Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 1, pp. 37-45, Warsaw 2015 DOI: 10.15632/jtam-pl.53.1.37 CFD COMPUTATION OF THE SAVONIUS ROTOR Krzysztof Rogowski, Ryszard Maroński Warsaw University of Technology, Faculty of Power and Aeronautical Engineering, Warsaw, Poland e-mail: krogowski@meil.pw.edu.pl; maron@meil.pw.edu.pl In this paper, Computational Fluid Dynamics (CFD) analysis of a two-dimensional two- -bucket Savonius rotor, using ANSYS Fluent solver, is presented. Computational methods of fluid dynamics are used to estimate the aerodynamic forces acting on the rotor, the torque and the power extracted by the Savonius rotor. In addition, an analysis of the results for different turbulence models is performed. Numerical investigations are compared with the experiment of the Sandia National Laboratories. Keywords: vertical axis wind turbine, simulation, Computational Fluid Dynamics 1. Introduction A wind turbine with the Savonius rotor is usually a vertical axis wind turbine (VAWT). It is a drag-type machine; however, because of an airflow between buckets (the bent sheets of the Savonius rotor), an additional lift force appears on the buckets. TheSavonius rotor is the biggest achievement of Finnish inventor Sigurd Johannes Savonius, who patented it in 1927 (Savonius, 1931). The typical Savonius rotor is equippedwith two buckets; however, three ormore buckets are possible. It usually has an S-shape cross-section. TheSavonius-typewind turbine can achieve even 30% of the maximum power coefficient comparing with 60% following from Betz theory (Hansen, 2008). However, this turbine needs 30 times greater surface of the buckets as compared with the wing area of a conventional wind turbine to give the same output power. Therefore, the Savonius rotors are small and low energy devices (Szuster, 2000). They can work as water pumps, drives for small electric generators or ventilators (e.g. FlettnerVentilator).Moreover, the Savonius wind turbines can provide water agitation to keep stock ponds ice-free during winter (Paraschivoiu, 2002;Backwell et al., 1977).Anemometers are oftenSavonius-type turbinesdue to low costs and reliability. Because of a very low starting torque of theDarrieus-typewind turbine, the Savonius-type wind turbine is sometimes used as a starter for these machines (Alam and Iqbal, 2010;Gupta andSharma, 2012).Undoubtedly, one of the advantages of theSavonius rotor is the possibility of using it by residents of Third World countries living without electricity. A worn out and empty oil barrel can be used as a low-power device. During rotation of the Savonuius rotor, the Magnus effect appears. This effect depends on the tip speed ratio of the rotor. Although the tip speed ratio of the Savonius rotor is very close to unity and the created lift forces are low, there are investigations of Savonius-type Magnus wind turbines where the Savonius rotor is used to create lift forces for a classical horizontal axis wind turbine (Tokumaru and Dimotakis, 1993; Komatinovic, 2006). The main advantages of the Savonius rotor are as follows: simple and cheap construction of the wind turbine; low noise level; very high starting torque; rotor operation independent of the wind direction. On the other hand, disadvantages of theSavonius-typewind turbineare as follows: lower efficiency in comparisonwith lift devices; low rotational speed; fluctuations in the torque during operation of the rotor; difficulty in designing high-power wind turbines because of large mass of the rotor. Many shapes and configurations of the Savonius rotor are currently used. Some rotors have the shape of a drill or the two-unit 38 K. Rogowski, R. Maroński two-bucket Savonius rotors rotated 90◦ apart. It decreases the pulsation of the torque with the azimuth. Flat circular discs are attached at the ends of the Savonius rotor. Sometimes, an additional shaftwhichmakes the structuremore stiff is used (Johnson, 2004;Komatinovic, 2006; Alam and Iqbal, 2010). In spite of the very simple structure of theSavonius rotor, its aerodynamics is verydifficult to analyze. There are notmany simplified aerodynamicmethods formodelingwind turbines of this type.Onemodel (perhaps the only one) noticed byParaschivoiu (2002) is amathematicalmodel proposedbyChauvin,basedon thepressuredropon each sideof theblades. It enables computing the power of the two-bucket Savonius rotor without any gap between the buckets. However, the Savonius rotor without the gap is only a drag-type device and its maximum power coefficient is much lower than that the one for the rotor with a specific gapwidth.There aremainly twoways to investigate the aerodynamic performance of the Savonius rotor: computational methods of fluid dynamics and/or via an experiment. In the 70’s, experimental research on a large scale of 2 and 3 bucket Savonius configurations were conducted by Blackwell, Sheldahl and Feltz in the Sandia National Laboratories (Blackwell et al., 1977). The aim of that study was to determine torque characteristics of the Savonius rotors. More detailed measurements of the static and dynamic