## Abstract

To improve the efficiency of a Savonius vertical axis wind turbine, this investigation proposes the use of expandable blades instead of rigid blades. The expandable blades have the ability to change their form during the turbine rotation. The expansion of the advancing blade increases the positive torque, and the contraction of the returning blade decreases the negative torque, which boosts the turbine efficiency. Two-dimensional numerical simulations have been carried out using the commercial code ansys fluent 18.0 with a deformable mesh to fit the changing shape of the blades during the rotation cycle. The paper involves the effect of the expansion amplitude as well as the effect of the blades gap and overlap ratios on the turbine overall performance. The numerical model is validated by comparison of predictions with experimental results. Results show that the torque coefficient is improved by about 24% for the lowest expansion amplitude and by about 90.6% for the highest expansion amplitude. A further improvement of 7% is recorded for an expandable turbine with a gap ratio of 1/20 the turbine diameter.

## 1 Introduction

Population growth and the industry boom have promoted the intensive consumption of fossil fuels. This fact has stimulated many countries to search for alternative energy sources. As a matter of fact, renewable energies are proposed as an alternative solution to cover the growing energy demand. They are also considered environmentally friendly sources compared to fossil energy sources [1,2]. Several incentives were imposed to help the world move towards clean and renewable energies. For example, the European parliament climate and energy package law has been employed since June of 2009. This law involves first that, during the next ten years, at least 20% of the European Union's energy consumption should be provided by renewable resources. Second, the ratio of greenhouse gas emissions should be reduced at least by 20% from 1990 levels [2]. Wind and solar energies are among the promising energy sources. A wind turbine can be classified relative to its axis of rotation as a vertical axis turbine (VAWT) or a horizontal axis turbine (HAWT). It can be classified also relative to the driven force as lift-driven turbine (Darrieus), drag-driven turbine (Savonius), and hybrid turbines driven by a resultant force (flapping wings) [3–6]. Despite their good starting torque and their simple design, Savonius wind turbines have low efficiency as compared to Darrieus turbines [7–10].

The typical Savonius turbine has an S-shape cross section made of two semicircular buckets. The buckets’ concave and convex surfaces have different pressure distributions, which generates a drag force that produces torque and induces rotation.

This drag-driven turbine can be used as a starter when it is coupled to another type of turbine because of its high starting torque. The lift force can also contribute to the turbine power extraction [11]. Typically, the advancing blade has a greater drag force than the returning blade. The returning blade generates almost negative torque that obstructs the turbine rotation; this loss of energy makes the turbine suffer from poor power in comparison to other types of turbines [12–15]. Improving the performance of Savonius turbines by modifying their design was intensively investigated. Several experimental and numerical researches highlighted the effects of the blade's number, the blade profile, and the gap between blades [16–18].

Novel blade shapes were proposed in the study by Ref. [19]. The originality of that work was based on the modification of blade fullness. A correlation between blade fullness and power enhancement was obtained. For a blade fullness equal to one, the power coefficient was improved by 10.98%. In Ref. [20], a blade based on a spline profile was proposed as an alternative to the traditional S-shape Savonius blade. The power output was boosted by 12–51% relative to the original turbine.

Other studies considered the effect of baffles and deflectors on the turbine performance. The wind guidance systems were typically used to redirect the flow path around the turbine. A simple deflector improved the turbine performance by 19% whereas a combined deflector was able to boost the output power up to 100% [21–24]. The study in Ref. [25] addressed experimentally the effect of deflector geometric parameters on the performance of a Savonius turbine. Two deflectors were used for both advancing and returning blades. A significant improvement of 100% was achieved when the first deflector was fixed at a distance equal to 0.5 *R* where *R* represents the turbine radius and the second deflector at a distance of 1.204 *R* far from the returning blade. Hossein et al. [26] performed a computational study on the effect of a circular barrier on the Savonius hydrokinetic turbine. The circular barrier changes the fluid flow from the returning blade convex side to the advancing blade concave side. The torque was boosted by 19% relative to the base turbine. Several flow control methods are available in the open literature. A multi-stage hydrokinetic turbine was proposed by Kumar et al. [27]. The effect of blade twisting was considered. Unlike the standard turbine, this model yielded a high power coefficient of 0.44.

