Computational fluid dynamics analysis of a modified Savonius rotor and optimization using response surface methodology

Abstract

This article aims to present a two-dimensional parametric analysis of a modified Savonius wind turbine using computational fluid dynamics. The effects of three independent parameters of the rotor, namely, shape factor, overlap ratio, and tip speed ratio on turbine performance were studied and then optimized for maximum coefficient of performance using response surface methodology. The rotor performance was analyzed over specific domains of the parameters under study, and three-variable Box-Behnken design was used for design of experiment. The specific parametric combinations as per design of experiment were simulated using ANSYS Fluent^®, and the response variable, coefficient of performance (C_p), was calculated. The sliding mesh model was utilized, and the flow was simulated using Shear Stress Transport (SST) k − ω model. The model was validated using past experimental results and found to predict parametric effects accurately. Minitab^® and ReliaSoft DOE++^® were used to develop regression equation and find the optimum combination of parameters for coefficient of performance over the specified parametric domains using response surface methodology.

Keywords

Vertical axis wind turbine modified Savonius wind turbine computational fluid dynamics parametric optimization response surface methodology

Introduction

Wind turbines are classified into two types based on their axis of rotation: horizontal axis wind turbines (HAWTs) and vertical axis wind turbines (VAWTs). HAWTs differ from VAWTs in their way of extraction of energy from the wind. HAWTs work completely on lift forces, whereas VAWTs can make use of drag forces as well to generate energy (Akwa et al., 2012b; Damak et al., 2013; Fujisawa and Gotoh, 1992; Rogowski and Maroński, 2015). HAWTs, due to their ability to extract wind energy from lift forces, are more efficient than their countertypes (Burton et al., 2001). However, HAWTs require yaw mechanism for continuous alignment due to the change in wind direction and thus are not suitable for fluctuating wind speeds and changing wind directions, as is the case for urban areas. VAWTs, however, do not require yaw mechanism, and additionally, their energy extraction is independent of the wind direction, thus making them preferable for urban sites (Akwa et al., 2012b; Saha et al., 2008; Savonius, 1931).

VAWTs are subdivided into two types: Savonius rotors and Darrieus rotors. A conventional Savonius rotor, invented by Finnish architect Savonius in 1924, is of bucket type construction with two or more blades. The Savonius rotor is primarily a drag-type turbine and has lower efficiency than Darrieus turbines (Damak et al., 2013; Fujisawa and Gotoh, 1992; Mohamed et al., 2011; Saha et al., 2008). However, Savonius rotors have a higher coefficient of static torque at lower wind speeds when compared to Darrieus rotors, and for this reason, they are frequently used for providing starting torque to pumps, motors, and so on (Blackwell et al., 1977; Damak et al., 2013; Fujisawa and Gotoh, 1992; Mohamed et al., 2011; Saha et al., 2008). Additional benefits like simple construction, ease of maintenance, and the ability of the rotor to accept winds from all directions make Savonius rotor preferable for operating in urban areas (Akwa et al., 2012b; Blackwell et al., 1977; Fujisawa and Gotoh, 1992; Kamoji et al., 2008; Saha et al., 2008; Savonius, 1931). Efficiency of a Savonius rotor is less than other wind turbines because of the retarding effect of negative torque generated by retreating blades.

