Influence of Reynolds Number on Aerodynamic and Performance Coefficients of a Novel Parabolic-Bladed Savonius Wind Rotor

The continuous utilization of fossil fuels has led to a considerable increase in greenhouse gases, particularly CO₂, resulting in a global warming trend [1]. The accelerated melting of polar ice caps has led to a rise in sea levels, resulting in floods in coastal regions. Given its vast coastline, India is especially vulnerable to these flooding occurrences. Besides the environmental impact, the escalating cost of fossil fuels further motivates the exploration of alternative energy sources [2]. In response to these challenges, renewable energy, known for its eco-friendly nature, has captured the attention of research communities. Wind energy, in particular, has gained popularity due to its cost-effectiveness, widespread, and eco-friendly [3]. The wind energy has emerged as the rapidly expanding renewable energy source globally. In this context, the wind turbines have undergone several design modifications to harness this wind energy efficiently for electric power production [3–6].

Wind turbines, in general, can be grouped into two categories, namely, vertical-axis and horizontal-axis wind turbines (VAWTs and HAWTs) [7]. These turbines can either be drag-driven (such as Savonius rotors) or lift-driven (such as Darrieus rotors) [8,9]. On the other hand, the HAWTs are composed of airfoil shaped blades and are lift-driven. With its inherent straightforward design, strong self-starting capabilities, minimal noise pollution, and ability to operate without a yaw control mechanism, the Savonius rotor appears to be the more promising of these two types of rotors [10–15]. Nevertheless, the Darrieus rotor demonstrates superior efficiency compared to the Savonius rotor and achieves higher rotational speeds [16–19]. Windmills and pumping devices commonly utilize the drag-driven rotors with low rotational speeds. Whereas the high-speed lift-based rotors are employed in electricity generating turbines. In order to generate power, the generator drive shaft needs to spin at a high rotational speed (rpm) of around 1000–1500. The drag-driven rotors are less advantageous than the lift-based rotors as they rotate at somewhat lower rpm. Even when the swept area is the same, the lift-driven rotors generally extract more power than drag-driven rotors [20,21]. To improve the efficiency of drag-based Savonius rotors, various geometric parameters such as aspect ratio (AR), overlap ratio (OR), number of blades, number of stages, and blade shapes have been considerably studied both computationally and experimentally [22–24]. While the previous studies primarily focused on evaluating the power coefficient (C_P) at specific flow environment, there has been limited exploration into the influence of Reynolds number (Re) on the aerodynamic coefficients, namely, drag and lift coefficients (C_D and C_L), and their influence on C_P.

A representative two-bladed Savonius rotor, with all the geometric and aerodynamic parameters defined, is shown in Fig. 1. The drag force (F_D) and lift force (F_L) being the functions of relative wind speed (V) at the surface of the rotor blades act perpendicular to each other. The Savonius rotor operates primarily due to F_D; however, there is a minor amount of F_L acting on it [25]. The F_D and F_L are directly influenced by the aerodynamic coefficients (C_D and C_L), air density (ρ), swept area (A), and the relative wind velocity (u = LΩ) [9]. These aerodynamic coefficients are denoted by Eqs. (5) and (6), correspondingly [9,10].

