MLP-Optimized Duct Design for Enhanced Hydrodynamic Performance in Tidal Turbines

Abstract

The duct, a crucial component of tidal energy power generation devices, is designed to enhance the environmental benefits of tidal energy by optimizing water flow paths and improving energy conversion efficiency. Traditional duct design methods are often considered overly complex, lacking precision, and exhibiting poor optimization efficiency and accuracy. In this study, computational fluid dynamics (CFD) and multi-layer perceptron (MLP) models are employed to investigate the impact of various duct designs on turbine power output and thrust. The MLP model is trained using numerical simulation results, which are then validated by comparing them with experimental data from the literature. Under optimized conditions—specifically, an attack angle of 20°, a blade tip distance of 8 mm, and a cubic curve X_m = 0.796—the power coefficient is found to increase by approximately 11.14% compared to the conventional duct 1, while thrust is reduced by about 52.11% compared to the conventional duct 2. Furthermore, energy loss in the wake vortex is minimized. Flow field analysis is conducted to further confirm the effectiveness of the optimized design, with the high-speed zone area being expanded and pressure extremes reduced by approximately 31.71%. These results demonstrate that machine learning methods can effectively be used to extract nonlinear relationships between complex parameters, offering more design options for duct development and facilitating the engineering application of tidal energy generation technology.

1. Introduction

With advancements in technology and increasing environmental awareness, the global energy landscape is undergoing a profound transformation. The shift towards clean and low-carbon energy has become the prevailing trend, driven by technological and financial innovations that are restructuring energy systems. Ocean energy encompasses various forms, including tidal current energy, tidal energy, and wave energy [1]. Among these, tidal current energy offers numerous advantages as a type of ocean energy [2]. It is a clean, renewable energy source that does not produce greenhouse gas emissions or other pollutants [3]. Tidal power generation devices are systems that convert the kinetic energy of tidal currents into electrical energy. Based on their structural composition, these devices are generally categorized into components such as support structures, energy capture mechanisms, power generation and control systems, and transmission systems [4]. Tidal turbines, a common type of tidal power generation device, currently exist in various forms [5]. In the rapid development of tidal power generation technology, the duct plays a critical role, directly influencing the overall performance and economic efficiency of the turbine. While traditional duct designs in the tidal turbine sector can assist in creating vanes that meet basic requirements, they also exhibit several shortcomings, primarily characterized by cumbersome processes and relatively low accuracy. The integration of machine learning technology into the optimization design of tidal turbine ducts represents a dynamic and innovative approach.

The flow guide hood is a water flow enhancement device designed based on the principles of wind turbine ducts. Its primary aim is to increase the power output of water turbines while reducing the startup flow velocity by enhancing the overall water flow velocity. Currently, researchers both domestically and internationally have conducted extensive studies on the design and optimization of tidal current turbine duct hoods. For instance, Coiro et al. [6] developed a new type of tidal current turbine duct hood and performed relevant model tests and hydrodynamic performance evaluations. Wang et al. [7] employed the lattice Boltzmann method within computational fluid dynamics to optimize various geometric configurations of tidal current turbine duct hoods. Ghassemi et al. [8] examined the impact of ducts and the number of blades on lake tidal energy turbines. Li Liangqian focused on the Savonius-type vertical-axis tidal energy turbine, utilizing numerical simulation methods to optimize the parameters of the unidirectional duct device he designed, followed by model test validation. Cui Baoyu et al. [9] applied CFD technology to explore the effects of different rotor solidity sizes on a duct-type horizontal-axis tidal current turbine, discovering that as rotor solidity increases, the turbine’s power coefficient decreases, thereby impairing the duct’s ability to accelerate internal water flow. Nunes et al. [10] evaluated the performance of turbines under various duct configurations. The study demonstrated the feasibility of using ducts to enhance turbine performance. Canadian Clean Current Company has developed a tidal turbine. With a maximum rotor diameter of 3.5 m, the duct used in this configuration has a length of 3.1 m, with a rated power output of 65 kW at a water flow velocity of 3 m/s [11]. This information is provided by Open Hydro Company [12]. The team developed an open-center tidal turbine equipped with a duct cover. A 1 MW unit was deployed in the Bay of Fundy in 2009. Watanabe [13] from Kyushu University in Japan proposed a floating tidal power generation system featuring a large duct. The feasibility of the system was demonstrated and validated through model experiments.