A number of researchers have performed experimental studies on Savonius rotor with different designs and operating conditions. Pressure distributions on Savonius rotor blades were measured to understand low- and high-pressure regions contributing to the overall torque generation of Savonius turbine (Fujisawa and Gotoh, 1992). Modifying blade shapes has experimentally been shown to improve the power coefficient of Savonius rotor wind turbine (Damak et al., 2013; Kamoji et al., 2009; Saha et al., 2008). The effects of single and multiple stages on power coefficient have been explored through wind tunnel tests (Hayashi et al., 2005; Kamoji et al., 2008). Increasing the number of stages reduces torque fluctuations and plays a vital role in rotor performance (Blackwell et al., 1977). Results indicate that two-bucket rotor design is aerodynamically superior to a three-bucket rotor (Alexander A and Holownia B, 1978); however, the starting torque for a three-bucket construction is more than that for the two-bucket rotor (Blackwell et al., 1977). The effect of endplate rotor construction was studied by Saha et al. (2008), who found that endplates help in reducing end losses and increase lift forces, thereby increasing the power coefficient to some extent. Experiments of Blackwell et al. (1977) complemented these findings. Placing an obstacle in front of retreating blade has also been shown to be effective through field experiments (Mohamed et al., 2011). Ogawa et al. (1989) provided guiding mechanism for directing inlet wind toward the advancing blade in order to reduce the negative torque on the returning blade. Their results revealed an increase in power coefficient; however, addition of guiding mechanism requires control system, and the rotor can no longer accept wind from all directions.

The effects of Reynolds number, height to diameter ratio (h/d), and overlap ratio on Savonius rotor have been investigated experimentally, showing positive impact on power coefficient (Blackwell et al., 1977). Helical-shaped rotors were studied and found to have improved power coefficient as compared to conventional semi-circular rotor shapes (Saha and Rajkumar, 2006; Zhao et al., 2009). Thus, changes made in the parameters such as number of rotor stages, number of buckets, blade shape, aspect ratio, overlap ratio, and flow guide have been found to significantly improve the aerodynamic performance of Savonius rotors.

Numerical analysis in the field of fluid dynamics has much eased the way it was dealt traditionally. With the development of computational fluid dynamics (CFD), numerous simulations can be performed with much lesser time and lower costs. CFD, for the past few decades, has extensively been employed to study the aerodynamic performance of conventional and modified Savonius rotors (Akwa et al., 2012a; Fernando and Modi, 1989; Hayashi et al., 2005; Kawamura and Sato, 2002; Rogowski and Maroński, 2015; Zhao et al., 2009; Tian et al., 2015; Kaprzak et al., 2013). Results obtained from CFD analyses are validated by comparing with experimental results and are found to be capable of accurately predicting the behavior of fluid flow. However, when comparing CFD results with the experimental ones, CFD sometimes overestimate or underestimate the value of variables under consideration, but the trends of overall effects of the parameters are same as that of experimental studies (Howell et al., 2010; Ivan and Fawaz, 2011; Svetlana, 2016).

The literature review presented submits that alteration in conventional design of Savonius wind turbine can lead to a better aerodynamic performance. Several modifications have been tried, but the most practical modification is to change the shape of the blade. However, most of the previous research works conducted on modifications have been on conventional Savonius rotors, and despite numerous experimental research works, studies employing numerical techniques in this field are required. Moreover, individual effects of various rotor parameters have been analyzed but the combined effect of different operational parameters on rotor performance has not yet been attempted. This study is aimed at analyzing a shape-modified blade and studying its three operational parameters, namely, tip speed ratio, overlap ratio, and blade shape factor. The effects of these three parameters are investigated and then optimized for turbine performance coefficient C_P. The analysis is done using CFD. The optimum parametric combination is determined using response surface methodology technique.

Methodology

Mathematical formulation and parameters

Parameters and geometry definition

A simple Savonius wind turbine consists of a two-blade rotor of semi-circular shapes fixed to a vertical central shaft about which the rotors rotate. The turbine works on the principle of torque produced because of difference of forces exerted on the blade rotors. The two blades of the rotor are said to be advancing and retreating depending upon their orientation with the wind direction (Figure 1). The blade moving in the wind direction is advancing, whereas the blade moving opposite to the wind direction is said to be retreating. The wind coming toward the advancing blade exerts drag force upon interaction with the blade shape due to change in momentum, which forces the blade to move along the wind direction, whereas wind coming onto the retreating blade hits and disperses around the blade shape producing a negative torque around the shaft. Although both blades generate drag force, the force produced by the concave shape of the advancing blade is much more than that produced by the convex shape of the retreating blade. This effect is due to the shape of the drag surface provided to the wind direction. This difference of forces which generates torque causes the rotor to rotate around its central axis.