The earliest study into the drag and lift behavior of Savonius rotor was conducted by Chauvin and Benghrib [25]. They estimated the C_D and C_L of a semicircular-bladed rotor by analyzing the pressure differential between its upper and lower blade surfaces. Subsequently, there was a noticeable gap in research in this area for nearly two decades, likely due to the advancement of newer blade profiles during and after the 1990s [26]. At a later stage, Irabu and Roy [27] identified the maximum C_D (C_Dmax) for a semicircular-bladed rotor to be 1.56 at an angle of attack (α) of 90 deg and 270 deg, with regions of negative C_D observed between α = 150–170 deg and α = 230–270 deg. The C_D was also found to reduce with a rise in OR. The C_Lmax and C_Lmin were found to be around 0.6 and 0.2, respectively, at OR = 0.0 and 1.0. Jaohandy et al. [28] conducted numerical investigations on a similar rotor, unveiling a C_Dmax of approximately 2.2 at α = 60–70 deg and tip speed ratio (TSR) = 0.6. The C_Lmax was observed to be roughly 1.72 at α = 30 deg and 210 deg, with this angle increasing with TSR. Akwa et al. [11] numerically evaluated the impact of increasing Re on a static rotor at α = 0 deg. They found that an increase in Re led to a rise of static torque coefficient (C_ST) of the Savonius rotor. Furthermore, the maximum averaged C_P decreased as the turbulence intensity increased with the rise in Re. Nasef et al. [29] carried out wind tunnel testing on two-bladed Savonius rotor by varying Re in the range of 1.2 × 10⁵–5 × 10⁵. In this study, the C_ST was found to increase gradually till α = 45 deg. Thereafter, the C_ST decreases to its minimum value at α = 157 deg. Additionally, the analysis demonstrated that the C_ST is not significantly affected by the Re. According to Kolekar and Banerjee [30], the hydrokinetic turbine's performance improves with increasing Re starting at Re = 0.28 × 10⁵. However, when the Re surpasses 2 × 10⁵, the C_P stays constant. Aliferis et al. [31] conducted wind tunnel testing on two-bladed Savonius rotor having helical blades and reported a slight decrease in values of C_D for Re = 1.5 × 10⁵–2.7 × 10⁵.

Few studies on the influence of Re on the axial-flow wind rotors were reported in the past [32–34]. These studies include the impact of Re variation on aerodynamics performance, scaling effects, and near wake regions. Nearly half a decade ago, Roy and Ducoin [35] utilized numerical methods to forecast the C_D and C_L associated with a novel blade profile for Savonius wind rotor, showing variations compared to conventional designs. In a recent study, Alom et al. [36] employed a 2D unsteady simulation to study the lift-drag traits of modified Bach and Benesh profiles. Few previously reported studies on the influence of Re of two-bladed Savonius rotor are outlined in Table 1, where the influence of Re on C_Pmax for a range of blade profiles has been elaborated. Furthermore, the performance pertaining to a particular blade profile needs to be expressed in terms of values of C_L, C_D along with C_P. Furthermore, a thorough insight into the influence of Re on C_Pmax along with C_D and C_L at various α is barely documented in the open literature. Thus, it is necessary to understand the influence of these coefficients over a wide range of Re to arrive at an optimum blade design of the Savonius rotor.

Enormous amount of research studies were conducted to raise the C_P of a Savonius rotor, where the main goal was mostly into the development of novel blade profiles/shapes and optimizing its various geometric parameters. Considering this, a unique parabolic Savonius rotor blade profile was recently devised by Mohan and Saha [41], and it was found to have potential over the traditional semicircular design. However, it is essential to address the influence of Re on performance of this novel blade design. In view of this, the present investigation is aimed at evaluating the performance of the parabolic-bladed rotor with the change of Re. To meet the proposed objective, a 2D computational fluid dynamics (CFD) study using ansys fluent [42,43] is carried out to calculate the C_D and C_L, and the C_P at various TSRs for the novel blade. The computation of C_D and C_L is conducted across α in the range of 0–360 deg. The performance of this novel blade profile is then compared with that of a semicircular blade profile under identical conditions. This quantitative investigation will improve the understanding on the influence of Re and hence the development of more effective Savonius rotor designs.

The parabolic profile being examined has been previously documented [41]. This profile is achieved through the optimization of the sectional cut angle (θ) along the parabolic curve, as depicted in Fig. 2. The angle θ is defined by the line AB relative to the x-axis of the parabola.

Furthermore, another semicircular blade profile with identical diameter (D = 200 mm) and thickness (t = 2 mm) as that of the parabolic profile is selected for the analysis. The schematic views of the tested blade profiles are shown in Fig. 3. The OR for both profiles is consistently maintained at 0.0 [44,45].

