availability and diversity of high-quality training data, which can constrain the model’s generalizability and predictive accuracy across varied operational conditions and design configurations. Many existing models rely on experimental results specific to one vessel, environment, or operating condition. To fully unlock the potential of ML in propeller design, there is a need to consolidate and utilize large-scale, heterogeneous datasets, including historical open-water test results and real-world operational feedback. By harvesting and curating such datasets, researchers can develop more generalizable models that not only improve accuracy but also extend applicability across different vessel types and operational profiles [38].

Ultimately, the advancement of marine propeller design is expected to rely on the intelligent integration of high-fidelity CFD simulations, machine learning techniques, and advanced optimization algorithms, underpinned by extensive empirical datasets. This multidisciplinary approach enables rapid, accurate, and environmentally sustainable decision-making, aligning with the growing demands for energy efficiency and emission reduction in the maritime industry. As ship designers continue to explore configurations that reduce cavitation, increase thrust, and lower emissions, advanced data-driven frameworks will be essential in driving sustainable innovation in the maritime sector [39,40].

Modern efforts to enhance marine propeller performance and mitigate adverse phenomena, such as cavitation, noise, and energy loss, have led to the incorporation of advanced geometrical modifications, particularly the modification of the face camber ratio (FCR) and the addition of a cup. These techniques have shown significant promise in optimizing propeller hydrodynamics, especially in high-load or high-speed operating conditions.

The FCR is a geometric parameter that quantifies the curvature applied to the pressure side, or face, of a propeller blade, typically expressed as a percentage of the local chord length, as shown in Figure 1. While traditional propeller design principles focus on cambering the suction side to enhance lift and thrust generation, recent developments in hydrodynamic optimization suggest that introducing a modest, well-distributed camber on the pressure side can yield improved performance under specific operating conditions. A higher FCR can be used to delay the onset of cavitation by redistributing pressure loads more evenly across the blade surface, thereby reducing peak suction pressures that typically initiate cavitation. This is particularly beneficial in highly loaded propellers operating at low advance ratios, such as those used in slow-speed or heavily loaded vessels. Additionally, the FCR can help improve blade strength and resistance to flow separation near the trailing edge, offering better control over the blade’s pressure distribution. However, excessive face camber may lead to increased drag and reduced efficiency at off-design conditions, indicating the importance of careful optimization using CFD, panel methods, or machine learning-assisted parametric studies. Experimental and numerical studies have demonstrated that propellers with optimized FCRs can outperform conventional counterparts in terms of thrust coefficient and cavitation inception margin, especially in demanding flow environments [34,41].

Cupping is another blade geometry modification strategy that involves adding a slight concave curvature or “cup” near the trailing edge of the propeller blade, typically on the pressure side, as shown in Figure 2. This geometric modification alters the local flow path, effectively increases the blade’s pitch, and consequently influences the chordwise loading distribution. Originally developed for high-performance outboard marine propellers, cupping has since demonstrated its utility across a broad range of marine propulsion applications. It is particularly advantageous for vessels operating under dynamic loading conditions or subject to frequent variations in engine speed and thrust demand, where improved flow attachment and enhanced thrust stability are essential. The primary hydrodynamic benefit of cupping is its ability to enhance the blade’s pressure recovery, which helps suppress trailing-edge cavitation and delay the detachment of flow under high loads.

Cupped blades also exhibit superior cavitation resistance, which extends their operational life and reduces noise and vibration. However, similar to FCR, the design and placement of the cup must be precisely tailored: excessive cupping can lead to overloading, increased torque demand, and higher propeller-induced pressure pulses. The degree and profile of cupping are therefore often optimized using iterative simulations and sea trials. When incorporated judiciously, cupping can significantly enhance thrust efficiency and robustness of the propeller, making it a valuable tool in performance-focused and noise-sensitive vessel applications [42,43,44].

While several machine learning-based frameworks, such as ML-BEM hybrids and multi-fidelity optimization models, have shown promise in accelerating propeller design, many remain constrained by limited datasets, narrow design scopes, or a lack of integration with real-world performance constraints. In contrast, the present study introduces three key advancements. First, it leverages a large-scale dataset comprising over 7000 systematically varied B-series propeller geometries, incorporating modifications in both cupping and FCR, thus enabling detailed coupled geometric analyses that are rarely addressed in the existing literature. Second, it develops an ANN-based surrogate model capable of accurately and rapidly predicting key performance metrics, including open-water efficiency, thrust, and torque, across a broad design space. Third, it embeds empirical cavitation thresholds and tip speed acoustic limits into a multi-variable constrained optimization framework, ensuring that the resulting designs are not only high-performing but also compliant with practical and regulatory constraints. While exploratory data analysis (EDA) can reveal intuitive trends, such as the efficiency gains from cupping or the modest efficiency trade-offs from increased FCR, EDA lacks the scalability and dynamic prioritization required for complex, multi-objective design problems. The integrated ANN and optimization framework presented here offers a robust, repeatable, and computationally efficient alternative. It not only quantifies nonlinear interactions among design variables but also identifies globally optimal configurations under realistic constraints, making it an effective decision-support tool for both the early-stage design and retrofitting of marine propellers.

