Effect Analysis of the V-Angle and Straight Edge Length on the Performance of V-Shaped Blades for a Savonius Hydrokinetic Turbine

Abstract

This study investigated the performance of Savonius hydrokinetic turbine blades through three-dimensional computational fluid dynamics simulations conducted at a fixed tip speed ratio of 0.87. A multi-factor experimental design was employed to construct 45 V-shaped rotor blade models, systematically examining the effects of a V-angle (30–140°) and straight-edge length (0.24 L–0.62 L) on hydrodynamic performance, where L = 25.46 mm (the baseline length of the straight edge). The results indicate that, as the V-angle and the straight-edge length vary independently, the performance of each blade first increases and then decreases. At TSR = 0.87, the maximum power coefficient (C_P) of 0.2345 is achieved by the blade with a 70° V-Angle and a straight edge length of 0.335 L. Pressure and velocity field analyses reveal that appropriate geometric adjustments can optimize the high-pressure zone on the advancing blade and suppress negative torque on the returning blade, thereby increasing net output. The influence mechanisms of the V-angle and straight-edge length variations on blade performance were further explored and summarized through a comparative analysis of the vorticity characteristics. This study established a detailed performance dataset, providing theoretical and empirical support for V-shaped rotor blade design studies and offering engineering guidance for the effective use of low-flow hydropower.

1. Introduction

With the intensification of the global energy crisis and environmental challenges, the unsustainability of traditional fossil fuels and their environmental hazards have prompted countries to accelerate the transition to clean and renewable energy sources [1]. Renewable energy sources, such as wind, solar, and hydro, are being widely utilized for green power generation. Compared to wind energy, water flow exhibits higher stability, providing a more consistent power generation capacity. Similarly, compared to solar energy, hydropower offers higher energy density and greater continuity. In comparison, hydropower emerges as a green energy source with greater development potential.

Hydrokinetic turbines can be broadly classified into horizontal-axis hydrokinetic turbines (HAHKT) and vertical-axis hydrokinetic turbines (VAHKT) based on the orientation of their main axis [2]. Vertical-axis turbines offer a more compact structure, increased installation flexibility, and potentially greater reliability. These characteristics make them particularly suitable for energy extraction from low-velocity rivers, tidal currents, and ocean currents. One of the important configurations of hydrokinetic turbines is the Savonius turbine, which was invented by engineer Sigurd J. Savonius in 1920 and was initially used for wind energy applications [3]. Later, the Savonius turbine was applied by researchers in the field of hydroelectric power generation, and it became known as the Savonius Hydrokinetic Turbine (SHT). Savonius-type hydrokinetic turbines (SHT) typically consist of two semi-cylindrical blades. It should be noted that the classical SHT design features a gap between the blades, and whether lift is generated depends on the blade geometry and the presence of this gap between rotor blades. In this study, we adopted a Savonius-type blade with an overlap ratio of zero, operating primarily based on the drag difference between the advancing and returning blades. Compared to lift-based turbines such as the Darrieus type, SHTs offer superior self-starting capabilities. However, on the other hand, the negative torque on the returning blades reduces the power output of the SHT, leading to a lower power coefficient [4], which severely restricts its further application.

Since the proposal of the SHT, research on its hydrokinetic applications and performance improvements has been ongoing. Researchers have employed various technical methods to enhance its performance. However, significant breakthroughs in SHT research were not achieved until the past decade.

Many researchers have attempted to enhance the hydrodynamic performance of SHT by optimizing the shape of the rotor blades. The study by Kerikous et al. [5] indicated that, compared to the standard blade design, the concave surface of the optimal Savonius blade is flatter but exhibits noticeable changes near the blade tip, forming a hook-like shape, while the convex surface remains almost unchanged, maintaining a near-semicircular profile. The optimal blade shape increased the average power coefficient (C_p) by 12% at a tip speed ratio (TSR) of 1.1, corresponding to the best value of the standard design, and by nearly 15% at TSR = 1.2. Talukdar et al. [6] conducted a parametric study on SHT based on the blade number, blade shape, and immersion level. The results showed that, at 60% and 80% immersion levels, the two-bladed semicircular SHT exhibited the highest maximum power coefficient (C_Pmax), followed by the three-bladed semicircular SHT and the two-bladed elliptical SHT. At the corresponding immersion levels of 60% and 80%, the three-bladed turbine outperformed the two-bladed elliptical turbine by 60% and 42.8%, respectively. At similar immersion levels, the two-bladed semicircular SHT demonstrated 28.6% and 30% better performance than the three-bladed semicircular SHT. Khan et al. [7] proposed a new blade profile to enhance the performance of conventional SHT blades. The new profile is based on the S1048 airfoil section, whose geometry consists of straight and curved segments. The straight section provides a larger moment arm and clearance flow, while the curved section reduces the adverse effects of negative torque on the rotor. The results showed that using the new blade profile increased the maximum power coefficient of the conventional design by 14% at a tip speed ratio of 1.

In addition, researchers have explored the influence of key characteristic parameters on the SHT performance by adjusting other variables. Yao et al. [8] investigated the effects of different overlap ratios on SHT performance through experiments and numerical simulations. The results indicated that the maximum power capture efficiency was achieved when the overlap ratio was 0.15, which was 19% and 22% higher than those with overlap ratios of 0 and 0.3, respectively. Jeon et al. [9] studied helical Savonius turbines with different end plates. The results demonstrated that the end plate effect was significant, with the use of upper and lower circular end plates increasing the power coefficient by 36% compared to designs without end plates. Similarly, Payambarpour et al. [10] investigated the aspect ratio (AR) of the turbine. Their findings revealed that increasing the aspect ratio generally reduces turbine leakage and increases pressure on the rotor blades, thereby improving overall efficiency.