Flexible blades, morphing blades, and active deformable blades are techniques previously used to improve the output power of horizontal and vertical (Darrieus) axis turbines [28–30]. Typically, flexible blades allow the pressure to distribute favorably along the blade, which enhances the amount of positive torque and thus acceleration. Flexible blades are less applied in Savonius turbines. Recently, Krzysztof et al. [31] studied the performance of a Savonius turbine with an elastic blade. A fluid– structure interaction (FSI) solver was used to assess the effect of deformation on the turbine performance. An improvement of 90% was recorded as compared to a rigid turbine. In the present work, self-expandable blades are proposed to improve the Savonius turbine efficiency. Unlike the elastic blade, the self-expandable blade moves rigidly with a prescribed motion. The blade has rigid parts with a guidance system that allows the turbine to expand passively relative to the turbine rotation angle. The guidance system controls the amplitude of the blade deformation. The elliptic configuration can be achieved by an elliptic guiding system or simply with a non-coaxial circular guiding system.

## 2 Description of the Suggested Model

*A*

_{S}, the wind power available for extraction can be expressed as

*C*

_{t}is

In nature, several aero/aqua animals use their flexible fins to control perfectly the surrounding flow to improve their propulsion performance. This extraordinary vortex control mechanism has stimulated researchers to use flexible blades in several applications [32,33]. By handling the leading-edge vortex (LEV), flexible wings can achieve a higher level of aerodynamic forces. For example, the bending and twisting motion of a flexible wing can effectively vary the vortex shedding mechanism and therefore the flying direction [33,34]. For wind turbines, the use of a flexible blade not only controls the near boundary layer fluid flow but also regulates the vortex interaction between each blade [27,28].

Figure 1 represents a schematic view of a two-bladed Savonius wind turbine with a gap and overlap distance; the wind velocity is denoted *U*_{∞}, *ω* is the turbine angular velocity, *c* is the blades chord length, *D* is the turbine diameter, and *θ* is the azimuth angle. The new proposed model is a Savonius vertical axis wind turbine with self-expandable blades (Fig. 2). First, the proposed turbine has no gap between blades. The blade shape changes continuously during turbine rotation. As shown in Fig. 3, the blade has rigid parts with a guidance system that allows the turbine to expand passively relative to the rotation angle. Thus, the coupled motion involves two different motion patterns, a pure rotation, and an elliptic deformation. The guidance system controls the amplitude and the frequency of deformation. The initial blade has a circular form where the turbine diameter before deformation is 1 m. The elliptic expansion changes the blade circular form to an elliptic form. The outer diameter of the turbine remains the same (1 m) because the expansion of the first blade and the contraction of the second blade has the same amplitude. In this investigation, the frequency of expansion is equal to the rotation frequency. Different amplitudes of expansion are studied.

The phase angle between the rotation and the expansion is set to zero, which means that the blade achieves its maximal amplitude (expansion) at *θ* = 90 deg. It returns to the initial form at *θ* = 180 deg. After that, it contracts to the minimal amplitude at *θ* = 270 deg. The turbine moves in a counter-clockwise direction. The rotor parameters are summarized in Table 1.

## 3 Numerical Method

### 3.1 Equations of Motion.

In the open literature, several works were performed using a two-dimensional (2D) solver. It was specified that a 2D computational fluid dynamics (CFD) simulation could reduce the processing time and computational requirements without losing accuracy [19,35,36]. The use of a 3D model to carry out a thorough study of a turbine efficiency is a challenge due to the high computational cost [37]. Furthermore, the present Savonius turbine model has two end plates like most small-scale Savonius turbines. This means that the tip vortex will be deleted, and the flow can be considered a 2D flow. The 2D numerical simulation is carried out using ansys fluent v18.1. The expansion of blades requires a grid deformation during the simulation. The grid deformation involves the turbine rotation and the blade deformation.