A comparison of B-series propellers with FCR and cupping techniques is essential for new propeller designs that aim to achieve higher efficiency, lower cavitation risk, and reduced noise. By incorporating cavitation and acoustic constraints into the optimization process and applying the method to a benchmark KRISO Container Ship (KCS), the study presents a practical and computationally efficient approach for designing high-performance, low-emission propellers that meet the demands of modern shipping. This study aims to assist ship designers and decision-makers in considering various techniques and selecting optimal solutions for their vessels, thereby improving energy efficiency and achieving net-zero goals.

The remainder of this paper is organized as follows: The dataset of the propeller series is presented in Section 2. The ANN model for predicting the performance of different propellers is presented in Section 3. The optimization model for propeller selection is described in Section 4. A case study is considered and the computed results based on the developed method are detailed in Section 5. Finally, a summary of the main findings and future recommendations are presented in Section 6.

2. Dataset of Propeller Series

In this study, a comprehensive dataset of standardized open propeller designs has been compiled to support the development and training of a machine learning model aimed at propeller performance prediction and optimization. All propellers in the dataset have open (non-ducted) configurations, selected to ensure consistency in hydrodynamic assumptions and boundary conditions during model training. The dataset is predominantly based on the well-established Wageningen B-series propellers, widely utilized in marine engineering owing to their systematic geometric variations and comprehensive experimental validation, which make them ideal for both benchmarking and performance analysis.

In addition to the standard B-series geometries, the dataset includes B-series propellers with systematic modifications, specifically incorporating different levels of cupping and varying percentages of FCR. These modifications enable the representation of a broader spectrum of hydrodynamic behaviors associated with contemporary propeller design approaches, which aim to improve resistance to cavitation, enhance propulsive efficiency, and optimize performance under off-design operating conditions. The proportion of cupping and the FCR are varied parametrically, thereby enabling the model to systematically assess the influence of these parameters on thrust generation, torque requirements, and overall efficiency.

A distinguishing attribute of the dataset is the consistent standardization in reporting the pitch-to-diameter ratio (P/D), which is conventionally measured at seventy percent of the propeller radius.

This convention simplifies the geometric characterization of propellers but does not fully account for radial variations in P/D across the blade span. While this simplification facilitates benchmarking and generalization across different propeller series, it may limit the detailed representation of flow. To mitigate this limitation, additional geometric descriptors and performance coefficients are integrated into the machine learning framework.

The open-water performance data—including the advance coefficient (J), thrust coefficient (K_T), and torque coefficient (K_Q)—are systematically compiled for all propellers. These hydrodynamic coefficients were sourced from authoritative references, including Carlton [14] and Hydrocomp [37], with some modifications and enhancements provided by the HydroComp team to broaden the operational coverage of the B-series dataset.

The dataset comprises over 7000 open propeller designs, encompassing a comprehensive and diverse range of performance characteristics influenced by variations in cupping and FCR. This extensive database serves as the foundation for the machine learning model developed in this study, enabling sophisticated analysis and accurate prediction of propeller behavior across a wide range of design conditions. More details about the main propellers in the dataset are presented in

Table 1. Main parameters of the propeller series.

Parameter[-]	Reference[-]	Number of Blades[-]	Blade Area Ratio[-]	Pitch/Diameter Ratio[-]	Cupping Percentage[%]	FCR Percentage[%]
Wageningen B Series	[37,45,46,47]	3–7	0.35–0.80 for 3 blades 0.40–1.00 for 4 blades 0.45–1.05 for 5 blades 0.50–0.95 for 6 blades 0.55–0.85 for 7 blades	0.5–1.4	0–1.5	0–1.5

.

