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Blade shape optimization of the Savonius wind turbine using a genetic algorithm
Applied Energy 213 (2018) 148–157
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Blade shape optimization of the Savonius wind turbine using a genetic algorithm
T
⁎
C.M. Chan, H.L. Bai , D.Q. He Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
H I G H L I G H T S is incorporated into CFD for the optimization of Savonius wind turbine’s blades. • GA improvement in power coefficient is observed for turbine with optimal blades. • 33% • Turbine with optimal blades outperforms conventional one at a wide range of TSR.
A R T I C L E I N F O
A B S T R A C T
Keywords: Wind energy Savonius wind turbine/rotor Genetic algorithm optimization
The Savonius wind turbine is one of the best candidates for harvesting wind energy in an urban environment, due to unique features such as compactness, simple assembly, low noise level, self-starting ability at low wind speed, and low cost. However, the conventional Savonius wind turbine with semicircular blades has a relatively low power coefficient. This work focuses on optimizing the shape of the blade of the Savonius wind turbine to further improve its power coefficient. An evolutionary-based genetic algorithm (GA) is incorporated into computational fluid dynamics (CFD) simulations, thereby coupling blade geometry definition with mesh generation and fitness function evaluation in an iterative process. Three variable points along the blade cross-section are used to define the geometry of the blade arc, and the objective function of GA is set to maximize the power coefficient. Two-dimensional flow around the wind turbine is modeled by the shear-stress transport (SST) k-ω turbulence model and solved through the finite-volume method in ANSYS Fluent. Three GA optimization runs with different population and genetic operations have been carried out to provide the optimal shape of the blade of the Savonius turbine. Compared to the wind turbine with semicircular blades, the wind turbine with optimal blades and a tip speed ratio (TSR) of 0.8 achieved significant improvement (up to 33%) on the time-averaged power coefficient. In addition, the Savonius turbine with optimal blades outperformed the one with semicircular blades at a wide range of TSR (= 0.6–1.2), suggesting that the Savonius wind turbine with optimal blades has great potential to be applied in the real urban environment. The aerodynamic forces and flow structures pertaining to both wind turbines with optimal and semicircular blades are compared and discussed, to improve our understanding on their underlying mechanisms and to further improve their performance.
1. Introduction Wind energy is an abundant, clean resource that greatly reduces the economic, social and environmental impact from energy consumption. So far, wind energy has mostly been harvested in such open environment as a suburban or offshore area, using large-scale propeller-type horizontal-axis wind turbines (HAWTs) [1]. There is no extensive utilization of wind energy in high-density high-rise urban environments [2]. Due to its unique features such as compactness, simple assembly, omni-directionality, low noise level, self-starting ability at low wind speed, and low cost, the Savonius wind turbine or rotor, one type of ⁎
vertical-axis wind turbines (VAWTs), serves as one of the wind power converters for distributed small-scale energy harvesting systems in urban environments [3]. Yet, the performance of the conventional Savonius turbine with two straight semicircular blades, in terms of power coefficient (Cp), is relatively low and requires further improvement. The performance of the conventional Savonius wind turbine can be improved using defectors or curtains [4–7]. These deflectors are basically designed to simultaneously gather the oncoming wind for the advancing blade and direct the oncoming wind away from the returning blade. Mohamed et al. [7] placed a flat plate upstream of the returning blade and optimized its position and orientation based on evolutionary
Corresponding author. E-mail address: [email protected] (H.L. Bai).
https://doi.org/10.1016/j.apenergy.2018.01.029 Received 21 August 2017; Received in revised form 2 January 2018; Accepted 9 January 2018 0306-2619/ © 2018 Elsevier Ltd. All rights reserved.
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Table 1 Recent work on the blade shape optimization for the Savonius turbine. Investigation
Blade shape
Cp improvement
Remarks
Kamoji et al. [8] Mohamed et al. [9] Kacprzak et al. [10]
Bach type Optimized shape using Genetic Algorithm Semi-elliptic type Bach type Various bucket arc angles
10.5% 38.9% (with a pre-installed deflector) 10.3% 16.1% –
Wind tunnel test Numerical simulation Numerical simulation
SR3345 Airfoil SR5050 Airfoil Semi-elliptic type Benesh type Modified Bach type Newly developed (Similar to Bach type) Newly developed (Similar to Bach type) Deflectable arc blade Twisted blade Optimized convex and concave sides of the blade using particle swarm optimization
10% at TSR < 0.5 Improvement only at TSR > 0.9 13.0% 26.1% 30.4% 34.8% 32.1% 41.1% 10% 4.41%
Driss et al. [11] Tartuferi et al. [12] Roy and Saha [13]
Roy and Ducoin [14] Yang et al. [15] Kumar & Saini [16] Tian et al. [17]
Both numerical simulation and wind tunnel test Numerical simulation Wind tunnel test
Numerical simulation Wind tunnel test and numerical simulation Numerical simulation Numerical simulation
performance of the Savonius turbine with two twisted blades was investigated by Kumar and Saini [16], based on a systematic procedure combining variable parameters setting, blade model generation and CFD simulation. A considerable improvement in Cp , about 10%, was achieved. Nevertheless, the aforementioned efforts mostly modified the shape of the turbine’s blades using trial-and-error; no advanced numerical optimization algorithms or approaches are involved. Although advanced numerical optimization approaches such as neuro-fuzzy (e.g., [18,19]) have been applied to the optimization of other types of wind turbine systems [20], only a few studies have focused on modifying the shape of the Savonius turbine’s blades using numerical optimization algorithms. Using evolutionary-based genetic algorithm (GA), Mohamed et al. [9] attempted to optimize the blade skeleton for the Savonius wind turbine. The Savonius turbine with optimal blades achieved a great improvement in Cp , about 39%, over the turbine with semicircular blades. However, a flat plate deflector was pre-installed upstream of the returning blade, rendering the turbine system complex and direction-dependent. Also, the shape of the blade could be different in the absence of the pre-installed deflector. A particle swarm optimization algorithm based on a response surface model was applied by Tian et al. [17] in optimizing the convex and concave surfaces of the blade for the Savonius turbine. The shape of the blade surfaces was constrained by a semi-ellipse and the length of the shortaxis was the only variable in their optimization. Improvement in Cp was about 4.4% for the Savonius turbine with optimal blades. The present work aims to apply the evolutionary-based GA in the optimization of the blade shape for the Savonius wind turbine working in a free stream. CFD simulations are incorporated into the procedure of GA optimization. Furthermore, the performance of the Savonius turbine with optimal blades is investigated at a wide range of tip speed ratio (TSR) to examine the feasibility of applying it in real-life applications. This work also aims to present the flow physics underlying the behaviour of the optimal blades, based on detailed numerical simulations, in order to improve our understanding of the aerodynamics of the Savonius turbine. The paper is organized as follows. Section 2 introduces briefly GA, followed by the description of GA-based optimization in Section 3. The numerical simulation aspects are given in Section 4. Results are presented and discussed in Section 5. Conclusions and remarks are given in Section 6.
algorithms. A considerable improvement of > 27% in the time-averaged power coefficient (Cp ) was achieved for the turbine with the plate deflector optimally-positioned, compared with that without a deflector. Experimental validation of this technique was conducted by Golecha et al. [6]. Altan & Atılgan [4] deployed two straight plates upstream of the Savonius turbine; as such, a convergent channel was formed to gather the oncoming wind. This design was further developed by ElAskary et al. [5], that is, one part of the oncoming wind was guided through a smooth channel toward the inner side of the returning blade, resulting in an enhanced negative pressure therein. A significant improvement was made, i.e., the maximum power coefficient was increased to Cp = 0.52 at TSR ≤ 0.82. Nevertheless, these deflectors introduce a highly turbulent wake and make the turbine system complex and direction-dependent [5]. Changing the shape of the Savonius turbine’s blades can improve its performance. Previous investigations are summarized in Table 1. Different types of blades have been used in the literature, such as the Bach type, Benesh type, semi-elliptic type, and airfoils. Kamoji et al. [8] and Kacprzak et al. [10] adopted the Bach-type blades for the Savonius turbine and studied the effects of geometrical parameters on the performance. Their wind tunnel and numerical simulations indicated that Cp was increased by up to 16% for the modified Savonius turbine given the optimal geometrical parameters, compared with that of the conventional turbine with semicircular blades. The semi-elliptic blades studied by Kacprzak et al. [10], achieved a considerable improvement of about 10% in Cp . Recently, based on the Bach-type blade, 13–14 made further modification to the blade shape and obtained a significant increase in Cp by up to 35%. Regarding the blades with variant thickness, Tartuferi et al. [12] used different types of airfoils as the turbine blades. Their numerical simulations indicated that Cp was improved by 10% for the SR3345 airfoil blade at a low TSR (< 0.5), compared with that of the semicircular blades. Aerodynamic forces and flow structures associated with the modified blades were examined in detail, with a comparison to that of the semicircular blades. Kacprzak et al. [10] and Roy and Ducoin [14] presented the force coefficients, pressure and velocity distributions for the developed blades. Roy and Ducoin [14] ascribed the enhanced performance of the turbine to the increased lift force and elongated momentum arms. Effects of the blade arc angle on the turbulent wake were conducted by Driss et al. [11], though without Cp improvement reported. Recently, Yang et al. [15] designed a turbine system with four deflectable arc blades, whose incident angle relative to the oncoming wind can be changed during rotation. The torque acting on the advancing blades was greatly increased while drag on the returning blades was greatly reduced, thus resulting in a significant improvement, up to 41.1%, in the maximum power coefficient. The
2. Genetic algorithm Genetic algorithms (GAs) are stochastic search algorithms inspired from biological evolution and based on the Darwin’s theory of the survival of the fittest [21]. As such, GAs present an intelligent approach 149
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Fig. 1. Flow chart of the GA process.