Large tidal energy hydro turbines have stringent requirements regarding water depth and flow velocity. In the shallow waters of most Chinese coastal areas, where flow velocities are relatively low, the large-scale application of tidal energy hydro turbines is somewhat limited. Additionally, the ducts of these turbines often require increased transverse and longitudinal dimensions, which results in reduced portability. It is important to note the higher installation and manufacturing costs, particularly concerning the trailing edge ducts, which have proven effective at low flow velocities in the field. However, the application of hydro turbines remains constrained. The presence of a duct enhances local flow velocity; however, the flow field within the duct is three-dimensional, highly turbulent, and exceedingly complex, especially when the turbine operates in conjunction with the duct. Consequently, further research into the hydrodynamic performance of duct turbines and enhancements to these hydraulic systems is essential. This paper will employ a multi-layer perceptron (MLP) model to optimize the duct and examine the effects of the optimized duct on the hydrodynamic characteristics of the turbine.

2. Theoretical Basis

2.1. The Governing Formulas

The following are the continuity and momentum conservation equations for incompressible fluids:

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

(1)

$$\rho {\displaystyle \frac{\partial u_i}{\partial t}}+\rho u_j{\displaystyle \frac{\partial u_i}{x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left(\mu {\displaystyle \frac{\partial u_i}{\partial x_j}}-\rho u_i{u}_j\right)-{\displaystyle \frac{\partial p}{\partial x_i}}$$

(2)

where $\rho$ denotes the density of the incompressible fluid in kg/m³; p represents the fluid pressure in Pa; and μ signifies the dynamic viscosity of the incompressible fluid in Pas.

A horizontal-axis turbine’s hydrodynamic performance is mainly characterized by dimensionless metrics like the axial thrust coefficient (C_t) and the power coefficient (C_p). While C_t shows the force applied to the turbine and its stability, C_p represents the efficiency of the energy conversion.

$$C_{\mathrm{p}}={\displaystyle \frac{M\omega }{0.5\rho A{U}^3}}$$

(3)

$$C_{\mathrm{t}}={\displaystyle \frac{Ft}{0.5\rho A{U}^2}}$$

(4)

$$\lambda ={\displaystyle \frac{\omega R}{U}}$$

(5)

Here, M denotes the torque exerted on the turbine shaft in Nm; ω represents the rotational angular velocity of the turbine in rad/s; $\rho$ signifies the fluid density in kg/m³; U represents the upstream flow velocity (undisturbed by the turbine) in m/s; A denotes the swept area of the turbine rotor (πR² for a rotor of radius R) in m²; and Ft signifies the axial thrust force acting on the turbine in N.

2.2. Design of Ducts

2.2.1. Principles of Duct Design

The following five fundamental principles should be universally adhered to during the design and manufacturing stages of a flow guide: 1. Flow Optimization Principle: This principle focuses on ensuring smooth fluid guidance while minimizing total pressure loss. 2. Structural Strength Principle: This principle guarantees the strength and stiffness of the flow guide under all operating conditions. 3. Manufacturing Process Principle: This principle takes into account the manufacturability of geometric shapes and materials. 4. System Compatibility Principle: This principle ensures that the flow guide can operate reliably within the entire system. 5. Conflict Resolution Principle: In cases where conflicts arise among the aforementioned principles, priority should be given to flow optimization and structural reliability [14], while also striving to balance manufacturability and system integration as much as possible.

2.2.2. Design of Duct Profiles

Currently, numerous scholars have conducted extensive research on the design of duct profiles, with double cubic curves being the most widely utilized in these designs. This paper offers a brief introduction to double cubic curves, focusing on axially symmetric duct profiles used in the study. Figure 1 illustrates the schematic diagram of the contraction section profile. During the preliminary design phase, this paper primarily examines the effects of various flow hood profile lines, angles, and blade tip distances on the flow hood.

Double cubic curves can be generated using different X_m values at the inflection points, which serve as the connection points between the two curves. The specific expressions are provided below:

$${\displaystyle \frac{h-H_o}{H_i-H_o}}=\left\{\begin{array}{c}1-{\displaystyle \frac{1}{X_m^2}}{\left(X/L\right)}^3\\ {\displaystyle \frac{1}{{\left(1-X_m\right)}^2}}{\left[1-\left(X/L\right)\right]}^3\end{array}\right.\begin{array}{c}\begin{array}{cc}& \end{array}\left(X/L\right)\le X_m\\ \begin{array}{cc}& \end{array}\left(X/L\right)\ge X_m\end{array}$$

(6)

where H_i and H_o represent the inlet and outlet cross-sectional radii of the contraction section of the guide hood (m), respectively; L is the length of the contraction section of the guide hood (m); and h is the cross-sectional radius at the axial distance X (m).