3. ANN Model for Performance Prediction

ANNs are inspired by how the human brain processes information. Originally proposed by McCulloch and Pitts [48] in 1943, these models are made up of layers of interconnected nodes, or “neurons,” that work together to find patterns and relationships in data. Each neuron takes in inputs, applies weights to them, adds a bias, and then passes the result through an activation function to determine the output. What makes ANNs particularly useful is their ability to learn complex, non-linear relationships through training, which involves adjusting the weights based on errors in the predictions using techniques like backpropagation and gradient descent. Because of this, ANNs are a powerful tool for problems where the data is large, varied, and too complex for traditional algorithms to handle effectively [49,50,51,52].

In this study, an ANN model is used to predict the performance of marine propellers based on a variety of design inputs. The structure of the network is shown in Figure 3. It is a multi-input, multi-output model, meaning it can handle multiple input variables and predict multiple outputs simultaneously. The main inputs include the thrust loading coefficient (K_T/J²), EAR, and P/D. From these, the ANN predicts key propeller performance metrics, such as open-water efficiency (η_o), J, K_T, and K_Q.

The ANN developed in this study employs a feedforward architecture consisting of 20 hidden layers, each containing 64 neurons. The output layer uses a linear activation function to support the regression nature of the problem. The network is trained using the Levenberg–Marquardt backpropagation algorithm, a quasi-Newton optimization method particularly suited for small to medium-sized datasets with rapid convergence properties.

The model architecture and training hyperparameters, such as the number of layers, neurons per layer, and learning configuration, are determined through a trial-and-error tuning process. This empirical approach involved varying network depth and width and selecting configurations that minimized the validation RMSE while maintaining generalization across unseen test data. The final selected architecture showed strong robustness and low prediction error across all subsets, as detailed in

Table 2. Evaluation results of ANN models by error metrics.

Propeller Series		B Series	B Series	B Series
Propeller blades		3	4	5
Cupping percentage		0.0	1.0	0.0
FCR		0.0	0.0	1.5
R2	Training	0.99995	0.99999	0.99999
	Validation	0.99995	0.99999	0.99999
	Test	0.99996	0.99999	0.99999
RMSE	Training	0.0026	0.0020	0.0027
	Validation	0.0025	0.0018	0.0025
	Test	0.0025	0.0020	0.0025

and Figure 4.

To compute the output in each neuron, the model calculates a weighted sum of the inputs, adds a bias term, and applies an activation function to produce the final value. This process is repeated across all layers of the network. To assess how well the model is performing, standard evaluation metrics are considered: the coefficient of determination (R²), which measures the proportion of variation in the output that the model can explain, and the root mean square error (RMSE), which quantifies the average prediction error. Given the large variety of propeller types in the entire dataset, a representative selection of propeller series—covering different blade numbers and design characteristics—is used to showcase the model’s accuracy.

The training data used for the ANN comes from open-water performance curves across a range of open propeller types, including the traditional B series, as well as B-series propellers with varying levels of blade cupping and FCR. The model is trained on a laptop equipped with a 13th Gen Intel(R) Core(TM) i7-1355U processor at 1.70 GHz and 16 GB of RAM (Intel, Santa Clara, CA, USA). Despite the size and diversity of the dataset, the ANN was trained in just 5 min. Once trained, it can predict the performance of new propeller designs in a few seconds, making it ideal for rapid evaluation in optimization tasks.

To manage the data efficiently, the open-water curve of each propeller is organized into Excel files with a standardized format. Each file name clearly indicates the propeller series, number of blades, EAR, and P/D, for example, “KTKQ_BSeries_3Z_0.35EAR_0.50PoD_1.50Cup_0.00FCR.csv”. Inside the files, the data is formatted consistently, with labeled columns (J, K_T, and K_Q) and clean numerical entries. This level of organization ensures smooth integration with MATLAB, where the ANN and optimization routines are run.

Overall, this ANN model provides a fast and reliable method for predicting propeller performance across a wide range of designs. Its speed and accuracy make it a valuable tool for exploring design options, running large-scale optimizations, or even supporting real-time decision-making in the early stages of propulsion system design.

After importing the curated dataset, the ANN model is developed within the MATLAB environment. To determine the optimal network architecture, a trial-and-error approach is employed, an established method for identifying suitable configurations in ANN development [53]. Through several iterations, it is found that a network with twenty hidden layers yields the best performance in terms of accuracy and generalization across the dataset.

To facilitate effective training and evaluation, the dataset is randomly partitioned into three subsets: 70% is allocated for training the neural network, 15% is designated for testing the model’s generalization on unseen data, and the remaining 15% is reserved for validation to mitigate overfitting. The training process is optimized using the Levenberg–Marquardt backpropagation algorithm, known for its efficiency in handling small to medium-sized datasets. Training is allowed to proceed for a maximum of 1000 epochs, with an early stopping condition if the validation performance fails to improve for six consecutive iterations.

