CFD Investigation of A New Elliptical-Bladed Multistage Savonius Rotors

The Savonius-conventional wind turbine is a class of wind turbines designed with a vertical axis. It has a good starting capacity and an insensitivity to wind direction. It works relatively at low wind speed in an easy installation. Savonius wind turbine faces major drawbacks, including some of the low efficiency and high negative torque created by the returning blade. Many attempts have been undertaken to optimize the blade’s shape to increase the performance of these wind turbines. The vertical axis is still under development. The elliptical-blades with a cut angle equal 47.50° have recently shown enhanced performance. In this study, we investigate the effect of Elliptical-bladed multistage Savonius Rotors (rotor aspect ratio, stage aspect ratio) on the performance by means of numerical simulation. The results obtained by comparison of one, two, and three-stage rotors indicate that the maximum power coefficient increase with a number of the stages (for the rotors with similar RAR of 0.7). Moreover, for the rotors with similar SAR of 0.7, the two stages have the highest performance than others. ©2020. CBIORE-IJRED. All rights reserved


Introduction
Wind energy is an inexhaustible source of energy and dependent on the wind. It is, therefore, a real renewable resource that does not cause pollution, no CO2 emission, and no toxic or dangerous substances freed into the environment. As wind energy depends on the cubic speed of the wind, a small increase in this parameter will increase the output power of the turbine. Furthermore, depending on the position of the rotor axis relative to the wind, wind turbines are generally classified into two types: wind turbines with the horizontal hub (HAWT) and wind turbines with the vertical hub (VAWT). The VAWTs are classified into two principal families: Darrieus-type (D-VAWT) and Savonius-type(S-VAWT). The most notable difference between the two is that the (D-VAWT) is named a lift turbine and the (S-VAWT) is named a drag turbine (ElCheikh et al. 2018). Savonius-type was developed by Sigurd Savonius, in 1925(Ushiyama & Nagai. 1988. The shape of the Savonius blade profile affects the rotor's performance significantly, in this context several studies attempt to enhance efficiency of conventional savonius turbines. Figure 1 illustrates different improved blade designs mainly, Elliptical blade tested by ( Banerjee et al. 2014;Zemamou et al. 2019;Alom and Saha. 2019), . Other researchers have investigated the performance of the savonius rotor formed of a straight segment and two arcs knowns as Benesh model (Roy & Saha. 2015;Benesh. 1988), and (Scheaua, 2020) Fig-1(c). In the same perspective, Bach model has been developed by ( Roy & Saha. 2015;Kacprzak et al. 2013) and (Kamoji et al. 2009) in Fig-1(d). In order to reduce the negative torque exerted on the returning blade, Nugroho et al. (2020) have developed a Slotted-blade as depicted in Fig.1(e). Zemamou et al. (2020) have introduced a novel blade design based on polynomial Bezier curves Fig.1(f).
Twisted blade have been developed by (Grinspan et al. 2001;Saha et al. 2008;and Mendoza et al. 2020), as shown in Fig-1(g). Saha & Rajkumar (2006) have compared the performances of the Savonius rotor by varying the torsion angles. The results presented that the CP of the twisted blades of a rotor could be changed by varying the twisted angle. The value of the optimal torsion angle is 15°, corresponding to maximum Cp = 0.14. Semicircular blades (Savonius, 1931) (a) Elliptical blade (Alom & Saha, 2018) Benech blades (Scheaua, 2020) Bach blades (Bach, 1931)  The performance coefficient of an (S-VAWT) is influenced by different parameters like overlap ratio OR, aspect ratio (AR), number of blades, shape of the blades and number of stages. Many researchers have carried out divers numerical and experimental researchers to improve the performance of an (S-VAWT), the overlap ratio of the Savonius turbines has a significant impact in the amelioration of the performance of these turbines, it is defined by the following equation (e/d) where (d) is the blade chord length and (e) represents the overlap.