2.2.3. Model and Parameters of Tidal Current Turbines

Based on the previously mentioned design principles for the duct hood, the contraction segment profile was modeled in UG using a double cubic curve with inflection point values of 0, 0.2, 0.4, 0.6, 0.8, and 1.0. The angle of attack was set to 16°, 20°, and 24°, while the blade tip spacing was configured to 8 mm and 12 mm. These models were utilized for numerical simulations and subsequently served as both the training and test sets for the machine learning model. Figure 2 and Figure 3 show the guide vane turbine models with different profile lines and tip pitches, respectively.

3. Duct Optimization Research Methodology

3.1. Multi-Layer Perceptron (MLP) Model

A multi-layer perceptron (MLP), also known as an Artificial Neural Network (ANN), consists of an input layer, an output layer [15], and one or more hidden layers in between. The simplest MLP contains only one hidden layer, resulting in a three-layer structure [16]. In this study, a machine learning method based on the MLP model was employed to optimize the duct design. The structure of the model is as follows: Input layer: The input layer consists of three features—angle of attack, tip clearance, and X_m. These features are derived from numerical simulation data and serve as the model’s inputs. Hidden layer: The model includes one hidden layer with eight neurons. The number of neurons can be adjusted based on specific requirements. During training, the hidden layer introduces nonlinearity through the ReLU (rectified linear unit) activation function, enhancing the model’s fitting capability. Output layer: The output layer contains a single neuron that predicts the target variable. Activation function: The hidden layer uses the ReLU activation function, which effectively mitigates the vanishing gradient problem and accelerates convergence. Optimizer and learning rate: The Adam optimizer (Adaptive Moment Estimation) was selected with a learning rate of 1 × 10⁻⁴. Adam offers strong adaptability, enabling rapid convergence at a low learning rate while preventing gradient explosion. Loss function: Mean Squared Error (MSE) is used as the loss function to measure the difference between the model’s predicted values and the actual values. MSE is commonly employed in regression problems and effectively optimizes prediction accuracy.

Hyperparameter settings: Batch size is set to 1, meaning each training update uses a single sample. The batch size can be adjusted based on computational resources and model requirements. Epochs were set to 50 to ensure the model learns sufficiently and converges. The random seed (SEED) was set to 42 to ensure reproducibility of the training process.

3.1.1. Evaluation Criteria and Error Analysis of Multi-Layer Perceptron (MLP) Models

In order to comprehensively evaluate the performance of multi-layer perceptron (MLP) models in predicting the hydrodynamic response of ducts, an evaluation system was introduced that encompasses three key dimensions.

• Mean Squared Error (MSE). Mean Squared Error (MSE) is the average of the squared differences between the predicted values and the actual values [17]. A smaller MSE indicates a better fit of the model to the data.
• Mean Absolute Error (MAE). The Mean Absolute Error (MAE) measures the average magnitude of absolute errors between predicted results and actual observations [18], and it is not sensitive to outliers.
• Coefficient of Determination (R²). R² is used to measure a model’s ability to explain the variance in data [19]. The closer the value is to 1, the stronger the model’s fit. The formula is as follows:

$$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_i-\overline{y_i}\right)}^2}$$

(7)

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|y_i-\overline{y_i}\right|}$$

(8)

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{\left(y_i-\overline{y_i}\right)}^2}}{{\displaystyle {\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}}}$$

(9)

where $y_i$ represents the actual value; $\overline{y_i}$ denotes the predicted value from the model; n indicates the total number of samples; and $\overline{y}$ represents the sample mean.

3.1.2. Multi-Round Cross-Validation and Robustness Testing

To further mitigate model overfitting and enhance generalization capabilities, this paper employs the k-fold cross-validation method (k = 5) to partition the entire dataset into training and validation sets [20]. In each iteration, a different subset is designated as the validation set, while the remaining data is utilized for training. The final model performance is assessed based on the average metrics obtained from each iteration.

