## Abstract

The global adoption of Savonius wind rotors as an eco-friendly means of small-scale power production is on the rise. However, their suboptimal performance remains a significant challenge due to the generation of higher unproductive torque. This paper aims to address this issue by obtaining an optimal blade profile considering the power coefficient (C_{P}) as an output function using optimization techniques. The objective function includes the overlap ratio, intermediate points on the curve, inlet velocity, and tip speed ratio (TSR) as the optimization geometric parameters. To achieve this, the simplex search method and the non-dominated sorting genetic algorithm II are opted to develop the blade profile. The blade profile is developed using a natural cubic spline curve with fixed end points and variable intermediate points along with other parameters. The computational analysis is done using ansys fluent software with shear stress transport k−ω turbulence model. The solver setup employs the finite volume method to simulate the transient 2D flow around the blade profile. A direct comparison is made between the optimized blade profile and the conventional semicircular one over a range of TSRs. The results clearly indicate the superior performance of the former, exhibiting a higher C_{Pmax} by 23% compared to the conventional one at TSR = 0.8. Finally, experiments have been conducted in a wind tunnel to find the practical feasibility of the optimized blade profile generated through the simplex search method.

## 1 Introduction

Wind power plays a crucial role in addressing the global energy crisis [1,2]. Wind turbines are the devices for capturing wind energy and converting it into electricity. Depending on the orientation of the rotational axis, these turbines are commonly categorized as horizontal-axis wind turbines (HAWTs) and vertical-axis wind turbines (VAWTs). The HAWTs have a rotation axis aligned parallel to the incoming wind direction, whereas the VAWTs rotate crosswise to the incoming wind direction. The HAWTs are commonly used in areas with higher energy demands and larger coverage as compared to VAWTs. Unlike VAWTs, the HAWTs require a yaw mechanism to operate and are not self-starting [3–6]. Additionally, there are on-shore and off-shore HAWTs designed to extract wind power from land and oceans, respectively. Although the HAWTs have seen significant advancements, they tend to be expensive and require extensive infrastructure for power transmission and distribution, making them less suitable for remote or inaccessible areas. Recently, the interest in VAWTs has grown due to their advantages, including wind direction independence, lower noise, ease of installation and maintenance, and potential for domestic energy solutions [7]. Among the VAWTs, the Savonius wind rotor stands out for its structural simplicity, requiring minimal initial torque and achieving an efficiency of around 30% [8]. The Savonius rotors offer cost-effective power generation compared to other VAWTs, such as H- and Φ type Darrieus rotors. The Savonius rotors, on the other hand, present an attractive solution for smaller and fluctuating energy demands. Figure 1 represents a typical Savonius rotor showing the torque produced by the advancing and the returning blades (front view), while the top view depicts the forces (lift and drag) acting on it along with geometric parameters such as separation gap (SG), overlap distance (e), and overlap ratio (OR).

In recent decades, researchers have tried to improve the design of Savonius wind rotors through experimental [9–12], numerical [13–17], and combined approaches [18–22]. While previous studies have examined specific parameters, such as the arc angle, which measures the circular arc angle of conventional semicircular blades and its influence on the rotor performance, it is essential to adopt a holistic optimization approach considering the interrelations of parameters. Several studies have also explored the optimal design of Savonius rotors using optimization algorithms [23–29]. Despite having lower performance, researchers have continuously strived to improve the efficiency of the Savonius rotors due to the associated benefits. Experimental investigations have been conducted to evaluate the impact of operating parameters, including geometric variables like aspect ratio (AR) [9–11], overlap ratio (OR) [25], tip speed ratio (TSR) [30–32], and Reynolds number (Re) [26,27]. Additional factors that are examined include the effect of end plates, the number of stages, the number of blades, and others [18]. Furthermore, studies have shown that a two-bladed rotor outperforms three- and four-bladed rotors. However, there is a need for a comprehensive framework that would lead to an efficient design through optimization, numerical simulations, and experiments.

In this context, the present study aims to maximize the power coefficient (*C _{P}*) by optimizing the Savonius rotor blade profile. This can be executed by optimizing various parameters, namely, OR, TSR, and inlet velocity (

*V*), through the use of optimization algorithms. The optimization is achieved using two different algorithms. The design validation of the rotor is performed using computational fluid dynamics (CFD) simulations, and the

*C*obtained is compared with a conventional semicircular blade profile. Additionally, wind tunnel tests are conducted to evaluate the blade generated through the optimization technique to find the practical feasibility. The process followed in the whole analysis is shown in Fig. 2.