Many researchers have also investigated the use of various forms of deflector plates within the flow path of SHTs to enhance their performance [11,12]. Another design of interest is the hybrid rotor, which combines the Darrieus and Savonius rotors in various configurations. This hybrid rotor integrates the excellent self-starting capability of the Savonius rotor with the high power advantages of the Darrieus rotor. In the study by Abdelsalam et al. [13] on the hybrid rotor, the effects of the attachment angle, radius ratio, and the number of Darrieus blades were tested and explored. The results showed that, at a low radius ratio of β = 0.27, the best attachment angle was found to be ϕ = 30° for both two-bladed and three-bladed hybrid rotors. The effect of the radius ratio was then tested at ϕ = 30°. It was observed that the influence of the radius ratio on hybrid rotor performance was greater than that of the attachment angle. The power coefficient at the optimal radius ratio of β = 0.43 at ϕ = 30° was found to be close to that of the Darrieus rotor.

Overall, the blade is the core component of a hydrokinetic turbine, and its performance and stability have a significant impact on the turbine’s overall performance. In general, optimizing the blade shape is particularly fundamental and universal compared to modifying other geometric parameters or employing enhancement techniques. To improve the performance of Savonius rotor blades, previous researchers have conducted numerous experimental, numerical, and theoretical studies aimed at reducing the fluid drag acting on the returning blade or increasing the force acting on the advancing blade.

In 2021, Shashikumar et al. [14] first proposed a rotor blade with a V-shaped profile, in which both the advancing and returning blades were composed of two straight edges and a small arc. This design aimed to reduce the negative torque on the returning blade by minimizing the arc width and introducing two tangent straight edges at the arc of the returning blade profile, thereby decreasing drag. Additionally, the modified V-shaped rotor blade facilitated fluid flow toward the advancing blade side, thereby enhancing the positive torque generated by the advancing blade. In their study, when the blade’s straight edge length was 0.43 L, the V-angle was 90°, the curvature radius was 0.56 L, and the tip speed ratio (TSR) was 0.87, the optimized blade achieved a Cp value of 0.22, surpassing the 0.176 of the conventional semi-circular rotor blade. Subsequently, they further investigated the effects of parameters such as the V-angle, aspect ratio, and overlap ratio of the V-shaped rotor blade on the hydrokinetic turbine’s performance. Overall, Shashikumar et al. conducted a relatively extensive investigation into the key parameters of their proposed novel V-shaped blade design, offering a simple, feasible, and effective blade configuration. Their findings demonstrate the significant research value and application potential of V-shaped blades in enhancing the performance of Savonius hydrokinetic turbines. Therefore, it is worthwhile pursuing further studies on V-shaped blades based on this foundation.

Regarding the V-angle, Shashikumar et al. investigated six V-shaped blades with a straight-edge length of 0.43 L, identical curvature radii, and varying V-angles, ranging from 40° to 90°. Their results indicated that a V-angle of 80° yielded superior performance. However, in their study, the blade diameter was altered while maintaining constant straight-edge length and curvature radius. Considering that blade diameter plays a more fundamental role in the rotor’s energy capture capability—directly determining the fluid interaction area and significantly influencing torque generation—its impact on performance may outweigh that of local changes in the curvature radius. Therefore, the present study maintained a fixed blade diameter to explore the influence of the V-angle on performance from an alternative perspective, ensuring continuity of parameters by covering a wider V-angle range from 30° to 140°.

Given the unknown coupling effect between the straight-edge length and V-angle, and in order to enhance the generalizability of the findings, this study selected five different straight-edge lengths to form various blade models in combination with different V-angles. This enabled the investigation of how the V-angle affects the blade performance under varying straight-edge lengths, as well as, conversely, how the straight-edge length influences the performance under different V-angles. Lastly, acknowledging the critical role of vortices in hydrokinetic turbine research, this study further attempted to elucidate and summarize the interaction mechanisms between the blade geometry and performance through a vorticity-based analysis.

This study focused on the profile design of V-shaped rotor blades, aiming to systematically investigate the effects of geometric parameters, such as the V-angle and straight-edge length on the performance of V-shaped rotor blades, without relying on other performance enhancement techniques (e.g., deflector plates and hybrid rotors) or modifying other key parameters (e.g., blade count, aspect ratio, overlap ratio, and endplate configuration). The research objectives included identifying the impact trends and mechanisms of these geometric parameters on the hydrodynamic performance of V-shaped rotor blades and seeking an optimized blade profile with superior hydrodynamic characteristics. Specifically, this study used the CFD software STAR-CCM+ (2019) to perform three-dimensional numerical simulations. Nine different V-angles (ranging from 30° to 140°) and five different straight-edge lengths (0.24 L, 0.335 L, 0.43 L, 0.525 L, and 0.62 L) were set for the V-shaped rotor blades, with the two sets of parameters cross-combined to produce 45 V-shaped rotor blades with different profile shapes. Due to the unique geometric configuration of V-shaped blades, the V-angle and straight edge length (L_v) were not suitable for independent investigation. The same V-angle can result in significantly different blade areas and structural forms when paired with different L_v values, and vice versa. Therefore, a combined analysis of these two parameters enables a more systematic and controlled research design, as well as facilitates the exploration of their potential coupled effects. Compared to previous studies on V-shaped rotor blade profiles, this study has a denser sample size and a broader parameter coverage, which is expected to reveal more universal design principles, providing theoretical support and engineering references for their optimization design. Crucially, this study further revealed the influence mechanisms of the two structural parameters of V-shaped blades on the rotor performance through vorticity analysis. In addition, the findings of this study on the V-shaped rotor blade profiles are expected to be applied in the design of other similar Savonius hydrokinetic turbine rotor blades.