*x*

_{old}and

*y*

_{old}are the blade node positions along the

*x*and

*y*-axes, respectively, in the previous time-step.

*ω*_{1} = 2*πf*_{1} where *f*_{1} is the expansion frequency.

*θ*= const =

*ωt*

The mathematical equations of the deformation of the blade have been executed within the DEFINE GRID MOTION macro, which is a c++ subroutine written by the user and hooked to ansys fluent.

### 3.2 Implementing Expandable Blades for Savonius VAWT.

As mentioned earlier, the expansion of blades requires a grid motion during the simulation. The grid motion involves turbine rotation and blade deformation. Two different methods within Fluent have been considered and tested.

*Method I: Sliding mesh combined with dynamic mesh*For this method and in order to provide the turbine motion, the DEFINE- ZONE- MOTION macro has to be used to provide the turbine angular velocity; however, this requires that the rotating zone mesh be declared as a rigid body within Fluent. Only Eq. (7) is used to deform the turbine blades. This strategy can be considered as a simple deformation inside a rotating zone, which provides a combined motion. The mesh updating is achieved by using the sliding technique for rotation and the dynamic mesh for deformation. Figure 4 shows the typical computational domain used in this method. The domain involves two subdomains. The rotating domain slides relative to the fixed outer domain. This technique was not able to provide satisfactory results [38].

*Method II: Pure Remeshing technique*Another dynamic mesh strategy is proposed in this work to avoid the use of sliding mesh combined with dynamic mesh. The idea of this alternate strategy is to find a universal mathematical equation that controls both, the blade deformation and the turbine rotation (Eq. (8)), and then only the dynamic Remeshing technique is used to update the meshin the computational domain (Fig. 5). Despite the cost of this method, the problem of unphysical results was fixed and accurate results were obtained.

### 3.3 Mesh Design.

The details of the computational domain used in Remeshing method are shown in Figs. 6(a)–6(d). The domain has a near wake domain with a high mesh density and an external domain with a moderate mesh density. The blade sub-domain in the present model meshes with triangular elements. The adaption tool is enabled. The adaption tool allows calculating the *y*^{+} value at each time-step and changing the mesh at the blade surface or any specific zone to meet the desired *y*^{+}. This strategy provides a fine grid to capture turbulent flow structures and energy.

To confirm the ability of the pure Remeshing method to provide accurate results, Fig. 7 displays a comparison of the torque coefficient obtained by Sliding and Remeshing methods. Using the Remeshing method, some further numerical errors interpreted as small random fluctuations are presented. However, the relative error in the average *C*_{t} value does not exceed 0.05%.

### 3.4 Boundary Conditions and Turbulence Model.

Practically, the same grid dimensions used by Ref. [31] are employed. The working far-field allows full development of the wake (Fig. 6(a)). It is extended 20 rotor diameters upwind and 40 rotor diameters downwind. Stationary wall boundary is selected—no-slip condition—for blades wall. For the external borders of the computational domain, the corresponding boundary conditions (BC) are as follows:

Pressure outlet condition is imposed on the outlet edge where the pressure is considered as gauge pressure.

Inlet velocity condition imposed at the inlet boundary is a Dirichlet condition specifying the velocity by one of the available ways in ansys fluent. The SIMPLE algorithm is used for the pressure-velocity coupling. The second-order scheme is used for all transport equations discretization. BC are described in detail in Table 2. The BC corresponds to the 2D computational domain depicted in Fig. 4.

Boundary | Parameter | Value | Note |
---|---|---|---|

Inlet | Constant velocity | 7 m/s | Inlet turbulent viscosity = 0.001 m^{2}/s |

Outlet | Gauge pressure | 0 Pa | Outlet turbulent viscosity = 0.001 m^{2}/s |

Sides | Symmetry | – | – |

Blades | Wall | – | No-slip wall condition |

Boundary | Parameter | Value | Note |
---|---|---|---|

Inlet | Constant velocity | 7 m/s | Inlet turbulent viscosity = 0.001 m^{2}/s |

Outlet | Gauge pressure | 0 Pa | Outlet turbulent viscosity = 0.001 m^{2}/s |

Sides | Symmetry | – | – |

Blades | Wall | – | No-slip wall condition |

For an accurate prediction of the flow around rotating machinery, the turbulence model should be selected carefully. The swirl effect, the adverse pressure gradient, and flow separation are the common physical phenomena observed during turbine rotation.