The results show strong predictive capability, with consistently high R² values and low RMSE, indicating minimal deviation between the ANN outputs and the known performance data. Table 2 summarizes the error metrics for the selected series, emphasizing the model’s reliability across various configurations. Furthermore, Figure 4 presents regression plots corresponding to the training, testing, and validation phases, illustrating the correlation between predicted and observed values for key open-water performance parameters. The close agreement between predicted and actual values, denoted as “Data” in the figure, underscores the robustness of the artificial neural network in accurately modeling propeller hydrodynamic performance, thereby supporting its applicability in future propeller selection and design optimization endeavors. Additionally, to visualize prediction uncertainty and residual distribution, error histograms for key output parameters are presented in Figure 5. These distributions exhibit narrow error margins centered around zero, further supporting the model’s ability to generalize across unseen configurations.

4. Optimization Model for Propeller Selection

To identify the most efficient propeller configuration for a given operational condition, an optimization approach is developed using surrogate models generated by the ANN. The goal is to maximize the η_o by selecting the best combination of EAR and P/D for a fixed KT/J². The optimization routine is executed in MATLAB using the fmincon function, which employs the interior-point algorithm [54]. This algorithm efficiently handles nonlinear optimization problems subject to constraints, as shown in Equation (1).

$$\underset{\mathrm{x}}{\min } \mathrm{f}(\mathrm{x}) \mathrm{such}\mathrm{that} \left\{\begin{array}{c}\mathrm{c}(\mathrm{x})\le 0\\ {\mathrm{c}}_{\mathrm{e}\mathrm{q}}(\mathrm{x})=0\\ \mathrm{A}\cdot \mathrm{x}\le \mathrm{b}\\ {\mathrm{A}}_{\mathrm{e}\mathrm{q}}\cdot \mathrm{x}={\mathrm{b}}_{\mathrm{e}\mathrm{q}}\\ \mathrm{l}\mathrm{b}\le \mathrm{x}\le \mathrm{u}\mathrm{b}\end{array}\right.$$

(1)

where f(x) is the optimization model objective, c is the inequality constraints, c_eq is the equality constraints, A as a matrix and b as a vector are the linear inequality constraints, Aeq as a matrix and beq as a vector are the linear equality constraints, and lb and ub are the lower and upper bounds, respectively.

In this model, EAR and P/D serve as design variables, while the objective is to maximize propeller efficiency. At the same time, operational constraints must be satisfied, particularly those related to cavitation and noise emissions. The cavitation limits are based on the empirical criterion introduced by Keller [55], as in equation (2), where the minimum EAR to avoid cavitation (EAR_min) is computed. The noise threshold is governed by tip speed limitations, V_tip, as in equation (3), that vary depending on the number of propeller blades and are detailed in [56].

$$EA{R}_{\min }={\displaystyle \frac{(1.3+0.3Z)T}{(P_{atm}+\gamma h-P_v)D^2}}+K$$

(2)

where D is the propeller diameter, Z is the blade number, T is the propeller thrust, P_atm is the atmospheric pressure, P_v is the vapor pressure, γ is the specific weight, h is the propeller centerline immersion, and k is a constant equal to 0 or 2.

$$V_{tip}={\displaystyle \frac{\pi DN}{60}}$$

(3)

where N is the propeller speed.

To streamline the optimization process, the objective function and all constraints are combined into a single fitness function, as shown in Equation (4).

$$Fitness Function=-{\eta }_o+R{\displaystyle \sum _{i=1}^{j}max(g_i(x),0)}$$

(4)

where η_o is the main objective of the study; the second part of the equation defines the nonlinear inequality constraints of the problem, where g(x) is the static penalty function, x is the number of variables, j is the number of constraints, and R is a penalty function.

This technique, initially designed for genetic algorithms [57], is versatile and also compatible with deterministic solvers, such as fmincon [58]. It allows for the simultaneous evaluation of the main objective (efficiency) and constraint violations (cavitation and noise), enabling faster convergence and reduced simulation time.

All constraints are implemented using a static penalty function approach, where violations are normalized and expressed as dimensionless values (absolute value less than 1). This allows constraint violations to be smoothly integrated into the fitness function, penalizing infeasible solutions without completely discarding them, thereby improving the optimizer’s ability to search the design space efficiently.