The main structure of this paper is as follows: Section 2 outlines the key equations used in this study and, based on a multi-factor experimental design method, proposes 45 V-shaped rotor blade models with different combinations of a V-angle and straight-edge length. Section 3 provides a detailed description and verification of the numerical simulation methods employed in this study, including domain decomposition, mesh generation strategy, boundary condition settings, and time step selection. The accuracy and reliability of the numerical model were validated through comparisons with existing experimental data, ensuring the credibility of the subsequent research results. Section 4 presents the numerical simulation results and provides a systematic analysis of the rotor blade performance based on the characteristics of the pressure and velocity fields. Section 5 discusses the vorticity contour plots, which are particularly important for Savonius turbines, in an effort to reveal the mechanisms by which structural parameter variations affect rotor blade performance.

2. Relevant Equations and Construction of the V-Shaped Rotor Blade Model

2.1. Relevant Equations

The tip speed ratio (TSR) is defined as the ratio of the rotational tangential velocity of the rotor blade to the incoming flow velocity:

$$TSR={\displaystyle \frac{\omega D_r}{2V_w}},$$

(1)

where ω represents the angular velocity of the rotor blade, D_r denotes the rotor blade diameter, and V_w signifies the incoming flow velocity.

The torque coefficient (C_T) and power coefficient (C_P) are key parameters for evaluating the performance of hydrokinetic turbines, and they are commonly used in turbine performance studies:

$$C_T={\displaystyle \frac{T_{Rotor}}{T_{Available}}}={\displaystyle \frac{T_{Rotor}}{\left(\frac{1}{2}{\rho }_w{A}_r{\left(V_w\right)}^2\right)\frac{D_r}{2}}}$$

(2)

$$C_p={\displaystyle \frac{p_{Rotor}}{p_{Available}}}={\displaystyle \frac{T_{Rotor}\omega }{\left(\frac{1}{2}{\rho }_w{A}_r{\left(V_w\right)}^3\right)}}$$

(3)

where A_r is defined as the swept area of the rotor blades, A_r = D_rH_r, T_Rotor represents the rotor torque, and ρ_w denotes the density of water.

2.2. Construction of the V-Shaped Rotor Blade and SHT

As shown in Figure 1, the advancing and returning blades of the V-shaped rotor blade consisted of two straight edges and a small circular arc. The SHT in this study was equipped with two circular end plates, with its fundamental geometric parameters detailed in

Table 1. Geometric parameters of the SHT.

Parameters	Values
Aspect ratio (AR) (dimensionless)	0.7
Diameter of the rotor (Dr) (mm)	72
Diameter of the endplates (Dep) (mm)	79.2
Height of the rotor (Hr) (mm)	50.4
L (mm)	25.46
Overlap ratio (OR) (dimensionless)	0.0
Thickness of rotor blade/thickness of the end plate (t) (mm)	1

.

Figure 2 presents the geometric profiles of the rotor blades selected in this study, with specific parameters detailed in

Table 2. The geometric parameters of the rotor blades that were selected in this study.

Blade Profiles	θ	Lv	Rb
V1	30°	0.24 L	0.667 L
V2	30°	0.335 L	0.642 L
V3	30°	0.43 L	0.616 L
V4	30°	0.525 L	0.591 L
V5	30°	0.62 L	0.565 L
V6	50°	0.24 L	0.668 L
V7	50°	0.335 L	0.623 L
V8	50°	0.43 L	0.579 L
V9	50°	0.525 L	0.535 L
V10	50°	0.62 L	0.490 L
V11	70°	0.24 L	0.695 L
V12	70°	0.335 L	0.628 L
V13	70°	0.43 L	0.561 L
V14	70°	0.525 L	0.495 L
V15	70°	0.62 L	0.438 L
V16	90°	0.24 L	0.769 L
V17	90°	0.335 L	0.664 L
V18	90°	0.43 L	0.569 L
V19	90°	0.525 L	0.474 L
V20	90°	0.62 L	0.379 L
V21	100°	0.24 L	0.813 L
V22	100°	0.335 L	0.700 L
V23	100°	0.43 L	0.587 L
V24	100°	0.525 L	0.474 L
V25	100°	0.62 L	0.360 L
V26	110°	0.24 L	0.889 L
V27	110°	0.335 L	0.754 L
V28	110°	0.43 L	0.618 L
V29	110°	0.525 L	0.482 L
V30	110°	0.62 L	0.347 L
V31	120°	0.24 L	0.998 L
V32	120°	0.335 L	0.833 L
V33	120°	0.43 L	0.669 L
V34	120°	0.525 L	0.504 L
V35	120°	0.62 L	0.340 L
V36	130°	0.24 L	1.158 L
V37	130°	0.335 L	0.954 L
V38	130°	0.43 L	0.750 L
V39	130°	0.525 L	0.547 L
V40	130°	0.62 L	0.343 L
V41	140°	0.24 L	1.407 L
V42	140°	0.335 L	1.146 L
V43	140°	0.43 L	0.885 L
V44	140°	0.525 L	0.624 L
V45	140°	0.62 L	0.363 L

. In the study by Shashikumar et al. on V-shaped blades, the V-angle did not exceed 90°. Although excessively large V-angles may lead to performance degradation, this study ultimately selected nine distinct V-angle parameters, ranging from 30° to 140°, in order to ensure both the completeness of performance evaluation and the breadth of pattern investigation.

As for the straight-edge length, Shashikumar adopted parameter intervals of 0.19 L. Considering that potentially superior straight-edge lengths may exist within this interval—and to further refine the scope of investigation—this study selected five straight-edge lengths centered around the previously reported optimal value of 0.43 L, namely 0.24 L, 0.335 L, 0.43 L, 0.525 L, and 0.62 L. These combinations ultimately yield a total of 45 distinct blade geometries.