*k*–

*ɛ*and shear stress transport (SST)

*k*−

*ω*models are among the most frequently used in Savonius turbine simulation [39,40]. The boundary layer is measured using the dimensionless wall distance parameter

*y*

^{+}

*y*is the distance from the wall to the cell center, $u\tau =\tau w/\rho $ is the friction velocity, and

*τ*

_{w}is the wall shear stress.

The wall *y*^{+} is a non-dimensional distance similar to the local Reynolds number, often used in CFD to describe how coarse or fine a mesh is for a particular flow. It is the ratio between the turbulent and laminar influences in a cell. A turbulent boundary layer consists of distinct regions for CFD. These regions with their corresponding wall *y*^{+} are: the viscous sublayer (*y*^{+} < 5), the buffer layer or blending region (5 < *y*^{+} < 30), and the fully turbulent or log-law region (30 < *y*^{+} < 300 ) [41].

Different turbulence models require different inputs depending on the method of resolving the viscous sublayer. For the wall modeling, ansys fluent offers two choices: using wall functions or resolving the viscous sublayer. Wall functions are generally not applicable for wind turbines because boundary layer separation is expected and the wall functions do not correctly predict the boundary layer profile in the separation region. The Enhanced Wall Treatment (EWT) option is used in the present solver because resolving the viscous sublayer with the mesh can provide accurate results. Furthermore, it can act as a wall function if the first grid point is in the log layer [41].

The RNG *k*–*ɛ* model was employed in this work. The near-wall treatment was utilized. The first grid cell needs to be at about *y*^{+} ≈ 1. To make sure that the first mesh cell is located at the proper distance from the blade surface, the Adaption tool is used. The Adaption tool allows calculating the *y*^{+} value at each time-step and changing the mesh at the blade surface to meet the specified condition. The *y*^{+} value had become stable after three turbine rotations, and no mesh cells were marked for refinement or coarsening.

The tip speed ratio varied from 0.4 to 1, and the amplitude of expansion varied from 0.25 *R*_{1} to *R*_{1}, where *R*_{1} is the turbine bucket radius (*R*/2). The time-steps (Δ*t*) were chosen to be consistent with 0.5deg of rotation every one Δ*t* [36].

The tip speed ratio is varying from 0.4 to 1, and the amplitude of expansion was varying from 0.25 *R*_{1} to *R*_{1}, where *R*_{1} is the turbine bucket radius (*R*/2). The time-steps (Δ*t*) were chosen to be consistent with 0.5 deg of rotation every one Δ*t* [36].

*ω*and consequently to the tip speed ratio by Eq. (10)

The relative time-steps are listed in Table 3.

## 4 Results

### 4.1 Grid Independence Test.

A grid independence test was accomplished to assess the effect of the grid deformation and resolution on the calculated torque. Simulations were conducted on a deformable blade Savonius turbine with the maximum amplitude of *A* = *R*_{1} (*R*_{1} is the Bucket radius = 0.25 m) at a tip speed ratio *λ* = 1. This case was selected to confirm also that the mesh updating method has no effect on the obtained results. The mesh density is taken in the same range as many previous works. In the work by Ref. [24], the mesh density varied from 14,000 to 220,000 elements by adding about 30,000 elements in each level. Comparing the calculated static torque for different grids showed that a mesh with approximately 150,000 elements was sufficient to obtain grid-independent results. In the work reported in Ref. [17], which considered the same flow parameters as the present work, the grid density was controlled by changing the size of elements along the blade and in the near wake domain. Three different grid densities, with approximately 60,000, 80,000, and 120,000 elements, were employed. It was found that grids with approximately 80,000 and 120,000 elements provided approximately the same results.