In an experimental investigation, (Mojola, 1985) studied the effect of OR on the rotor performance, it is found that the maximum power of coefficient corresponds to an overlap ratio of about 0.25. According to (Akwa et al. 2012) the optimal value for the overlap ratio is 0.15. The latter corresponds to an average power coefficient equal to 0.3161. From a numerical investigation, (Alom & Saha, 2017) also produced that the overlap ratio value of 0.15 gives a higher performance.
An experimental analysis has been conducted by (Roy & Saha. 2013a) based on the variation of the aspect ratio. Results have shown that the rotor with the aspect ratio equal to 0.8 gives a higher coefficient of power Cpmax=0.178. Kamoji et al. (2009) concluded experimentally that the power and torque coefficients are maximum for an aspect ratio of 0.7. The effect on a number of the blade has been investigated by (Blackwell et al. 1977), they found that two-blade Savonius turbines have a better power coefficient when compared with three-blade Savonius, (Mahmoud et al. 2012) , also noticed that the two-blade gave a better power coefficient than three and four ones. Indeed, the addition of blades disturbs the flow near the returning blade.
The use of multi-staging reduces the high fluctuation of torque without significant performance loss of the turbine, operating with cycles lagged relatively to one another. In the experimental analysis Mahmoud et al. (2012) concluded that the two stages rotor have a higher specific power. Hayashi et al. (2005) found that the values of CPmax and CTmax of the rotor with one-stage were much higher than those of the three-stage rotor. Saha et al. (2008) asserted that the rotor with two-stage and twoblade has a better aerodynamic performance than the others. However, from this study, a Cpmax of 0.31 and 0.29 is reached for twisted and Semicircular geometry blade profile respectively. Frikha et al. (2016) studied the effect of the number of stages on performance. The results showed an improvement of the dynamic torque coefficient and the power coefficient as a function of the number of stages, they found that the Cpmax = 0.132 for the five-stage rotor. Kamoji et al. (2008) found that by increasing Reynolds number the power and torque coefficients increase. However, the rotors with three and two stages are the same performance for the same SAR of 1. In an experimental investigation, Chen et al. (2016) found for two-stage that the azimuth angle range of negative CTS decreases by increasing GR. According to Menet (2004), the rotor with two-stage has a superior power coefficient (Cp) over the one stage rotor.
To reduce the converse force on the returning blade (Golecha et al. 2011) proposed to use the deflecting plate in front of the returning blade. Their result showed in optimal position the power coefficient improved by 50%. In addition, Ogawa et al. (1989) attained a maximum power of 0.27 when the plate with A =0.5R and B = R is placed at θ= 30°. In the same perspective, (Altan & Atılgan. 2010) has investigated the effect of curtain arrangement on performance. They found that the greatest performance has been obtained from curtain l1 = 0.34 m and l2 = 0.39 m at its position θ=60° for the angles α =45°, β=15°, Fig-1 It is during the last few years many studies have been carried out to demonstrate the influence of modified savonius design on the performance. The general objective of this study to optimize the different parameters of the modified multistage Savonius rotor (Table 1) to reach at an appropriate design configuration. Savonius modified wind rotor comprises two elliptical blades, placed between two end plates as illustrated in Fig. 2. Based on the work of a previous study (Alom et al. 2016) the value of the aspect ratio AR = (H/D) is chosen 0.7.