of random searches to solving optimization problems. Due to their generality, GAs have been applied to a wide range of optimization problems, especially those with discrete design variables and discontinuous and nondifferentiable objective functions [21]. A typical GA-based optimization problem requires a genetic representation of the solution domain and a fitness function to evaluate the solution domain. The following fundamental procedures are involved in the GA optimization, viz., initialization, evaluation, selection, crossover and mutation. Fig. 1 illustrates the iterative process for the GA approach. GA starts with a randomly generated population of individuals. The individuals in the initial population are characterized by genomes containing a string of chromosomes, which is composed of design variable values generated randomly. Then the evaluation process follows, where each individual is evaluated based on the fitness function expressed in terms of the objective function. At the end of the evaluation process, all the individuals are ranked according to their fitness values. The GA operators (i.e., selection, crossover and mutation) are then applied. The selection, which is usually the first operator to be applied, selects individuals with the best performance to enable the crossover and mutation operators to produce new offspring. Typical techniques in the selection operator are elitism, roulette wheel and ranking selection [21]. The crossover operator combines two selected individuals to produce a new offspring. The chromosomes of individuals with larger fitness values have a higher probability to reproduce, and the offspring, with combined traits from the selected parents, may achieve higher fitness values. Finally, the mutation operator creates new offspring through small random changes of the information contained in the design variables of a selected individual. As such, the evolution is not only determined by inherited traits, but also the genetic diversity is promoted. Typical mutation techniques include bit inversion, order changing, and adaptive mutation. Once the GA operator procedure finishes, a new generation with evolved individuals is produced and, if a stopping criterion is not met, the iteration process continues. Instead, the optimal solution is obtained when the stopping criterion is satisfied. The stopping criterion can be convergence, time, and the number of generation or iteration, etc.
Fig. 2. Geometry and symbol designations of the conventional Savonius wind turbine with semicircular blades.
The blade is called a “returning” blade when its movement is opposite to the oncoming wind at the rotational angle θ = 0–180°, while the blade is called an “advancing” blade when its movement is along the oncoming wind at θ = 180–360°. The two blades have a spacing S and an overlap O. The oncoming wind speed is Uo = 7.30 m/s, corresponding to a Reynolds number Re = UoD/ν = 1.0 × 105, where ν is the kinematic viscosity of air. The coordinate system origin is defined at the rotation center o, with x-coordinate along the oncoming wind (streamwise) direction and y-coordinate along the cross-stream direction. Since the focus of the present study is on the blade shape optimization, we simplify the turbine configuration, i.e., S = O = 0; further, the tip speed ratio is set to be a constant TSR = 0.8, corresponding to ωz = 58.43 rad/s, at which the conventional Savonius turbine achieves the best performance in terms of the power coefficient [3]. 3.2. Blade shape optimization problem formulation
3. GA-based optimization The evolutionary-based GA is employed to optimize the blade shape for the Savonius wind turbine, in order to maximize the power coefficient. The objective function of the optimization problem is the timeaveraged power coefficient, Cp , and the coordinates to define the blade shape are the design variables. As shown in Fig. 3, three variable points P1(x1, y1), P2(x2, y2) and P3(x3, y3), together with two fixed end points o (0, 0) (i.e., the rotation center) and A(L, 0) (i.e., the blade tip), are used to define the blade geometry in the present study. With the six coordinate design variables (i.e., x1, x2, x3, y1, y2, and y3), the skeleton
3.1. Savonius wind turbine model Fig. 2 presents schematically the conventional Savonius wind turbine, together with key parameter definitions. The Savonius turbine has two identical semicircular blades with a chord length L = 100 mm and a uniform thickness t = 2 mm. The turbine blades rotate periodically around their center, with a diameter D (= 2r). The angular velocity is denoted by ωz and the tangential velocity at the blade tip by Vt (=ωzr). 150
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solutions are not degraded from one generation to the next. The elite number per generation is chosen to be 1 for Run 1 and 2 for Runs 2 and 3. For the crossover operator, an arithmetic function is adopted, with a crossover rate of 0.5 for Runs 1 and 2, and 0.2 for Run 3. Further, instead of using a binary string, the real values of coordinate variables are used to present the chromosome of an individual, and the chromosome values are assigned as the arithmetic mean of two real-valued candidates. An adaptive mutation operator is used for the present GA optimization. 4. Numerical simulation aspects The conventional Savonius wind turbine with semicircular blades is taken as the baseline model in the present study; similar numerical simulation details are applied to the wind turbines generated during the GA process. 4.1. Turbulence model Two-dimensional (2D) transient simulations are conducted in ANSYS Fluent, with unsteady Reynolds-Averaged Navier-Stokes (RANS) equations solved using the finite-volume method. The shear-stress transport (SST) k-ω turbulence model is employed to calculate the turbulent viscosity terms in the RANS equations. The SST k-ω turbulence model is a two-equation eddy-viscosity model, combining the advantages of both the k-ε model for free-stream flows and the k-ω model for boundary-layer flows, thus ensuring a highly accurate prediction of flow separation with adverse pressure gradients. This turbulence model has been used by, e.g., Roy & Saha [22], Kacprzak et al. [10], Roy & Ducoin [14], to simulate the Savonius turbine turbulent flows. For the spatial discretization of the governing equations, a secondorder upwind scheme is used for the discretization of momentum, turbulent kinetic energy and specific dissipation rate, while the Pressure-Staggering Option (PRESTO!) scheme is used for the discretization of pressure. For the temporal discretization, a second-order implicit scheme is applied. The Pressure-Implicit with Splitting of Operators (PISO) algorithm is adopted in the pressure-velocity coupling.
Fig. 3. The skeleton line of the turbine blades with the variable points.
line of the blade can be interpolated using a natural cubic spline curve. Therefore, the design objective of the blade shape optimization of the turbine can be formulated as
Maximize Cp (→→ x , y ),
(1)
y are the coordinate vectors, each containing the three where → x and → design variables (x1, x2 and x3) and (y1, y2 and y3), respectively. In order to produce a reasonable blade shape, defined by two fixed end points and three variable points, the six coordinate design variables are to be bounded as follows: x1: x2: x3: y1: y2: y3:
0.05 < x1/L < 0.3 0.3 < x2/L < 0.7 0.7 < x3/L < 0.95 0.1 < y1/L < 0.6 0.1 < y2/L < 0.75 0.1 < y3/L < 0.6.
3.3. Optimization procedure and parameters
4.2. Computational domain, boundary conditions and mesh generation
As presented in Fig. 4, an automated process is deployed for the blade shape optimization, which couples the blade geometry definition, mesh generation and objective function evaluation with CFD simulations. MATLAB is employed as the workflow platform, calling all codes sequentially. The process begins with an initialization step of generating six coordinate variables in GA. Based on the variable points, the skeleton line of the blade is fitted by a natural cubic spline curve. Then the formatted data points representing the blade inner and outer surfaces are interpolated and saved into a script file. The ICEM package in ANSYS is used to generate the meshes for subsequent CFD simulations that are carried out in Fluent using a journal file. A post-processing of numerical results is then conducted to calculate the power coefficient of the wind turbine, which represents the fitness function in the GA process. Finally the optimization process terminates if a convergence criterion is satisfied; otherwise, the process goes back to the GA optimization process to produce a new generation of variables, and the workflow repeats. Here, we take the standard deviation (ε) of the power coefficients in each generation as the stopping criterion; the GA process terminates once ε < 0.2%. Table 2 lists the GA parameters used in the present study. Three runs of different population sizes as well as GA operation parameters have been conducted to verify the robustness of the GA optimization. The population size is 10 for Run 1 and 20 for Runs 2 and 3. For the selection operator, an elitism strategy is used, ensuring that the GA
Fig. 5(a) illustrates the computational domain and the boundary conditions. The computational domain is rectangular, with the Savonius wind turbine resided at the symmetric centerline along the x direction. The upper and lower sides of the computational domain are 7.5D away from the turbine center, resulting in a lateral distance of 15D. The inlet is located at 7.5D upstream of the turbine center while the outlet is at 15D downstream of the turbine center. The size of the computational domain guarantees negligible effects of the boundaries on the turbine performance. The computational domain is further divided into two sub-zones, that is, a rotational zone around the turbine and a non-rotational outer zone. The former has a diameter of 2D. There is an interface between the two sub-zones. The ‘velocity inlet’ boundary condition is imposed at the inlet, with a uniform velocity Uo = 7.30 m/s and a turbulent intensity It = 5%, while the ‘pressure outlet’ boundary condition is used at the outlet. The ‘symmetry’ boundary conditions are set for the lateral sides. The ‘no-slip’ boundary condition is defined over the blade surface. Structural quadrilateral meshes are generated in the computational domain, with a sliding mesh at the interface between the rotational and non-rotational zones (Fig. 5b and c). The minimum size of the mesh cells in the rotational zone is 0.0025D; the mesh cells at the interface is uniformly distributed, with the cell size of 0.017D (Fig. 5c). Furthermore, in order to accurately capture flow separation and ensuing vortex shedding, refined meshes of boundary-layer type are adopted over the 151
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Fig. 4. Flow chart illustrating the automated optimization procedure.
5. Results and discussion
Table 2 GA parameters.