_{P}## 2 Optimization Algorithms

Optimization algorithms are iterative procedures that compare different solutions until a satisfactory solution is achieved. Ogawa [33] presented the first model using an optimization algorithm on the Savonius wind rotor with a discrete vortex method; however, the results obtained did not agree with the experimental data quantitatively. Nevertheless, the idea of implementing the optimization algorithms opened up new vistas of research, and this has led to many advancements [19,23,27,32,34].

In the process of optimizing a design, the primary goal can involve minimizing the production costs or maximizing the production efficiency. As the optimization process relies on software tools such as ansys fluent for further calculations, the objective function calculations cannot be determined directly [23,24,28,33]. This happens by coupling the software with an automated procedure to achieve the objectives. Thus, it is important to select the algorithms to achieve the objective functions accordingly [32]. Algorithms such as the simplex search method (SSM) and non-dominated sorting genetic algorithm II (NSGA-II) can be invoked in the optimization process.

### 2.1 Simplex Search Method.

This method begins by creating a hypercube or simplex in the N-dimensional variable space, consisting of (*N* + 1) points with a non-zero volume. Figure 3 illustrates the process of generating new points using the SSM [34–36]. The objective function is evaluated at these points. Depending on the objective function, these points are segregated as the worst point (*x _{h}*), the best point (

*x*), and next to the worst point (

_{b}*x*). In the context of maximizing the objective function,

_{g}*x*represents the point with the lowest value, while

_{h}*x*corresponds to the point with the highest function value among all

_{b}*N*points. Since the method aims to move away from the worst case,

*x*is calculated as the centroid of all points to determine the worst point, and

_{c}*x*is defined as 2

_{r}*x*−

_{c}*x*.

_{h}The objective function is then evaluated at this specified point *x _{r}*. If the objective function at

*x*is better than

_{r}*x*, a new point is generated by extending

_{b}*x*as

*x*= (1 +

*γ*)

*x*−

_{c}*β x*, where

_{h}*γ*is a constant greater than one. In case

*x*is worse than

_{r}*x*, and conversely, if

_{h}*x*is better than

_{r}*x*but worse than

_{h}*x*, the new point is then obtained by contraction as

_{g}*x*= (1−

*β*)

*x*+

_{c}*β x*, where

_{h}*β*is a constant ranging between 0 and 1 (Fig. 4). When

*x*falls within either of these cases, the new point is added to the SSM by replacing

_{r}*x*. This process continues until a predetermined number of iterations are reached or the difference between the new point and the best point becomes smaller than a specified threshold value (

_{h}*ɛ*). Although this approach initially functions as an unconstrained optimization procedure, there are certain limitations regarding flexible parameters. To confine the search within a specified region, the SSM is combined with the bracket operator penalty method.

### 2.2 Non-Dominated Sorting Genetic Algorithm II.

The non-dominated sorting genetic algorithm II (NSGA-II) is a search algorithm commonly employed to address multi-objective optimization problems. It is done by identifying solutions in terms of non-dominated or lies on the Pareto front. This algorithm incorporates various techniques such as non-dominated sorting, elitism, and a crowding distance mechanism [23]. These mechanisms work together to ensure quick convergence and preserve diversity in the generated solutions. NSGA-II method is initiated by randomly generating an initial population of size *N*. The objective function is evaluated for this initial generation, and subsequently, the population is sorted using a non-dominated sorting approach. In non-dominated sorting, individuals are divided into different fronts (F1, F2, F3, and so on) based on their non-domination levels, as shown in Fig. 5. A lower non-domination level indicates a higher quality solution. Next, a binary tournament selection is employed to generate a population of parents from the current population. Two different solutions are randomly selected, and the one with a better non-domination rank is chosen. If the solutions have the same non-domination rank, their selection is based on the crowding distance, which measures the density of solutions in the neighborhoods, as illustrated in Fig. 6. Genetic operators such as crossover and mutation are then applied to create an offspring population of size *N*.