Methodology
The multistage savonius rotors with the elliptical blade are tested numerically at various stage number (SN). The 3D unsteady simulation is realized by using pressurebased finite volume method (FVM) solver ANSYS Fluent. The power and torque coefficients are evaluated with respect to TSR. The SN is varied from 1 to 3 as illustrated in Figures 3 and Figure 4.
The dimensions of the elliptical blade are dictated by the ratio OB/OP=2/3 ( Figure 5), similar to that the overall diameter of the turbine is 0.514 m, whereas the blade thickness is taken as 0.0066 m. The ratio OM/OA is taken as 0.54, to determine the required blade chord length (d) (Fig 5). Recently, the elliptical blade with a cutting angle θ = 47.5° showed improved performance than the traditional savonius rotor (Alom et al. 2016). The value of overlap ratio is taken as 0.2 (Alom & Saha 2017).

Mathematical formulation
The power of wind depends on the air density (ρ), the swept area (A), and the cubic wind speed (V 3 ) is given by (Roy & Saha. 2013b): (1) The coefficient of power (CP) the rotor is defined from the relationship below (Roy & Saha. 2013b). (2) Where Pturbines: power actually provided by the turbine. The relationship between the aerodynamic coefficients, is represented by the following equations. (3) (4)

Simulation domain and Boundary Condition
The three dimensions computational domain and Boundary condition for the Savonius rotor are shown in Figure 6. Indeed, the simulation domain has two subdomains separated by a sliding interface: a rotate and fluid domain. the dimension of the fluid domain is 10D × 5D × 4D. The Savonius rotors are located at the center of the rotate domain. The computational domain has been discretized by unstructured triangular grids mesh (Fig. 8). y + is an important parameter for calculating the value of the distance from the first node to the wall. The thickness of the first layer from the rotor surface is calculated using the following relation: The frictional velocity is calculated from the following equation: Where 3 is wall shear stress, y distance of first cell from rotors wall and µ= 1.81·10 5 Ns.m -2 is the dynamic viscosity. In this work the first layer is set to be 5·10 -5 m and the growth rate of 1.2 (Fig. 8). For this study, we fix five complete rotations of the turbine. The maximum iteration by time step are 30. Absolute criteria are used to monitor the convergence of the solution with a value of 10 -3 . The grid-independent test has been performed by varying the grids between 1147050 and 5826499 (Fig. 7). The simulation results show that there is no significant change in the coefficient of torque beyond 3439589 elements. The grid with 3439589 cells is adopted for all simulations. Based on the work of a previous study (Alom 2019) the value of time step size for the rotor is chosen 1°/step rotation. The inlet is given as uniform and a constant velocity of V = 6.2 m/s with a 5% turbulence intensity. This value is calculated by: (7) Where: v: is the mean velocity (m/s) u': is the root-mean-square of the turbulent velocity fluctuations calculated from the following equation: Where k represents the turbulence kinetic energy its value is 0.14415 m 2 /s 2 .
Moreover, the pressure outlet is set as Gauge pressure of 0 Pa and sliding wall conditions are utilized at the boundary wall. The CFD codes used the average of the Reynolds Navier Stokes Rans equations for the calculation of turbulent flows. The instantaneous continuity and momentum equations are given by: The Boussinesq hypothesis is used to solve the Reynolds stresses. This hypothesis corresponds to an alignment between the Reynolds stress and the mean strain tensors. Boussinesq equation is expressed as follow: where µt is the turbulent viscosity.
The turbulence is considered to be critical and must be precisely taken into account in the case of CFD simulation. Many different formulations are available for solving turbulent flow problems in the simulation such as standard k-e realizable k-e, RNG k-e, k-ω, SST k-ω, BSL k-ω. In this study, the SST k-ω turbulence model has been utilized (Menter 1994). The SST k-ω model has two partial equations where 'k' refer to turbulence kinetic energy and ω refer to a specific dissipation.
• The kinetic energy of turbulence Specific dissipation rate (13) Where: • + = Turbulent eddy viscosity • A = Generation of turbulence kinetic energy owing to mean velocity gradients • 3 = Generation of • D and 3 = dissipation of k and ω owing to turbulence • 3 = the GROSS-diffusion term 3. Results and discussion.

Numerical simulation validation
In this part, validation is carried out to evaluate the proposed numerical system. The particular results of the simulation of the typical flow around the classical Savonius rotor have been compared to the experimental results obtained by (Sanusi et al. 2016). The variation of the power coefficient and torque coefficient, as well as experimental data for different TSR are shown in Figure  9. For both experimental and simulation curves, the highest power coefficient is obtained at TSR = 0.8. For experimental data the CPmax was 0.24 while for simulation results, the CPmax was found 0.243. The maximum error value between the experimental and numerical data about 8.8 % for all ranges of TSR. The computational results have a better accord with the experiment test. As a result, it is admissible to perform the numerical method in order to foresee the performance of Savonius rotor.

Savonius rotors with the similar RAR of 0.7
Figure 10 explains the variations of CP and CT with TSR for various stage numbers. In fact, for all the cases the CT decreases as TSR increases from 0.2 to 1.2, It is observed that the maximum is located at TSR = 0.2. The value of CT of the three-stage savonius rotor is approximately 27 % higher than that of the two-stage savonius rotor for the range from 0.2 to 0.4 of TSR, while the value of CT gets closer when TSR > 0.4. The three-stage shows a peak CP of 0.19 compared to a peak of about 0.14 for the two-stages as illustrated in Fig 10. Thus the CT value of the two-stage rotor is approximately 15 % higher than that of the onestage rotor at TSR = 0.2 (and 21% at TSR = 0.4), while the CT values for one stage is higher than two when TSR > 1. On the other side, the Cp curves of the single and two stage rotors are quite close in TSR range of 0.2 to 0.3 and of 0.9 to 1.2. Indeed, the Cp value of two-stage is higher than that of one stage for TSR lower than 1 as illustrated in Figure 10. However, when TSR > 1, the two stage tends to show a very low decrease compared to other cases. The Cp curve of three-stage is the highest when compared to other stages for all of the range of TSR as illustrated in Figure 10. Additionally, the highest power coefficient (Cpmax) happens nearly at TSR = 0.8 for all the rotors.