This study focused on two key parameters—the V-angle and straight-edge length—due to their pronounced influence on the blade profile. The arc curvature radius was then defined by constructing a circular arc tangential to the straight edge. Discussions regarding the curvature radius of the arc and the blade diameter have been addressed earlier and will not be repeated here.

3. Simulation Method

This study employed the CFD software STAR-CCM+ (2019) for numerical simulations, utilizing a three-dimensional approach to enhance simulation accuracy.

3.1. Three-Dimensional Computational Domain and Solver Settings

The computational domain used in the simulation must be sufficiently large to accurately capture the wake effects and flow characteristics around the hydro turbine. At the same time, it should not be excessively large so as to avoid unnecessary mesh elements and computational costs [15,16]. In this study, the computational domain was divided into a stationary region and a rotating region. Based on prior research [17], the size of the rotating region was set to 1.5 times the blade diameter.

The turbulence model employed was the unsteady SST k-ω model. The specific parameters of the computational domain and the boundary conditions are detailed in Figure 3 and

Table 3. The geometric parameters and boundary conditions of the computational domain.

Parameters	Values
Width of the domain (mm)	215
Depth of the domain (mm)	70
Length of the domain (mm)	612
Front face	Velocity inlet
Rear face	Pressure outlet
Left, right, and bottom face	No-slip wall
Top face	Symmetry
Rotor blade	No-slip wall

. The velocity inlet was defined as a uniform inflow with a velocity of V_w = 0.3090 m/s, while the pressure outlet was set to zero gauge pressure. The left, right, and bottom boundaries of the computational domain, as well as the rotor blade surfaces, were designated as no-slip walls, whereas the top boundary was set as a symmetry condition. The density and dynamic viscosity of water were taken as ρ_w = 998.2 kg/m³ and μ_w = 0.001003 kg/(m·s), respectively.

Previous studies [18] have indicated that, within the same range of the tip speed ratio (TSR), hydrokinetic turbines equipped with rotor blades of varying geometric profiles exhibit similar performance trends. This study primarily investigated the influence of different profile parameters on the rotor blade performance and sought to uncover the underlying mechanisms, rather than focusing solely on identifying the blade geometry that yields the maximum power coefficient. To enhance computational efficiency, a uniform TSR of 0.87 was adopted throughout this study.

3.2. Mesh Settings and Grid Independence Analysis

As shown in

Table 4. The relationship between the C_T value and turbine revolutions.

Number of Revolutions	CT
1	0.20295154
2	0.26368339
3	0.25707207
4	0.25828949
5	0.25760530
6	0.25737543
7	0.25727734

, when the hydro turbine reached the fifth revolution, the computational results stabilized. The torque coefficient (C_T) differed by only 0.1274% compared to the seventh revolution and by 0.089% compared to the sixth revolution. To enhance computational efficiency and reduce simulation time, this study selected data from the fifth revolution for analysis.

As shown in Figure 4, this study adopted a polyhedral mesh, which offers advantages over traditional tetrahedral or hexahedral meshes by effectively reducing the total number of mesh elements, minimizing discretization errors and improving both computational accuracy and solution efficiency. To improve the quality of the mesh generation, three concentric cylindrical control volumes with gradually increasing radii were created around the rotating domain where the blades were located, enabling a progressively refined mesh. As a result, the base mesh size within the rotating domain was reduced to 0.0011 m. A total of 15 boundary layers were generated on the blade surface, with the first layer thickness set to 0.00004 m and a growth rate of 1.2, ensuring that the dimensionless wall distance y+ remained below 1. This is significant for the SST k-ω model simulations of the high separation rates and adverse pressure gradients around the blades [19]. The y+ values in this study were verified as remaining below 1 to ensure simulation accuracy.

Initially, this study employed a consistent meshing strategy to generate five sample cases with varying mesh base sizes for the same rotor blade (

Table 5. The relationship between the C_T value and grid resolution.

Number of Mesh Elements	CT
848,371	0.25376821
976,787	0.25664925
1,104,061	0.25760530
1,261,968	0.25664066
1,324,779	0.25810950

). These cases were analyzed comparatively to determine the optimal mesh resolution for subsequent simulations. The results indicate that, as the mesh number increased from 848,371 to 976,787 and then to 1,104,061, the variation in the torque coefficient (C_T) was significant, demonstrating that improved mesh resolution had a substantial impact on the computational accuracy. However, further increasing the mesh number to 1,261,968 and 1,324,779 yielded negligible improvements in accuracy while significantly increasing computational time. Therefore, to balance solution accuracy and computational efficiency, this study ultimately selected the mesh configuration corresponding to 1,104,061 elements for subsequent simulations.

3.3. Time Step Independence Test

As shown in Figure 5, five different time step sizes (TSS) were selected in this study to conduct a time step independence test. As the time step size decreased from 10°/step to 1°/step, the C_T value exhibited significant variations, indicating a continuous improvement in computational accuracy. However, when the time step size was further reduced to 0.5°/step, no noticeable change in the C_T value was observed, while the computational time doubled. Therefore, after balancing computational accuracy and efficiency, a time step size of 1°/step was chosen for subsequent simulations. This result is consistent with the findings of Paniagua-García et al. [20].

3.4. Validation Test

Finally, in this simulation, the tip–speed ratio (TSR) was uniformly set to 0.87, and the analysis was conducted based on the fifth revolution of the turbine. The SST k-ω model was used as the turbulence model, along with polyhedral meshes and a time step of 1°/step. To validate the results of this study, a comparison was made with the numerical simulation results and experimental data for Savonius turbines published by Shashikumar et al. in 2021 [14], as shown in Figure 6. The results exhibit similar patterns and trends, and the error was within an acceptable range.