After evaluating the objective function for the offspring population, it is sorted using the non-dominated sorting approach. The next generation is created by copying the best solutions (elitism) [28,29] or the first *N* individuals from the mixed population of parents and offspring. The solutions that are not copied are discarded. The selection criterion is based on the non-domination rank first, followed by the crowding distance. If the size of the first non-dominated front (F1) is smaller than *N*, all members of F1 are included in the new population. The rest of the slots are filled with solutions from subsequent non-dominated fronts (F2, F3, and so on) until the available slots are exhausted. The crowding distance is then calculated for each potential solution from the last rank. This process continues until a user-defined number of generations are reached, where the algorithm terminates [23]. If the maximum number of generations is not attained, the process of generating offspring population continues, as depicted in Fig. 7.

### 2.3 Optimization Problem Formulation.

Due to the variability in the design objectives and the parameters across different engineering problems, it is impractical to apply a single formulation procedure universally. Usually, various techniques are employed for different design problems. The aim of the formulation process is to develop a mathematical model representing the optimal design that can subsequently be solved using an optimization algorithm [34]. The formulations for SSM and NSGA-II are described in Secs. 2.2.1 and 2.2.2.

#### 2.3.1 Simplex Search Method Formulation.

*C*represents the time-averaged torque coefficient and can be expressed as

_{T}*T*is the torque (N.m) produced by the blade profile,

*ω*is the angular velocity (rad/s),

*ρ*is the air density (kg/m

^{3}),

*A*is the swept area (m

^{2}), and

*V*is the inlet air velocity (m/s).

Further, *x* and *y* in Eq. (1) are coordinates of the intermediate point. The two end points of the blade profile are kept fixed. Hence, the coordinates of one of the intermediate points *x* and *y*, and OR are taken as the design variable to generate the blade profile using a natural cubic spline curve.

#### 2.3.2 NSGA-II Formulation.

The optimization problem aims to obtain an efficient (optimum) Savonius rotor using design variables that are the coordinate of three points (*x*_{1}, *y*_{1}), (*x*_{2}, *y*_{2}), and (*x*_{3}, *y*_{3}) along with TSR and *V*. The time-averaged *C _{P}* is the objective to be maximized. To get the blade profile, a natural cubic spline curve is used with three intermediate variable points, while the SG and OR are neglected (SG = OR = 0).

### 2.4 Process Flow Chart.

A matlab code is deployed as a workflow platform to execute the codes sequentially using design variables. The process flowchart is depicted in Fig. 8, where the process followed in both the optimization algorithms remains the same. The SSM requires four initial random points with different spacing within the search space. Those points are defined by conclusions (-*e*/2, 0) and (*L*-*e*/2, 0) to describe a single blade profile. Now, using these three points of a natural cubic spline curve, the blade skeleton is generated, as shown in Fig. 9. Assigning a thickness ‘*t*” by determining the inner and outer points of the blade profile, the geometry of the turbine blade is created. The points on the blade surface are extracted and saved in a specific format within a .txt file. This process is repeated for the other three variable points to generate the geometry of the remaining blade profiles.

The generated data points representing different geometries of the blade are imported into ansys icem cfd software to create the corresponding geometries. Once the geometry is imported, it is meshed further and is saved in a format compatible with ansys fluent for simulations [37,38]. The entire process is saved as a script file to facilitate geometry generation and meshing in subsequent iterations. The ansys fluent is launched, and the simulations are performed with predefined solution setups. The results are saved as text files for subsequent iterations. The entire process in ansys fluent is saved as a journal file [37,38]. The matlab code reads the text file containing the simulation results and the time-averaged *C _{T}*. The SSM is united with the bracket operator penalty method to create new points and improve the optimization function until the terminus. Once the final optimal blade shape is obtained, a direct comparison is made with the existing semicircular blade profile at various TSRs.

