14èmes Journées de l’Hydrodynamique
18-19-20 novembre 2014
DGA Techniques hydrodynamiques Chaussée du Vexin
27105 Val de Reuil
ANALYSE NUMERIQUE DE L'HYDRODYNAMIQUE ET DU CONTROLE DE LA CIRCULATION AUTOUR DES PALES
D’UNE TURBINE DARRIEUS
NUMERICAL ANALYSIS OF HYDRODYNAMICS AND CIRCULATION CONTROLLED BLADES FOR DARRIEUS
TURBINE
J.M.R.GORLE*, L.CHATELLIER*, F.PONS*, M.BA*
* Institut PPRIME, UPR CNRS,
Département Fluide Thermique Combustion, ENSMA 86962 FUTUROSCOPE Cedex
Résumé
La recherche intensive à travers le monde sur les sources d'énergie renouvelable et l'analyse des performances des systèmes de récupération de cette énergie demande une amélioration de la performance et de l'efficacité de ces systèmes. Dans ce but, la présente étude se concentre sur les implications de la circulation contrôlée autour de la pale dans le cas de l’hydrolienne à axe vertical de type Darrieus, fonctionnant à faible nombre de Reynolds.
Des lois de commande robustes sont déterminées pour piloter le mouvement de tangage de la pale de sorte à imposer des plages à circulation constante autour de la pale pendant le fonctionnement de la turbine. Le développement de ces lois de commande est basé sur les concepts d’écoulement potentiel et considère une loi de conservation de la circulation autour de profils en mouvement. La loi de commande est injectée dans le solveur CFD incompressible du logiciel Star CCM+ et associée à des maillages "chimères" pour intégrer le mouvement du corps solide dans le cadre du calcul.
Summary
The ever-growing worldwide search for renewable energy sources and the performance analysis of energy-harvesting systems leads to further improvement of performance and efficiency of such systems. To this end, the present study focuses on the implications of controlled circulation around the blade in the case of vertical axis water turbines of Darrieus type operating at low Reynolds number. Robust control laws were derived to govern the blade’s pitching motion so that a constant circulation was obtained around the blade during the turbine’s operation. The development of these control laws were based on the potential flow concepts and consider the law of conservation of circulation. The control law was coupled with the incompressible CFD solver of Star CCM+ software and overset meshes were used to incorporate the rigid body motion within the computational frame.
I – Introduction
Following the 1970s energy crisis, green energy sources in the renewable forms have been receiving increased attention in both industrial development and academic research.
Adverse environmental effects from fossil fuels and availability issues in hydrocarbons still push this concern about inexhaustible energy sources such as water, wind and solar.
Consistent growth in product designing competences, simulation technologies and prototyping skills are likely to transform the renewable energy sources into major ones. An implication is that European Union aims to get 20% of its entire energy from green systems by 2020 and 30% from wind sources on global scale by 2030. Looking particularly at the energy extraction from water currents, vertical axis water turbines are popular in highly turbulent flows, as noted by Salter et al [14] and provide better operating range than horizontal ones [11]. Numerous studies have been conducted in analyzing, understanding and improving different designs like Darrieus, helicoidal etc... While looking at the improvement issues and performance optimization techniques, one possible way is the so-called “controlled blade pitching”, which provides an integrated platform for different problems like blade dynamic stall, torque improvement and blade vortex interaction.
At PPRIME Laboratory in ISAE-ENSMA, major efforts have been devoted to the environmental and hydrodynamic flows. As a part of it, new methodologies to incorporate blade pitching in the design, development and performance improvement of a vertical axis water turbine were investigated. This study presents the application of pitch control laws applied to the classical Darrieus turbine, its torque analysis and performance improvement.