In Eqs. (1)–(6), L, Ω, and V represent the effective rotor radius (m), the angular velocity (rad/s), and the wind velocity (m/s); whereas T and A denote the total torque (N m) and the area swept by the rotor (m²), respectively. Furthermore, ρ and μ indicate the density (kg/m³) and the dynamic viscosity (kg/m s) of air, respectively.

In this section, the computational domain dimensions and boundaries, meshing, turbulence model/solver setup, as well as grid and time independence tests are elaborated.

For proper simulation of the rotor, two separate zones, as depicted in Fig. 4, are considered: inner circular rotating zone containing the rotor model and the stationary rectangular zone in the immediate vicinity. An interface separates the two zones, including the rotating zone containing the rotor spins relative to the stationary zone. The diameter of the rotating zone is taken as twice of the rotor diameter (2D) [44,45]. No-slip boundary condition (BC) is applied on the rotor blades. The flow enters the stationary zone from the left and leaves from the right side. Both top and bottom walls of the stationary zone are positioned at a distance 7.5D from the rotational axis and are designated as symmetry boundaries [44]. The inlet BC (V = constant) is assigned at the upstream of the rotor and at a distance of 7.5D from the rotational axis. Likewise, the downstream right vertical edge is subjected to an outlet pressure BC at a distance of 15D from the rotational axis.

The mesh structure of the domain is displayed in Fig. 5, utilizing the ansys meshing to discretize the domain. Small control volumes are employed to discretize the domain, facilitating the solution of the Reynolds-averaged Navier–Stokes (RANS) equations. Given the subsonic flow nature, comparable accuracy is anticipated in both structured and unstructured meshes [46,48]. However, due to its adaptability in handling complex geometries and faster grid generation, the unstructured triangular elements are employed to discretize the rotor zone [48,50]. The outer stator zone is discretized using coarse quadrilateral elements [41].

It is to be noted that very fine grids are generated near the rotor blades to capture the flow physics accurately. The grid sizes remain consistent at the interface between the stationary and rotating zones. In order to improve accuracy and effectively resolve the boundary layer flow around the rotor blades, inflation layers are introduced based on the dimensionless wall distance (y⁺) with a growth rate of 1.2. The y⁺ along all walls is maintained at values below unity [35]. The mesh surrounding the rotor blades is illustrated in Fig. 5. The first layer has a thickness of 0.05 mm, and up to a maximum of 10 layers are applied.

The sliding mesh technique has been applied to facilitate the rotation of the rotor models under investigation. The time-step size is determined based on a rotation of 1 deg per step of the rotor at TSR = 0.80. A total of 1800 time-steps are considered, with each time-step lasting 0.00027299 s. Moreover, simulations are carried out for five complete rotations of the rotor blades. Although the quasi-steady behavior is nearly achieved after the first rotation, however, the simulation is continued till the fifth rotation of the rotor for a better convergence. A residual convergence criterion of 10⁻⁴ is employed, along with a maximum of 20 iterations/time-step, which has been found adequate for modeling flows with significant flow detachment [52,53]. The second-order upwind scheme is utilized for spatial discretization of the conservative equations, while the temporal terms of the conservative equations are discretized using a second-order fully implicit temporal scheme. Pressure–velocity coupling is confirmed through the semi-implicit method for pressure-linked equations (SIMPLEs). A list of various input data for solving the equations has been enlisted in Table 2.

Both grid and time independence tests are performed for four refinement levels as shown in Figs. 6 and 7. The mesh density is varied from 71,790 to 203,372. As the grid elements varied from 71,790 to 120,849, the C_T increases by 11.2% while the refinement levels with higher number of grid elements of 159,839 and 203,372 shows negligible change in C_T with increased computational efforts. Therefore, the mesh density with grid elements of 120,849 is selected for the current simulations. The variation of C_T with the level of refinement is listed in Table 3.

Similarly, the time independence analysis is performed to assess the temporal consistency. The variation of C_T with time-step (degrees of rotation per time-step) is listed in Table 4. It is observed that with higher degrees/steps (1.5 deg and 2 deg), the average C_T decreases. Meanwhile, the time-step sizes of 0.5 deg and 1 deg demonstrate almost the same magnitude of C_T. Therefore, a step size of 1 deg is set for the present computation because of its lesser computational time.