Figure 1.

Flow physics of Savonius rotor.

The rotor under study is a modified Savonius turbine blade with one end curved at a specific radius of curvature and the second end straight and overlapping as shown in Figure 2. The overlap in Savonius turbine reduces the negative torque generated by the retreating blade in a way that the wind coming from the advancing blade finds its way, through the overlap, into the closed side of the retreating blade (Figure 1), thus applying drag force on the concave side of the retreating blade as well and supporting the overall rotation of the rotor. The modified rotor used in this study increases the air flow through the overlap, thus increasing the rotor efficiency.

Figure 2.

Modified Savonius rotor construction.

The performance of a Savonius wind turbine can be measured in terms of coefficient of power (C_p) and coefficient of torque (C_t). The parameters under observation are shape factor (p/q), overlap ratio (G), and tip speed ratio (TSR). The ranges of these parameters are shown in Table 1.

Table 1.

Range of parameters.

Parameter	Range
Shape factor (p/q)	0.2–0.6
Overlap ratio (G)	0–0.2
Tip speed ratio (TSR)	0.4–1.2

The study is a two-dimensional (2D) analysis of the rotor, so the rotor is considered to be of unit height (h = 1 m). The diameter D is taken to be 2.05 m with angle of curvature (Ψ) equal to 124°. The blade thickness is 0.025 m. The fluid domain is air with free stream velocity u = 5 m/s, density = 1.225 kg/m³, and dynamic viscosity = 1.7894 × 10⁻⁵ kg/ms.

Mathematical formulation and numerical approach

C_p and C_t are obtained as functions of shape factor and TSR, respectively, and overlap ratio using CFD. Torque is calculated at each instant and is used to calculate C_t and C_p using the relations C_t = Q/V²_inA_SR and C_P = C_t·TSR where TSR = ω·Ρ/V_in. For unit height, A_S = S·1 and aspect ratio = 1/D. The overlap ratio for the rotor is G = m/2R, and Reynolds number is defined as Re = ρU = uD/µ.

The software used for calculating torque using CFD is ANSYS Fluent^®. The software uses Reynolds-averaged Navier–Stokes (RANS) equations for solving and simulating turbulent flows. The equations are basically an expression of conservation of momentum and equation of continuity. The major difference between various turbulence models used in CFD analysis is the methodology which is used to solve these equations.

CFD model

The computational domain determination is one of the most important steps in CFD analysis, particularly for a wind turbine analysis. The effect of wind tunnel has to be kept in mind when verifying CFD results against experimental results (Rogowski and Maroński, 2015) since it can cause major variations in the simulation results if the domain size is insufficient. However, a much larger domain is undesirable since it significantly increases the simulation time. So, a balance has to be struck between the two extremes. The rotor size for this study was 2 m which is represented by D (Figure 3).

Figure 3.

Computational domain dimensions.

This study ignores the wind tunnel wall effect by considering the parallel walls to present zero shear force to the fluid flow. The domain size before inlet is kept small since it least effects the results of the analysis with the wake region kept significantly larger. The blades of the turbine were kept at 0.025 m thick, and the condition of no slip was applied to these blades.

This study does not involve mesh deformation which requires a dynamic mesh model which is very computationally expensive. Instead, sliding mesh model was used in its place which is much more efficient. This methodology uses a fluid domain around the rotor which rotates with it (Figure 4). This circular mesh domain slides with the stationary fluid domain.

Figure 4.

Sliding mesh domain.