Population size Selection operator (Elite number per generation) Crossover operator (Crossover rate) Mutation operator
Run 1
Run 2
Run 3
10 Elitism (1) Arithmetic (0.5) Adaptive
20 Elitism (2) Arithmetic (0.5) Adaptive
20 Elitism (2) Arithmetic (0.2) Adaptive
5.1. Optimal blade geometry The optimal solutions, i.e., the GA-optimized coordinate variables defining the geometry of optimal blade, are given in Table 4, together with the corresponding power coefficients and their improvements over that of the conventional Savonius turbine. It can be seen that there is only a slight difference between the results from Run 1 and Run 2 and there is no difference between the results from Run 2 and Run 3. Although somewhat different elitism selection, crossover rate and population size have been implemented in the three runs, the similar optimal solution results obtained from the three runs indicate the robustness of the GA optimization. With the most optimal blades obtained from Runs 2 and 3, the Savnonius turbine has resulted in a significant increase of 33.4% in the power coefficient than that of the conventional turbine with semicircular blades. Fig. 6 presents the convergence history of the fitness values (i.e., the power coefficients of the wind turbines) of Run 3. The power coefficients are largely scattered for the initial generations of Run 3 (Fig. 6a). The power coefficients range from 6.82% to 21.25% for generation 1, where the 20 blade shape individuals of the first generation were randomly generated. The corresponding blade skeleton lines for generation 1 are presented in Fig. 7(a). One can immediately see the variety of the blade skeleton lines due to the random initialization for the first generation. However, the power coefficients converge rapidly in the subsequent generations, particularly from generation 2–8 (Fig. 6a). The best fitness value, i.e., the maximum power coefficient in each generation also escalates progressively from generation 1–14, and then approaches a constant value after generation 14 (Fig. 6b). The blade skeleton lines in generation 3, 6 and 14 (Fig. 7b, c and d, respectively) clearly indicate this rapid convergence in the GA optimization. For generation 3 (Fig. 7b), we can see a large part of the blade skeleton lines converge to a similar pattern corresponding to the relatively high power coefficient, whereas the blade skeleton lines of the relatively low power coefficients are eliminated. This trend is evident in generation 6 (Fig. 7c), with more blade skeleton lines converging close to the pattern corresponding to the relatively high power coefficient. Finally, in
blade surface (Fig. 5b). There are ten layers of the boundary-layer type meshes, corresponding to a depth of 0.013D, which progressively grow up at a ratio of 1.2 away from the blade surface.
4.3. Grid independence tests The grid independence tests are conducted for the conventional Savonius wind turbine with semicircular blades. As shown in Table 3, three sets of meshes, with the total cell numbers from 48,347 to 166,630, are generated for the grid-independent tests. At the time-step size of Δt = 2.9871 × 10−4 s, the time-averaged power coefficient Cp increases from Cp = 15.33% (Mesh 1) to Cp = 16.90% (Mesh 2), while the cell number increases from 48,346 (Mesh 1) to 97,643 (Mesh 2); however, further increase in the cell number to 166,630 (Mesh 3) causes the Cp value to slightly increase to Cp = 16.96%. Compared with the Cp value for Mesh 3, there is only a discrepancy of 0.354% in Cp for Mesh 2. The size of time step is halved to be Δt = 1.4935 × 10−4 s for Mesh 2, and the corresponding Cp of 17.04% is only 0.472% higher than that for Mesh 3. Therefore, Mesh 2 and the size of time step Δt = 2.9871 × 10−4 s are employed in this study, so as to save the computational effort. Due to the highly time-consuming nature of the CFD simulations, a multi-process procedure with 10 parallel evaluations was carried out simultaneously. A computing time of approximately 2 h was required for the CFD simulation of each blade shape on a workstation with 16 processing cores. Thus, around 100 blade shapes were evaluated per day based on the developed automated process (see Section 3.3). 152
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Fig. 5. Computational domain, boundary conditions and mesh details.
5.2. Effects of tip speed ratio
Table 3 Grid independence tests based on the Savonius turbine of semicircular blades. Mesh
Cell number
Time step size (s)
Power coefficient Cp
Discrepancy (%)
1 2
48,347 97,643 97,643 166,630
2.9871 × 10−4 2.9871 × 10−4 1.4935 × 10−4 2.9871 × 10−4
15.33% 16.90% 17.04% 16.96%
9.61 0.354 0.472 –
3
Fig. 8 presents the effects of tip speed ratio (TSR) on the performance of the Savonius turbine with optimal blades from the present GA optimization. A wide range of TSR = 0.6–1.2 is considered, with a view of examining the feasibility of applying it in urban environments where wind conditions are relatively unsteady. The time-averaged power coefficient (Cp ) for the Savonius turbine with the semicircular blades decreases gradually as TSR increases at TSR > 0.7. On the other hand, Cp for the Savonius turbine with optimal blades changes slightly with TSR, Cp is not reduced at TSR = 1.2. As a result of this, the Savonius turbine with optimal blades from the present GA optimization outperforms the convention Savonius turbine at a wide range of TSR. These observations indicate that the Savonius turbine with optimal blades has great potential to be applied in the real urban environment.
Table 4 Optimal solutions of three GA runs. Parameters
Run 1
Run 2
Run 3
(x1/L, y1/L) (x2/L, y2/L) (x3/L, y3/L) Power coefficient Cp
(0.2058, 0.1675) (0.6047, 0.3527) (0.8648, 0.2499) 22.48%
(0.2065, 0.1683) (0.6084, 0.3565) (0.8633, 0.2425) 22.55%
(0.2065, 0.1683) (0.6084, 0.3565) (0.8633, 0.2425) 22.55%
CP improvement
33.0%
33.4%
33.4%
5.3. Force characteristics The aerodynamic forces acting on the optimal blades at TSR = 0.8 are examined in terms of the coefficients of torque (CT), drag force (CD), lift force (CL), and static pressure (Cpre), with a comparison to that on the conventional semicircular blades. The force coefficients CT, CD, CL and Cpre are defined as follows, respectively,
generation 14 (Fig. 7d), all the blade skeleton lines have converged to the optimal one that produces the maximum power coefficient of 22.55%. 153
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Fig. 8. Effects of tip speed ratio (TSR) on the turbine performance.
Cpre =
T , 0.5ρDUo2 L
(2)
CD =
FD , 0.5ρDUo2
(3)
CL =
FL , 0.5ρDUo2
(4)
(5)
where T, FD, FL and P are the torque, drag force, lift force and static pressure acting on the blades, and ρ is the density of air. The drag and lift forces are along the x- and y-directions, respectively (Fig. 2). To facilitate the following discussion, the positive lift force is defined to be along the negative y-direction. The positive and negative torques are counter-clockwise and clockwise, respectively. The pressure is normal to the blade surface, with positive pressure toward the surface and negative pressure away from the surface. Note that the forces may refer to that associated separately with the advancing or returning blade, or both blades. Fig. 9 presents the variations of CT, CD and CL with the rotor angle θ over one period of rotation (i.e., 0° < θ < 180°), for both the optimal and the conventional blades at TSR = 0.8. It can be seen that, in Fig. 9(a) positive CT indicating counter-clockwise torques are produced by both optimal and semicircular advancing blades at 0° < θ < 140°. Compared with that by the semicircular advancing blade, a larger CT (positive) is produced by the optimal advancing blade at 0° < θ < 140°. That is, the optimal advancing blade is driven by a larger counterclockwise torque than the semicircular advancing blade at
Fig. 6. Convergence history of fitness values in Run 3.
CT =
P , 0.5ρUo2
Fig. 7. Convergence history of blade shapes in Run 3.
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blade. The maximum CD on the optimal advancing blade is attained at θ = 60° (Fig. 9b). Meanwhile, a greater CL on the optimal advancing blade than that on the semicircular blade is observed at 0° < θ < 80° (Fig. 9c). The largest CL on the optimal advancing blade is identified at θ = 20° (Fig. 9c). The increased drag and lift forces acting on the optimal advancing blade are connected with the enhanced torque on the same blade (Fig. 9a). Furthermore, CT on the optimal advancing blade (Fig. 9a) is found to gradually reduce from its maximum at θ > 32° with CD increasing (Fig. 9b) and CL decreasing (Fig. 9c) at 32° < θ < 60°. That is, the reduced torque on the advancing blade is mainly ascribed to the drop of the lift force on the same blade at this range of 32° < θ < 60°. This suggests that the lift force may play a more significant influence than the drag force on the torque acting on the optimal advancing blade. For the returning blade, a larger CD on the optimal blade is observed at 35° < θ < 130° (Fig. 9b), compared with that on the semicircular blade. The maximum CD on the optimal returning blade is attained at θ = 85° (Fig. 9b). And CL on the optimal returning blade has a significantly low magnitude at 0° < θ < 120° (Fig. 9c). Therefore, the lowered CT (i.e., more negative) on the optimal returning blade at 40° < θ < 140° (Fig. 9a) may be mainly ascribed to the increased drag force acting against the same blade. Since the torque acting on the wind turbine is a product of the rotational driving force and its corresponding lever arm, it could be straightforward to better understand the mechanism of large torque generation through examining the behaviour of the driving force and lever arm. As illustrated in Fig. 10, the aerodynamic forces acting on the blade can be decomposed into two components, with one component (Ft) along the tangential direction of the rotation, associated with a lever arm LFt, and another component (Fn) along the chord direction. The positive and negative Ft are along and opposite to the tangential direction of the rotation, respectively. As such, the turbine rotation is driven and dragged by a positive and a negative Ft, respectively. Fig. 11 presents the variations of the driving force coefficient (CFt ) and its corresponding lever arm (LFt) with the rotation angle (θ). It can be seen in Fig. 11a that, at 0° < θ < 140°, CFt > 0 is associated with the optimal advancing blade while CFt < 0 with the optimal returning blade (Fig. 11a). This observation of CFt indicates that the driving force on the optimal advancing blade has a positive contribution to the turbine rotation, but that on the optimal returning blade has a negative contribution to the turbine rotation, at the range of 0° < θ < 140°. Opposite effect is observed at 140° < θ < 180°, that is, the driving force on the optimal advancing blade has negative contribution to the turbine rotation, but that on the optimal returning blade has a positive contribution to the turbine rotation. Compared with the semicircular
Fig. 9. Variations of the force coefficients with the rotational angle θ: (a) torque coefficient CT, (b) drag coefficient CD, (c) lift coefficient CL. Solid curves are for the optimal blades and dashed curves are for the semicircular blades. TSR = 0.8.