4. Results and Analysis

This study focused on investigating the influence and underlying mechanisms of blade structural variations on the hydrodynamic performance rather than identifying the optimal performance conditions under varying tip speed ratios (TSR). To ensure consistency in operating conditions, the TSR was fixed at 0.87, and performance differences were solely evaluated under this constant TSR scenario. Due to space constraints,

Table 6. The torque coefficient of each rotor blade.

Blade Profiles	CT	CTmax	CTmin
V1	0.242566337	0.6546328	−0.089582825
V2	0.247855216	0.661787884	−0.083964303
V3	0.239921559	0.640023203	−0.11053581
V4	0.229249273	0.638748754	−0.13568807
V5	0.222587409	0.663160452	−0.184714392
V6	0.252616648	0.614070257	−0.137484522
V7	0.25892544	0.669489001	−0.155511112
V8	0.257668234	0.679214801	−0.149879528
V9	0.25004171	0.680787943	−0.156906798
V10	0.245951342	0.700596528	−0.172529828
V11	0.258713344	0.705168723	−0.231825159
V12	0.26962249	0.708495215	−0.238413719
V13	0.262207084	0.685037761	−0.241734023
V14	0.254714614	0.675066559	−0.258228473
V15	0.244683801	0.676022745	−0.268634885
V16	0.257216686	0.786804497	−0.299764761
V17	0.260257436	0.785154444	−0.304685629
V18	0.25760531	0.784082547	−0.317457482
V19	0.245568179	0.754261913	−0.332825954
V20	0.237418791	0.728473908	−0.351694012
V21	0.255133368	0.811616406	−0.344916672
V22	0.253861862	0.822238634	−0.351238653
V23	0.244987914	0.809294746	−0.359846362
V24	0.23118938	0.801896429	−0.37124379
V25	0.221350677	0.776664799	−0.384322705
V26	0.252078993	0.831104896	−0.397780171
V27	0.252443845	0.839225322	−0.396788569
V28	0.239789872	0.838231181	−0.402157824
V29	0.223845783	0.843362153	−0.407815823
V30	0.208301795	0.833826221	−0.418427912
V31	0.24956291	0.83964562	−0.453250485
V32	0.252489608	0.84583865	−0.452953251
V33	0.247674471	0.860534661	−0.451795269
V34	0.231104078	0.856631407	−0.457099904
V35	0.207037673	0.849671646	−0.456080883
V36	0.231356401	0.826635033	−0.507867922
V37	0.239114543	0.830458193	−0.509754419
V38	0.238852366	0.843556394	−0.512434478
V39	0.230569656	0.857179842	−0.516457376
V40	0.216954745	0.854256357	−0.519259147
V41	0.207713008	0.83399491	−0.554213854
V42	0.216039071	0.826543197	−0.555307727
V43	0.216386358	0.824920885	−0.560569322
V44	0.211740453	0.822442551	−0.565422754
V45	0.205452226	0.819598824	−0.570955838

only includes the key parameters related to the torque coefficient (C_T) for each rotor blade. Although the power coefficient (C_P) is mathematically the product of C_T and TSR—and the TSR remained constant in this study—the C_P values for several representative blades were briefly discussed to enable comparison with the findings from other studies. For instance, the best-performing blade configuration, V₁₂, exhibited a C_P of 0.2345, while the worst-performing blade, V₄₅, recorded a C_P of 0.1787, indicating a 31% improvement. This result demonstrates that targeted structural parameter optimization can substantially enhance the hydrodynamic performance of V-shaped blades. Based on the above considerations, the torque coefficient C_T was adopted as the primary performance metric in subsequent analyses. Under a fixed tip speed ratio (TSR), the torque coefficient (C_T), like the power coefficient (C_P), effectively captured the performance differences and variation patterns among different blades. Meanwhile, by keeping the TSR constant and avoiding the introduction of the tip speed ratio as an additional factor, the blade performance can be more directly represented under the same TSR conditions—making them better aligned with the analytical focus and objectives of this study.

In this paper, C_P and C_T refer to the average power coefficient and average torque coefficient over one full rotation of the SHT, respectively, and C_Tmax and C_Tmin represent the maximum and minimum instantaneous torque coefficients within a single rotation cycle of the SHT.

This section will investigate the impact of rotor blade profile variations on its performance from the two dimensions of V-angle and the length of the V-edge, as well as attempt to explore the reasons behind these effects by analyzing pressure contour plots and velocity contour plots.

As shown in Figure 7, a coordinate system for the rotor blades was defined for the analysis in Section 4 and Section 5. In this system, the return blade was defined as the end of the rotor blade where the rotation direction is opposite to the inflow direction, while the advancing blade refers to the end where the rotation direction is consistent with the inflow direction.

4.1. The Effect of Variations in the V-Angle on the Performance of the Rotor Blade

Shashikumar et al.’s previous study on the V-angle of V-shaped rotor blades indicates that, for a V-shaped rotor blade with L_v = 0.43 L, optimal performance occurs at a V-angle of 80°. Additionally, the effect of the V-angles of 70° or 80° on performance is minimal across different TSR values. As shown in Figure 8, among the five sets of rotor blades with varying straight edge lengths in this study, the optimal V-angle was 50° for the rotor blade with L_v = 0.62 L, while the optimal V-angle for the other groups was consistently at 70°. Integrating the findings of this study with those from previous research, it is evident that a V-angle of 70° is an optimal choice for V-shaped rotor blades. However, due to the limitations of the sample size in this study, the actual optimal V-angle for rotor blades with different straight edge lengths should be close to the aforementioned theoretical optimal angle. Moreover, the minimum V-angle of 30° and maximum V-angle of 140° selected in this study exhibited relatively weaker performances. Therefore, in future research on the performance of V-shaped rotor blades, the value of studying V-angles outside this range appears diminished; thus, the research focus on V-angles can be further refined based on these results.