The results of multi-round cross-validation indicate that the mean R² value of the MLP model across various subsets is 0.962, with no significant fluctuations in Mean Squared Error (MSE) and Mean Absolute Error (MAE). This suggests that the model demonstrates strong adaptability to different sample distributions. On the test set, the R² value for power coefficient prediction reaches 0.965, with an MSE of 0.00052, and an MAE of 0.017. Additionally, the R² value for thrust prediction is 0.957. Please refer to

Table 1. MLP model key performance indicator statistics table.

Dataset	MSE	MAE	R2
Training Set	0.00048	0.016	0.967
Test Set	0.00052	0.017	0.965
5-Fold Mean	0.00054	0.018	0.962

for further details.

3.2. Comparative Analysis of Machine Learning Models

To comprehensively evaluate the effectiveness of the multi-layer perceptron (MLP) model in predicting the hydrodynamic characteristics of fairings, this paper introduces two widely used regression algorithms—Support Vector Regression (SVR) [21] and Random Forest Regression (RFR)—as baseline models for comparison. All three models are trained and optimized using the same dataset, preprocessing techniques, and feature engineering processes. Additionally, 5-fold cross-validation is employed to ensure the fairness of the evaluation and the robustness of the results.

3.2.1. Support Vector Regression (SVR)

Support Vector Regression (SVR) is a supervised learning model that operates on the principle of minimizing structural risk. It is particularly well suited for nonlinear regression modeling of small to medium-sized datasets. The fundamental concept behind SVR is to map input data into a high-dimensional space using a kernel function, allowing for the identification of the optimal regression hyperplane within that space, ultimately minimizing generalization error.

3.2.2. Random Forest Regression (RFR)

Random Forest Regression (RFR) is an ensemble learning method that constructs multiple decision trees and combines their predictions to reduce variance and enhance robustness [22]. It is widely utilized in scenarios involving nonlinear feature interactions. Its advantages include strong adaptability to outliers and multicollinearity, as well as the capability to automatically evaluate feature importance.

3.2.3. Performance Metrics and Experimental Results

With the power coefficient (C_p) and thrust coefficient (C_t) as the primary prediction targets, three metrics—Mean Squared Error(MSE), Mean Absolute Error(MAE), and Coefficient of Determination (R²)—were employed for quantitative comparison. The key experimental results are summarized in

Table 2. Comparison of errors in predicting the hydrodynamic response of ducts using different machine learning models.

Model	Cp Prediction (MAE)	Cp Prediction (MSE)	Cp Prediction (R2)	Ct Prediction (MAE)	Ct Prediction (MSE)	Ct Prediction (R2)
MLP	0.0075	0.00009	0.965	0.0183	0.00047	0.958
SVR	0.0121	0.00021	0.945	0.0247	0.00085	0.926
RFR	0.0109	0.00015	0.956	0.0212	0.00062	0.942

below.

As shown in the table, the MLP model achieved the lowest Mean Absolute Error (MAE) and Mean Squared Error (MSE) in predicting the power coefficient and thrust coefficient. Additionally, it attained the highest R² coefficient, indicating its superiority over the Support Vector Regression (SVR) and Random Forest Regression (RFR) models in terms of fitting accuracy and generalization capability. This advantage is further illustrated by the distribution of actual prediction residuals; the residuals of the MLP model are more concentrated, and extreme errors are significantly minimized.

3.3. Optimization Process

A machine learning approach based on a multi-layer perceptron (MLP) model was employed to optimize the design of the duct. The optimization framework consists of three primary stages: data preprocessing, model training, and prediction and optimization. During data preprocessing, the raw dataset obtained from numerical simulations was cleaned, normalized, converted into PyTorch 3.19 tensor format, and split into training and testing sets with an 8:2 ratio. In the model training stage, the MLP model utilized a fully connected layer architecture. Hyperparameters such as random seed, number of hidden layers, training iterations, batch size, and learning rate were configured to ensure training stability and reproducibility. The training process involved forward propagation, loss calculation, backward propagation, and optimization. Finally, hyperparameters were fine-tuned using a validation set to enhance model performance.