Overall, this integrated optimization framework enables a rapid and reliable selection of propeller geometries that satisfy performance and operational requirements, ensuring efficient, low-noise, and cavitation-safe designs suitable for various vessel types and mission profiles.

5. Case Study and Computed Results

5.1. Ship Characteristics

The KCS is the ship considered to perform the simulation in this study. It is a widely used benchmark model in naval architecture and marine hydrodynamics research, as shown in Figure 6. It was developed by the Korea Research Institute of Ships and Ocean Engineering (KRISO) as a reference container ship for CFD validation and experimental studies [59]. The characteristics of the ship are presented in

Table 3. Main characteristics of KCS.

Specification	Symbols	Units	KCS Ship
Length water line	LWL	[m]	232.4
Breadth	B	[m]	32.2
Draft	T	[m]	10.8
Wetted surface area	S	[m2]	9645
Displacement	Δ	[tonne]	52,030
Ship speed	Vs	[knot]	24

.

5.2. Design Methodology

The machine learning-based propeller design methodology presented in this study is developed to assist naval architects and marine engineers in identifying the most efficient propeller configuration for a specific vessel. By integrating empirical data from a range of standardized propeller series with artificial intelligence techniques and performance optimization routines, the methodology enhances decision-making during the early design phase, where time is limited but critical design choices must be made.

The process begins with the estimation of the ship’s total resistance (R_T) at its design speed (V_s), a foundational parameter for any propulsion analysis computed using the method of Holtrop [60]. Once R_T is established, the next step involves calculating the key propulsive coefficients: the thrust deduction factor (t) and the wake fraction (w). These can be estimated using well-established empirical approaches based on the work of Holtrop and Mennen [61], which shows a good agreement with higher-fidelity CFD simulations and experimental measurements from towing tank tests [62,63].

After the resistance and propulsive coefficients are determined, the propeller’s geometric constraints are defined. This involves identifying the available space at the stern of the vessel to select an appropriate propeller diameter.

With these parameters defined, the next step is to compute the required propeller thrust and the corresponding advance velocity (V_A). From here, the K^T/J² is calculated, which characterizes the loading condition of the propeller and serves as a critical input for the optimization process.

The core of the methodology lies in the optimization of the EAR and the P/D for each candidate propeller series and blade number. Using the trained ANN, these parameters are tuned to maximize η_o, while simultaneously adhering to practical design constraints.

Once the optimization is complete, the ANN model predicts the open-water performance coefficients, including K_T, K_Q, and η_o. All candidate configurations are then ranked according to their efficiency under the defined design conditions.

The outcome is a rapid, adaptable, and data-driven tool that enables engineers to approach complex design decisions with enhanced confidence and precision. A flowchart depicting the complete methodology is presented in Figure 7, providing a visual reference for implementing the procedure within practical design workflows.

5.3. Computed Results

Following the detailed methodology outlined earlier, the estimation of ship resistance and propulsive coefficients is carried out using the well-established Holtrop empirical formulas. For this study, NavCad v2021 [37] is employed as a reliable tool to perform these calculations. The process begins by importing the 3D geometry of the ship’s hull into the software environment, where the vessel’s draft and design speed are specified. NavCad then computes the total resistance, wake fraction, and thrust deduction factor under the defined operating conditions. The resulting hydrodynamic coefficients are subsequently exported and used as input in a MATLAB v2025a-based optimization routine, which integrates the machine learning model for propeller selection.

To ensure the validity of the developed approach, a comparison is made against a benchmark propeller, the KP505, which is part of the well-known B series of standard propellers widely used in marine propulsion. According to SIMMAN [59], the KP505 features modified NACA66 profile sections, optimized for favorable hydrodynamic performance. The decision to maintain a five-bladed configuration for the KCS in all baseline studies stems from its widespread adoption as a balanced design in commercial ship propulsion. Five-blade propellers offer a compromise between efficiency, vibration control, and structural robustness, especially under off-design conditions [64,65,66]. While fewer blades (e.g., three or four) may yield slightly higher peak efficiency under ideal conditions, five-blade configurations are generally more robust across varying speeds and loading conditions, making them a preferred choice for vessels with dynamic operational profiles; the same concept has been presented in [67]. Owing to its well-established characteristics and extensive application, this propeller serves as a reliable benchmark for assessing the accuracy and robustness of machine learning predictions.

In the optimization phase, boundary conditions for the EAR and P/D of the KP505 are set according to the values reported in [68,69]. Specifically, the EAR is constrained to a minimum of 0.8, and the P/D is constrained to 0.95. After that, the machine learning model is applied across multiple standard propeller series with varying blade numbers, cupping percentages, and FCRs. For each case, several optimal propeller geometries are generated and ranked based on open-water efficiency, with only those exceeding 60% efficiency and complying with the limits of constraints being included. The results of the computation are presented in

Table 4. Suggested B-series propeller solutions for KCS.