In the NSGA-II approach, the blade profile is defined using fixed endpoints (0,0) and (*L*,0) along with three intermediate points (*x*_{1}, *y*_{1}), (*x*_{2}, *y*_{2}), and (*x*_{3}, *y*_{3}). An offset “*t*” is applied to the inner and outer surface points to create the blade geometry, as illustrated in Fig. 10. The resulting points on the blade surface are extracted and saved in a specific format of a .txt file. Furthermore, three additional blade geometries are generated using the initially chosen variable points. Once the formatted data points of various blade geometries are stored, the ansys icem cfd software is employed to import the data points to generate the corresponding geometries. The meshing is done after importing the blade geometry, and then the meshed file is saved as a format compatible with ansys fluent, which further allows simulations. The entire process is automated and is then saved as a script file to streamline the geometry and meshing steps. Subsequently, the ansys fluent is launched, and the simulation setup is defined. The simulation results are saved in text files, and all the operations performed in ansys fluent are recorded in a journal file, enabling automation for future iterations. The matlab code reads the text file containing the simulation results and calculates the time-averaged *C _{T}*. The optimization algorithm is then executed to generate new blade profiles. This iterative process continues until the termination conditions are met.

## 3 Blade Geometry

The chord length (*L*) of a single blade is measured as 100 mm, while the thickness (*t*) is 2 mm. The rotor blade rotates around the center point (0, 0), having a diameter (*D*) equal to *2L* and undergoing periodic angular velocity (*ω _{z}*). The blades are separated by a gap known as overlap distance (

*e*). On the

*x*-axis, we set the inlet wind speed,

*V*= 7.30 m/s [23]. The corresponding angular velocity is represented on the

*y*-axis as

*ω*= 58.4 rad/s.

_{z}## 4 Computational Methodology

The computational domain and its boundary conditions, meshing, turbulence model, solver setup, numerical model validation, and grid and time independence tests are discussed in this section.

### 4.1 Computational Domain and Boundary Conditions.

The domain in this analysis is divided into two distinct sections. The first section corresponds to the surface where the rotor blades rotate (rotational zone), while the second section represents the stationary surface without the rotor blades (non-rotational zone), as shown in Fig. 11. The rotating surface is specifically designed to accommodate the turbine blades and is centered at the origin. The chord length of the turbine blade is denoted as *L*, and the rotational zone spans a length of 2*L*. The non-rotational surface surrounding the rotating zone has a rectangular shape. The upper and lower edges are located at a distance of 7.5*L* from the origin and are referred to as the “Symmetry” at the boundary. The vertical left edge is known as the “Inlet,” positioned 7.5*L* from the origin, while the right edge is referred to as the “Outlet,” located at a distance of 15*L* from the origin. The inlet boundary condition specifies an input velocity of *V* = 7.30 m/s [23] with a maximum turbulence intensity of 1%. The pressure outlet serves as the output boundary condition and has identical turbulence intensity. The blade surface is treated as a “wall” with rotational motion and non-slip conditions. An interface “*Int*” is defined amid the rotational and non-rotational sections. The outer zone of the turbine is selected to ensure that the boundary remains unaffected [34,35].

### 4.2 Meshing.

The finite volume method (FVM)-based ansys fluent solver is used. The computational domain is divided into small control volumes and discretized to solve the Reynolds-averaged Navier–Stokes (RANS) equations. Due to the subsonic flow, equal accuracy can be expected in both structured and non-structured meshes. However, the unstructured mesh can be easily made in the case of complex geometry [35]. In addition, due to the faster generation of the grids, the unstructured mesh with all tri-elements is used to discretize the computational domain [36]. The sliding mesh is chosen at the interface amid rotational and non-rotational zones with a rotational speed of *ω _{z}* = 58.4 rad/s and TSR = 0.8. The meshing of the rotational and non-rotational zones is shown in Figs. 12 and 13, respectively.

### 4.3 Turbulence Model.

The transient simulations in this study are conducted using ansys fluent, a solver based on FVM. The solver solves the unsteady RANS equations for 2D cases. The mathematical formulation of these equations is provided in Eqs. (5)–(10):

*k−ω*turbulence model is employed. The SST

*k−ω*turbulence model combines the strengths of the

*k−ɛ*model for freestream flows, and the

*k−ω*model for boundary layer flows, ensuring accurate prediction of flow separation with adverse pressure gradients. The transport equations for turbulent kinetic energy “

*k*” and specific dissipation rate “

*ω*” are used to obtain their values in Eqs. (11) and (12):

*ω*. Γ

_{k}and $\Gamma \omega $ indicate the effective diffusivity of

*k*and

*ω*, respectively. $S\omega $ denotes the dissipation of

*k*due to turbulence, which is a user-defined source term. The effective diffusivities for the

*k−ω*model are determined by Eqs. (13) and (14):

*σ*

_{k}and $\sigma \omega $ are the turbulent Prandtl numbers for

*k*and

*ω*, respectively.