Figure 8 depicts the torque coefficients of the turbine for one period with three different grid densities; the grids have about 80,000, 100,000, and 120,000 elements, respectively. After a preliminary test, the blade size element is fixed to 0.00129 which corresponds to 630 nodes on the turbine blade. This provides an average *y*^{+} = 0.3, which is similar to *y*^{+} obtained by Ref. [42]. More specifications of the present grid test are listed in Table 4.

Mesh density | Coarse | Medium | Fine |
---|---|---|---|

Total number of cells | 80,000 | 100,000 | 120,000 |

Number of cells in near wake zone | 50,000 | 60,000 | 70,000 |

Number of cells in far wake zone | 30,000 | 40,000 | 50,000 |

First layer thickness (mm) | 0.5 | 0.1 | 0.05 |

Number of nodes on the turbine blade | 260 | 420 | 630 |

Average y^{+} | 2.1 | 0.4 | 0.3 |

Mesh density | Coarse | Medium | Fine |
---|---|---|---|

Total number of cells | 80,000 | 100,000 | 120,000 |

Number of cells in near wake zone | 50,000 | 60,000 | 70,000 |

Number of cells in far wake zone | 30,000 | 40,000 | 50,000 |

First layer thickness (mm) | 0.5 | 0.1 | 0.05 |

Number of nodes on the turbine blade | 260 | 420 | 630 |

Average y^{+} | 2.1 | 0.4 | 0.3 |

### 4.2 Solver Validation.

In order to validate the present CFD solver, the simulation results were compared with the experimental data from [43]. Figure 9(a) shows the CFD and experimental averaged torque coefficients as a function of tip speed ratio. It can be seen that the simulation results agree well with the experimental data, except for the two cases of TSR = 0.6 and 0.7. The maximum relative error in these cases is about 5%. These two cases are confined between high and low TSR. The difference in results could be attributed to the flow behavior (several detached vortices). One can conclude that the present solver is able to predict tolerably the performance of a Savonius wind rotor in the selected range of TSR.

Figure 9(b) illustrates the torque coefficient of a rigid Savonius turbine through eight rotation periods. It can be noticed that the *C*_{t} curve achieves the periodic solution after five rotation periods. The next three rotation periods have approximately the same average value. The last period is used to present the evolution of different time-varying quantities and to predict the turbine performance.

A comparative examination of the torque coefficients, obtained with four commonly used eddy-viscosity turbulence models, is depicted in Fig. 10. The models are detached eddy simulation, *k*-epsilon, *k*-omega, and 5-equation Reynolds-stress model. The examination reveals the low sensitivity of the results to the turbulence model. The results show that the detached eddy simulation (DES) model fails in predicting the correct peak of the torque coefficients between *θ* = 0 deg and *θ* = 50 deg. When the same grid resolution was applied, the free shear layer originating from the blade tip was not well reproduced by DES.

This may be attributed to the modified definition of the filter width typically applied for DES, leading to strongly increased values of the eddy viscosity. The two models *k*-epsilon and *k*-omega are capable of displaying reasonable agreement with the experimental data. The CFD simulations were carried out on a workstation, equipped with an Intel core i7 Intel^{®} Core™ i7-4790 K CPU, 16 GB of the random access memory (RAM), and NVidia Geo force GTX 960 graphics card. The computational cost can be measured in units of time, like real universe time, or core-time. It can also be measured in actual real-world money, like USD/EUR, needed to perform the simulation.

In Table 5, a simple comparison between the required running times for the last two turbine rounds is provided for different cases. All cases are carried out for TSR = 0.4 and *A* = 0.5*R*_{1}. It can be seen that the Reynolds stress (Eq. (5)) model has the longest running time while the other models have practically the same running time. The comparison of the computational costs between Sliding mesh and dynamic mesh Remeshing for rigid cases demonstrates the large difference between their required running times. This is because for sliding mesh all boundaries move rigidly with respect to each other, while in the Remeshing method, they are updated each time-step.