0° < θ < 140°. The maximum CT by the optimal advancing blade is attained at θ = 32°, larger than the maximum CT for the semicircular advancing blade which is achieved at θ = 20°. Meanwhile, negative CT indicating clockwise torque are produced by both optimal and semicircular returning blades at 0° < θ < 140°. Compared to that by the semicircular returning blade, a more negative CT by the optimal returning blade is observed at 40° < θ < 140° with the lowest CT achieved at θ = 100°. As a result, a larger CT sum (positive) is produced by the two optimal blades than that by the semicircular blades at 0° < θ < 90°, while a lower CT sum (negative) is produced by the optimal blades than that by the semicircular blades at 90° < θ < 140°. In other words, the rotor with optimal blades is driven by a larger counterclockwise torque than the rotor with semicircular blades at 0° < θ < 90°, but the former is driven by a larger clockwise torque than the latter at 90° < θ < 140°. At θ > 140°, both optimal and semicircular advancing blades are associated with a negative CT corresponding to a clockwise torque, whereas both returning blades with a positive CT corresponding to a counter-clockwise torque. However, the difference in CT between the optimal and semicircular blades is small at this range of θ > 140°. Therefore, in view of the proportional relationship between the power coefficient and torque (i.e., Cp = CT × TSR), we can conclude that the significant improvement in the timeaveraged power coefficient Cp of the Savonius turbine with optimal blades by GA is mainly attributed to the larger torque produced by the optimal advancing blades relative to its semicircular counterpart. We now discuss the behaviour of drag and lift forces and pressure distributions on the optimal blades in order to gain deeper insight into the underlying mechanisms by which the large torque is generated. In Fig. 9(b), a larger CD on the optimal advancing blade is observed at 20° < θ < 120°, compared with that on the conventional semicircular
Fig. 10. Definitions of the driving force and its corresponding lever arm on the optimized blades. The superscripts ‘a’ and ‘r’ denote advancing and returning blades, respectively.
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comparison to that of the corresponding semicircular blades. At the angle θ = 20° (Fig. 12a), positive Cpre distributions are observed over the entire inner sides of the optimal and semicircular advancing blades and over a large part of the outer sides of the returning blades (close to the rotational axis). Similar patterns of the positive Cpre distributions are identified for the optimal and semicircular blades. The maximum Cpre is 0.83 and is found at the tips of the advancing blades. Concentrated negative Cpre distributions are detected in the areas close to the rotational axis. On the other hand, negative Cpre distributions are observed over the entire outer sides of the advancing blades as well as over the entire inner sides of the returning blades for both turbines. The most negative Cpre distributions occur over the advancing blades. However, the optimal blade has a minimum Cpre = −4.96, lower than that (Cpre = −3.67) for the semicircular blade. This difference in the pressure distributions results in the larger lift (or suction force) and torque, in terms of magnitude, acting on the optimal advancing blade, compared with that on the semicircular advancing blade at the rotational angle θ = 20° (Fig. 9c). At θ = 80° (Fig. 12b), similar Cpre distributions are observed over the optimal and semicircular blades; that is, there are positive stagnation pressure distributions over the outer sides of the returning blades and negative pressure distributions over the inner sides of the same blade. For the advancing blades, negative pressure distributions are over its inner and outer sides, with concentrated negative pressure distributions attached on the outer sides close to the blade tips. The minimum Cpre is −5 for these concentrations for both optimal and semicircular blades. However, we can see the inner side of the optimal advancing blade is driven by a lower Cpre than the inner side of the semicircular advancing blade. For the returning blades, the inner side of the optimal one is driven by a more negative Cpre distribution than the inner side of the semicircular one. Therefore, larger drag forces are acting on the optimal blades than on the semicircular blades at this rotational angle θ = 80° (Fig. 9b).
Fig. 11. Variations with θ of the driving force (a) and the corresponding lever arm (b) of the turbine blades. TSR = 0.8.
blades, CFt of both optimal advancing and returning blades have a larger magnitude at 0° < θ < 140° (Fig. 11a). However, in Fig. 11(b) the lever arm LFt of the optimal advancing blade is longer than that of the semicircular blade at 0° < θ < 140°, and the lever arm LFt of the optimal returning blade is shorter than that of the semicircular blade at 0° < θ < 90°. Therefore, a larger total torque is obtained at 0° < θ < 90° for the turbine with optimal blades, compared with the turbine with semicircular blades (Fig. 9a). Note that the abrupt change of LFt at 130° < θ < 145° is resulted from the fact that a nearly-zero driving force, which is the denominator in the calculation of LFt, is found at this θ range. Pressure distributions around the optimal blades at two typical rotational angles θ = 20° and 80° are presented in Fig. 12, with a
5.4. Vortical structures Fig. 13 presents the distributions of vorticity (ωz*) around the optimal and semicircular blades (TSR = 0.8) at four different rotational angles, i.e., θ = 20°, 60°, 100° and 160°. The behaviour of flow structure is intrinsically connected with the force characteristics discussed in the preceding section. One can see that, in Fig. 13 strong shear layers with concentrated vorticity are generated from the blade tips and then roll up into vortices. Vortices with oppositely signed vorticity are alternately shedding from the blades, displaying a staggered pattern in the wake of both the optimal and semicircular turbines. For the optimal turbine, spots of positive vorticity are identified in the proximity area of the blade surface, and they appear being attached upon the blades during the entire rotation period. As indicated by the streamlines, these highly-concentrated vorticity spots are associated with detached or separated flows occurring immediately over the blade surface. For the semicircular blades, a pair of the vorticity spots are located on its both sides and close to the rotational center. During the rotation, the vorticity spot on the outer side of the advancing blade extends to the blade tip, resulting in the detached flow on the leeward side. On the contrary, the concentrated vorticity distributions are largely flattened over the optimal blade surface, suggesting reduced flow separation therein. As a matter of fact, the delayed flow separation over the outer side of the optimal advancing blade is manifested by the streamline distributions in Fig. 13a–c. This results in the enhanced negative pressure distribution over the outer side of the advancing blades, thus partially contributing to the large forces. 6. Conclusions and remarks
Fig. 12. Distributions of the pressure coefficient over the optimal and semicircular blades at (a) θ = 20°, (b) θ = 80°. TSR = 0.8. ‘*’ denotes normalization by Uo and/or D.
The evolutionary-based GA has been employed in this study to optimize the blade shape geometry of the Savonius wind turbine, with the 156
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Fig. 13. Distributions of vorticity and streamlines for the optimal and semicircular blades at (a) θ = 20°, (b) θ = 60°, (c) θ = 100° and (d) θ = 160°. TSR = 0.8.
References
objective of maximizing the power coefficient. The Savonius turbine was operated at a constant TSR = 0.8 and Re = 1 × 105. 2D CFD simulations of the turbulence flows around the turbine blades were conducted to evaluate the objective function, based on the automated process coupling the blade geometry definition, mesh generation and fitness function evaluation with numerical iterations. The robustness of GA optimization on the turbine blade shape is demonstrated. A significant improvement, up to 33%, in the time-averaged power coefficient has been found in the modified Savonius turbine with optimal blades, when compared to that of the conventional Savonius turbine with the semicircular blades. Aerodynamic forces and flow structures associated with the optimal blades were investigated in detail to gain insight into the flow physics underlying the improvement in performance. A larger lift force and ensuing torque are generated by the optimal blades at 0° < θ < 90°, compared with the conventional semicircular blades. It has been observed, compared with that on the semicircular blades, the rotational driving force on the optimal advancing blade is associated with a longer lever arm, whereas the driving force on the optimal returning blade with a short lever arm. Furthermore, at this θ range the flow separation over the outer side of the optimal advancing blade is postponed, as indicated by the streamline and vorticity distributions. The Savonius wind turbine with optimal blades performs much better than that with semicircular blades at a wide range of TSR (= 0.6–1.2), indicating a great potential of the Savonius turbine with optimal blades to be applied in the urban environments where wind conditions are relatively complex. Furthermore, it is worth noting that the optimal blade from the present work has such merits as compactness, simple assembly, low cost and easy manufacturing, similar to the conventional semicircular blade. For example, the optimal blade can be easily manufactured using 3D printing technique. All of these features are favorable for its real applications.
[1] Herbert GMJ, Iniyan S, Sreevalsan E, Rajapandian S. A review of wind energy technologies. Renew Sustain Energy Rev 2007;11(6):1117–45. [2] Ishugah TF, Li Y, Wang RZ, Kiplagat JK. Advances in wind energy resource exploitation in urban environment: a review. Renew Sustain Energy Rev 2014;37:613–26. [3] Akwa JV, Vielmo HA, Petry AP. A review on the performance of Savonius wind turbines. Renew Sustain Energy Rev 2012;16(5):3054–64. [4] Altan BD, Atılgan M. An experimental and numerical study on the improvement of the performance of Savonius wind rotor. Energy Convers Manage 2008;49(12):3425–32. [5] El-Askary WA, Nasef MH, AbdEL-hamid AA, Gad HE. Harvesting wind energy for improving performance of Savonius turbine. J Wind Eng Ind Aerodyn 2015;139:8–15. [6] Golecha K, Eldho TI, Prabhu SV. Influence of the deflector plate on the performance of modified Savonius water turbine. Appl Energy 2011;88(9):3207–17. [7] Mohamed MH, Janiga G, Pap E, Thévenin D. Optimization of Savonius turbines using an obstacle shielding the returning blade. Renew Energy 2010;35(11):2618–26. [8] Kamoji MA, Kedare SB, Prabhu SV. Experimental investigations on single stage modified Savonius rotor. Appl Energy 2009;86(7–8):1064–73. [9] Mohamed MH, Janiga G, Pap E, Thévenin D. Optimal blade shape of a modified Savonius turbine using an obstacle shielding the returning blade. Energy Convers Manage 2011;52(1):236–42. [10] Kacprzak K, Liskiewicz G, Sobczak K. Numerical investigation of conventional and modified Savonius wind turbines. Renew Energy 2013;60:578–85. [11] Driss Z, Mlayeh O, Driss S, Driss D, Maaloul M, Abid MS. Study of the bucket design effect on the turbulent flow around unconventional Savonius wind rotors. Energy 2015;89:708–29. [12] Tartuferi M, D’Alessandro V, Montelpare S, Ricci R. Enhancement of Savonius wind rotor aerodynamic performance: a computational study of new blade shapes and curtain systems. Energy 2015;79:371–84. [13] Roy S, Saha UK. Wind tunnel experiments of a newly developed two-bladed Savoniusstyle wind turbine. Appl Energy 2015;137:117–25. [14] Roy S, Ducoin A. Unsteady analysis on the instantaneous forces and moment arms acting on a novel Savonius-style wind turbine. Energy Convers Manage 2016;121:281–96. [15] Yang M-H, Huang G-M, Yeh R-H. Performance investigation of an innovative vertical axis turbine consisting of deflectable blades. Appl Energy 2016;179:875–87. [16] Kumar A, Saini RP. Performance analysis of a single stage modified Savonius hydrokinetic turbine having twisted blades. Renew Energy 2017;113:461–78. [17] Tian W, Mao Z, Zhang B, Li Y. Shape optimization of a Savonius wind rotor with different convec and concave sides. Renew Energy 2018;117:287–99. [18] Petkovic D, Cojbasic Z, Nikolic V. Adaptive neuro-fuzzy approach for wind turbine power coefficient estimation. Renew Sustain Energy Rev 2013;28:91–195. [19] Petkovic D, Cojbasic Z, Nikolic V, Shamshirband S, Kiah MLM, Anuar NB, et al. Adaptive neuro-fuzzy maximal power extraction of wind turbine with continously variable transmission. Energy 2014;64:868–74. [20] Chehouri A, Younes R, Ilinca A, Perron J. Review of performance optimization techniques applied to wind turbines. Appl Energy 2015;142:361–88. [21] Goldberg DE. Genetic Algoriths in Search, Optimization, and Maching Learning. 1 ed. Addison-Wesley Professional; 1989. [22] Roy S, Saha UK. Review on the numerical investigations into the design and development of Savonius wind rotors. Renew Sustain Energy Rev 2013;24:73–83.