*μ*

_{t}is the turbulent viscosity.

### 4.4 Solver Setup.

The turbulence kinetic energy and specific dissipation rate are discretized using a second-order upwind scheme. The pressure is discretized using a second-order scheme, and the gradient is calculated using the least-squares cell-based scheme. The pressure-velocity coupling is implemented using the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) scheme.

### 4.5 Numerical Model Validation.

The experimental data obtained by Blackwell et al. [39] are widely recognized as benchmark data in various numerical studies [32,34]. A direct comparison between the present simulation and the experimental data is made in the range of TSR from 0.6 to 1 (Fig. 14). The test model is selected to match the aspect ratio (AR *=* 1) of the numerical model. To replicate the experimental conditions, the Re in the numerical model is adjusted to 4.32 × 10^{5}. The numerical results are found to be consistent with the experimental data.

### 4.6 Grid and Time Independence Tests.

To assess the grid independence, the profile generated using one of the optimal points [23], specifically (60.84, 35.65) at OR = 0.10, is utilized. Three different mesh elements are tested, as shown in Table 1. It is seen that the *C _{T}* improves by 1.5% from mesh 1 to 2, and the additional time required does not exceed 2 h. From mesh 2 to 3, the

*C*increases by 0.21%, while the simulation time increases by 2.5 h. Mesh 2 is therefore chosen for this study to reduce the computational effort. For the time independence test, multiple points with different ORs are evaluated for up to 15 rotations of the rotor. The total time required for all rotor rotations is approximately 45 min. Figure 15 illustrates the relationship between the change in

_{T}*C*and the number of rotor rotations. It is observed that after 13 rotor rotations, the change in

_{T}*C*becomes less than 0.001. Hence, the

_{T}*C*is calculated depending on the sum of 13 rotor rotations. The time-averaged

_{T}*C*for all three blade profiles takes approximately 10 h to compute.

_{T}## 5 Results and Discussion

This section deals with the results obtained from optimizing the OR of the blade profile and analyzes the effect of TSR on the rotor's performance. The SSM is employed, with commonly used parameters *γ*, *β*, and *ɛ* set to 0.5, 2, and 10^{−4}, respectively. Apart from this, the NSGA-II application is also seen to have a multi-objective function optimization. The computational analysis is performed on a system with a 3.7 GHz Intel Xeon processor, 16 GB RAM, and Windows 10 Pro operating system (64-bit).

### 5.1 Results of Simplex Search Method.

To ensure a robust solution using the SSM, the process is repeated three times with different input values. This is necessary because it does not always guarantee the global maximum solution. Table 2 presents the results obtained from the three runs, with run 3 demonstrating superior performance among them. After 25 iterations, the process is terminated. The decision variable values for the optimum profile are determined to be (79.63, 33.6301, and 7.7), as depicted in Fig. 16. Consequently, the *C _{P}* achieved at this optimal point is the highest for this profile, as shown in Fig. 17, and reaches a plateau after 25 iterations.

Particulars | Test run 1 | Test run 2 | Test run 3 |
---|---|---|---|

Input 1 | (75,54,0.20) | (70,50,0.10) | (75,48,0.12) |

Input 2 | (50,58,0.28) | (72,49,0.11) | (70,60,0.15) |

Input 3 | (60,65,0.10) | (74,48,0.12) | (75,57,0.16) |

Input 4 | (50,60,0.15) | (60,65,0.25) | (80,65,0.21) |

Optimum profile | (51.07,68.52,9.2) | (74.56,45.7,11.2) | (79.63,33.63,7.7) |

C_{P} | 0.263 | 0.268 | 0.283 |

Particulars | Test run 1 | Test run 2 | Test run 3 |
---|---|---|---|

Input 1 | (75,54,0.20) | (70,50,0.10) | (75,48,0.12) |

Input 2 | (50,58,0.28) | (72,49,0.11) | (70,60,0.15) |

Input 3 | (60,65,0.10) | (74,48,0.12) | (75,57,0.16) |

Input 4 | (50,60,0.15) | (60,65,0.25) | (80,65,0.21) |

Optimum profile | (51.07,68.52,9.2) | (74.56,45.7,11.2) | (79.63,33.63,7.7) |

C_{P} | 0.263 | 0.268 | 0.283 |

#### 5.1.1 Effect of Tip Speed Ratio.

Figure 18 shows the effect of TSR over a range of 0.6–1. It is the ratio of the tangential speed of the rotor to the incoming air velocity. Thus, a direct comparison in terms of *C _{P}* with respect to TSR for the optimum profile and the semicircular blade is done. It is noted that the optimum profile performs better throughout the given TSR with a

*C*

_{P}_{max}of 0.283 at TSR = 0.8.

