Aerodynamic and Mechanical System Modeling of a vertical axis wind turbine (VAWT) Placide JAOHINDY1 Laboratoire de Physique et Ingénierie Mathématique appliquée à l’Energie et à l’EnvironnemeNT (PIMENT)1, Campus universitaire sud, Université de La Réunion, 117 Avenue du Général Ailleret, 97430, Le Tampon, France. mail:
[email protected]
François GARDE1 Laboratoire de Physique et Ingénierie Mathématique appliquée à l’Energie et à l’EnvironnemeNT (PIMENT)1, Campus universitaire sud, Université de La Réunion, 117 Avenue du Général Ailleret, 97430, Le Tampon, France. mail:
[email protected]
Alain BASTIDE1 Laboratoire de Physique et Ingénierie Mathématique appliquée à l’Energie et à l’EnvironnemeNT (PIMENT)1, Campus universitaire sud, Université de La Réunion, 117 Avenue du Général Ailleret, 97430, Le Tampon, France. mail:
[email protected] maximum power coefficient Cp of 30%, an average static torque coefficient Cm of 33% and an overlap ratio ȕ=e/d with a value between 0.2 to 0.3 [5]. e is the main overlap of the rotor. d is the diameter of the blade. After a two-dimensional study of a conventional rotor, it is found that the optimal value of ȕ is 0.242 [6, 7]. Blackwell et al. studied the performance of fifteen configurations of a Savonious rotor in wind tunnel conditions [8]. They affirmed that the increase of the Re number increases the performance of the Savonious rotor. The aim of this article is to provide a brief overview of steady and unsteady computational strategies to validate our numerical models. In this study, the governing equations of the Savonious rotor will be concisely presented. The steady and unsteady studies were achieved with the standard k-İ and SST k-Ȧ models. The dynamic study has permitted to identify the behaviors of the Savonious rotor.
Abstract— The Savonious rotor is a slow running vertical axis wind turbine (VAWT) which gets stabilized after a certain period of functioning. This work aims to assess the results of an aerodynamic study concerning a Savonious rotor. The computational fluid dynamics (CFD) is a technique used to solve the airflow around the wind turbine rotor. In this paper, the turbulence models standard k-İ and shear stress transport (SST) k-Ȧ are the turbulences models. A three-dimensional study of the problem has been conducted. Firstly, the moment of static forces acting on the rotor blades have been compared to with experimental results. Secondly, the integration of the equation of motion of the Savonious rotor with one degree of freedom (1DOF) rigid body dynamics model. Implicit schemes for unsteady Euler equations have been used to perform dynamic simulation. Keywords-VAWT; CFD; Rigid-rotor; 1-DOF;
I.
INTRODUCTION
II.
The Savonious rotor is a vertical axis wind turbine (VAWT) originally invented by a Finnish engineer Sigurd J. Savonius in 1922 [1]. The concept of a Savonious rotor is based on the principle developed by Flettner [2]. The Savonious rotor’s operation is based on the differential drag between its two blades. The Savonious rotor is considered as a drag machine, but Modi and Fernando; Kamoji et al. reported that it is not only a pure drag machine [2, 3]. The lift force contributes to torque production. Le Gourières studied the geometrical parameters which affect the low efficiency of the basis Savonious rotor of 20% [1, 4]. This rotor can have a
This work was partially supported by European Union (EU) and the Regional Council of Reunion Island.
978-1-4244-8165-1/11/$26.00 ©2011 IEEE
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NUMERICAL METHOD
RANS solver for incompressible flow was used with standard k-İ and SST k-Ȧ approach, because these models have rapid convergence with interesting results. The threedimensional RANS equations were discretized in the computational domain by a finite volume method. The time integration was accomplished by the Euler implicit scheme. This integration gives a temporal variation of unsteady parameters of the simulation using 1-DOF method. In this study, two models are tested. They include the standard k-İ and SST k-Ȧ models [9, 10]. The SST k- Ȧ is an hybrid model based on the standard k-İ and k-Ȧ models [10]
and developed by Menter. The first Menter’s interest was to improve the performances for adverse pressure gradient (APG) flows and the second was to solve the problem of spurious sensitivity to free-stream conditions [9, 12]. Christopher et al. (2002) argued that the k-Ȧ model performed poorly for wall bounded separated flows caused by the APG [13]. The improvements of SST k-Ȧ approach in predicting many aeronautical flows motivate us to use this model in our study. III.
0.01 to 0.0021 meters are considered. For the steady state study, ep=0.01 meter was chosen.
COMPUTATIONAL DOMAIN
The computational domain used in this study at Fig. 1 is a parallelepiped, with a size of (20 meters × 10 meters × 3 meters) and has two regions. A cylinder of one meter radius and three meters height is placed at the center of this domain. The Savonious rotor is interdependent to the cylindrical region and placed at one point five meter high along the rotational axis Oz. The parallelepiped and the cylinder are centred at (0, 0, 1) meter. The Savonious rotor is centred at (0, 0, 1.5) meter. The Savonious rotors, the external and internal regions of the computational domain are modeled with polyhedral elements.