	Propeller Geometry			Propeller Performance							Cavitation and Noise Limits
Rank [-]	Z [-]	EAR [-]	P/D [-]	ηo [%]	J [-]	KT [-]	KQ [-]	Cup [%]	FCR [%]	Vtip [m/s]	EARmin [-]	Vtip-max [m/s]
1	6	0.89	1.40	71.7	1.017	0.443	0.101	1.5	0.0	28.5	0.89	46
2	6	0.89	1.06	68.1	0.824	0.292	0.057	1.0	0.0	35.1	0.89	46
3	5	0.89	0.99	67.9	0.778	0.259	0.048	1.0	0.0	37.2	0.83	46
4	3	0.80	1.40	67.1	0.953	0.392	0.090	1.5	0.0	30.4	0.69	53
5	6	0.89	1.06	67.0	0.777	0.254	0.047	0.5	0.0	37.3	0.89	46
6	5	0.83	0.96	66.9	0.715	0.217	0.037	0.5	0.0	40.5	0.83	46
7	5	0.83	0.97	66.6	0.818	0.285	0.056	1.5	0.0	35.4	0.83	46
8	4	0.76	0.98	66.1	0.759	0.249	0.045	1.0	0.0	38.2	0.76	53
9	4	0.76	0.93	66.1	0.685	0.200	0.033	0.5	0.0	42.2	0.76	53
10	4	0.76	1.01	65.4	0.818	0.287	0.057	1.5	0.0	35.4	0.76	53
11	3	0.69	0.91	65.1	0.714	0.214	0.038	1.0	0.0	40.5	0.69	53
Ref.	5	0.8	0.95	65.0	0.679	0.195	0.032	0.0	0.0	42.66	0.813	46
12	5	0.83	1.03	64.9	0.732	0.228	0.042	0.0	1.5	39.5	0.83	46
13	5	0.83	1.04	64.7	0.726	0.224	0.041	0.0	1.0	39.9	0.83	46
14	6	0.89	1.10	64.4	0.755	0.242	0.046	0.0	0.5	38.4	0.89	46
15	4	0.76	1.00	64.4	0.702	0.207	0.037	0.0	1.5	41.2	0.76	53
16	3	0.69	0.85	64.4	0.640	0.169	0.026	0.5	0.0	45.3	0.69	53
17	5	0.83	1.03	64.4	0.712	0.214	0.038	0.0	0.5	40.6	0.83	46
18	6	0.89	1.11	64.3	0.786	0.260	0.051	0.0	1.5	36.8	0.89	46
19	6	0.89	1.08	64.2	0.758	0.243	0.046	0.0	1.0	38.2	0.89	46
20	4	0.76	1.01	64.2	0.697	0.203	0.036	0.0	1.0	41.6	0.76	53
21	4	0.76	0.99	63.7	0.682	0.195	0.034	0.0	0.5	42.5	0.76	53
22	3	0.69	0.94	62.9	0.656	0.179	0.030	0.0	1.5	44.1	0.69	53
23	3	0.69	1.00	62.9	0.677	0.191	0.034	0.0	1.0	42.8	0.69	53
24	3	0.69	1.00	62.2	0.670	0.187	0.032	0.0	0.5	43.2	0.69	53

.

To ensure consistent benchmarking across all evaluated propeller configurations, the efficiency comparisons presented in this study are made under fixed operational conditions. Specifically, the ship speed is held constant at 24 knots, corresponding to the design speed of the KCS vessel. All propellers are designed to generate the same thrust, as determined by the ship resistance estimation using the Holtrop method, resulting in identical KT/J² values across all optimization runs. This setup guarantees that the variations in η_o observed between different propeller designs, particularly with varying cupping and FCR values, are solely attributable to geometric modifications and not to external operating condition changes. It is also important to note that this study focuses on open-water propeller performance. While cupping may influence local flow characteristics near the blade tips and trailing edges—potentially affecting hull–propeller interaction or total system resistance—such effects are not captured within the current surrogate model. Future work will incorporate high-fidelity CFD simulations and viscous-flow coupling to quantify the influence of cupping geometry on overall ship resistance and wake flow behavior under realistic operational conditions.