#### 5.1.2 Effect of Overlap Ratio.

The OR has a significant effect on the rotor performance. However, to ensure the degree of overlap can be used as a design variation, some simulations were performed where three different blade geometries were selected. The time-averaged *C _{T}* values are calculated for these geometric blades, within a range of OR = 0.05–0.2. With an increase of OR, the

*C*increases, and over time, it decreases, as shown in Fig. 19. However, the maximum OR for the maximum

_{T}*C*is different for all the geometries. From the simulations, it can be argued that the maximum OR for a maximum

_{T}*C*is different for different blade profiles. Therefore, it can be concluded that the OR can be used as a design variable for the Savonius rotor.

_{T}### 5.2 Results of NSGA-II.

To obtain an optimal solution using NSGA-II, both OR and SG are neglected, and the maximum population generation is set to 15. Here, *x*_{1}, *y*_{1}, *x*_{2}, *y*_{2}, *x*_{3}, *y*_{3}, TSR, and *V* are used as design variables, and objectives are set to maximize *C _{P}* and minimize

*V*. The process is terminated after 30 iterations. Table 3 presents the outcome; it can be seen that

*C*

_{P}_{max}= 0.286 is obtained at

*V*= 9.88 m/s, and the lowest

*C*= 0.233 is obtained at

_{P}*V*= 5.24 m/s. It can be observed that the value of TSR is around 0.8 in all the Pareto solutions. Therefore, a constant value of TSR is used for further research. Pareto-front solution is shown in Fig. 20. The

*C*

_{P}_{max}obtained from NSGA-II is 0.286, which is better than the optimum profile obtained from the SSM with a

*C*

_{P}_{max}of 0.283.

x_{1} | x_{2} | x_{3} | y_{1} | y_{2} | y_{3} | TSR | V | C_{p} |
---|---|---|---|---|---|---|---|---|

29.2648 | 31.273 | 86.9925 | 22.7141 | 13.5067 | 14.1910 | 0.8863 | 9.8825 | 0.28618 |

19.0833 | 43.878 | 82.6744 | 40.0843 | 49.8752 | 47.1893 | 0.9885 | 9.4994 | 0.27431 |

19.0525 | 47.2039 | 77.9454 | 42.7601 | 42.2905 | 46.0381 | 0.8845 | 9.2214 | 0.26572 |

29.1222 | 57.1494 | 70.8611 | 15.9498 | 26.7380 | 55.8596 | 0.8022 | 8.8139 | 0.25160 |

7.69722 | 50.0513 | 90.8805 | 24.0750 | 42.7457 | 30.1745 | 0.7821 | 8.4460 | 0.24830 |

18.7260 | 43.4606 | 74.9722 | 32.6226 | 65.4435 | 52.1598 | 0.9576 | 8.1756 | 0.24621 |

21.6664 | 63.2047 | 70.5717 | 49.7066 | 55.3950 | 40.5927 | 0.8417 | 7.8955 | 0.24538 |

27.8344 | 67.3597 | 89.88 | 44.9538 | 45.7320 | 36.4266 | 0.8649 | 7.5662 | 0.24531 |

9.82530 | 68.5732 | 85.8518 | 15.0353 | 55.2058 | 38.4007 | 0.8812 | 7.5194 | 0.24226 |

15.6694 | 43.5403 | 74.8353 | 31.0676 | 14.2709 | 41.3493 | 0.8122 | 7.0040 | 0.23933 |

16.4633 | 46.5600 | 84.6357 | 42.3958 | 63.9693 | 43.6390 | 0.8020 | 6.5056 | 0.23860 |

27.8344 | 61.7017 | 71.1542 | 45.3999 | 55.4399 | 39.3270 | 0.8815 | 5.9460 | 0.23423 |