The structural differences in rotor blades cause the forces exerted by the fluid on them to change continuously during rotation [21]. One manifestation of this variation is that the C_T values of the rotor blades change with the rotation azimuth angle. For rotor blades with different geometries, the azimuth angles at which the C_Tmax and C_Tmin values occur may differ [22,23].

As shown in Figure 9, for the V-shaped rotor blades in this study, the C_Tmax values were observed around azimuth angles of 30° and 210°, while the C_Tmin values occurred at approximately 120° and 300°. The performance of the rotor blades showed a similar relationship between the azimuth angles of 120° to 300° and 300° to 120°, which is related to the antisymmetry of the rotor blade structure.

In this study, the five sets of rotor blades with different straight edge lengths demonstrate that C_Tmax increases with the increase in the V-angle, while C_Tmin decreases as the V-angle increases. Notably, for the 45 rotor blades with different geometries, the C_T values were close to zero when the azimuth angle was 90°, which may also be related to the antisymmetry of the rotor blade structure.

4.2. The Effect of L_v Variation on Rotor Blade Performance

Compared to the previous studies on V-shaped rotor blades by Shashikumar et al., this study identified 0.335 L as the straight edge length that provides the greatest performance advantage for V-shaped rotor blades. As shown in Figure 10, for nine sets of rotor blades with different V-angles, the performance of the rotor blades showed a trend of first increasing and then decreasing as the L_v increased, with the C_T peak generally occurring at L_v = 0.335 L and the trough at L_v = 0.62 L. It can be observed that a moderate increase in the straight edge length helps improve rotor blade performance, which aligns with the initial hypothesis proposed by Shashikumar et al. when constructing V-shaped rotor blades. However, excessive increases in straight edge length lead to significant negative impacts on the rotor blade performance.

Through the above discussion, this study preliminarily identified the turning point in the effect of L_v on the performance of V-shaped rotor blades to be around L_v = 0.335 L. The following sections will further investigate the mechanisms by which L_v influences the performance of V-shaped rotor blades through pressure and velocity contour maps, explaining the underlying reasons for this trend and providing a theoretical basis and support for further optimization of the straight edge length of V-shaped rotor blades.

As shown in Figure 11, for the two sets of rotor blades with V-angles of 30°and 50°, the variation in L_v significantly affected the relationship between the C_T values and the rotational azimuth angle. However, for the other seven sets of rotor blades with different V-angles, the variation in L_v had a smaller impact on the relationship between the C_T values and the rotational azimuth angle.

4.3. Pressure Contour Plots

From the previous analysis, it is evident that the C_Tmax of the rotor blade occurred at approximately α = 30°, while C_Tmin was observed at approximately α = 120°. When α = 90°, the rotor blade was perpendicular to the flow direction, and the C_T value of the rotor blade approached zero. These specific angles have been widely studied in previous research [24,25]. The selection of these distinctive rotational azimuths for study was instrumental in elucidating how changes in the blade profile influence the performance of rotor blades.

The Savonius turbine is a drag-type turbine. When water flow impacts the blades, a high-pressure zone forms on the leading surface of the blades and a low-pressure zone forms on the trailing surface, generating a shape-induced drag difference. This results in an imbalance of forces on both sides of the blade, generating rotational torque. As shown in Figure 12, the pressure in the upstream region of the flow field was significantly higher than that in the downstream region. This pressure differential produces energy, which, in turn, drives the rotation of the rotor [26]. Specifically, under water flow, the leading surface of the advancing blade forms a high-pressure zone, while the trailing surface forms a low-pressure zone. The pressure difference between these two surfaces generates a positive torque, which favors blade rotation. In contrast, especially at α = 90° and 120°, the combined influence of the water flow impact and blade rotation results in the formation of a high-pressure region on the leading (inflow-facing) surface of the returning blade and a low-pressure region on its trailing (downstream) surface. The pressure differential between these two sides generates a torque opposite to the blade’s rotational direction, which negatively affects blade performance. This phenomenon is, therefore, referred to as negative torque. The difference between the positive and negative torques directly reflects the performance of the rotor blade and is termed the net torque.

By comparing the pressure contour plots of the V₁₂ and V₄₂ rotor blades shown in Figure 12, both blades have a straight edge length of 0.335 L, with V-angles of 70° and 140°, respectively. As the V-angle increases, the profile of the rotor blade becomes flatter. At α = 30°, compared to V₁₂, V₄₂’s advancing blade formed a higher and more extensive high-pressure zone on the leading surface, resulting in a stronger positive torque in V₄₂. Consequently, the C_T value at this rotational azimuth was higher for V₄₂ than for V₁₂.

At α = 90°, both rotor blades exhibited a distinct low-pressure zone near the blade tip on the trailing surface of the advancing blade. A noticeable high-pressure zone was also observed on the leading surface of the returning blade. This resulted in both blades generating strong positive and negative torques. Due to the antisymmetry of the rotor blade structure, the magnitudes of the positive and negative torques were similar but opposite in sign, which caused the C_T values of both rotor blades to approach zero at this rotational azimuth. Consequently, the effect of variation in the V-angle on the torque coefficient was minimal at this point.

At α = 120°, both the V₁₂ and V₄₂ rotor blades exhibited a clear high-pressure zone on the leading surface of the returning blade, which generated a negative torque that opposed rotor blade rotation. Due to the flatter profile of V₄₂, the contact area between the blade and the incoming flow was larger, which made it easier for the water flow to generate a stronger and longer-lasting impact, resulting in a stronger negative torque. This explains why, at this rotational azimuth, both V₁₂ and V₄₂ exhibited negative torque coefficients, with V₄₂ having a lower torque coefficient.

Overall, the variation in the V-angle significantly affected the relationship between the rotor blade’s C_T value and the rotational azimuth.