While focusing on suppressing or reducing the vortex shedding from the blade during the operation thereby controlling the blade-vortex interaction, this study aims at determining the optimal blade pitching regime in order to maximize the torque produced by the device. The developments proposed in this paper are based on the seminal work of Couchet [4] about the non-stationary potential flow around moving bodies which was adapted to a rotating profile by Vincendet et al [15]. Considering a distribution of vortices in a potential flow, a proper control of vortex shedding can optimize the interaction between the blades by limiting the mutual influence of their turbulent wakes. Several scholars have performed the aerodynamic and hydrodynamic studies of Darrieus turbine with H-shaped blades to enhance the efficiency of the device. Results of Hwang et al [7] and Paillard et al [12] with pitching blade and Antheaume et al [2] with fixed blade suggested the adverse influence of blade wake and dynamic stall on the turbine’s overall performance due to the unsteady forces exerted on the blades. Without taking vortex triggering into account, these types of turbines are subject to wake interactions over a large part of the rotation, which can result in cyclic disturbances related to each blade. The importance of vortex formation during the fluid dynamic analysis was identified by the studies of Ahmadi et al [1], Khalid et al [8], Paraschivoiu et al [13] and Wang et al [16] applied the potential flow concepts in the vertical tidal turbine analysis.
This study aims to propose the kinematic theory of pitching blades ab-initio for a vertical axis water turbine based on the Couchet potential theory, and its validation against CFD computations over a wide range of operating conditions. Novelty of this study is associated with the development of ‘hybrid’ pitching laws of the turbine blade in order to enhance the propulsive characteristics without losing the essence of ideal flow application to the hydrofoil to ensure an effective vorticity control and no blade-vortex interaction. Extending the work of Gorle et al [6], this study starts with hydrodynamic analysis of classical turbine and the derivation of control law. CFD methodology implemented to incorporate multiple motions in the computational domain is detailed. The performance of various transition laws for variable pitching to be applied to the blades is then explained, followed by the conclusion and further path of the research in this subject.
II – Analysis of Darrieus H turbine with fixed blades
Figure 1. Velocity and force vectors acting on the blade at various azimuth positions Blade incidence (θ) and its azimuth angle (α) are the two fundamental parameters that influence the kinematics of classical vertical axis water turbine which is characterized by the fixed geometry. In addition, two other functional parameters for parametric analysis performed in this study are free-stream velocity (V0) and tip-speed ratio (λ). The fluid dynamic characteristics of the turbine kept in a free-stream velocity V0 and rotating at an angular velocity ‘ω’ are clearer through the Figure 1, where the relative velocity magnitude (W) is equal to
sin α cos α ... (1)
where θ is the azimuth angle and R is the radius of the rotor. The tip speed ratio (λ) is defined as
... (2) The local angle of attack (θ) as a function of azimuth angle (α) is defined as
tan sin
cos ... (3)
Figure 2. Blade incidence (θ) as a function of azimuth (α) for different tip-speed ratios
Figure 2 shows the relationship between local incidence of the blade (θ) and azimuth angle (α) where the value of λ is confined to the interval [1.5, 3.5], the range within which the optimal tip-speed ratio is expected [10]. Moreover, this range ensures a smooth profile whereas λ less than or equal to 1 provides non-smooth profiles and greater than 3.5 results in lower power output.
III – Numerical methodology
Numerical studies using Star CCM+ commercial CFD software were performed to evaluate the flow physics around the Darrieus turbine with or without pitching blades, assuming a two-dimensional computational domain with infinitely long blades. The fundamental shortcomings of such two-dimensional computations including the underprediction of drag force and omission of free surface effects are not taken into account due to the main purpose of the study being limited to the kinematics of the device.
Vertical axis turbine model for numerical simulations consists of a four-blade rotor system with NACA 0015 profiles. The rotor radius (R) is 0.3m, chord length (2l=4a) of each profile is 0.08m and the far-field boundary is placed at a distance of 1m from the centre of the rotor. A geometrically progressive stretch function with a factor of 1.08 was used to stretch the grid spacing in radial direction around the profile resulting in y+ in the order of 1 for the first layer of cells. The computational domain was so chosen that the all of the important flow characteristics around the model could be well captured. In order to save the computational effort, the IID geometry without rotor was used.