To validate the present CFD model, a validation study is carried out against the findings of Alom et al. [36] and Mohan and Saha [41], as shown in Fig. 8. The chosen test model aligns its AR with that of the referenced numerical model, both are set at AR = 1.0. To emulate the parameters of the published work [36,41], a Re of 0.89 × 10⁵ is employed in our validation study. The present CFD model goes well with the reported data.

This section discusses the outcomes of the CFD simulations on the influence of Re on C_T, C_P, C_D, and C_L.

The existing literature suggests that the C_P of a conventional Savonius rotor usually increases with rise in Re [39,40,42]. In view of this, the Re study is carried out for both the parabolic and semicircular profiles to determine the approximate optimal Re for practical applications. The simulations are conducted at TSR = 0.80. The C_T and C_P go on increasing up to an optimum value and then decreases as shown in Fig. 9. The average C_T of the semicircular blade profile is found to be 0.37 at Re = 8.6 × 10⁴; while the parabolic blade profile shows an average C_T of 0.4 at Re = 10 × 10⁴ (Fig. 9(a)). The parabolic profile shows C_Tmax of 1.11 and C_Tmin of −0.24 at Re = 10 × 10⁴. Moreover, Fig. 9(b) suggests a C_Pmax of 0.30 for the semicircular profile, while the parabolic profile reveals a C_Pmax of 0.35. It can also be inferred here that for each design, there are different optimum Re. The results as shown in Fig. 9 suggest that the parabolic blade profile reaches the optimum value of C_P at higher Re, while the semicircular profile reaches the optimum value comparatively at lower Re. The optimum C_P for the semicircular profile is at Re = 8.6 × 10⁴, while the parabolic profile shows an optimum C_P at Re = 10 × 10⁴. It can therefore be said that different blade profiles perform differently at the identical Re. It is evident that the optimum C_T and C_P are different for both the profiles. Table 5 represents the maximum and minimum C_T obtained for a range of Re. As it is somewhat difficult to understand the shifting of optimum Re, a direct comparison of C_D is therefore made to get an idea of this shift in the values of C_T and C_P. This is elaborated in Sec. 5.2.

As the Savonius rotors are drag-based devices, the study of aerodynamic coefficients (C_D and C_L) of the blade profile is crucial. In this section, an attempt has been made to examine the C_D and C_L exerted on the novel parabolic blade profile. Results are also generated for the semicircular blade profile for a direct comparison. The evaluation of C_D and C_L is conducted across a complete turbine rotation (α = 0–360 deg) at TSR of 0.8. The analyses of C_D and C_L determine the range of α where the amplitude provides the understanding of gaining the optimum C_P.

The geometry of novel parabolic profile is characterized by a nearly straight trailing edge (TE) and a leading edge (LE) with a greater curvature than the semicircular profile, and this primarily induces a variation in pressure drag. During the initial quarter of the rotational cycle (α < 40 deg), the advancing side of the parabolic profile experiences a higher magnitude of pressure drag compared to the semicircular-bladed profile. Similar findings of Savonius rotors with other types of blade profiles have also been observed in the past [27,35]. Additionally, the straight TE of the parabolic blade profile contributes positively by enhancing the rotor's moment.