Acknowledgements The work described in this paper was partially supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (project no. 16200714). 157
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Blade shape optimization of the Savonius wind turbine using a genetic algorithm
T
⁎
C.M. Chan, H.L. Bai , D.Q. He Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
H I G H L I G H T S is incorporated into CFD for the optimization of Savonius wind turbine’s blades. • GA improvement in power coefficient is observed for turbine with optimal blades. • 33% • Turbine with optimal blades outperforms conventional one at a wide range of TSR.
A R T I C L E I N F O
A B S T R A C T
Keywords: Wind energy Savonius wind turbine/rotor Genetic algorithm optimization
The Savonius wind turbine is one of the best candidates for harvesting wind energy in an urban environment, due to unique features such as compactness, simple assembly, low noise level, self-starting ability at low wind speed, and low cost. However, the conventional Savonius wind turbine with semicircular blades has a relatively low power coefficient. This work focuses on optimizing the shape of the blade of the Savonius wind turbine to further improve its power coefficient. An evolutionary-based genetic algorithm (GA) is incorporated into computational fluid dynamics (CFD) simulations, thereby coupling blade geometry definition with mesh generation and fitness function evaluation in an iterative process. Three variable points along the blade cross-section are used to define the geometry of the blade arc, and the objective function of GA is set to maximize the power coefficient. Two-dimensional flow around the wind turbine is modeled by the shear-stress transport (SST) k-ω turbulence model and solved through the finite-volume method in ANSYS Fluent. Three GA optimization runs with different population and genetic operations have been carried out to provide the optimal shape of the blade of the Savonius turbine. Compared to the wind turbine with semicircular blades, the wind turbine with optimal blades and a tip speed ratio (TSR) of 0.8 achieved significant improvement (up to 33%) on the time-averaged power coefficient. In addition, the Savonius turbine with optimal blades outperformed the one with semicircular blades at a wide range of TSR (= 0.6–1.2), suggesting that the Savonius wind turbine with optimal blades has great potential to be applied in the real urban environment. The aerodynamic forces and flow structures pertaining to both wind turbines with optimal and semicircular blades are compared and discussed, to improve our understanding on their underlying mechanisms and to further improve their performance.
1. Introduction Wind energy is an abundant, clean resource that greatly reduces the economic, social and environmental impact from energy consumption. So far, wind energy has mostly been harvested in such open environment as a suburban or offshore area, using large-scale propeller-type horizontal-axis wind turbines (HAWTs) [1]. There is no extensive utilization of wind energy in high-density high-rise urban environments [2]. Due to its unique features such as compactness, simple assembly, omni-directionality, low noise level, self-starting ability at low wind speed, and low cost, the Savonius wind turbine or rotor, one type of ⁎
vertical-axis wind turbines (VAWTs), serves as one of the wind power converters for distributed small-scale energy harvesting systems in urban environments [3]. Yet, the performance of the conventional Savonius turbine with two straight semicircular blades, in terms of power coefficient (Cp), is relatively low and requires further improvement. The performance of the conventional Savonius wind turbine can be improved using defectors or curtains [4–7]. These deflectors are basically designed to simultaneously gather the oncoming wind for the advancing blade and direct the oncoming wind away from the returning blade. Mohamed et al. [7] placed a flat plate upstream of the returning blade and optimized its position and orientation based on evolutionary
Corresponding author. E-mail address: [email protected] (H.L. Bai).
https://doi.org/10.1016/j.apenergy.2018.01.029 Received 21 August 2017; Received in revised form 2 January 2018; Accepted 9 January 2018 0306-2619/ © 2018 Elsevier Ltd. All rights reserved.
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Table 1 Recent work on the blade shape optimization for the Savonius turbine. Investigation
Blade shape
Cp improvement
Remarks
Kamoji et al. [8] Mohamed et al. [9] Kacprzak et al. [10]
Bach type Optimized shape using Genetic Algorithm Semi-elliptic type Bach type Various bucket arc angles
10.5% 38.9% (with a pre-installed deflector) 10.3% 16.1% –
Wind tunnel test Numerical simulation Numerical simulation
SR3345 Airfoil SR5050 Airfoil Semi-elliptic type Benesh type Modified Bach type Newly developed (Similar to Bach type) Newly developed (Similar to Bach type) Deflectable arc blade Twisted blade Optimized convex and concave sides of the blade using particle swarm optimization
10% at TSR < 0.5 Improvement only at TSR > 0.9 13.0% 26.1% 30.4% 34.8% 32.1% 41.1% 10% 4.41%
Driss et al. [11] Tartuferi et al. [12] Roy and Saha [13]
Roy and Ducoin [14] Yang et al. [15] Kumar & Saini [16] Tian et al. [17]
Both numerical simulation and wind tunnel test Numerical simulation Wind tunnel test
Numerical simulation Wind tunnel test and numerical simulation Numerical simulation Numerical simulation
performance of the Savonius turbine with two twisted blades was investigated by Kumar and Saini [16], based on a systematic procedure combining variable parameters setting, blade model generation and CFD simulation. A considerable improvement in Cp , about 10%, was achieved. Nevertheless, the aforementioned efforts mostly modified the shape of the turbine’s blades using trial-and-error; no advanced numerical optimization algorithms or approaches are involved. Although advanced numerical optimization approaches such as neuro-fuzzy (e.g., [18,19]) have been applied to the optimization of other types of wind turbine systems [20], only a few studies have focused on modifying the shape of the Savonius turbine’s blades using numerical optimization algorithms. Using evolutionary-based genetic algorithm (GA), Mohamed et al. [9] attempted to optimize the blade skeleton for the Savonius wind turbine. The Savonius turbine with optimal blades achieved a great improvement in Cp , about 39%, over the turbine with semicircular blades. However, a flat plate deflector was pre-installed upstream of the returning blade, rendering the turbine system complex and direction-dependent. Also, the shape of the blade could be different in the absence of the pre-installed deflector. A particle swarm optimization algorithm based on a response surface model was applied by Tian et al. [17] in optimizing the convex and concave surfaces of the blade for the Savonius turbine. The shape of the blade surfaces was constrained by a semi-ellipse and the length of the shortaxis was the only variable in their optimization. Improvement in Cp was about 4.4% for the Savonius turbine with optimal blades. The present work aims to apply the evolutionary-based GA in the optimization of the blade shape for the Savonius wind turbine working in a free stream. CFD simulations are incorporated into the procedure of GA optimization. Furthermore, the performance of the Savonius turbine with optimal blades is investigated at a wide range of tip speed ratio (TSR) to examine the feasibility of applying it in real-life applications. This work also aims to present the flow physics underlying the behaviour of the optimal blades, based on detailed numerical simulations, in order to improve our understanding of the aerodynamics of the Savonius turbine. The paper is organized as follows. Section 2 introduces briefly GA, followed by the description of GA-based optimization in Section 3. The numerical simulation aspects are given in Section 4. Results are presented and discussed in Section 5. Conclusions and remarks are given in Section 6.