Turbulence model | Deformation amplitude | Time-step (s) | Running time for the last two turbine round |
---|---|---|---|

Reynolds stress (Eq. (5)) | A = 0.5R_{1} | 0.5 deg/step | 5 h 19 min |

SST-k-omega | A = 0.5R_{1} | 0.5 deg/step | 3 h 49 min |

RNG k–ɛ | A = 0.5R_{1} | 0.5 deg/step | 3 h 45 min |

Detached eddy simulation (DES) | A = 0.5R_{1} | 0.5 deg/step | 3 h 50 min |

RNG k–ɛ (sliding mesh) | Rigid | 0.5 deg/step | 40 min |

RNG k–ɛ (remeshing) | Rigid | 0.5 deg/step | 3 h 30 min |

Turbulence model | Deformation amplitude | Time-step (s) | Running time for the last two turbine round |
---|---|---|---|

Reynolds stress (Eq. (5)) | A = 0.5R_{1} | 0.5 deg/step | 5 h 19 min |

SST-k-omega | A = 0.5R_{1} | 0.5 deg/step | 3 h 49 min |

RNG k–ɛ | A = 0.5R_{1} | 0.5 deg/step | 3 h 45 min |

Detached eddy simulation (DES) | A = 0.5R_{1} | 0.5 deg/step | 3 h 50 min |

RNG k–ɛ (sliding mesh) | Rigid | 0.5 deg/step | 40 min |

RNG k–ɛ (remeshing) | Rigid | 0.5 deg/step | 3 h 30 min |

### 4.3 Mechanism of Performance Improvement.

In order to comprehend the mechanism of performance improvement, a deep examination of the total and individual time-varying torque and pressure distribution along the blade during the turbine rotation is provided. Different instants are displayed in Fig. 11, which are sufficient to describe the flow behavior during the turbine rotation. The main vortices trajectories inside the turbine are presented in Fig. 10 as a function of the blade azimuthal position. The comparison between the rigid and the deformable turbines is conducted for the case A = *R*_{1} at TSR = 1.

Figure 11(a) shows that for the rigid turbine at *θ* = 0 deg, a large section zone is installed on the convex side of the advancing blade *A*. This section zone assists the blade motion. It is formed when the flow separates due to the curvature of the blade. A small pressure stagnation zone is also formed at blade *A* trailing edge. For the deformable turbine (Fig. 11(b)), when the blade *B* begins to deform (contraction), the flow resists this deformation and high-pressure stagnation zone is formed at the trailing edge. This behavior makes the rigid turbine perform better than the deformable turbine at *θ* = 0 deg. At *θ* = 90 deg, for the deformable turbine (Fig. 11(d)), the positive effect of the higher arm on the pressure distribution can be noticed along the blade chord.

For a rigid turbine (Fig. 11(c)), the section zone moves toward the blade tip and the pressure difference between blade sides decreases, which decreases the torque coefficient. At *θ* = 120 deg, the rigid case (Fig. 11(e)) shows that the tip vortex leaves the blade tip. The turbine loses energy to the flow, due to the equivalence in pressure between the concave and convex blade sides. The pressure contour of the deformable turbine (Fig. 11(f)) confirms that the contraction of the returning blade generates high resistant pressure inside the convex side of the blade.

The evolution of the individual torque of rigid blade A (Fig. 12(a)) shows that: At *θ* = 15 deg, the rigid blade reaches the peak of its power output, which is attributed to the large section zone installed in the concave side of the advancing blade A. The pressure difference is interpreted as a peak in the power output. After this peak, the section zone moves toward the blade tip (*θ* = 90 deg) and the blade loses the pressure difference which decreases the torque coefficient. Between *θ* = 120 deg and *θ* = 320 deg, the tip vortex begins to separate from the blade tip. Thus, negative torque is generated. Between *θ* = 320 deg and *θ* = 360 deg, the interaction between the returning blade concave side and the incoming flow generates a small section zone near the turbine axis, which increases the blade torque. Typically, the same behavior is recorded (Fig. 12(b)) for the second rigid blade *B* where the torque peak was shifted right by 180 deg.