By comparing the pressure contour plots of V₁₀ and V₄₅, a similar pattern to the analysis of V₁₂ and V₄₂ was observed.

The V-angles of the three rotor blades, V₃₆, V₃₇, and V₄₀, were all 130°, with straight edge lengths of 0.24 L, 0.335 L, and 0.62 L, respectively. As shown in Figure 13, the pressure contour plots of the three rotor blades illustrated their pressure distributions at different rotational azimuth angles. Since the differences in the torque coefficients of these three rotor blades were mainly observed between α = 50–70° and α = 230–250°, α = 60° was chosen here instead of the commonly used α = 90° to better analyze the performance differences of the three rotor blades. At α = 30° and 120°, the pressure contour plots of the three rotor blades showed no significant differences in the distribution range and intensity of high-pressure and low-pressure regions. This also explains why, in Figure 11, the C_Tmax and C_Tmin values of the nine groups of rotor blades with the same V-angle but different straight edge lengths remained almost identical, respectively. Compared to the variation in V-angle, the changes in straight edge length had a limited effect on the relationship between the C_T values of the rotor blades and the rotational azimuth angle.

At α = 60°, the torque coefficients C_T for V₃₆, V₃₇, and V₄₀ were 0.55609, 0.56207, and 0.38869, respectively. As the straight edge length L_v increased from 0.24 L to 0.335 L, no significant changes were observed in the high-pressure region on the upstream side of the returning blade. When L_v further increased from 0.335 L to 0.62 L, the extent of this high-pressure region slightly expanded, but the change remained inconspicuous and had minimal impact on performance. Meanwhile, with L_v extended to 0.62 L, a new and relatively distinct low-pressure region formed on the downstream surface of the returning blade V₄₀, which intensified the negative torque that opposes rotation, thereby further reducing the blade’s performance.

4.4. Velocity Contour Plots

As shown in Figure 14, two main characteristic regions can be clearly observed in the velocity contour plots: one is the high-speed region near the leading edge of the advancing blade, which contributes significantly to torque and power output; the other is the wake region caused by flow separation, typically occurring at the tip of the return blade. When the flow separation region becomes too large, it increases energy losses, thereby affecting the performance of the rotor blades. A similar method for delineating characteristic regions in velocity contour plots was also employed in the study conducted by Sarma et al. [27].

At α = 30°, the wake region of V₁₂ was larger than that of V₄₂, leading to greater energy losses and, consequently, reduced rotor blade performance. However, at α = 120°, the tip of the advancing blade of V₄₂ formed a larger and higher-value high-speed region compared to V₁₂, which helped improve the rotor blade performance [28]. Nevertheless, at this angle, the flow separation at the tip of the return blade of V₄₂ was more severe, resulting in a larger downstream wake region and increased energy loss. The negative impact of the flow separation outweighed the positive effect of the high-speed region at the blade tip, causing the performance of V₄₂ to be lower than that of V₁₂ at this rotational azimuth.

Figure 15 presents the velocity contour plots of the three rotor blades with the same V-angle but different straight-edge lengths, showing no significant differences in the characteristic regions.

5. Discussion

5.1. Vorticity Contour Plots and Mechanism Analysis

Vorticity, defined as the curl of the velocity vector, characterizes the local rotational intensity of a fluid. The operation of a Savonius rotor inherently involves strong flow separation, vortex formation, and shedding—key mechanisms that significantly influence its performance. Therefore, compared to analyzing the pressure and velocity fields around the rotor, the study of vorticity offers irreplaceable insights. Vorticity contour plots can directly reveal the generation, interaction, and shedding of vortices. It is, thus, essential to examine how structural modifications affect rotor performance through the vorticity contour plots of different rotor configurations. This analysis can guide structural optimization by aiming to control vortex distribution, suppress detrimental vortices, promote favorable flow structures, and, ultimately, enhance the performance of rotor blades.

Vortices are not solely a matter of intensity; their structural forms and evolutionary behaviors are equally critical. Generally, stronger vorticity indicates higher levels of turbulence in the water flow [29]. On the upstream-facing surface of the blade, a lower-intensity and smoother vorticity distribution is beneficial for generating effective and stable torque output, while large areas of high-intensity vortices tend to reduce torque. As the fluid flows around the blade, a sharp pressure drop can lead to the formation of a low-pressure zone, which is often accompanied by prominent high-vorticity regions on the downstream side of the rotor blade. These concentrated vorticity zones are typically the primary locations of energy dissipation. Unfavorable distributions of high-intensity vortices may prevent the flow from producing thrust on the rotor blade [30], resulting in significant energy losses, torque fluctuations, reduced efficiency, and degraded inflow quality for downstream blades. On the other hand, under certain conditions, high-intensity vortices may induce suction through negative pressure, thereby enhancing blade performance and delaying flow separation. However, the formation of large-scale, strong, and disordered vortices near the blade tips often causes energy loss and compromises the stability of the wake.

The vorticity contour plots of selected rotor blades are presented in Figure 16. At α = 30°, a distinct region of high vorticity intensity was observed on the downstream surface of Blade V₁₂, i.e., near the upper middle portion of the blade (denoted as Region A in the figure), in contrast to Blade V₄₂. This phenomenon is attributed to the increased curvature of the blade caused by a smaller V-Angle. As water flows over the more curved blade surface, flow deceleration becomes more pronounced, promoting the development of adverse pressure gradients. These gradients make vortex formation and flow separation more likely, ultimately diminishing the hydrodynamic performance of the blade. A similar pattern can be clearly observed in the comparison between the vorticity fields of V₁₀ and V₄₅. These results provide a physical explanation for the superior torque coefficients exhibited by rotor blades with larger V-Angles at α = 30° in water flow conditions.