algorithms. A considerable improvement of > 27% in the time-averaged power coefficient (Cp ) was achieved for the turbine with the plate deflector optimally-positioned, compared with that without a deflector. Experimental validation of this technique was conducted by Golecha et al. [6]. Altan & Atılgan [4] deployed two straight plates upstream of the Savonius turbine; as such, a convergent channel was formed to gather the oncoming wind. This design was further developed by ElAskary et al. [5], that is, one part of the oncoming wind was guided through a smooth channel toward the inner side of the returning blade, resulting in an enhanced negative pressure therein. A significant improvement was made, i.e., the maximum power coefficient was increased to Cp = 0.52 at TSR ≤ 0.82. Nevertheless, these deflectors introduce a highly turbulent wake and make the turbine system complex and direction-dependent [5]. Changing the shape of the Savonius turbine’s blades can improve its performance. Previous investigations are summarized in Table 1. Different types of blades have been used in the literature, such as the Bach type, Benesh type, semi-elliptic type, and airfoils. Kamoji et al. [8] and Kacprzak et al. [10] adopted the Bach-type blades for the Savonius turbine and studied the effects of geometrical parameters on the performance. Their wind tunnel and numerical simulations indicated that Cp was increased by up to 16% for the modified Savonius turbine given the optimal geometrical parameters, compared with that of the conventional turbine with semicircular blades. The semi-elliptic blades studied by Kacprzak et al. [10], achieved a considerable improvement of about 10% in Cp . Recently, based on the Bach-type blade, 13–14 made further modification to the blade shape and obtained a significant increase in Cp by up to 35%. Regarding the blades with variant thickness, Tartuferi et al. [12] used different types of airfoils as the turbine blades. Their numerical simulations indicated that Cp was improved by 10% for the SR3345 airfoil blade at a low TSR (< 0.5), compared with that of the semicircular blades. Aerodynamic forces and flow structures associated with the modified blades were examined in detail, with a comparison to that of the semicircular blades. Kacprzak et al. [10] and Roy and Ducoin [14] presented the force coefficients, pressure and velocity distributions for the developed blades. Roy and Ducoin [14] ascribed the enhanced performance of the turbine to the increased lift force and elongated momentum arms. Effects of the blade arc angle on the turbulent wake were conducted by Driss et al. [11], though without Cp improvement reported. Recently, Yang et al. [15] designed a turbine system with four deflectable arc blades, whose incident angle relative to the oncoming wind can be changed during rotation. The torque acting on the advancing blades was greatly increased while drag on the returning blades was greatly reduced, thus resulting in a significant improvement, up to 41.1%, in the maximum power coefficient. The
2. Genetic algorithm Genetic algorithms (GAs) are stochastic search algorithms inspired from biological evolution and based on the Darwin’s theory of the survival of the fittest [21]. As such, GAs present an intelligent approach 149
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Fig. 1. Flow chart of the GA process.
of random searches to solving optimization problems. Due to their generality, GAs have been applied to a wide range of optimization problems, especially those with discrete design variables and discontinuous and nondifferentiable objective functions [21]. A typical GA-based optimization problem requires a genetic representation of the solution domain and a fitness function to evaluate the solution domain. The following fundamental procedures are involved in the GA optimization, viz., initialization, evaluation, selection, crossover and mutation. Fig. 1 illustrates the iterative process for the GA approach. GA starts with a randomly generated population of individuals. The individuals in the initial population are characterized by genomes containing a string of chromosomes, which is composed of design variable values generated randomly. Then the evaluation process follows, where each individual is evaluated based on the fitness function expressed in terms of the objective function. At the end of the evaluation process, all the individuals are ranked according to their fitness values. The GA operators (i.e., selection, crossover and mutation) are then applied. The selection, which is usually the first operator to be applied, selects individuals with the best performance to enable the crossover and mutation operators to produce new offspring. Typical techniques in the selection operator are elitism, roulette wheel and ranking selection [21]. The crossover operator combines two selected individuals to produce a new offspring. The chromosomes of individuals with larger fitness values have a higher probability to reproduce, and the offspring, with combined traits from the selected parents, may achieve higher fitness values. Finally, the mutation operator creates new offspring through small random changes of the information contained in the design variables of a selected individual. As such, the evolution is not only determined by inherited traits, but also the genetic diversity is promoted. Typical mutation techniques include bit inversion, order changing, and adaptive mutation. Once the GA operator procedure finishes, a new generation with evolved individuals is produced and, if a stopping criterion is not met, the iteration process continues. Instead, the optimal solution is obtained when the stopping criterion is satisfied. The stopping criterion can be convergence, time, and the number of generation or iteration, etc.
Fig. 2. Geometry and symbol designations of the conventional Savonius wind turbine with semicircular blades.
The blade is called a “returning” blade when its movement is opposite to the oncoming wind at the rotational angle θ = 0–180°, while the blade is called an “advancing” blade when its movement is along the oncoming wind at θ = 180–360°. The two blades have a spacing S and an overlap O. The oncoming wind speed is Uo = 7.30 m/s, corresponding to a Reynolds number Re = UoD/ν = 1.0 × 105, where ν is the kinematic viscosity of air. The coordinate system origin is defined at the rotation center o, with x-coordinate along the oncoming wind (streamwise) direction and y-coordinate along the cross-stream direction. Since the focus of the present study is on the blade shape optimization, we simplify the turbine configuration, i.e., S = O = 0; further, the tip speed ratio is set to be a constant TSR = 0.8, corresponding to ωz = 58.43 rad/s, at which the conventional Savonius turbine achieves the best performance in terms of the power coefficient [3]. 3.2. Blade shape optimization problem formulation
3. GA-based optimization The evolutionary-based GA is employed to optimize the blade shape for the Savonius wind turbine, in order to maximize the power coefficient. The objective function of the optimization problem is the timeaveraged power coefficient, Cp , and the coordinates to define the blade shape are the design variables. As shown in Fig. 3, three variable points P1(x1, y1), P2(x2, y2) and P3(x3, y3), together with two fixed end points o (0, 0) (i.e., the rotation center) and A(L, 0) (i.e., the blade tip), are used to define the blade geometry in the present study. With the six coordinate design variables (i.e., x1, x2, x3, y1, y2, and y3), the skeleton
3.1. Savonius wind turbine model Fig. 2 presents schematically the conventional Savonius wind turbine, together with key parameter definitions. The Savonius turbine has two identical semicircular blades with a chord length L = 100 mm and a uniform thickness t = 2 mm. The turbine blades rotate periodically around their center, with a diameter D (= 2r). The angular velocity is denoted by ωz and the tangential velocity at the blade tip by Vt (=ωzr). 150
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solutions are not degraded from one generation to the next. The elite number per generation is chosen to be 1 for Run 1 and 2 for Runs 2 and 3. For the crossover operator, an arithmetic function is adopted, with a crossover rate of 0.5 for Runs 1 and 2, and 0.2 for Run 3. Further, instead of using a binary string, the real values of coordinate variables are used to present the chromosome of an individual, and the chromosome values are assigned as the arithmetic mean of two real-valued candidates. An adaptive mutation operator is used for the present GA optimization. 4. Numerical simulation aspects The conventional Savonius wind turbine with semicircular blades is taken as the baseline model in the present study; similar numerical simulation details are applied to the wind turbines generated during the GA process. 4.1. Turbulence model Two-dimensional (2D) transient simulations are conducted in ANSYS Fluent, with unsteady Reynolds-Averaged Navier-Stokes (RANS) equations solved using the finite-volume method. The shear-stress transport (SST) k-ω turbulence model is employed to calculate the turbulent viscosity terms in the RANS equations. The SST k-ω turbulence model is a two-equation eddy-viscosity model, combining the advantages of both the k-ε model for free-stream flows and the k-ω model for boundary-layer flows, thus ensuring a highly accurate prediction of flow separation with adverse pressure gradients. This turbulence model has been used by, e.g., Roy & Saha [22], Kacprzak et al. [10], Roy & Ducoin [14], to simulate the Savonius turbine turbulent flows. For the spatial discretization of the governing equations, a secondorder upwind scheme is used for the discretization of momentum, turbulent kinetic energy and specific dissipation rate, while the Pressure-Staggering Option (PRESTO!) scheme is used for the discretization of pressure. For the temporal discretization, a second-order implicit scheme is applied. The Pressure-Implicit with Splitting of Operators (PISO) algorithm is adopted in the pressure-velocity coupling.
Fig. 3. The skeleton line of the turbine blades with the variable points.
line of the blade can be interpolated using a natural cubic spline curve. Therefore, the design objective of the blade shape optimization of the turbine can be formulated as
Maximize Cp (→→ x , y ),
(1)
y are the coordinate vectors, each containing the three where → x and → design variables (x1, x2 and x3) and (y1, y2 and y3), respectively. In order to produce a reasonable blade shape, defined by two fixed end points and three variable points, the six coordinate design variables are to be bounded as follows: x1: x2: x3: y1: y2: y3:
0.05 < x1/L < 0.3 0.3 < x2/L < 0.7 0.7 < x3/L < 0.95 0.1 < y1/L < 0.6 0.1 < y2/L < 0.75 0.1 < y3/L < 0.6.
3.3. Optimization procedure and parameters
4.2. Computational domain, boundary conditions and mesh generation
As presented in Fig. 4, an automated process is deployed for the blade shape optimization, which couples the blade geometry definition, mesh generation and objective function evaluation with CFD simulations. MATLAB is employed as the workflow platform, calling all codes sequentially. The process begins with an initialization step of generating six coordinate variables in GA. Based on the variable points, the skeleton line of the blade is fitted by a natural cubic spline curve. Then the formatted data points representing the blade inner and outer surfaces are interpolated and saved into a script file. The ICEM package in ANSYS is used to generate the meshes for subsequent CFD simulations that are carried out in Fluent using a journal file. A post-processing of numerical results is then conducted to calculate the power coefficient of the wind turbine, which represents the fitness function in the GA process. Finally the optimization process terminates if a convergence criterion is satisfied; otherwise, the process goes back to the GA optimization process to produce a new generation of variables, and the workflow repeats. Here, we take the standard deviation (ε) of the power coefficients in each generation as the stopping criterion; the GA process terminates once ε < 0.2%. Table 2 lists the GA parameters used in the present study. Three runs of different population sizes as well as GA operation parameters have been conducted to verify the robustness of the GA optimization. The population size is 10 for Run 1 and 20 for Runs 2 and 3. For the selection operator, an elitism strategy is used, ensuring that the GA
Fig. 5(a) illustrates the computational domain and the boundary conditions. The computational domain is rectangular, with the Savonius wind turbine resided at the symmetric centerline along the x direction. The upper and lower sides of the computational domain are 7.5D away from the turbine center, resulting in a lateral distance of 15D. The inlet is located at 7.5D upstream of the turbine center while the outlet is at 15D downstream of the turbine center. The size of the computational domain guarantees negligible effects of the boundaries on the turbine performance. The computational domain is further divided into two sub-zones, that is, a rotational zone around the turbine and a non-rotational outer zone. The former has a diameter of 2D. There is an interface between the two sub-zones. The ‘velocity inlet’ boundary condition is imposed at the inlet, with a uniform velocity Uo = 7.30 m/s and a turbulent intensity It = 5%, while the ‘pressure outlet’ boundary condition is used at the outlet. The ‘symmetry’ boundary conditions are set for the lateral sides. The ‘no-slip’ boundary condition is defined over the blade surface. Structural quadrilateral meshes are generated in the computational domain, with a sliding mesh at the interface between the rotational and non-rotational zones (Fig. 5b and c). The minimum size of the mesh cells in the rotational zone is 0.0025D; the mesh cells at the interface is uniformly distributed, with the cell size of 0.017D (Fig. 5c). Furthermore, in order to accurately capture flow separation and ensuing vortex shedding, refined meshes of boundary-layer type are adopted over the 151
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Fig. 4. Flow chart illustrating the automated optimization procedure.