Figures 12(a) and 12(b) display the individual torques of deformable blades *A* and *B* as well as the contribution of each blade for the case *A* = 0.5 *R*_{1}_{.} The hatched areas between deformable and rigid blade lines in both figures indicate positive or negative effects of the deformable blade on the power coefficient. From Fig. 12(c), an increase in the power coefficient of the deformable blade can be observed during all portions of the revolution cycle except a small zone of slight negative effect between *θ* = 140 deg and *θ* = 200 deg.

Figures 13(a) and 13(b) display respectively the difference between elliptic and rigid power coefficients for blade *A* and blade *B*. This factor is used to assess the effect of deformation on the total power coefficient. Negative Δ*C*_{P} means that rigid turbine performs better than deformable turbine and vice versa. The area between curves and the line *y* = 0 denotes the relative loss/gain of energy in terms of power coefficient. It can be understood that blade A loses energy to the flow in the first half of the cycle exactly between 0 deg and 60 deg. This energy is required to deform the blade, which provides negative Δ*C*_{P}. Between 70 deg and 260 deg, the *C*_{P} of deformable blade exceeds that of a rigid blade. This behavior is attributed to a greater arm (radius) which ensures positive Δ*C*_{P}. For $\theta >260deg$ the deformable blade *A* starts in contraction and the flow resists this contraction, which decreases Δ*C*_{P}. The contribution of the second blade B (Fig. 13(b)) can be divided into three portions: a positive contribution between *θ* = 0 deg and *θ* = 120 deg due to the contraction of the returning blade B which means lower resistant force. Negative contribution between *θ* = 120 deg and *θ* = 240 deg due to the expansion of blade *B* and finally a portion of high positive torque attributed to a greater arm.

In general, the torque improvement is due to the technique by which the blade changes the strength and the size of the section zone.

### 4.4 Effect of the Expansion Amplitude on the Turbine Performance.

The effect of the expansion amplitude on the torque coefficient is presented in Fig. 14. It is investigated by varying the expansion amplitude from *A* = 0.25 *R*_{1} to *A* = *R*_{1}. Recall that, the phase angle between the rotation and the expansion is set to zero, which means that the blade achieves its maximal amplitude at *θ* = 90 deg. Then, it returns to its original form at *θ* = 180 deg, after that the blade contracts to the minimal amplitude at *θ* = 270 deg. The turbine moves in a counter-clockwise direction. The results indicate that the individual torques of the deformable blades *A* and *B* experience three phases as compared to rigid blades (Figs. 14(a) and 14(b)).

For the advancing blade *A*, the first zone is situated between *θ* = 0 deg and *θ* = 70 deg corresponds to an energy loss due to the blade expansion. The lowest value of *C*_{t} was obtained for *A* = *R*_{1} because this case has the largest deformation amplitude which requires more energy for deformation. In this zone, *C*_{t} gradually decreases with the increment of the expansion amplitude. The second zone is situated between *θ* = 70 deg and *θ* = 300 deg corresponding to a high torque mostly due to the greater blade arm. In this zone, the torque increases by increasing the expansion amplitude. The last zone corresponds to the blade contraction. *C*_{t} regularly decreases with the increment of the contraction amplitude. The same behavior was recorded for the returning blade *B*, and the lowest value of *C*_{t} was obtained in the second zone due to the lower blade arm.

The results of the total time-varying torque (Fig. 14(c)) clearly indicate that the deformable blades improve the turbine performance despite the consumed energy due to the expansion/contraction mechanism.

As mentioned previously, *C*_{t} and *C*_{P} were assessed with respect to TSR for all configurations. In Fig. 15(a), *C*_{t} decreases quasilinearly with the increase of TSR for all cases due to the increase of the angular velocity. All cases with deformable blades improve the average *C*_{t} as compared to rigid blades. The average torque coefficient is improved by about 24% (from rigid 0.2544 to deformable 0.31587) for the lowest deformation amplitude *A* = 0.25 *R*_{1} and by about 90.6% (from rigid 0.2544 to deformable 0.485) for the highest expansion amplitude *A* = *R*_{1}. Furthermore, the magnitude of improvement increases with the increase of *A*. From Fig. 15(b), it can be noted that the average power coeffcient goes up with the increase of TSR. The gap of improvement between cases also improved with the increase of TSR. The enhancement became more considerable at TSR = 1. At TSR = 0.4, the lowest *C*_{P} of 0.2 corresponds to a rigid case, and the highest values of 0.275 is displayed by the case *A* = *R*_{1}. This means that the increase in expansion amplitude will increase *C*_{P}.