torque were performed by Ushiyama and Nagai in the 80’s (Ushiyama and Nagai, 1988). More detailed numerical investigation of the performance of the Savonius wind turbine, based on the discrete vortex method, was performed by Fernando andModi (1989). Numerical computations of the flowaround the rotor require integration of theNavier-Stokes equations. Direct integration of these equations (Direct Numerical Simulation – DNS) requires very fine mesh and very efficient supercomputers. It gave an impulse for further development of the CFD methods based on time-averaged equations (Reynolds-averaged Navier-Stokes – RANS) or spatially averaged equations (Large Eddy Simulation – LES). Using them, large scale problems canbe solved and they are satisfactory formanypractical applications (ANSYS, 2013). The Savonius rotor is shown in Fig. 1, wherein the symbols are as follows: HP is the height of the Savonius rotor; d is the diameter; R is the rotor radius;DP is the diameter of the plate; s is the bucket gap width. Fig. 1. Savonius rotor 2. Mathematical formulation The computational fluid dynamics (CFD) software, ANSYSFluent, is used to obtain the power coefficient CP as a function of the tip speed ratio, TSR. The power coefficient CP , and the tip speed ratioTSR are as follows CP =CQ ·TSR TSR= ωR Vin (2.1) where: ω is the turbine rotational speed;R is the rotor radius of rotation; Vin is the freestream velocity; CQ is the torque coefficient which is given by CQ = Q 1 2 ρV 2inASR (2.2) CFD computation of the Savonius rotor 39 where: ρ is the air density;Q is the turbine torque;AS is the rotor swept area which is defined as AS =(4r−s) ·1 (2.3) where: r is the bucket radius; s is the bucket gap width; 1 is a unit height of the Savonius rotor. Moreover, the side forceFy, and the downwind force (drag)Fx, acting on the Savonius rotor as a function of the azimuth position are investigated (see Fig. 2). These forces, expressed as dimensionless coefficients, can be written as CFx = Fx 1 2 ρV 2inAS CFy = Fy 1 2 ρV 2inAS (2.4) Fig. 2. The side forceFy and downwind force Fx exerted on the Savonius rotor 3. Basic fluid flow CFDcodes use theReynolds-averagedNavier-Stokes (RANS) equations for computing turbulent flows. The instantaneous continuity andmomentum equations can be written as ∂ρ ∂t + ∂ ∂xi (ρui)= 0 ∂ ∂t (ρui)+ ∂ ∂xj (ρuiuj)=− ∂p ∂xi + ∂ ∂xj [ µ (∂ui ∂xj + ∂uj ∂xi − 2 3 δij ∂ul ∂xl )] + ∂ ∂xj (−ρu′iu ′ j) (3.1) There are many hypotheses which allow one to solve the Reynolds stresses −ρu′iu ′ j. One of them is the hypothesis of Boussinesq which can be written as −ρu′iu ′ j =µt (∂ui ∂xj + ∂uj ∂xi ) − 2 3 ( ρk+µt ∂uk ∂xk ) δij (3.2) where µt is the turbulent viscosity. This hypothesis is commonly used to determine theReynolds stresses. It is considered in the following turbulencemodels: the Spalart-Allmaras, the k-ε and the k-ω. Thedifferences between thesemodels is thewayof computing the turbulentviscosityµt.TheSpalart-Allmaras turbulence model solves only one transport equation fromwhich the turbulent viscosity is computed, while the k-ε and the k-ω turbulence models introduce two transport equations, for the turbulence kinetic energy k, and for its dissipation rate ε in the case of the k-εmodel, or specific dissipation rate ω in the case of the k-ω turbulence model. For the k-ε and the k-ω models, the turbulent viscosity µt is a function of k and ε or k and ω. One of the disadvantages of the Boussinesq hypothesis is the constant isotropic scalar, µt. Therefore, this hypothesis cannot be used for flowswith anisotropic turbulence. However, formany flow classes, this hypothesis gives accurate results (Hinze, 1975). 40 K. Rogowski, R. Maroński 3.1. Numerical approach Thegeometricalmodel of the examinedwind turbine (Fig. 3a), used forCFDanalysis, consists of two buckets (blades of the Savonius wind turbine). This two-dimensional model of the Savonius rotor is developed based on Blackwell et al. (1977). Usually, the Savonius rotor has upper and lower plates (Fig. 1) so that the energy loss associatedwith the finite length of the buckets is not large. Therefore, the use of the two-dimensional model is justified. There is the gap s between the buckets of the Savonius rotor (see Fig. 3a). The dimension s, one of themain parameters of the Savonius rotor, is defined in the presented study as a distance between the bucket chamber lines. The bucket thickness is 2mm. The computations presented in this article are performed for the same buckets but for two different lengths of the gap which are expressed as a ratio of the gap length s to the diameter of the bucket d. The most important geometrical parameters of the Savonius rotor, which are used during computations, are presented in Table 1. Fig. 3. (a) The basic geometrical parameters of the Savonius wind turbine, (b) the virtual wind tunnel (domain) Table 1.Basic parameters of the Savonius rotor Parameter Value Bucket diameter d 0.5m Gap width ratio s/d 0.1 and 0.2 Rotor radiusR 0.47m and 0.45m Theproblemconsidered in thework does not involvemesh deformation, therefore the sliding meshmodel has been used. This approach is simpler andmore efficient in comparison with the dynamic mesh model. The area around the Savonius rotor is modeled as a large square. The rotor is located in the middle of this square area. Moreover, a smaller circle zone around the rotor is created. This circle zone can rotate with the rotor during simulation. Themodel of the Savonius rotor and the virtual tunnel is presented inFig. 