The circular rotating domain was kept at 1.75 D, larger than the rotor diameter, to keep the mesh around the corners of the rotor homogeneous. The mesh edges of the rotor were inflated to 10 layers for homogenization and resolving turbulent conditions along blade wall effectively (Figures 5 and 6). Both domains were discretized using triangular mesh elements. The interface of the rotating and stationary domains was edge sized for smooth transition between the two layers.

Figure 5.

Meshed geometry.

Figure 6.

Inflated layers adjacent to blade surface.

Mesh independence was verified by deteriorating the quality of the refined mesh which was eventually used for final simulations by 75%, that is, decreasing element number of original mesh from 224,950 to 149,238 as well as increasing element size by decreasing blade edge and domain interface edge divisions. The face size of the rotating domain was also increased for the validation mesh. The statistics for both meshes are shown in Table 2. It was inferred that if an inferior mesh passed the independence test by comparing with a refined mesh, using the refined mesh for the actual study would yield results strictly mesh independent. The validation mesh was simulated using the same conditions as the refined mesh, and average rotor torque results were calculated for both meshes by averaging torque generated over three complete rotations. The difference between the calculated average torques for both meshes was <0.38%, and the average absolute difference between rotor torques at any instant is <0.082 N/m (Table 2). These results were deemed to be within the acceptable range, and the refined mesh was reasoned to be sufficiently refined to yield mesh independent results. The y+ value of mesh cells adjacent to blade walls was also verified and was in the range of 0.01–4.8 which is <5. The wall mesh resolution near the blade was thus considered to be of sufficient quality to model turbulent velocity independently.

Table 2.

Mesh independence validation.

Mesh	Refined (Original) Mesh	Mesh For Independence Validation
Meshing Method	All Triangles	All Triangles
Number Of Elements	224950	149238
Minimum Edge Length (m)	2.50E-02	2.50E-02
Mesh Growth Rate	1.2	1.2
Blade Edge Sizing Divisions	100	75
Inflation Layers At Blade Edge	10	10
Rotating Domain Face Sizing	0.01	0.0125
Domain Interface Edge Sizing	800	600
Average Mesh Skewness	5.81E-02	6.21E-02
Average Mesh Orthogonal Quality	0.96579	0.96331
Average Torque Calculated (N/m)	20.95776052	21.03595547
Relative Torque Difference (%)	0.373107374
Average Absolute Difference Between Instantaneous Torques (N/m)	0.081911355

SST k − ω model was used in this study for modeling turbulence. It was chosen for its efficiency and suitability to this problem. SST k − ω model successfully predicts wind turbine performance and is computationally inexpensive for low fluid speeds (ANSYS, 2006). This model introduces two transport equations for turbulent viscosity by solving for k (turbulent kinetic energy) and ω (specific dissipation rate).

The viability of using SST k − ω model for modeling turbulence in a 2D Savonius turbine study was checked by simulating a conventional Savonius rotor and comparing the results obtained with experimental values. The study chosen for this purpose was one by Blackwell et al. (1977). The two-bucket conventional Savonius turbine configuration used for simulation was same as in the experimental study with wind velocity 7 m/s and 1 m rotor diameter. The simulation results were correlated to the experimental results and found to be in good agreement with those of the experimental values, as shown in Figure 7, at low to medium rotational speeds. At very high rotational speeds and wind velocities, flow separation occurs, which is a limiting characteristic of k − ω model. The turbulence model and the simulation settings were considered to be adequate for the simulation of a modified Savonius rotor based on these results since the turbulence conditions for both rotors are essentially the same. The current analysis focuses on low wind speed and rotational speeds which are accurately predicted by the model.

Figure 7.

Comparison of CFD and experimental results.