27.7727 | 53.5156 | 94.4424 | 13.0602 | 60.1835 | 21.0550 | 0.832 | 5.7618 | 0.23405 |

5.9501 | 46.6966 | 84.6357 | 42.7563 | 68.4505 | 43.6390 | 0.8020 | 5.6236 | 0.23402 |

11.517 | 44.6211 | 79.5488 | 26.7564 | 64.0544 | 53.0017 | 0.8325 | 5.2482 | 0.23331 |

x_{1} | x_{2} | x_{3} | y_{1} | y_{2} | y_{3} | TSR | V | C_{p} |
---|---|---|---|---|---|---|---|---|

29.2648 | 31.273 | 86.9925 | 22.7141 | 13.5067 | 14.1910 | 0.8863 | 9.8825 | 0.28618 |

19.0833 | 43.878 | 82.6744 | 40.0843 | 49.8752 | 47.1893 | 0.9885 | 9.4994 | 0.27431 |

19.0525 | 47.2039 | 77.9454 | 42.7601 | 42.2905 | 46.0381 | 0.8845 | 9.2214 | 0.26572 |

29.1222 | 57.1494 | 70.8611 | 15.9498 | 26.7380 | 55.8596 | 0.8022 | 8.8139 | 0.25160 |

7.69722 | 50.0513 | 90.8805 | 24.0750 | 42.7457 | 30.1745 | 0.7821 | 8.4460 | 0.24830 |

18.7260 | 43.4606 | 74.9722 | 32.6226 | 65.4435 | 52.1598 | 0.9576 | 8.1756 | 0.24621 |

21.6664 | 63.2047 | 70.5717 | 49.7066 | 55.3950 | 40.5927 | 0.8417 | 7.8955 | 0.24538 |

27.8344 | 67.3597 | 89.88 | 44.9538 | 45.7320 | 36.4266 | 0.8649 | 7.5662 | 0.24531 |

9.82530 | 68.5732 | 85.8518 | 15.0353 | 55.2058 | 38.4007 | 0.8812 | 7.5194 | 0.24226 |

15.6694 | 43.5403 | 74.8353 | 31.0676 | 14.2709 | 41.3493 | 0.8122 | 7.0040 | 0.23933 |

16.4633 | 46.5600 | 84.6357 | 42.3958 | 63.9693 | 43.6390 | 0.8020 | 6.5056 | 0.23860 |

27.8344 | 61.7017 | 71.1542 | 45.3999 | 55.4399 | 39.3270 | 0.8815 | 5.9460 | 0.23423 |

27.7727 | 53.5156 | 94.4424 | 13.0602 | 60.1835 | 21.0550 | 0.832 | 5.7618 | 0.23405 |

5.9501 | 46.6966 | 84.6357 | 42.7563 | 68.4505 | 43.6390 | 0.8020 | 5.6236 | 0.23402 |

11.517 | 44.6211 | 79.5488 | 26.7564 | 64.0544 | 53.0017 | 0.8325 | 5.2482 | 0.23331 |

### 5.3 Wind Tunnel Results.

The purpose of the wind tunnel experiment is to compare the performance of an optimized rotor with a rotor featuring semicircular blades. The experiments were conducted in a wind tunnel (Fig. 21) with an open test section that has dimensions of 700 mm × 500 mm × 500 mm (length, width, and height). The rotor is mounted on a shaft with a diameter of 12 mm and a height of 600 mm. Two bearing houses, each measuring 140 mm × 140 mm, support the shaft and provide a hole at their centers to accommodate the ball bearings. An accurate precision anemometer with a ±2% accuracy is used to measure the wind speed. The rotational speed of the shaft is measured using a digital tachometer with a ±1% accuracy. To measure the torque, a mechanical load (F) is applied to the rotor, and this load can be varied [40].

The test rotor is manufactured using galvanized iron using the optimum decision variable obtained through SSM (79.63, 33.63) with an aspect ratio of 0.7 (Fig. 22). The velocity measurement is done using a weighted average technique at 5 × 5 matrix positions to get the required velocity. The area-weighted wind speed of 7.3 m/s was adopted. Moreover, a spring balance ranging from 0 kg to 2 kg with an accuracy of ±% is used to measure the torque [40,41]. Hence, the output power is calculated with respect to various TSRs.