### 4.5 Effect of Gap Ratio.

It is well known that the difference in torque between the advancing and returning blades is the source of turbine rotation. By adding a gap between blades, a small amount of air stream is forced to pass through the gap from the concave face of the advancing blade, which is a high-pressure region to the low-pressure region of the returning blade. Such behavior is believed to reduce the negative contribution of the returning blade. The investigation of the gap ratio effect on the expandable turbine performance is carried out for three different values, which are 0, 0.05, and 0.1 at an amplitude of expansion *A* = 0.5 *R*_{1}. Figure 16(a) shows the effect of the gap ratio on the instantaneous blade torque coefficient. A further improvement is observed on the blade torque coefficient for various expandable blades. The comparison of various blades on the time-mean power coefficient variation with gap ratio is plotted in Fig. 16(b). A slight improvement is recorded for the case *s*/*D* = 0.05, where the average power coefficient is varied from 0.372 to 0.399, which is about 7% better than the expandable blade without a gap.

The above-mentioned observations and the pressure contour plots in Fig. 17 indicate that the gap effect on the present model of the expandable turbine does not play a significant role (improvement by 7%) on the turbine performance as compared to the initial improvement (improvement by 90%) due to the use of expandable blade instead of a rigid blade.

### 4.6 Effect of Overlap Ratio.

In this section, a further investigation on the effect of overlap ratio is carried out for a fixed gap ratio *s*/*D* = 0.05. The time-mean power coefficient and the instantaneous blade torque coefficient shown in Figs. 18(a) and 18(b) indicate a slight improvement for the case *e*/*D* = 0.1. It is believed that the use of overlap is not beneficial to the expandable turbine performance; this is attributed to the fact that the amount of air passing through the gap is not enough to make a clear change in the behavior of the returning blade. The stream traces from Fig. 19 confirm the appearance of a vortex near the turbine axis that prevents the passage of air.

## 5 Conclusion

In the present study, a novel concept of expandable blades was proposed to enhance the efficiency of the Savonius wind turbine. Numerical results were validated against experimental data, and good agreement was noted. The main conclusions found from this investigation can be summarized as follows:

The elliptic deformation improved the turbine performance by more than 90.6% for the appropriate operating conditions.

All cases with deformable blades improved

*C*_{t}as compared to rigid blades.A further improvement by 7% in the time-mean power coefficient is recorded for a turbine with a gap of 1/20 the turbine diameter.

Concerning the flow behavior, elliptic deformation ensures higher curvature of the deformed blade, which increases the flow acceleration.

All elliptic deformation cases display positive starting torque at the beginning of the rotation, which confirmed the self-starting ability.

The present CFD results provide a new concept with low manufacturing cost and promising output power. Other parameters could be investigated such as deformable blade number, the behavior of the turbine in the presence of deflectors, and may be other modes of deformation.

## Nomenclature

*c*=blades chord length (m)

*f*=rotation frequency (Hz)

*x*=blade position in

*x*-axis (m)*y*=blade position in

*y*-axis (m)*A*=deformation amplitude (m)

*D*=turbine diameter (m)

*T*=tangential force (N)

*f*_{1}=expansion frequency (Hz)

*x*_{old}=blade previous position in

*x*-axis (m)*y*_{old}=blade previous position in

*y*-axis (m)*A*_{S}=swept area (m

^{2})*C*_{P}=power coefficient

*C*_{t}=torque coefficient

*P*_{m}=energy extracted by the turbine (W)

*P*_{w}=total power (W)

*R*_{1}=bucket radius (m)

*U*_{∞}=wind velocity (m/s)