The complete domain was defined to be composed of air with the circular domain, given constant mesh motion, sliding with respect to the stationary mesh interface. Pressure–velocity coupled system was used for solving equations with residual monitors set at 10⁻³ and 100 iterations for each 0.01 s time step. The solution was simulated for 19 s (1900 time steps) and convergence was reached before the end of iterations. For each time step, instantaneous torque and moment coefficient on the blade surface were calculated. Instantaneous torque was recorded using macros since Fluent does not have a predefined function for recording it. These values were consequently used to calculate the C_p and C_t of the turbine rotor by calculating the average torque on the blade over a complete rotation. Instantaneous torque values of the rotor blade over whole rotation were determined and the sum was then averaged to get the rotor torque for the complete rotation.

Response surface methodology and optimization

To establish the relationship between test parameters, response surface methodology was used. The design of experiment (DoE) model used for this was Box-Behnken method. This model places the independent variables into three distinct, equally spaced levels. It ignores the test combinations that are the endpoints of the experimental space edges and is very efficient for estimation of first- and second-degree coefficients and is not affected by extreme combinations. For this reason, this methodology is much more computationally viable as compared with the full factorial design and was used in this study.

The three independent variables, overlap ratio, shape factor, and TSR, were combined using Box-Behnken methodology in their respective domains, as shown in Table 1, and were discretized into three equally spaced levels. The total number of combinations with two extra midpoints is 15 for three independent variables. The combinations are shown in Table 3.

Table 3.

CFD simulation results.

Experiment no.	TSR	Shape factor	Overlap ratio	p actual (W)	Torque (Nm)	C_p	C_t
1	0.8	0.6	0	83.683	20.921	0.403	0.504
2	1.2	0.6	0.1	32.347	5.391	0.166	0.138
3	0.8	0.6	0.2	56.400	14.100	0.305	0.381
4	1.2	0.4	0.2	31.654	5.276	0.160	0.133
5	0.8	0.4	0.1	80.923	20.231	0.387	0.484
6	0.4	0.4	0.2	55.271	27.635	0.279	0.699
7	0.8	0.2	0.2	84.735	21.184	0.390	0.488
8	1.2	0.4	0	14.827	2.471	0.067	0.056
9	0.8	0.4	0.1	80.923	20.231	0.387	0.484
10	0.8	0.4	0.1	80.923	20.231	0.387	0.484
11	0.8	0.2	0	94.787	23.697	0.389	0.487
12	0.4	0.4	0	70.002	35.001	0.315	0.787
13	1.2	0.2	0.1	43.088	7.181	0.188	0.157
14	0.4	0.6	0.1	56.562	28.281	0.290	0.725
15	0.4	0.2	0.1	77.467	38.734	0.338	0.844

CFD: computational fluid dynamics.

The results of the simulations were then optimized over specific domains by first developing a regression model using Minitab^®. The terms of the regression model were evaluated for their effect on response variable C_P using R² and p value statistics. The equation formed was again judged for its goodness of fit after identifying the significant terms. The equation was then optimized for C_P using Visual Studio^®. The C++ optimization code was designed to sequentially plug in all the possible value combinations of the three independent variables over their specific domains and calculate the value of C_P. The combination which maximizes C_P was identified as the optimum solution. The results obtained by CFD can accurately predict the parametric effects of the variables involved although they may overestimate or underestimate the effects of those variables when compared with experimental results (Howell et al., 2010; Ivan and Fawaz, 2011; Svetlana, 2016). The optimization results were validated using ReliaSoft DOE++^® RSM software, and the parametric optimization solutions were found to be exactly the same for both methodologies.

Results and discussion

Results of 15 runs of ANSYS Fluent simulation are shown in Table 3. Results of each combination shown in the table are averaged for one revolution. Power and torque coefficients are presented against values of three independent variables, namely TSR, shape factor, and overlap ratio.

The pressure and velocity contours, obtained by ANSYS CFD-Post^®, were used to check for flow separation. The contours were checked at the end of complete simulation for 90° intervals. The contours for 0°/180° and 90°/270° are shown for fluid velocity with respect to stationary frame in Figure 8(a) and (b) and pressure distribution in Figure 9(a) and (b).

Figure 8.