3b.The area of the virtualwind tunnel has been discretized using triangular elements. An irregularmeshwithout inflation layers on the blade surfaces is used. The mesh interface is situated between the circle zone and the square zone to allow the exchange of data. There are 41336 cells and 21039 nodes in the presented mesh. Figure 4 presents the mesh used during computations. 4. Results and discussions 4.1. Turbulence models There is not any single turbulence model which gives satisfactory results for all classes of flow. The choice of the turbulence model depends on many factors, for example: physics of the flow, accuracy of the solution, computational resources and the amount of time available for simulation. The ANSYS Fluent solver provides many turbulence models. During the authors’ CFD computation of the Savonius rotor 41 Fig. 4. Example of mesh (the buckets – bold line) Fig. 5. Power coefficient as a function of the tip speed ratio investigations, the following turbulencemodels are used: the SpalartAllmaras (SA), the k-ε, the realizable k-ε, the RNG k-ε and the k-ω. The CFD results and the experimental results of the power coefficient as a function of the tip speed ratio are presented in Fig. 5. Most of the used turbulence models except for the standard k-ε and the standard k-ω models give satisfactory results in the low tip speed ratio range. For a range of tip speed ratios between 0.8 and 2, two turbulencemodels: the SpalartAllmaras and theRNG k-ε present results similar to those of the experiment. The results for the SpalartAllmaras turbulence model seem to be overestimated in comparison to theexperimental ones.However, theCFDmodel is two-dimensional andtheeffects of the finite aspect ratio of the Savonius buckets are neglected. All aerodynamic characteristics of the Savonius rotor, presented in the next Sections, are performed using the SpalartAllmaras turbulencemodel because it is the simplestmodel and the accuracy of the solution is satisfactory for the purposes of this work. 4.2. Dimensions of the virtual wind tunnel Appropriate size of the virtual wind tunnel (domain) is very important from the viewpoint of CFD computation. Too small size of the virtual wind tunnel causes too high results of the power coefficientsCP , whereas too large size causes an increase in computation time. To find an optimal size of the domain, an analysis of the following three dimensions is performed:W-width of the virtual wind tunnel, Z-length between the inlet of the tunnel and the axis of rotation of the Savonius rotor, and T-length between the axis of the Savonius rotor and the outlet (Fig. 6a). 42 K. Rogowski, R. Maroński Fig. 6. (a) Dimensions of the domain, (b) the influence of the dimensions on the rotor performance The results of the power coefficient as a function of length of the domain are presented in Fig. 6b. For the diameter of the rotor of 0.97m, the dimensions:W ,Z andT shouldbeminimum 10 times larger than the rotor diameter. If these lengths are two times larger than the diameter, the power coefficients are overestimated even by 30%. All CFD results of the Savonius rotor presented in this article are performed for a square domain with the side of 60m. 4.3. Power coefficient Unsteady flow of the air has been considered in the work. During the computations, aerody- namic forces and the torque are determined at each time step. The time step size corresponds to the azimuth of 0.5deg. The simulation of 4-6 revolutions of the Savonius rotor is required to eliminate the effects associated with initial conditions. Therefore, all simulations presented in this article are performed for 10 revolutions of the rotor. The results of the torque and the forces from the last computed revolution of the rotor are taken for the performance analysis of the Savonius rotor. Figure 7presents the torque coefficientCQ as a function of the azimuthposition for six values of the tip speed ratio. As it can be observed, the maximum values of the torque coefficient are approximately 0.6 for each tip speed ratio except for the torque coefficient for the tip speedof 0.6. However, the average value of the torque coefficient decreases as the tip speed ratio increases. Fig. 7. Torque coefficient as a function of the azimuth position The mean values of the torque coefficient as a function of the tip speed ratio are presented in Fig. 8a. The obtained results are compared with the experimental torque coefficient. As it can be seen, for the range of the tip speed ratio between 0.4 and 1.2, the results of the torque coefficient are acceptable. However, the power coefficient CP depends on the tip speed ratio (see Eq. (2.1)1). Figure 8b presents the power