Figure 10 presents the polar representation of C_D for both the profiles over the tested range of Re. The polar representations show the actual variation of C_D over a complete 360 deg rotation of the rotor with an increment of 1 deg. For a given Re, the C_D increases with respect to angular position and reaches a maximum at around α = 40–55 deg in the first quarter of rotation. Likewise, the C_D decreases to a minimum value in the second quarter at around α = 150–155 deg. At about α = 50 deg and 220 deg, the wind incident on the advancing blade profile is more concentrated toward the LE due to parabolic profile's increasing curvature. Similar behavior is seen in the third and fourth quarter of the complete rotation. This pattern is observed for all the tested range of Re. The C_Dmax and C_Dmin obtained at various Re are depicted in Table 4. It is observed that with an increase of Re, there is an increase of C_D values and it reaches a peak value (optimum) for the parabolic blade profile and then it starts decreasing beyond an inversion point. It can be noted that the C_Dmax and C_Dmin obtained are lower for both the profiles at the lower Re (Figs. 10(a) and 10(b)). The C_Dmax and C_Dmin increase with the increase of Re for both the blade profiles and reach the optimum value at different Re. It can therefore be inferred that the optimum Re value to produce the C_Dmax is different for both the profiles. The semicircular profile has showcased the maximum performance at Re = 6.6 × 10⁴ as shown in Fig. 10(c). However, the parabolic profile shows an improved performance in terms of C_D (Figs. 10(d)–10(i)) in the tested range of Re except Re = 6.6 × 10⁴. Thus, it is clear that the semicircular profile performs better at the lower Re. As the C_Dmax obtained corresponding to the optimum Re is found to be different, the average C_D for the tested profiles is compared. The C_Davg of the parabolic and the semicircular profiles is found to be 1.47 and 0.99, respectively. There is an improvement of C_Davg of parabolic profile by 49% as compared to its semicircular counterpart. The complete data of C_D in the tested range of Re are shown in Table 6.

A notable enhancement in the C_L values is observed throughout the entire rotational cycle of the novel parabolic blade profile. This profile is similar to that of an aircraft wing and therefore has the primary advantage over the semicircular profile. In the parabolic profile, the C_L increases up to α = 25 deg (first quarter of the blade rotation), and then decreases to a minimum in the range of α = 110–120 deg (second quarter of the blade rotation), followed by an increase till α = 200 deg (third quarter of the blade rotation). Conversely, for the semicircular profile, the C_L increases to a maximum in the range of α = 10–15 deg, followed by a decrease till it reaches a minimum at α = 100 deg. Similar pattern is observed for both the profiles in the tested range of Re. It can therefore be said that the semicircular profile reaches a maximum C_L at a relatively lower α, and this allows the rotor to rotate earlier than its parabolic counterpart. This is because of the higher curvature of the parabolic profile that significantly increases the pressure gradient on the suction side. This leads to a vertical orientation of the normal pressure and consequently an increase of lift. Hence, the suction side of the advancing blade acts as the primary surface for the lift generation of the parabolic profile. The Savonius rotor with other types of blade profiles have shown similar observations [27,35].

Figure 11 depicts the polar variation of C_L over one complete rotation of the rotor. It can be observed that there is not much change in C_L for both the profiles (Figs. 11(a) and 11(b)). However, with a slight increase of Re, the C_L of the semicircular profile increases and attains an optimum value at Re = 8.6 × 10⁴ (Fig. 11(c)). Apart from this, C_L of the semicircular profile is found to be prominent at lower Re as compared to its parabolic counterpart. At Re = 8.6 × 10⁴, the C_Lmax of semicircular profile is obtained and is found to be 2.25 at α = 13 deg and 185 deg and the C_Lmin of 0.45 is attained at α = 106 deg. Beyond this Re, the C_Lmax observed is significantly higher in case of the parabolic profile (Figs. 11(d)–11(i)). The parabolic profile at Re = 10 × 10⁴ has demonstrated a C_Lmax of 2.87 at α = 16 deg and a C_Lmin of 0.00009 as compared to its semicircular counterpart (Fig. 11(h)). The plots of the C_L also suggest that the optimum Re is different for the two test profiles. Thus, it is difficult to compare the C_Lmax to have a direct comparison of the test profiles, and therefore, the C_Lavg is calculated. The C_Lavg for the parabolic and semicircular profiles is found to be 1.41 and 1.25, respectively. The parabolic profile thus shows an improvement of C_Lavg by 12.8%. Table 7 shows the variation of C_L with Re for the parabolic and semicircular blade profiles.