Figure 1. The computational domain used in this present study. The Savonious rotor is positioned at the half of the domain. The dimension of the radius of the internal region is 2.22R. The wind flow direction at the computational domain is along the Ox axis.
The computational domain of the steady state study is composed with 61, 198 cells and the minimum polyhedral cells width of this domain near the wind turbine rotor is equal to 0.015 meter. The computational domain of the dynamical study includes 242, 198 cells and has minimum polyhedral cells width of 0.0074 meter. A. Geometry of the rotor The Savonious rotor used in this study in Fig. 2a and 2b has the following features: height H = one meter, radius R=0.4512 meter, ȕ=0.2 and the second overlap a=0. Before the Savonious rotor of Ushiyama and Nagai [5], the Savonious rotors of Blackwell et al. are the most studied low efficiency wind turbine [8]. The Savonious rotor of Run 35 and Run 37 of Blackwell et al. was used because the uncertainty analysis of the dynamic torque coefficient Ct and the power coefficient Cp curves of this rotor was measured in their experiments [8]. The thickness ep of the Savonious rotor is not mentioned in their manuscript but a ep of 0.0021 meter was used after a several tests. The steady studies of the Savonious rotor is independent of the thickness of the rotor and a ep in range of
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Figure 2. Sketch and geometric parameters of the Savonious rotor. Polyhedral cells size are uniform for the blades and endpaltes of the Savonious rotor. For the height H=one meter, the minimal Savonious rotor cells thickness ǻx was 0.015 meter. =ڧ0 deg was the initial position of the Savonious rotor take into account in this study.
B. Boundary conditions To set up this study, we had to determine the thermophysical properties of air at the experiments. Knowing the Reynolds number Re, we estimated the density ȡ and the dynamic viscosity ȝ of the air. For a Re of 4.32×105 we founded that ȡ=1.2157 kg/m3 and ȝ=1.777×105 Ns/m2. The exact value of ȡ and ȝ of the air is necessary to solve the incompressible CFD problem. The variation of these two parameters affects the numerical simulation results. The inlet boundary conditions include the free stream velocity U=7 m/s at Re=4.32×105. Pressure outlet boundary conditions specify the static pressure at flow outlets. The outlet boundary is far enough from the rotor with a constant static pressure which can be measured experimentally. A symmetry plane was used for the top, lower, left and right parts of the computational domain. The Savonious rotor was manufactured using aluminum with a ȡ of 2700 kg/m3. The rotor mass Mr=17.74 kg and the componements of the inertia matrix J of the rotor IOz=2 kgm2 was estimated analytically. IV.
RESOLUTION METHODS
The static torque coefficient Cm, dynamic torque coefficient Ct and the power coefficient Cp developed by the VAWT is given by the following expressions:
Cm =
Ct =
Cp =
M 1 ρDAU 2 2 Mt 1 ρDAU 2 2 M tω 1 ρAU 3 2
(eq.3)
(eq.4)
(eq.5)
Where A is the projected area of the rotor, D is the diameter of the rotor, U is a free stream velocity. The static torque M is the output of the steady simulation. The angle ڧ, the angular velocity Ȧ and the dynamic torque Mt are the output of the unsteady strategies. The aerodynamic power P acting on the Savonious rotor blades is given by MtȦ. The CFD method can be used to conduct an unsteady simulation with implicit or explicit schemes. Few authors have opted for the variable rotational speed for the wind turbine modeling [15]. The implicit scheme was used for the unsteady simulation because the implicit methods have a better convergence than the explicit method. Fig. 3 below covers the steady and unsteady simulation solving algorithm of the Savonious rotor. This algorithm shows that the unsteady simulation has an iteration loop and the steady simulation has an open loop.
V.
RESULTS AND DISCUSSIONS
The Run 35 test case of Blackwell et al. was taken into account in the steady study [8]. The static torque coefficient Cm of the Savonious rotor is obtained at different angles of the rotor ڧ. In this study, ڧvaried from 0 to 360 deg with an angular step of 15 deg. It was found that 30 deg is the optimum position for the Savonious rotor [4, 7]. Among these four curves, the minimum of the present study curves are close to the Blackwell et al. curve [8].
Figure 4. Comparison of the experimental and the numerical static torque of the Savonious rotor
Fig. 5 describe the rotor behaviors in range of moment of external forces Mf between -4.2 to -0.1 Nm depending on the tip speed ratio Ȝ. Between Mf of-4 Nm to -3.8 Nm, the standard k-İ models is in agreement the Blackwell et al. curve with a small perturbation on the point corresponding to Mf=3.9 Nm [8]. From Ȝ=1.2 to the end, the standard k-İ model curve underestimate the Blackwell et al. curve [8]. Figure 3. Flow chart of the steady and unsteady simulation method algorithm
A. Equation of motion The Savonious rotor motion was considered as a rigid-rotor rotation around a fixed axis Oz. The equation of motion of the rotor is a second order differential equation and its expression is given by the following equation:
I Ozθ = M Oz + GOz
Figure 5. Tip speed ratio depending on the dynamic torque coefficient of the Savonious rotor.