In the optimization phase, a genetic algorithm (GA) was employed to optimize the geometric parameters of the duct. Initially, a multi-layer perceptron (MLP) model predicted the water turbine’s performance under various geometric parameters—namely, angle of attack, tip clearance, and duct profile—including the power coefficient (C_p) and thrust coefficient (C_t). Subsequently, the genetic algorithm conducted a global search within the geometric parameter space to identify the optimal combination that maximizes C_p while minimizing C_t. The optimization process of the genetic algorithm involves the following steps: 1. Initialize the population: Begin with a randomly generated initial population, where each individual represents a specific set of geometric parameters (angle of attack, tip clearance, and duct profile). 2. Fitness evaluation: Calculate the fitness of each individual using the MLP model. Fitness values are based on a combination of the turbine’s power coefficient (C_p) and thrust coefficient (C_t), with the objective of maximizing C_p and minimizing C_t. 3. Selection: Select individuals with favorable fitness values to proceed to the next generation. 4. Crossover and mutation: Generate new individuals through crossover (exchanging gene segments between two individuals) and mutation (randomly altering parts of an individual’s genes), introducing new solutions and preventing premature convergence to local optima. 5. Termination criteria: The algorithm terminates and returns the optimal solution when the fitness reaches a predefined threshold or when the number of generations exceeds the maximum allowed. 6. Reason: The revision improves clarity and readability by breaking down the optimization process into numbered steps and refining sentence structure. Technical terms are clearly defined, and punctuation is corrected for better flow. The vocabulary is enhanced to maintain precision and professionalism without altering the original meaning.

The optimization process is illustrated in Figure 4. According to this process, the optimized duct data predicts an angle of attack of 20°, a blade tip distance of 8 mm, and a duct profile characterized by an X_m value of 0.796 in a double cubic curve. This configuration enables the water turbine to achieve a high power coefficient while minimizing the water thrust on the duct.

4. Physical Model

4.1. Computational Domain Creation and Meshing

Figure 5 shows a schematic diagram of the computational domain. The computational domain encompasses 5D in the forward direction and 15D in the backward direction, as well as 3D in the left–right and front–back orientations. Numerical simulations were conducted using STAR-CCM+ 17. The computational model is divided into two components: the turbine and the channel. Given the advantages of polyhedral grids in enhancing computational convergence and solution accuracy and considering the degree of distortion on the surface of the duct, this study employed the cut-body mesh feature in STAR-CCM+ fluid analysis software for mesh partitioning. Mesh refinement was applied to the surfaces of the turbine blades and the duct. Additionally, rectangular regions were defined within the following ranges from the center of the outlet cross-section of the duct contraction section: 1.0D forward/backward, 1.25D left/right, and 1.25D up/down; 3.5D forward/backward, 1.5D left/right, and 1.5D up/down; and 5.0D forward/backward, 2D left/right, and 2D up/down. Here, D represents the diameter of the tidal energy turbine runner, with D = 20 cm.

4.2. Grid Independence Verification

In numerical simulations, the accuracy of computational results is closely linked to the number of grid cells used. An inadequate number of grid cells may fail to capture the intricate details of the flow field or physical phenomena, resulting in significant numerical errors [23]. While increasing the number of grid cells can reduce these errors and enhance the accuracy of the results, it also escalates computational costs, including the demand for resources and time. More grid cells necessitate additional memory and computational power for both storage and processing [24]. Therefore, it is essential to strike a balance between the accuracy of computational results and the associated costs, selecting an appropriate number of grid cells accordingly. Figure 6 shows the mesh partitioning diagram.

To verify mesh independence, we analyzed the effect of varying mesh counts on the horizontal thrust exerted on the duct, using the same mesh partitioning method, as shown in

Table 3. Mesh independence verification.

Scheme	Number of Cells (Million)	Ft (N)	∆Ft (%)
1	2.075	1.8931	2.19
2	3.204	1.9122	1.16
3	4.871	1.9253	0.48
4	5.256	1.9346	-

. The results indicate that the difference in horizontal thrust between Scheme 3 and Scheme 4 is less than 1.0%. Given the consideration of computational efficiency, the parameters of Scheme 3 were chosen for subsequent mesh partitioning.