A detailed examination of the optimized propeller series reveals that cupping, particularly in the range of 1.0% to 1.5%, plays a critical role in enhancing open-water efficiency. The majority of the top 11 optimized designs feature a cupped blade surface, and these configurations outperform the reference propeller (η_o = 65.0%) by margins of up to 9.3%, with peak efficiencies reaching 71.7%.

Cupping modifies the blade’s trailing edge curvature, effectively increasing the effective camber and pressure-side loading, which can enhance lift without significantly increasing drag, especially under conditions prone to cavitation. This design feature not only contributes to higher thrust generation but also improves cavitation resistance by delaying the onset of suction-side cavitation at higher loading conditions.

The observed performance gains are not solely due to cupping but rather the result of a synergistic optimization with other geometric parameters. In all high-performing cupped propellers, the EAR varies between 0.69 and 0.89, with most designs closely matching or slightly exceeding the minimum EAR thresholds required to suppress cavitation. Notably, Cases 3 and 4 show EARs higher than the minimum needed, reflecting a design emphasis on robust cavitation margins.

Furthermore, the P/D ratios in these cupped configurations are consistently higher than the reference value of 0.95, ranging up to 1.40, indicating an increase in blade pitch that boosts thrust. This is especially effective in combination with cupping, as the additional lift generated by the cupped section complements the enhanced pitch angle without excessively increasing the risk of cavitation on the blade surface.

The number of blades in these designs ranges from three to six, with Z equal to 6 appearing most frequently among the top performers. A higher blade count improves load distribution and reduces individual blade loading, which, when combined with cupping, results in smoother operation and reduced susceptibility to vibrational stresses.

Importantly, all cupped propellers maintain V_tip well below their respective maximum allowable limits (ranging from 46 m/s to 53 m/s, depending on Z). The lowest tip speed, recorded in Case 1, is 28.5 m/s, illustrating that even the most efficient designs emphasize acoustic compliance and structural integrity. This reduced tip speed also contributes to effective cavitation control by diminishing the local pressure gradients that initiate cavitation near the blade tips.

In summary, cupping emerges as a key enabler of high-efficiency propeller design when integrated with optimized EAR, P/D ratio, and blade count. Although cupping by itself does not inherently ensure performance enhancement, it accentuates the advantages of well-balanced geometry by enhancing lift characteristics, mitigating cavitation, and maintaining low noise and vibration levels, thereby constituting a valuable component in contemporary propeller optimization strategies.

In contrast to the top-performing designs, a subset of propellers in the series incorporates non-zero FCR values, ranging from 0.5% to 1.5%. These cases show a slight to moderate reduction in efficiency compared to both the optimized zero-FCR designs and the reference propeller. Specifically, their efficiencies decrease by as little as 0.1% and reach a maximum reduction of 4.3% relative to the reference.

The observed efficiency reduction associated with increasing FCR is primarily evident when compared to cupped propeller designs, which demonstrate superior performance gains in this study. However, it is important to note that FCR-modified propellers still offer an efficiency improvement relative to the standard B-series baseline with the same blade number, EAR, and P/D but without any geometric enhancements (i.e., no FCR or cupping). This trend can be explained by underlying hydrodynamic mechanisms observed in prior CFD studies, such as the work of Tadros et al. [34]. Specifically, adding FCR modifies the pressure distribution on the pressure side of the blade, increasing local curvature and affecting boundary layer behavior. While moderate camber can help redistribute pressure loads and delay cavitation onset, higher FCR values may trigger earlier flow separation near the trailing edge, increasing pressure drag and torque demand. This leads to a net reduction in open-water efficiency compared to cupping-enhanced blades.

This performance degradation suggests that although the inclusion of FCR may be beneficial for certain hydrodynamic or structural purposes (e.g., improving cavitation behavior or load distribution), it does not inherently enhance propulsive efficiency.

However, the impact of FCR on efficiency is not strictly negative. Some of the designs with FCR, particularly Cases 12, 13, 14, 17, and 19, remain competitively efficient, with η_o values between 64.2% and 64.9%, only slightly below the reference value of 65.0%. These results suggest that the efficiency penalty associated with FCR can be mitigated by carefully optimizing other geometric parameters. For instance, these designs frequently incorporate higher blade counts (Z = 5–6), EAR equal to or exceeding 0.83, and P/D of 1.00 or greater. Such geometric modifications are likely to enhance hydrodynamic performance by promoting improved load distribution, attenuating pressure peaks, and increasing thrust generation.