At α = 120°, compared to V₄₂, Blade V₁₂ exhibited a higher degree of curvature due to its smaller V-Angle. As a result, vortices on the downstream surface formed earlier at the arc of the advancing blade (Region B in the figure). These vortices belong to suction vortices and superimpose on the positive torque generated by the advancing blade, thereby increasing the net torque. In contrast, the blade profile of V₄₂ was relatively flat, causing vortices on the downstream surface to form later near the mid-to-upper section of the returning blade (Region C in the figure). The suction induced by vortices in this position had a limited effect on the rotor performance due to its proximity to the mid-section, and it may have even generated adverse torque opposing the blade rotation because of its relatively elevated location. This phenomenon was also clearly observed in the comparison of the vorticity contours between V₁₀ and V₄₅. These observations provide an explanation for why rotors with smaller V-Angles demonstrate superior torque coefficients at α = 120°.

For Blades V₃₆, V₃₇, and V₄₀, no significant differences in performance were observed at α = 30° and α = 120°, nor were notable differences in their vorticity distributions detected, as shown in Figure 17. However, at α = 60°, the torque coefficients of the three blades diverged, necessitating an analysis of the vorticity contours at this rotational angle. The straight edge lengths of V₃₆, V₃₇, and V₄₀ were 0.24 L, 0.335 L, and 0.62 L, respectively. The performances of V₃₆ and V₃₇ were similar, whereas V₄₀ showed a marked decrease in performance. Correspondingly, the vorticity region in Area D of the figure gradually enlarged with increasing straight edge length. This phenomenon can be attributed to the elongation of the flow path as the straight edge length (L_v) increases, causing the fluid to traverse a longer and steeper angle around the advancing blade before reaching the returning blade. Consequently, the fluid must cover a greater distance to reattach to the returning blade. In V₃₆, this recirculation was relatively smooth, with the main flow quickly passing by and there being minimal residence on the downstream surface of the returning blade. In contrast, the excessively long straight edge of V₄₀ led to flow separation over the advancing blade and eventual retention on the downstream surface of the returning blade, forming a localized recirculation zone. Overall, the excessively long straight edge disrupted the flow attachment behavior, resulting in the significant performance degradation of V₄₀.

5.2. Mechanism Discussion

The analysis of the influence mechanism of the V-angle on the blade performance revealed that an increase in the V-angle exerted both positive and negative effects on the blade. It is, therefore, necessary to balance these opposing mechanisms to identify a more optimal V-angle parameter. According to prior research by Shashikumar, the optimal V-angle for V-shaped blades is 80°. To facilitate comparative discussion of different V-angles, this study, based on previous computations, further investigated a rotor blade with a straight edge length of 0.335 L and a V-angle of 80°, which was designated V₄₆.

The calculated torque coefficient of V₄₆ was 0.26336, which was slightly lower than that of V₁₂. As shown in Figure 18. A comparison of their vorticity distributions showed that V₄₆ exhibited a slight increase in vorticity in Region A at α = 30° and in Region B at α = 120°. However, due to the relative positions of the vorticity increments, the negative impact of the increased vorticity in Region A slightly outweighed the positive effect of the suction vortex generated in Region B. As a result, the overall performance of V₄₆ was marginally inferior to that of V₁₂. Nonetheless, this difference was not significant, and under different operating conditions, the performance rankings of the two V-angles may vary.

To investigate the influence of the straight edge length on vorticity distribution, a rotor blade with a straight edge length of 0.38 L and a V-angle of 130° was selected for computation, designated as V₄₇. The calculated torque coefficient for V₄₇ was 0.23792.

A comparative analysis of the vorticity contour maps of V₄₇ and V₃₇ revealed a slight increase in vorticity within Region D of V₄₇. Combined with the previously observed vorticity increase in Region D of Blade V₄₀, it can be concluded that, when the straight edge length (L_v) exceeds 0.335 L, an excessively long straight edge tends to promote the formation of detrimental vortices on the pressure side of the blade, thereby leading to a decline in blade performance.

6. Conclusions

This study systematically investigated the impact of the V-angle and straight-edge length of V-shaped rotor blades on their performance through three-dimensional numerical simulations, and the following conclusions were obtained:

When the V-angle is 70°, the hydrodynamic performance of the rotor blade generally performs the best. By observing five sets of rotor blades with the same straight-edge length, it was found that C_Tmax increases as the V-angle increases, while C_Tmin decreases with an increasing V-angle, showing a certain regularity. The variation in V-angle introduces both beneficial and detrimental changes in vorticity distribution, which must be carefully balanced to optimize blade performance.

Properly increasing the straight-edge length helps the water flow smoothly off the blades, reducing drag. However, excessively long straight-edge lengths alter the distribution of the low-pressure area, increasing the negative torque that hinders blade rotation, which results in a decline in the rotor blade performance. An excessively long straight edge delays the formation of vortices on the suction side of the blade, intensifying the distribution of detrimental suction vortices and thereby negatively impacting the blade performance. The peak value of C_T for the rotor blade occurred at L_v = 0.335 L, while the trough value occurred at L_v = 0.62 L.

The coupling between the V-angle and the straight-edge length was relatively weak. Under varying relevant parameters, a V-angle of 70° combined with a straight-edge length of 0.335 L consistently demonstrated superior performance across most conditions. Under the condition of a fixed tip speed ratio (TSR) of 0.87, the rotor blade with the best performance was V₁₂ with a straight-edge length of 0.335 L and a V-angle of 70°, producing a C_T value of 0.2696 and a C_p value of 0.2345.

The variation in the blade profile has a certain impact on the relationship between the torque coefficient and the rotational azimuth angle. It is recommended to focus on rotational azimuth angles with significant performance differences when analyzing the hydrodynamic performance of the rotor blades rather than being limited to the rotational azimuth angles corresponding to C_Tmax and C_Tmin.