coefficient as a function of the tip speed ratio. The maximum power coefficient CPmax of 0.24 is computed for the tip speed ratio of 0.8. The experimental results are confirmed numerically. CFD computation of the Savonius rotor 43 Fig. 8. (a) The torque coefficient as a function of the tip speed ratio, (b) the power coefficient as a function of the tip speed ratio 4.4. Gap length Blackwell et al. (1977) tested different configurations of buckets of the Savonius rotor to find the optimal one. They recommended a configuration consisting of two buckets positioned as it is shown in Fig. 3a, with dimensionless gap width of 0.1-0.15. In this Section, the results of the power coefficient as a function of the tip speed ratio for two dimensionless gap widths are presented. The torque coefficient versus the tip speed ratio is presented in Fig. 9b whereas the power coefficient as a function of the tip speed ratio is shown in Fig. 9a. The CFD results of the torque coefficient in comparison with the experimental one seem to be optimistic. Larger differences are observed for the power coefficient. However, it seems to the authors that the computational analysis confirms the results taken from literature. Thepower coefficient is higher for the dimensionless gapwidth of 0.1 in comparisonwith the gap width of 0.2. This is themost important conclusion for constructors. Fig. 9. (a) The torque coefficient as a function of the tip speed ratio, (b) the power coefficient as a function of the tip speed ratio 4.5. Effect of the Reynolds number During experimental tests of Blackwell et al. (1977), it turned out that the power coefficient depends on the Reynolds number. The performance of the Savonius rotor is computed for two velocities of wind, Vin of 7m/s and 14m/s. The computed results are shown in Fig. 10. As can be seen, the efficiency of the Savonius rotor is slightly larger for the wind speed of 7m/s in comparison with the results for the wind speed of 14m/s. 44 K. Rogowski, R. Maroński Fig. 10. Power coefficient as a function of the tip speed ratio 4.6. Influence of the side force In this Section, the aerodynamic forces acting on the Savonius rotor as a function of the azimuth position are presented. The reason for which the authors of this article decided to perform these computations is the following sentence from the article byBlackwell et al. (1977): Recent analytical studies, along with unpublished data of the authors, indicate that the Savonius rotor experiences side forces that are of the same order of magnitude as the downwind (drag) force. Fig. 11. (a) Drag force coefficient as a function of the azimuth position, (b) the side force coefficient as a function of the azimuth position Figures 11a and 11b present coefficients of the aerodynamic force: the drag force coeffi- cientCFx and the side force coefficientCFy, respectively. Figure 2 shows the vectors of the drag forceFx and the side forceFy. The aerodynamic forces acting on the Savonius rotor vary perio- dically in time. The results of the drag force coefficient and the side force coefficient for both tip speed ratio of 0.8 and 1.0 differ slightly. For these tip speed ratios CFx is approximately in the range of 0.5-1.8 whereasCFx is approximately in the range of−2.0-−0.4. For the tip speed ratio of 0.4CFx varies in the range of 0.35-2.0 andCFy varies in the range of−1.4-0.15. The average values of the force coefficients are shown in Table 2. As it can be seen, the mean value of CFx is similar for all tip speed ratios while the average value of the side force coefficient increases with an increase in the tip speed ratio. For the tip speed ratio of 0.4, themean value of the side force coefficient is about two times lower in comparison with the drag force coefficient, while for the tip speed ratio of 0.8 and 1.0 the side force coefficients are in the same order of magnitude as the drag force coefficients. This conclusion is important from the viewpoint of the design of bearings and dynamics of the shaft. CFD computation of the Savonius rotor 45 Table 2.The average values of the drag force coefficient and the side force coefficient for three tip speed ratios Tip speed ratio CFx CFy 0.4 1.1436 0.6405 0.8 1.1372 1.0824 1.0 1.1383 1.1974 5. Conclusions In the present investigation, the aerodynamic efficiency of the Savonius rotor using computa- tional methods of fluid dynamics is studied. The obtained CFD results are compared with the experiment. The conducted analyses have shown that the one-dimensional turbulence model Spalart-Allmaras andmesh without inflation layers can be used for the Savonius rotor applica- tions in the two-dimensional case. The optimum size of the domain has been investigated. This study has demonstrated that the CFD methods confirm the experimental results and can be used to optimize the shape of buckets of the Savonius rotor. 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Ushiyama I.,NagaiH., 1988,OptimumdesignconfigurationsandperformanceofSavonius rotors, Wind Engineering, 12, 1, 59-75 Manuscript received May 25, 2014; accepted for print July 11, 2014