Moreover, all FCR-enabled designs maintain tip velocities below their respective maximum limits (typically under 46–53 m/s), suggesting that noise and cavitation-related acoustic concerns remain manageable. This reflects a design compromise: while FCR does not directly boost efficiency, it may be utilized for its favorable impact on blade strength, delayed cavitation onset, and enhanced pressure recovery, particularly under off-design conditions.

In summary, while the inclusion of a non-zero face camber ratio tends to reduce maximum achievable efficiency, the negative effects can be largely offset through coordinated tuning of other design variables. These findings support the concept of a multi-parameter optimization strategy, wherein the fineness of the cavitation region is not an isolated performance enhancer but an integral design parameter contributing to a robust and balanced propeller performance.

While some of the observed efficiency variations are relatively modest, it is essential to interpret these differences within the context of the surrogate model’s predictive accuracy and inherent uncertainty. The ANN developed in this study exhibits consistently high predictive performance, with R² exceeding 0.9999 and RMSE on the order of 0.002–0.003 across the training, validation, and testing datasets. Despite this strong performance, variations in efficiency within the lower range may still fall within the model’s confidence interval and should therefore be evaluated with appropriate caution.

Importantly, the influence of FCR on propeller performance has been independently validated through high-fidelity CFD simulations, as presented in the work of Tadros et al. [34]. This prior validation supports the physical plausibility of the observed trends and reinforces the credibility of the ANN’s predictions, even in cases where the improvements are relatively small. Thus, while the absolute efficiency gains may appear marginal, the consistent trends observed across the dataset, especially in relation to cavitation behavior, thrust generation, and robustness under off-design conditions, are of practical significance for propeller selection and design.

Overall, the results demonstrate that increasing cupping, combined with higher pitch-to-diameter ratios and carefully controlled expanded area ratios, significantly improves propeller efficiency, contingent upon maintaining cavitation within acceptable limits. Additionally, blade number plays a critical role, with higher blade counts facilitating enhanced thrust generation and more effective cavitation management. Such design configurations provide a comprehensive improvement in propulsion performance, rendering them well-suited for practical marine applications where efficiency, noise reduction, and cavitation control are paramount.

6. Conclusions

This study presents a machine learning-based propeller design and optimization framework aimed at evaluating the effects of the FCR and the cupping technique on marine propeller performance. By integrating a high-fidelity dataset of B-series propellers, incorporating various geometric modifications, into an ANN model, the framework enables rapid and accurate predictions of critical performance metrics such as open-water efficiency, thrust, and torque. The ANN and optimization framework successfully demonstrated its ability to explore the design space quickly, identify optimal geometric configurations, and enhance propeller performance without the need for computationally expensive simulations, thereby streamlining the early-stage design process. The results from applying this framework to the KCS case study highlight several key findings:

• Cupping in the range of 1.0% to 1.5% significantly improves propeller performance, increasing η_o by up to 9.3% compared to the reference design, while reducing tip speed and maintaining cavitation safety.
• FCR introduces a minor efficiency penalty (0.1–4.3%) but can still yield acceptable performance when carefully balanced with other parameters, such as EAR, P/D, and blade number.
• High blade count (Z = 5–6), elevated EAR (up to 0.89), and increased P/D (up to 1.40) are common characteristics among the top-performing designs.
• The ANN model, trained on a diverse dataset, demonstrated high prediction accuracy (R² > 0.9999) and extremely fast execution time (<20 s per design), making it suitable for real-time optimization and early-stage design evaluations.

One of the limitations of the present study is that all propeller designs are based on the standardized B-series geometries, which assume a fixed blade section shape and hub ratio. These geometric characteristics are held constant to isolate the effects of FCR and cupping modifications, allowing for focused analysis and more straightforward interpretation of their individual contributions to performance. Recent work by Ge et al. [70] has shown that interactive optimization of geometric features, such as throat curvature and pressure amplitude, can lead to substantial reductions in cavitation intensity through synergistic design tuning. In real-world propeller design, the blade section profile and hub diameter can have a significant influence on hydrodynamic efficiency, cavitation behavior, and structural performance. Therefore, future studies should extend the proposed machine learning framework to include these additional design parameters, potentially through expanded datasets or high-fidelity CFD simulations, to enhance generalizability and accuracy in more complex scenarios.

This study demonstrates the efficacy of integrating data-driven models with domain-specific constraints to navigate complex propeller design spaces efficiently. The findings have direct implications for the selection and retrofitting of energy-efficient propellers in contemporary commercial vessels, thereby contributing to the International Maritime Organization’s decarbonization objectives. Future research will extend this methodology to encompass propeller designs across a wider array of vessel types and operational profiles.