4.3. Boundary Conditions Setup

In the process of fluid flow, a complex turbulent flow occurs. In turbulent flow, parameters such as fluid pressure and velocity are not constant; they vary with changes in time and space. Therefore, in computational fluid dynamics (CFD), it is essential to introduce relevant turbulence models to approximate and simulate actual fluid flow. Common turbulence models in CFD technology include the standard K-ε model, the RNG K-ε model, the standard K-ω model, the SST K-ω model, the Reynolds Stress (RS) model, and the Large Eddy Simulation (LES) method. Among these turbulence models, the LES method typically incurs relatively high computational costs in practical applications. The SST K-ω model, proposed by Menter in 1994, incorporates cross-diffusion from the equations and accounts for the propagation of turbulent shear stress [25]. This model provides more accurate results for near-wall regions, wake flows, and flows around bodies [26], making it more widely used in engineering practice. This paper utilizes the SST k-ω turbulence model [27] to simulate turbulent flow. The incoming flow velocity in the water channel is set at 0.35 m/s, with a turbulence intensity of 7%. The boundary conditions are depicted in Figure 7: the inlet and outlet boundaries of the water channel are designated as a velocity inlet and a pressure outlet, respectively; the relative atmospheric pressure is established at 0 Pa; and the boundary condition for the top surface of the water channel is defined as a pressure outlet. The side boundaries of the water channel are configured as symmetric planes, while both the bottom surface of the water channel and the duct boundaries are treated as fixed walls. Furthermore, the boundary conditions at the interface between the turbine runner’s rotational domain and the experimental water channel are defined using overlapping grids.

4.4. Verification of Numerical Simulation Methods

To validate the accuracy of the selected simulation method, this study utilized experimental data from a horizontal-axis tidal current turbine obtained during wave flume experiments conducted by Li Yanwei et al. [28] at the Port and Waterway Hydrodynamics Laboratory of Shandong Jiaotong University. During the simulation process, the study maintained the same turbulence intensity as the experimental data and selected an appropriate tip speed ratio (TSR). Additionally, data from reference [28] was used to reconstruct the turbine experimental model. This approach facilitated the simulation of the relationship between the energy conversion coefficient (C_p) of the turbine and the tip speed ratio (TSR), with the simulation results compared to the experimental data from reference [28] (see Figure 8). The results indicate that the overall increasing trend of the energy conversion coefficient in relation to changes in the tip speed ratio, as obtained from the simulation, closely aligns with the experimental data. The error between the simulation values and the experimental values is within 12%, and the trend of the simulation results curve is largely consistent with the trend of the experimental results curve presented in reference [28], thereby verifying the reliability of the numerical simulation method.

As illustrated in the figure, the power coefficient (C_p) of the turbine initially increases and then decreases as the tip speed ratio (TSR) rises. It reaches its peak at a TSR of approximately 3.8, after which it declines steadily with further increases in TSR.

5. Numerical Simulation and Results Analysis

Under a water flow velocity of 0.35 m/s, the power coefficient (C_p) of the duct-type horizontal-axis tidal current turbine, as calculated by numerical simulation, varies with the tip speed ratio (TSR), as illustrated in the figure. The figure indicates that the power coefficient initially increases and then decreases as the tip speed ratio rises. The optimal tip speed ratio determined by the simulation is approximately 3.8–4. Consequently, the impeller speed is established at 14 rad/s, corresponding to a TSR of 4.

According to the optimized process described above, the predicted data for the duct cover are as follows: angle of attack 20°, tip pitch 8 mm, and a duct cover profile line X_m = 0.796, defined by a double cubic curve. In this chapter, it is compared with two duct covers obtained through the traditional design process: traditional duct cover 1, with an angle of attack of 20°, tip spacing of 12 mm, and a double cubic curve with a fairing profile of X_m = 0.796; and traditional fairing 2, with an angle of attack of 20°, tip spacing of 8 mm, and a double cubic curve with a fairing profile of X_m = 0.

5.1. Turbine Thrust Analysis

Axial water thrust is primarily caused by the pressure difference between the high-pressure and low-pressure sides of the turbine impeller, as shown in

Table 4. Thrust forces and thrust coefficients of different hydraulic turbine types.

Turbine Configuration	Horizontal Thrust (N)	Thrust Coefficient Ct
Conventional Duct Turbine 1	2.378	0.9916
Conventional Duct Turbine 2	4.679	2.1782
Optimized Duct Turbine	2.171	0.9745

. The optimized duct significantly reduces both horizontal thrust and the thrust coefficient. Comparing the three schemes in Table 4 reveals that the blade angle has minimal impact on thrust; the reduction in thrust is mainly due to changes in the duct contour. Specifically, compared to traditional duct 2, the modified duct contour reduces the turbine’s thrust by approximately 52.11%. In contrast, compared to traditional duct 1, the change in blade pitch reduces the turbine’s thrust by about 1.72%. Based on the previously discussed principles of duct design, this paper will focus on analyzing the flow field in the outlet section of the duct to further validate the performance improvements achieved through duct optimization.