5. Results and discussion
Table 2 GA parameters.
Population size Selection operator (Elite number per generation) Crossover operator (Crossover rate) Mutation operator
Run 1
Run 2
Run 3
10 Elitism (1) Arithmetic (0.5) Adaptive
20 Elitism (2) Arithmetic (0.5) Adaptive
20 Elitism (2) Arithmetic (0.2) Adaptive
5.1. Optimal blade geometry The optimal solutions, i.e., the GA-optimized coordinate variables defining the geometry of optimal blade, are given in Table 4, together with the corresponding power coefficients and their improvements over that of the conventional Savonius turbine. It can be seen that there is only a slight difference between the results from Run 1 and Run 2 and there is no difference between the results from Run 2 and Run 3. Although somewhat different elitism selection, crossover rate and population size have been implemented in the three runs, the similar optimal solution results obtained from the three runs indicate the robustness of the GA optimization. With the most optimal blades obtained from Runs 2 and 3, the Savnonius turbine has resulted in a significant increase of 33.4% in the power coefficient than that of the conventional turbine with semicircular blades. Fig. 6 presents the convergence history of the fitness values (i.e., the power coefficients of the wind turbines) of Run 3. The power coefficients are largely scattered for the initial generations of Run 3 (Fig. 6a). The power coefficients range from 6.82% to 21.25% for generation 1, where the 20 blade shape individuals of the first generation were randomly generated. The corresponding blade skeleton lines for generation 1 are presented in Fig. 7(a). One can immediately see the variety of the blade skeleton lines due to the random initialization for the first generation. However, the power coefficients converge rapidly in the subsequent generations, particularly from generation 2–8 (Fig. 6a). The best fitness value, i.e., the maximum power coefficient in each generation also escalates progressively from generation 1–14, and then approaches a constant value after generation 14 (Fig. 6b). The blade skeleton lines in generation 3, 6 and 14 (Fig. 7b, c and d, respectively) clearly indicate this rapid convergence in the GA optimization. For generation 3 (Fig. 7b), we can see a large part of the blade skeleton lines converge to a similar pattern corresponding to the relatively high power coefficient, whereas the blade skeleton lines of the relatively low power coefficients are eliminated. This trend is evident in generation 6 (Fig. 7c), with more blade skeleton lines converging close to the pattern corresponding to the relatively high power coefficient. Finally, in
blade surface (Fig. 5b). There are ten layers of the boundary-layer type meshes, corresponding to a depth of 0.013D, which progressively grow up at a ratio of 1.2 away from the blade surface.
4.3. Grid independence tests The grid independence tests are conducted for the conventional Savonius wind turbine with semicircular blades. As shown in Table 3, three sets of meshes, with the total cell numbers from 48,347 to 166,630, are generated for the grid-independent tests. At the time-step size of Δt = 2.9871 × 10−4 s, the time-averaged power coefficient Cp increases from Cp = 15.33% (Mesh 1) to Cp = 16.90% (Mesh 2), while the cell number increases from 48,346 (Mesh 1) to 97,643 (Mesh 2); however, further increase in the cell number to 166,630 (Mesh 3) causes the Cp value to slightly increase to Cp = 16.96%. Compared with the Cp value for Mesh 3, there is only a discrepancy of 0.354% in Cp for Mesh 2. The size of time step is halved to be Δt = 1.4935 × 10−4 s for Mesh 2, and the corresponding Cp of 17.04% is only 0.472% higher than that for Mesh 3. Therefore, Mesh 2 and the size of time step Δt = 2.9871 × 10−4 s are employed in this study, so as to save the computational effort. Due to the highly time-consuming nature of the CFD simulations, a multi-process procedure with 10 parallel evaluations was carried out simultaneously. A computing time of approximately 2 h was required for the CFD simulation of each blade shape on a workstation with 16 processing cores. Thus, around 100 blade shapes were evaluated per day based on the developed automated process (see Section 3.3). 152
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Fig. 5. Computational domain, boundary conditions and mesh details.
5.2. Effects of tip speed ratio
Table 3 Grid independence tests based on the Savonius turbine of semicircular blades. Mesh
Cell number
Time step size (s)
Power coefficient Cp
Discrepancy (%)
1 2
48,347 97,643 97,643 166,630
2.9871 × 10−4 2.9871 × 10−4 1.4935 × 10−4 2.9871 × 10−4
15.33% 16.90% 17.04% 16.96%
9.61 0.354 0.472 –
3
Fig. 8 presents the effects of tip speed ratio (TSR) on the performance of the Savonius turbine with optimal blades from the present GA optimization. A wide range of TSR = 0.6–1.2 is considered, with a view of examining the feasibility of applying it in urban environments where wind conditions are relatively unsteady. The time-averaged power coefficient (Cp ) for the Savonius turbine with the semicircular blades decreases gradually as TSR increases at TSR > 0.7. On the other hand, Cp for the Savonius turbine with optimal blades changes slightly with TSR, Cp is not reduced at TSR = 1.2. As a result of this, the Savonius turbine with optimal blades from the present GA optimization outperforms the convention Savonius turbine at a wide range of TSR. These observations indicate that the Savonius turbine with optimal blades has great potential to be applied in the real urban environment.
Table 4 Optimal solutions of three GA runs. Parameters
Run 1
Run 2
Run 3
(x1/L, y1/L) (x2/L, y2/L) (x3/L, y3/L) Power coefficient Cp
(0.2058, 0.1675) (0.6047, 0.3527) (0.8648, 0.2499) 22.48%
(0.2065, 0.1683) (0.6084, 0.3565) (0.8633, 0.2425) 22.55%
(0.2065, 0.1683) (0.6084, 0.3565) (0.8633, 0.2425) 22.55%
CP improvement
33.0%
33.4%
33.4%
5.3. Force characteristics The aerodynamic forces acting on the optimal blades at TSR = 0.8 are examined in terms of the coefficients of torque (CT), drag force (CD), lift force (CL), and static pressure (Cpre), with a comparison to that on the conventional semicircular blades. The force coefficients CT, CD, CL and Cpre are defined as follows, respectively,
generation 14 (Fig. 7d), all the blade skeleton lines have converged to the optimal one that produces the maximum power coefficient of 22.55%. 153
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Fig. 8. Effects of tip speed ratio (TSR) on the turbine performance.
Cpre =
T , 0.5ρDUo2 L
(2)
CD =
FD , 0.5ρDUo2
(3)
CL =
FL , 0.5ρDUo2
(4)
(5)
where T, FD, FL and P are the torque, drag force, lift force and static pressure acting on the blades, and ρ is the density of air. The drag and lift forces are along the x- and y-directions, respectively (Fig. 2). To facilitate the following discussion, the positive lift force is defined to be along the negative y-direction. The positive and negative torques are counter-clockwise and clockwise, respectively. The pressure is normal to the blade surface, with positive pressure toward the surface and negative pressure away from the surface. Note that the forces may refer to that associated separately with the advancing or returning blade, or both blades. Fig. 9 presents the variations of CT, CD and CL with the rotor angle θ over one period of rotation (i.e., 0° < θ < 180°), for both the optimal and the conventional blades at TSR = 0.8. It can be seen that, in Fig. 9(a) positive CT indicating counter-clockwise torques are produced by both optimal and semicircular advancing blades at 0° < θ < 140°. Compared with that by the semicircular advancing blade, a larger CT (positive) is produced by the optimal advancing blade at 0° < θ < 140°. That is, the optimal advancing blade is driven by a larger counterclockwise torque than the semicircular advancing blade at
Fig. 6. Convergence history of fitness values in Run 3.
CT =
P , 0.5ρUo2
Fig. 7. Convergence history of blade shapes in Run 3.