(eq.6)
Where IOz is the component of the matrix J along Oz axis. MOz is the moment of forces acting on the Savonious rotor. GOz represents the external forces acting on the Savonious rotor. Assuming that the Savonious rotor makes a perfect pivot with his rotational axis Oz, this implies that GOz is equal to zero. The equation of motion of the Savonious rotor is then:
I Ozθ = M Oz
Figure 6. Tip speed ratio depending on the power coefficient of the Savonious rotor.
(eq.7)
Where Mp is the pressure moments, MIJ is the viscous stress moment, Mf is the external forces moment. fr is the ramping factor and it was applied to MOz and GOz on the body and used to minimize the shock caused by the coupling between structural and fluid dynamics [14].
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The power coefficient Cp in Fig.6 contributes in the wind turbine design. A Savonious rotor is often designed for a tip speed ratio Ȝ near 1 [19]. The optimum Ȝ obtained by the present study turn around 0.86. The standard k-İ model, the SST k-Ȧ model and the Blackwell et al. (1977) curves are close and have a common maximum of 0.21[8]. From Ȝ=0 to 1, the standard k-İ and the SST k-Ȧ models curves
overestimate slightly the Blackwell et al. (1977) curve but they remains in the uncertainty intervals except at Ȝ=0.21 [8]. From Ȝ=1 to 1.47, the standard k-İ model curve underestimate the Blackwell et al. (1977) curve but it remains in the uncertainty intervals [8]. The same behaviors is observed for the SST k-Ȧ model where the Ȝ=1.54 was achieved. Generally, at each unsteady simulation, the angular velocity Ȧ of the rigid-rotor starts at 0 rad/s before reaching its equilibrium state. Each rigid-rotor equilibrium state corresponds to a specific Ȧ. The time step ǻt was fixed by the experimental dynamic torque coefficient data. The Ȧ and the dynamic torque Mt curves of the rigid-rotor corresponding to moment of external forces Mf=2.6 Nm are shown in the following Fig. 5.
of -0.1 Nm to -4.2 Nm. (b) The SST k-Ȧ model curves was obtained with Mf in range of -0.1 Nm to -4.2 Nm. CONCLUSIONS
In order to have a Savonious rotor model valid for our study, the modeling and simulation of a VAWT of experiments has been carried out. The numerical models used are based on the Navier Stokes equations with standard k-İ and SST k-Ȧ models. The results of the steady and unsteady simulations of the Savonious rotor provide that the best approximation of the static torque coefficient, the dynamic torque coefficient and the power coefficient of the Savonious rotor with the SST k-Ȧ model in comparison to the standard k-İ model. The dynamic study of the Savonious rotor made it possible to identify the phenomenon which occurs with the application of the external force moments on the rigid-rotor.
REFERENCES
Figure 7. Time-dependent dynamic torque and angular velocity of the Savonious rotor.
Fig. 8 report the evolutions of the dynamic torque coefficient Ct curves in polar coordinate with moment of external forces Mf in range of -0.1 Nm to -4.2 Nm for the standard k-İ and SST k-Ȧ models. The first full rotation of the rigid-rotor in Fig. 6a with Mf=-4 Nm and a time step ǻt=0.001 s was obtained with the standard k-İ model. The first full rotation of the rigid-rotor where it is subject to Mf=-3.95 Nm and a ǻt=0.001 s in Fig. 8b was obtained with the SST k-Ȧ model. These constatations are in conformity of the high maximum value of the Ct=0.64 obtained with the standard k-İ model and 0.61 obtained with the SST k-Ȧ model. Using the standard k-İ model and Mf=-4.2 Nm, the rigid-rotor oscillates between angle ڧof 73.48 deg and 77.75 deg and tends to infinity in =ڧ75.44 deg. With the SST k-Ȧ model and Mf=-4 Nm, the rigid-rotor oscillates between =ڧ10.10 deg and 90 deg and tends to infinity in =ڧ41.27 deg. 100 90 80 -0.1 Nm 110 70 -2 Nm 120 60 -2.6 Nm 130 50 -3 Nm -3.8 Nm + ++++++++++ 140 40 +++ +++ + + ++ -3.9 Nm ++ ++ ++ + + + 150 -4 Nm 30 ++ +++ ++ -4.2 Nm +++ ++ ++++ ++ ++++ + 160 20 + + ++ 170 180 190 200 210
++ + +++ ++ ++ ++ ++ 10 ++ + ++ + + ++ ++ ++ + + 0 + + -0.6 -0.4 -0.2 0 0.2 0.4 ++ 0.6 ++ + + + + ++ + 350 + ++ ++ ++ + + ++ ++ +++ ++ ++++ 340 +++++ ++ ++++ ++ +++ ++ ++ 330 ++ +++ ++ ++ + +++ + +++++++++++++++
220
320
230
310 240
250
260
270
280
290
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Figure 8. Dynamic torque coefficients of the Savonious rotor in a polar coordinate. (a) The standard k-İ model curves was obtained with Mf in range
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