Fig.11
Power coefficient and Torque coefficient variation The maximum CP and corresponding CT increase (for rotors with a similar RAR of 0.7) as a function of the stage numbers as depicted in Table 2. It is more appropriate to use the three stage rotors as they can start automatically due to the uniform static torque coefficient (Kamoji et al. 2008).

Savonius rotors with the similar SAR of 0.7
Figures 11 describes the variations of CT and CP with TSR for various stage number (the effect of RAR on the performance). In general, for all the cases the CT decreases by increasing the TSR. The results reveal that the value of CT for three -stage is 13.7% higher than the two-stage rotor at 0.2 of the TSR, whereas the CT values are quite close when TSR varying in the range from 0.4 to 1.2. Moreover, the two-stage rotor shows a CP peak equal to 0.195, compared to a peak of 0.188 for the three-stage rotors. However, the comparison between one and twostage rotors proves that the CT value of the two-stage rotor is higher in the range from 0.2 to 1.1 of TSR, while the values of CT for two-stage are lower than one-stage when TSR > 1.1.
The Cp curves of the one and two stage rotors are quite approached in the range from 1 to 1.2. The Cp value of two-stage is better than that of one stage for TSR lower than 1.1. In fact, when TSR > 1.1, the two stage rotor tends to show a very low decrease. Further, the two-stage Cp value is the highest in comparison to other stages along with the range 0.3 to 1.2. The maximum power of the coefficient occurs for all rotors between (TSR = 0.6 -0.8).
In addition, the greatest power coefficient and corresponding CT increases (for rotors with a similar SAR of 0.7) as the RAR decreases from 2.1 to 1.4. It increases when the RAR increase from 0.7 to 1.4. The Cp of the three and two-stage rotors are quite close in TSR range from 0.3 to 1. For the rotors with a similar SAR of 0.7 (rotors aspect ratio 0.7, 1.4 and 2.1), it has been found in this paper that the two-stage rotor is more efficient than others. (Table 3).

4 Performance analysis for two-stage elliptical savonius rotors
Every stage of two-stage rotors was placed to one other using a phase shift associated with 90° to one other. Two rotors with a RAR of 0.7 (SAR of 0.35) and RAR of 1.4 (SAR of 0.7) were studied. The Figure 12 shows the comparison in the variation of power coefficient and torque coefficient by varying the TSR for two-stage elliptical rotor wind turbine (RAR of 0.7 and 1.4), the highest power coefficient of the rotor with a RAR of 1.4 is 28.2 % bigger than the savonius rotor with a RAR of 0.7. When TSR > 0.8 the Cp curves associated with two stages rotors are quite close. The coefficient of torque CT decreases with the increases in the TSR. The highest torque coefficient CTmax is noticed at a TSR as 0.2, when increasing the TSR the CT curves associated with two stage rotors are quite close. Two-stage rotor's performance differ after augmenting the SAR and RAR by a factor of 2, which leads to the fact that the SAR has a marginally influence on the performance of the twostage rotors when the TSR is lower than 1.

Performance analysis for three-stage elliptical savonius rotors
Every stage of three-stage rotors was placed to one other using a phase shift associated with 60° to one other. Figure  13 shows the comparison in the variation of power coefficient and torque coefficient by varying the TSR three-stage elliptical rotor wind turbine (RAR of 0.7 and 2.1). In the range of TSR between 0.2 and 0.75, the variation of power coefficient shows that the rotor with RAR = 2.1 has a higher performance than that of the rotor with RAR = 0.7, however, after reaching the highest power coefficient, the rotor with RAR = 0.7 tends to show higher performance compared to the rotor with RAR = 2.1. In the range of TSR between 0.2 and 0.75 the variation of torque coefficient shows that the rotor with RAR = 2.1 has higher CT values than that of the rotor with RAR = 0.7. When the TSR value increases, the rotor with RAR = 0.7 starts to show higher torque coefficient values.