5.2. Turbine Flow Field Analysis

The optimized duct utilizes a contraction section profile optimization design, with its contraction and outlet geometries refined through iterative machine learning algorithms. This approach ensures that the main flow stream transitions smoothly from the inlet to the impeller region along axisymmetric flow lines. The design effectively prevents flow separation, allowing the fluid to maintain a high velocity and low vorticity before entering the impeller. This significantly reduces local recirculation and vortex generation caused by streamline bending or abrupt changes in cross-section. Thanks to the guiding and contraction effects of the duct, the water flow entering the impeller zone is more uniform and directional, with a marked reduction in turbulence intensity. The optimized structure guarantees that the water flow primarily follows a tangential and axial velocity distribution, minimizing radial components and lateral interference.

5.2.1. Duct Velocity Analysis

Figure 9 presents the velocity contour plot of the traditional duct turbine 1, Figure 10 shows the velocity contour plot of the traditional duct turbine 2, and Figure 11 displays the velocity contour plot of the optimized duct turbine. By comparing and analyzing these velocity contour diagrams of different duct tidal energy turbines, the impact of various designs on flow characteristics can be observed. In the YZ-plane view, Figure 10 reveals a distinct vortex structure, with high-velocity regions concentrated around the turbine center, while the surrounding areas exhibit significantly reduced flow velocities. This indicates that the duct profile substantially influences the distribution of turbulence and local velocities. In contrast, the flow patterns in Figure 9 and Figure 11 demonstrate a more uniform velocity distribution with fewer turbulent phenomena, resulting in smoother overall flow. This suggests that the optimized duct profile achieves superior flow control performance. Figure 11 shows the most uniform velocity distribution, with minimal variation in flow velocity around the turbine, indicating that this design may offer advantages in enhancing flow stability and reducing energy losses. From the comparison of the XZ-plane views, although all three designs exhibit a ring-shaped flow structure, the vortices in Figure 10 are more pronounced, indicating greater flow instability, while the flows in Figure 9 and Figure 11 are more uniform, suggesting better performance in terms of turbine efficiency and stability. This highlights the influence of blade pitch on the performance of tidal energy turbines, particularly regarding flow control and energy conversion efficiency.

The standard deviation of the outlet velocity of a hydro turbine refers to the variability in the outlet water flow velocity during stable operation [29]. It reflects the stability of the outlet water flow velocity; a smaller standard deviation indicates greater stability, while a larger standard deviation indicates increased variability [30]. The center point velocity at the outlet of a hydro turbine is the velocity measured at the center of the water flow velocity distribution across the outlet cross-section during operation. This velocity value reflects the turbine’s efficiency in converting water flow energy.

Table 5. Standard deviation of outlet velocity, velocity at outlet section center, and power coefficient for various hydraulic turbines.

Turbine Configuration	Outlet Velocity SD	Outlet Center Velocity (m/s)	Power Coefficient Cp
Conventional Duct Turbine 1	0.1627	0.1746	0.2944
Conventional Duct Turbine 2	0.1499	0.1832	0.2163
Optimized Duct Turbine	0.1384	0.2061	0.3272

presents the standard deviation of the outlet velocity, the velocity at the center point of the outlet cross-section, and their corresponding power coefficients. From Table 5, it is evident that the tip clearance has a more significant impact on the power coefficient. After optimization, compared to the traditional diffuser 1, the standard deviation of the duct outlet velocity decreased by approximately 14.94%, resulting in a more stable velocity across the outlet cross-section and a higher velocity at the center point. This enhancement improved the water flow energy conversion efficiency, increasing the power coefficient of the optimized duct fan by approximately 11.14%. Compared to the traditional diffuser, the standard deviation of the duct outlet velocity was reduced by approximately 8%, and the power coefficient of the optimized duct fan increased by approximately 51.27%.

5.2.2. Turbine Surface Pressure Distribution

Figure 12 shows the pressure distribution cloud map of the traditional duct turbine 1, Figure 13 shows the pressure distribution cloud map of the traditional duct turbine 2, and Figure 14 shows the pressure distribution cloud map of the optimized duct turbine. As can be seen from the figures, the pressure distributions on the upstream and downstream sides of the three types of duct turbines are similar but exhibit significant differences. Compared to the traditional duct turbines, the optimized duct reduces pressure values and achieves a more uniform pressure distribution. In addition, as shown in

Table 6. Runner pressure limits for different hydraulic turbine types.