Velocity contours with respect to stationary frame at (a) 0°/180° rotor angle and (b) 90°/270° rotor angle.

Figure 9.

Pressure contours with respect to stationary frame at (a) 0°/180°rotor angle and (b) 90°/270°rotor angle.

Since power coefficient is dependent upon the three independent variables, TSR, shape factor, and overlap ratio, a relationship is needed to predict the behavior of the rotor. A regression model is developed using Minitab to develop the relationship between the said variables. Regression analysis is performed in which null hypothesis is rejected for variables with p value near to zero, which means those values have significant relationship with the predicted variable. The regression calculation summary is given in Table 4.

Table 4.

Regression data summary.

Term	Coefficient	Standard error	Low confidence	High confidence	T value	p value
Intercept	0.394777	0.012625	0.370857	0.418696	31.268852	8.837961E–9
A: TSR	−0.0802	0.009292	−0.097805	−0.062596	−8.631188	0.000056
B: Shape Factor	−0.017654	0.009292	−0.035258	−0.00005	−1.89992	0.099218
C: Overlap Ratio	−0.004889	0.009292	−0.022493	0.012715	−0.52614	0.615043
A • C	0.032197	0.013141	0.007301	0.057093	2.450175	0.044096
B • C	−0.024761	0.013141	−0.049658	0.000135	−1.884322	0.101519
A • A	−0.155213	0.013637	−0.181049	−0.129377	−11.381901	0.000009
C • C	−0.028544	0.013637	−0.05438	−0.002708	−2.093141	0.074623

As a result of the regression analysis, a relationship between the dependent and independent variables is formed. This is presented in the form of equation as follows

C_{p} = 1.2036 A + 0.0131 B + 0.223 C - 0.9349 A^{2} - 2.74 C^{2} + 0.88 A C - 1.089 B C

where, A = TSR, B = Shape Factor, and C = Overlap Ratio.

In order to check the goodness of fit for the equation developed, R² and R² (adjusted) values are analyzed. These show whether the predicted relationship is a good fit for the values given.

R² = 96.90%.

R² (predicted) = 72.28%.

R² (adjusted) = 93.81%.

After forming the equation depicting the relationship between variables in use, a C++ optimization code is generated for the purpose of determining optimum solution for C_p. It is done by plugging in the values of predictor variables from the specified domain of each in the generated code and finding the maximum value of the response variable produced. The optimum parametric combination value obtained for the maximum value of the response variable against the predictor variables is shown in Table 5.

Table 5.

Optimization result summary.

TSR	Shape factor	Overlap ratio	C_P predicted
0.70	0.2	0.12	0.424

Conclusion

This article presents a 2D CFD parametric analysis of a modified Savonius rotor. The turbine parameters TSR, overlap ratio, and shape factor are studied with respect to rotor coefficient of performance. Box-Behnken method was used to formulate the DoE model of the study. The simulations were carried out using ANSYS Fluent. The computations were carried out in a 2D environment with unit rotor height and using sliding mesh and k − ω turbulence model. The turbulence model was verified by validating against experimental results and was found to be sufficiently accurate. Instantaneous torque on the blade surface was computed for the whole rotation for each 0.01 s interval and then averaged to get average torque over the whole rotation. This, in turn, was used to calculate the coefficient of performance (C_P) of the rotor which was the response variable for this study.

Parametric optimization was carried out using response surface methodology. Minitab was used to obtain regression model for the experiment. The model was optimized for parametric combination for the maximum C_P value using C++. The optimum parametric combination values are as follows: shape factor = 0.2, overlap ratio = 0.12, and TSR = 0.70 for the parametric domains. These optimization results were verified by ReliaSoft DOE++ and found to be accurate. This optimum value of rotor parameters for maximum response variable can be used for developing modified Savonius turbine rotors for optimum performance of the specific parametric domain studied.

Footnotes

Appendix 1 Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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