#### 5.3.1 Performance Calculation.

*M*and

_{t}*M*) are recorded. The output torque (

_{s}*T*) can be calculated using the following equation:

_{O}*M*and

_{t}*M*are mass on tight and slack sides,

_{s}*g*is the gravitational acceleration, and

*R*is the radius of the pulley. The total input torque to the system is calculated as

_{p}*ρ*is the density of air,

*A*is the frontal area (

*D*×

*H*),

*V*is the velocity inlet, and

*R*is the radius of the turbine blade.

*C*) is calculated as

_{T}*C*is obtained, the

_{T}*C*can be calculated as

_{P}To take into account the uncertainties in the experiments, a sequential perturbation technique for *C _{P}*,

*C*, and

_{T}*T*is given by Moffat [42] is adopted that are 4.5%, 4.8%, 2.9%, respectively. The wind tunnel experiments for the optimized bladed rotor show a

*C*

_{P}_{max}of 0.128 at TSR = 0.67, while for the semicircular, the

*C*

_{P}_{max}is 0.112 at the same TSR, as shown in the Fig. 23. Thus, the optimized bladed rotor is showing its superiority by 14.28%.

## 6 Conclusions

The Savonius wind rotor is known for its ability to harness wind energy in compact spaces where large land areas are not available. It is favored for its simple design, small size, and lower installation and maintenance costs. However, its relatively lower efficiency has prompted extensive research efforts to improve its performance. Numerous numerical and experimental studies have been conducted over the years, resulting in the development of various blade profiles.

In this study, the objective is to design a blade profile for the Savonius wind rotor using a simplex search method combined with transient 2D CFD simulations. The obtained blade profile is then compared to the conventional semicircular profile at specific TSR levels to assess its suitability for practical applications. The optimization process involves 25 iterations of the chosen algorithm to arrive at an optimal blade profile. The optimal blade profile can be obtained by defining the natural cubic spline using the following points: (−96, 0), (−75.63, −32.63), (4, 0), (−4, 0), (75.63, 32.63), and (96, 0). The *C _{P}*

_{max}for the optimal and semicircular profiles are found to be 0.283 and 0.23, respectively. It is evident that the performance of the optimal profile surpasses that of the semicircular profile within the TSR range of 0.6–1. In the simplex search technique, only one variable point is used to define a blade geometry. If more variable points are given to define the blade geometry, a better solution could be obtained. Therefore, NSGA-II is applied with three intermediate points, TSR, and

*V*as design variables. It is observed that the

*C*

_{P}_{max}in the case of NSGA-II is 0.286 at

*V*= 9.88 m/s. While in

*SSM*, the

*C*

_{p,}_{max}is 0.283 at

*V*= 7.3 m/s. Thus, it can be concluded that the procedure used in the present work can be used to optimize the blade profiles of a Savonius rotor using different optimization algorithms.

Finally, the wind tunnel experiments have been conducted to validate the results of the generated blade profile. It is observed that the optimized blade profile shows a *C _{P}*

_{max}of 0.128 at TSR = 0.67, while for semicircular, the

*C*

_{P}_{max}is 0.112. Thus, the spline blade is found to be better in terms of performance by around 14%.

In the future, researchers can pursue different optimization strategies for the development of newer blades with spline curves. Additionally, the stability of the generated blades can be an interesting topic of future research.

## Acknowledgment

During the period of this study, the scholarship extended to the first author by the Indian Institute of Technology Guwahati, India, is gratefully acknowledged.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## Nomenclature

*e*=overlap distance, Mm

*t*=rotor thickness, mm

*u*=rotor speed, m/s

*A*=swept area, m

^{2}*D*=diameter of the rotor, mm

*L*=chord length of the blade profile, mm

*N*=rotational speed of a rotor, rpm

*V*=free stream wind speed, m/s

*C*=_{D}drag coefficient

*C*=_{L}lift coefficient

*C*=_{P}power coefficient

*C*=_{T}torque coefficient

*F*=_{L}lift force, N

*F*=_{D}drag force, N

- AR =
aspect ratio

- OR =
overlap ratio

- SG =
separation gap

- TSR =
tip speed ratio

### Greek Symbols

### Abbreviations

## References

*Ptilosarcus gurneyi*)