Turbine Configuration	Max Pressure (Pa)	Min Pressure (Pa)	Extreme Pressure Difference (Pa)
Conventional Duct Turbine 1	4366.9	1359.0	3007.9
Conventional Duct Turbine 2	3894	1534.1	2360.1
Optimized Duct Turbine	3698.3	1644.1	2054.2

, both the maximum pressure and the maximum pressure difference of the optimized duct are lower than those of the two traditional duct water turbines. Compared to traditional diffuser 1, the maximum pressure difference of the optimized duct is reduced by approximately 31.71%.

5.2.3. Turbine Wake Vortex Analysis

Wake vortices significantly impact the performance of tidal energy turbines, primarily by creating uneven flow fields and increasing energy loss [31]. The generation of wake vortices induces flow disturbances that lead to unstable turbine operation, affecting power output and potentially causing fluctuations in output power [32]. Furthermore, the morphology of wake vortices directly influences the rate of energy dissipation. Loose and fragmented vortices exacerbate turbulent mixing, resulting in over 30% of kinetic energy being converted into ineffective thermal energy through viscous dissipation, which considerably reduces the turbine’s output. Therefore, controlling the generation and expansion of wake vortices is crucial for enhancing efficiency, stability, and performance during the design and optimization of tidal current energy turbines.

Figure 15, Figure 16 and Figure 17 illustrate the vortex structures of different duct turbines using the Q criterion (Q = 0.2). Comparative analysis reveals significant differences in the stability and flow efficiency of the wake vortices. By examining the vortex flux contour maps of various duct tidal energy turbines, the impact of different designs on flow characteristics becomes evident. In Figure 15, the vortex distribution shows strong vortex activity, particularly in the region behind the turbine blades, indicating intense turbulence and unstable flow. This flow pattern results in higher energy losses and reduced turbine efficiency. In contrast, the vorticity distribution in Figure 16 is more uniform, with fewer vortex regions, smoother flow, and reduced turbulence, suggesting that this design offers improved flow control and decreased energy losses. However, compared to Figure 17, the flow stability in Figure 16 still has potential for further optimization. Figure 17, representing the optimized design, exhibits a more uniform vortex distribution, significantly reduced vortex intensity, smoother flow, and diminished turbulence, indicating superior flow control and enhanced turbine efficiency. In summary, the optimized design clearly demonstrates advantages in reducing vortices, improving flow stability, and increasing turbine efficiency. The optimized duct turbine outperforms the traditional design in terms of tail vortex stability, energy loss, and flow efficiency. The optimized vortex region is well maintained, with slower tail flow dispersion and lower energy loss, resulting in more ideal flow stability and reduced turbulence intensity, making it suitable for the efficient operation of tidal energy turbines. Conversely, traditional duct turbines experience faster tail flow dispersion, rapid dissipation of vortex energy, lower efficiency, and unstable flow.

6. Conclusions

This paper presents an optimization study of the duct utilizing a multi-layer perceptron (MLP) model. It applies the optimization results to analyze the hydrodynamic characteristics of a tidal energy turbine and evaluates the impact of the optimized duct on energy conversion efficiency and axial thrust of the turbine. The study draws the following key conclusions:

Through machine learning techniques, the optimal parameters for duct design were automatically explored, resulting in a significant enhancement of the power coefficient of the water turbine. The optimization results indicate that, compared to traditional methods, the optimized design of the double cubic curve inflection point on the deflector hood profile has increased the water turbine’s energy utilization coefficient by approximately 51.27%.

The optimized duct effectively directs and concentrates water flow. This is accomplished through the contraction at the inlet and the expansion at the outlet of the duct, which significantly reduces water thrust and enhances energy capture efficiency. The data show that the axial thrust coefficient (C_t) is reduced by 52.11% compared to traditional design 2. These results demonstrate dual advantages in minimizing water thrust and enhancing energy capture efficiency.

The optimized design significantly enhances the uniformity of the water turbine flow field distribution and the pressure stability at the outlet section. The maximum pressure difference in the optimized duct is reduced by 31.71%. These improvements demonstrate that the design can increase operational reliability and performance stability under complex operating conditions.

Optimizing the energy loss reduction of the tail vortex in the duct hood decreases the influence range of the vortex area and slows down the vortex breakup speed, thereby enhancing the flow’s orderliness. This improvement directly reduces the kinetic energy dissipation caused by turbulent mixing, which is crucial for enhancing the overall efficiency of the water turbine.

This study demonstrates that machine learning-based optimization methods provide significant advantages in duct design, offering innovative solutions to enhance the performance of tidal power generation devices and improve design efficiency.