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blade. The maximum CD on the optimal advancing blade is attained at θ = 60° (Fig. 9b). Meanwhile, a greater CL on the optimal advancing blade than that on the semicircular blade is observed at 0° < θ < 80° (Fig. 9c). The largest CL on the optimal advancing blade is identified at θ = 20° (Fig. 9c). The increased drag and lift forces acting on the optimal advancing blade are connected with the enhanced torque on the same blade (Fig. 9a). Furthermore, CT on the optimal advancing blade (Fig. 9a) is found to gradually reduce from its maximum at θ > 32° with CD increasing (Fig. 9b) and CL decreasing (Fig. 9c) at 32° < θ < 60°. That is, the reduced torque on the advancing blade is mainly ascribed to the drop of the lift force on the same blade at this range of 32° < θ < 60°. This suggests that the lift force may play a more significant influence than the drag force on the torque acting on the optimal advancing blade. For the returning blade, a larger CD on the optimal blade is observed at 35° < θ < 130° (Fig. 9b), compared with that on the semicircular blade. The maximum CD on the optimal returning blade is attained at θ = 85° (Fig. 9b). And CL on the optimal returning blade has a significantly low magnitude at 0° < θ < 120° (Fig. 9c). Therefore, the lowered CT (i.e., more negative) on the optimal returning blade at 40° < θ < 140° (Fig. 9a) may be mainly ascribed to the increased drag force acting against the same blade. Since the torque acting on the wind turbine is a product of the rotational driving force and its corresponding lever arm, it could be straightforward to better understand the mechanism of large torque generation through examining the behaviour of the driving force and lever arm. As illustrated in Fig. 10, the aerodynamic forces acting on the blade can be decomposed into two components, with one component (Ft) along the tangential direction of the rotation, associated with a lever arm LFt, and another component (Fn) along the chord direction. The positive and negative Ft are along and opposite to the tangential direction of the rotation, respectively. As such, the turbine rotation is driven and dragged by a positive and a negative Ft, respectively. Fig. 11 presents the variations of the driving force coefficient (CFt ) and its corresponding lever arm (LFt) with the rotation angle (θ). It can be seen in Fig. 11a that, at 0° < θ < 140°, CFt > 0 is associated with the optimal advancing blade while CFt < 0 with the optimal returning blade (Fig. 11a). This observation of CFt indicates that the driving force on the optimal advancing blade has a positive contribution to the turbine rotation, but that on the optimal returning blade has a negative contribution to the turbine rotation, at the range of 0° < θ < 140°. Opposite effect is observed at 140° < θ < 180°, that is, the driving force on the optimal advancing blade has negative contribution to the turbine rotation, but that on the optimal returning blade has a positive contribution to the turbine rotation. Compared with the semicircular
Fig. 9. Variations of the force coefficients with the rotational angle θ: (a) torque coefficient CT, (b) drag coefficient CD, (c) lift coefficient CL. Solid curves are for the optimal blades and dashed curves are for the semicircular blades. TSR = 0.8.
0° < θ < 140°. The maximum CT by the optimal advancing blade is attained at θ = 32°, larger than the maximum CT for the semicircular advancing blade which is achieved at θ = 20°. Meanwhile, negative CT indicating clockwise torque are produced by both optimal and semicircular returning blades at 0° < θ < 140°. Compared to that by the semicircular returning blade, a more negative CT by the optimal returning blade is observed at 40° < θ < 140° with the lowest CT achieved at θ = 100°. As a result, a larger CT sum (positive) is produced by the two optimal blades than that by the semicircular blades at 0° < θ < 90°, while a lower CT sum (negative) is produced by the optimal blades than that by the semicircular blades at 90° < θ < 140°. In other words, the rotor with optimal blades is driven by a larger counterclockwise torque than the rotor with semicircular blades at 0° < θ < 90°, but the former is driven by a larger clockwise torque than the latter at 90° < θ < 140°. At θ > 140°, both optimal and semicircular advancing blades are associated with a negative CT corresponding to a clockwise torque, whereas both returning blades with a positive CT corresponding to a counter-clockwise torque. However, the difference in CT between the optimal and semicircular blades is small at this range of θ > 140°. Therefore, in view of the proportional relationship between the power coefficient and torque (i.e., Cp = CT × TSR), we can conclude that the significant improvement in the timeaveraged power coefficient Cp of the Savonius turbine with optimal blades by GA is mainly attributed to the larger torque produced by the optimal advancing blades relative to its semicircular counterpart. We now discuss the behaviour of drag and lift forces and pressure distributions on the optimal blades in order to gain deeper insight into the underlying mechanisms by which the large torque is generated. In Fig. 9(b), a larger CD on the optimal advancing blade is observed at 20° < θ < 120°, compared with that on the conventional semicircular
Fig. 10. Definitions of the driving force and its corresponding lever arm on the optimized blades. The superscripts ‘a’ and ‘r’ denote advancing and returning blades, respectively.
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comparison to that of the corresponding semicircular blades. At the angle θ = 20° (Fig. 12a), positive Cpre distributions are observed over the entire inner sides of the optimal and semicircular advancing blades and over a large part of the outer sides of the returning blades (close to the rotational axis). Similar patterns of the positive Cpre distributions are identified for the optimal and semicircular blades. The maximum Cpre is 0.83 and is found at the tips of the advancing blades. Concentrated negative Cpre distributions are detected in the areas close to the rotational axis. On the other hand, negative Cpre distributions are observed over the entire outer sides of the advancing blades as well as over the entire inner sides of the returning blades for both turbines. The most negative Cpre distributions occur over the advancing blades. However, the optimal blade has a minimum Cpre = −4.96, lower than that (Cpre = −3.67) for the semicircular blade. This difference in the pressure distributions results in the larger lift (or suction force) and torque, in terms of magnitude, acting on the optimal advancing blade, compared with that on the semicircular advancing blade at the rotational angle θ = 20° (Fig. 9c). At θ = 80° (Fig. 12b), similar Cpre distributions are observed over the optimal and semicircular blades; that is, there are positive stagnation pressure distributions over the outer sides of the returning blades and negative pressure distributions over the inner sides of the same blade. For the advancing blades, negative pressure distributions are over its inner and outer sides, with concentrated negative pressure distributions attached on the outer sides close to the blade tips. The minimum Cpre is −5 for these concentrations for both optimal and semicircular blades. However, we can see the inner side of the optimal advancing blade is driven by a lower Cpre than the inner side of the semicircular advancing blade. For the returning blades, the inner side of the optimal one is driven by a more negative Cpre distribution than the inner side of the semicircular one. Therefore, larger drag forces are acting on the optimal blades than on the semicircular blades at this rotational angle θ = 80° (Fig. 9b).
Fig. 11. Variations with θ of the driving force (a) and the corresponding lever arm (b) of the turbine blades. TSR = 0.8.
blades, CFt of both optimal advancing and returning blades have a larger magnitude at 0° < θ < 140° (Fig. 11a). However, in Fig. 11(b) the lever arm LFt of the optimal advancing blade is longer than that of the semicircular blade at 0° < θ < 140°, and the lever arm LFt of the optimal returning blade is shorter than that of the semicircular blade at 0° < θ < 90°. Therefore, a larger total torque is obtained at 0° < θ < 90° for the turbine with optimal blades, compared with the turbine with semicircular blades (Fig. 9a). Note that the abrupt change of LFt at 130° < θ < 145° is resulted from the fact that a nearly-zero driving force, which is the denominator in the calculation of LFt, is found at this θ range. Pressure distributions around the optimal blades at two typical rotational angles θ = 20° and 80° are presented in Fig. 12, with a
5.4. Vortical structures Fig. 13 presents the distributions of vorticity (ωz*) around the optimal and semicircular blades (TSR = 0.8) at four different rotational angles, i.e., θ = 20°, 60°, 100° and 160°. The behaviour of flow structure is intrinsically connected with the force characteristics discussed in the preceding section. One can see that, in Fig. 13 strong shear layers with concentrated vorticity are generated from the blade tips and then roll up into vortices. Vortices with oppositely signed vorticity are alternately shedding from the blades, displaying a staggered pattern in the wake of both the optimal and semicircular turbines. For the optimal turbine, spots of positive vorticity are identified in the proximity area of the blade surface, and they appear being attached upon the blades during the entire rotation period. As indicated by the streamlines, these highly-concentrated vorticity spots are associated with detached or separated flows occurring immediately over the blade surface. For the semicircular blades, a pair of the vorticity spots are located on its both sides and close to the rotational center. During the rotation, the vorticity spot on the outer side of the advancing blade extends to the blade tip, resulting in the detached flow on the leeward side. On the contrary, the concentrated vorticity distributions are largely flattened over the optimal blade surface, suggesting reduced flow separation therein. As a matter of fact, the delayed flow separation over the outer side of the optimal advancing blade is manifested by the streamline distributions in Fig. 13a–c. This results in the enhanced negative pressure distribution over the outer side of the advancing blades, thus partially contributing to the large forces. 6. Conclusions and remarks
Fig. 12. Distributions of the pressure coefficient over the optimal and semicircular blades at (a) θ = 20°, (b) θ = 80°. TSR = 0.8. ‘*’ denotes normalization by Uo and/or D.
The evolutionary-based GA has been employed in this study to optimize the blade shape geometry of the Savonius wind turbine, with the 156
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Fig. 13. Distributions of vorticity and streamlines for the optimal and semicircular blades at (a) θ = 20°, (b) θ = 60°, (c) θ = 100° and (d) θ = 160°. TSR = 0.8.
References
objective of maximizing the power coefficient. The Savonius turbine was operated at a constant TSR = 0.8 and Re = 1 × 105. 2D CFD simulations of the turbulence flows around the turbine blades were conducted to evaluate the objective function, based on the automated process coupling the blade geometry definition, mesh generation and fitness function evaluation with numerical iterations. The robustness of GA optimization on the turbine blade shape is demonstrated. A significant improvement, up to 33%, in the time-averaged power coefficient has been found in the modified Savonius turbine with optimal blades, when compared to that of the conventional Savonius turbine with the semicircular blades. Aerodynamic forces and flow structures associated with the optimal blades were investigated in detail to gain insight into the flow physics underlying the improvement in performance. A larger lift force and ensuing torque are generated by the optimal blades at 0° < θ < 90°, compared with the conventional semicircular blades. It has been observed, compared with that on the semicircular blades, the rotational driving force on the optimal advancing blade is associated with a longer lever arm, whereas the driving force on the optimal returning blade with a short lever arm. Furthermore, at this θ range the flow separation over the outer side of the optimal advancing blade is postponed, as indicated by the streamline and vorticity distributions. The Savonius wind turbine with optimal blades performs much better than that with semicircular blades at a wide range of TSR (= 0.6–1.2), indicating a great potential of the Savonius turbine with optimal blades to be applied in the urban environments where wind conditions are relatively complex. Furthermore, it is worth noting that the optimal blade from the present work has such merits as compactness, simple assembly, low cost and easy manufacturing, similar to the conventional semicircular blade. For example, the optimal blade can be easily manufactured using 3D printing technique. All of these features are favorable for its real applications.
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Acknowledgements The work described in this paper was partially supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (project no. 16200714). 157