Analysis of pressure and velocity contours
The streamlines of the flow around the rotor blades at a velocity of 6.2 m/s are shown in Figure 14. When the fluid hits the frontal area of the blades, it generates complex flow in front of the blades. That's why we restrict on the two-dimensional analysis. The total pressure contours on the transverse planes defined in the middle for each stage of the RAT = 0.7 and RAR = 1.4, respectively are shown in Figure 15.
From these results, it has been noted that the higher total pressure is observed on the advancing blade of the elliptical profile (RAR = 1.4) in comparison to the elliptical profiles with (RAR = 0.7) at α = 90°. On the concave side of the returning blade, the total pressure for the elliptical profile (RAR = 1.4) is higher than the elliptical profile (RAR = 0.7). The pressure on the concave side of the blade can contribute to the increase of the positive torque, while the convex side has the possibility to reduce the rotor torque. The depression on the tip of the blade is very important for the elliptical profile (RAR = 1.4) compared to the elliptical profile (RAR = 0.7) at α = 90°, this negative value is due to reverse flow at the tip of the blade. The region of higher pressure on the convex side of the returning elliptical profile for (RAR = 1.4) is lesser as compared to elliptical profile for (RAR = 0.7), and from this result, the torque is higher in the elliptical profile for (RAR = 1.4). For = 0° the magnitude of total pressure near the concave side of the advancing profile is found to be around 2-28 N/m 2 for elliptical profile (RAR = 1.4), but for the elliptical profile (RAR = 0.7), the range of total pressure is −17 to -25 N/m 2 at a similar α. As well, it noticed that the concave side of the advancing blade for (RAR = 1.4) shows a higher total pressure compared to the (RAR=0.7). Also, the low-pressure zone in the convex side of the returning blade is higher in RAR = 1.4 than the RAR = 0.7. The velocity contours of on the transverse planes defined in the middle for each stage of RAR = 0.7 and RAR = 1.4 respectively are shown in Figure 16. The magnitude of velocity near the surface of the elliptical profile (RAR = 1.4) is approximately in the range of 4-5.75 m/s, whereas in the elliptical profile (RAR = 0.7). The velocity magnitude lies in the range of 3-4.5 m/s. at α =90°. Furthermore, it has been noted that the smaller velocity region on the convex side of the returning elliptical profile (RAR = 1.4) indicates lesser negative drag as compared to the other profile. It is also observed that at the tip of the advancing elliptical blade for (RAR = 1.4), the magnitude of velocity lies in the range of 7-10 m/s, whereas for the other, the velocity magnitude lies in the range of 8-12 m/s at α = 90°. This implies that the chances of flow separation are lower in the case of the elliptical blade for (RAR = 1.4) and this obviously brings an improvement in its performance coefficients. Also, there is a bigger velocity region on the convex side of the advancing elliptical profile (RAR = 1.4) indicating an enhanced positive drag as compared to the elliptical profile (RAR = 0.7). At α = 0°. According to these results, it has been observed the dragging flow on the convex side of the advancing elliptical profile (RAR = 1.4) is more important compared with the elliptical profile (RAR = 0.7). The total pressure contours on the transverse planes defined in the middle for each stage of the RAT = 0.7 and RAR = 2.1 respectively are shown in Figure 17. For α = 90°, the magnitude of the total pressure near the concave side of the advancing blade is around 2-12 Pa for the elliptical profile (RAR = 0.7), however, for the elliptical profile (RAR = 2.1) the total pressure range is -10 to 4 Pa at a similar α, whereas, on the convex side of the returning blade, the total pressure for the elliptical profile ( RAR = 2.1) is higher than the elliptical profile ( RAR = 0.7). For α = 30°, the low-pressure area behind the advancing blade that forms the recirculation is higher in the elliptical profile (RAR = 2.1) than in the elliptical profile (RAR = 0.7). In addition, the higher pressure area on the convex side of the returning blade for (RAR = 0.7) is less than the other, and therefore the resulting torque is higher in the elliptical profile for (RAR = 0.7).
At the rotor angle of 150° ,on the basis of these results, it was found that the concave side of the advancing blade for (RAR = 0.7) shows a higher total pressure compared to the (RAR = 2.1). Also, the pressure distribution around the convex side of returning blade for a (RAR = 2.1) is almost similar to that of a (RAR = 0.7).

Conclusion
In this numerical investigation, the effect of stage number on the efficiency of a savonius wind turbine is studied by developing a 3 D numerical simulation using the SST k-ω turbulence model. The wind speed is 6.2 m/s and the flow is considered as three-dimensional, incompressible and turbulent. The results of the simulation around the conventional blade of savonius wind turbine reveal good accordance with the previous works. The Comparison between one, two and three stage rotors with the similar RAR which equals 0.7 indicates that the maximum of the coefficient of power increases as a function of the number of stages. For rotors with a similar SAR of 0.7 (rotors aspect ratio 0.7, 1.4 and 2.1), it has been found that the two-stage rotor is more efficient than others