On the other hand, a direct comparison of the present aerodynamic coefficients with the published data is shown in Table 8. The C_Dmax of the parabolic blade profile is compared with the published data of other blade profiles to get an understanding of the optimum Re. At Re = 1.0 × 10⁵, the novel parabolic blade profile has showcased a C_Dmax of 2.73 and a C_Lmax of 2.87. The other blade profiles such as Modified Bach [36], Benesh [36], and elliptical [49] have shown almost similar optimum Re to get the C_Dmax.

Here, the influence of Reynolds number on aerodynamic and performance coefficients of a novel parabolic-bladed Savonius wind rotor is studied. Unsteady 2D computation is carried out using RANS equations by adopting SST k–ω turbulence model. The range of Re is varied between 5.3 × 10⁴ and 10.6 × 10⁴; whereas the TSR of the rotor is kept at 0.80. The key findings of the study are summarized below:

• The increase of Re leads to higher C_T and C_P for both the test profiles. Each profile reaches an optimum performance at different Re.

• The parabolic profile shows improved C_P against the semicircular counterpart at higher Re (>7.3 × 10⁴). This happens due to fact that the parabolic profile comprises of an almost straight TE and a LE having more curvature compared to the semicircular profile, which mainly induces variations in pressure drag. During the initial quarter of the rotational cycle (α < 40 deg), the advancing blade of the parabolic profile experiences a higher magnitude of pressure drag compared to the semicircular profile. Additionally, the TE of the blades contributes positively by enhancing the rotor's moment. Moreover, between α = 220 and 260 deg, the wind flow impacting on the advancing blade is more concentrated toward the LE due to the increased curvature of the parabolic profile.

• The novel parabolic profile reaches a C_Pmax of 0.35 whereas the semicircular profile achieves a C_Pmax of 0.30.

• The parabolic blade profile reaches optimum value of C_P at higher Re, while the semicircular profile reaches an optimum value comparatively at a lower Re. The C_Pmax of the parabolic profile is obtained at Re = 10 × 10⁴, while the C_Pmax of the semicircular profile is obtained at Re = 8.6 × 10⁴. As compared to the semicircular profile, the parabolic profile shows an improvement of C_Pmax by 16.6%.

• The C_D and C_L values are found to be greater in case of the parabolic profile than the semicircular profile. The parabolic profile shows the C_Dmax of 2.7 at Re = 10 × 10⁴; whereas the semicircular profile shows the C_Dmax of 1.86 at Re = 8.6 × 10⁴.

• The C_Davg for the parabolic and the semicircular profiles is found to be 1.47 and 0.99, respectively. The parabolic profile shows an improvement of C_Davg by 49% as compared to its semicircular counterpart.

• The C_Lavg for the parabolic and semicircular profiles is found to be 1.41 and 1.25, respectively. The parabolic profile thus shows an improvement of C_Lavg by 12.8%.

As a future scope of research, the evaluation of instantaneous aerodynamic coefficients can be taken up. Furthermore, for a more accurate evaluation of the aerodynamics performances, a three-dimensional CFD analysis can be pursued. The study of fatigue and stress analysis of the Savonius rotor is to be conducted at much higher Re where the wind is likely to exert a greater force on the blade surfaces. In the present study, two-bladed rotor with the parabolic profile is considered. Thus, the researchers can explore the performance of three-bladed rotor with the parabolic profile. Other research study may include the optimization or soft-computing techniques to enhance the blade design.

The first author is thankful to the Indian Institute of Technology Guwahati, India, for providing scholarship during the period of study.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

u =

relative wind speed, m/s

A =

swept area, m²

D =

rotor diameter, m

L =

radius of the rotor, m

T =

torque, N m

V =

inlet wind velocity, m/s

C_D =

drag coefficient

C_L =

lift coefficient

C_T =

torque coefficient

C_P =

power coefficient

F_D =

drag force, N

F_L =

lift force, N

P_available =

available wind power, W

P_rotor =

rotor shaft power, W

Re =

Reynolds number

TSR =

tip speed ratio

α =

angle of attack, deg

θ =

sectional cut angle, deg

ρ =

density of air, kg/m³

Ω =

angular speed, rad/s