Figure 3. Spatial discretization of the computational domain for boundary layer approximation and overset mesh for superimposed motion
O-type Chimera meshes m1, m2, m3 and m4 around respective blade profiles were created to apply individual superposed motion to the blades in order to simulate both the rotation of the turbine and blade-pitching within the computational domain. Each of the overset mesh around the hydrofoils was structurally meshed with 40x40 diametrical grid points. Background rectangular mesh m6 consists of 575x375 cells in x- and y-directions with double refined cylindrical volumetric control ‘m5’ that accommodates the four overset meshes. Critical cell connectivity information pertained to overset and background meshes was obtained by implicit hole cutting scheme. Works of [3], [9] provide more details on this.
Control volumes are grouped into active, passive and acceptor cells as shown in Figure 3.
Acceptor cells Passive cells Active cells
Conservation equations are solved within the acceptor cells and passive cells remain inactive except when the overset mesh moves. The solutions between these regions are coupled through acceptor cells by means of distance weighted interpolation.
Despite the capability of assessing the pressure fields, every turbulence model has some problems with boundary layer modeling and viscous drag analysis. Therefore, at lower incidences where the viscous drag is a dominant player, the numerical investigations usually provide underestimated values. As the incidence increases, pressure drag predominates hence better accuracy in the CFD results. Considering the available computational resources and feasibility of the study in terms of computational effort, a two-dimensional viscous solver with SST k-ω model was chosen within the incompressible solver. The incompressible flow field is analyzed using implicit unsteady solver with a suitable time step size depending on the flow configuration. The convergence criterion is supplemented by the default under- relaxation factors.
III – CFD results of classical Darrieus turbine
CFD predictions of the torque evolution of a single blade as it travels around the azimuth for a free stream velocity of 1 m/s over a range of tip-speed ratios (λ) are shown in the left part of Figure 4. Blade kinematics provides the information about the blade’s local incidence with respect to resultant velocity which is the key in calculating the torque. Positive torque represents the power extraction resulting in propulsion from the flow and negative does the power loss. If the blade’s motion and the tangential hydrodynamic force experienced by the blade are in same direction, then the torque will be positive, otherwise torque will be negative which will lead to power lost to the flow. It is interesting to notice that most propulsive and power loss features are attributed to the front half of the blade’s rotation (00 < α < 1800) while the rear half is comparatively silent in torque fluctuations. The product of instantaneous torque produced by the blade acting on the blade and rotational velocity of the device (ω) gives the power (P) produced by the blade at a given instance. With swept area (A) of the device and fluid density (ρ) being known, the performance coefficient is then calculated from
12 ! " ... (4)
Figure 4. CFD analysis of classical Darrieus turbine with fixed blades.
Instantaneous torque plot for a single blade for a complete cycle (left), and Coefficient of performance (COP) vs tip speed ratio ‘λ’ (right)
0 0.1 0.2 0.3 0.4 0.5 0.6
1 1.5 2 2.5 3 3.5 4
COP
Tip-speed ratio (λ) V = 2 m/s V = 2.5 m/s V = 3 m/s Power loss
Propulsive
Another important design parameter is tip-speed ratio (λ) whose effect is critical in identifying the optimal operating conditions to obtain the maximum performance. Right part of Figure 4 shows the plot of coefficient of performance of the device under study calculated using CFD studies for different velocities where the optimal value for tip-speed ratio (λ) was found to be 2.5 beyond which the performance characteristics deteriorate. The mean torque proportionally increases with tip-speed ratio (λ) and reaches the maximum at λ equal to 2.5 and then falls down. In the limiting case, the turbine will rotate just neutrally without producing energy. Portion of the characteristic curve left side of the peak is controlled by the blade stall, which is associated with lower values of λ and higher values of relative incidence.
On the other hand, relative incidence decreases on the right side of the peak which leads to lower power output. Overall Reynolds effect is not very significant for classical Darrieus case.
(a) (b)
Figure 5. Distribution of (a) pressure coefficient, and (b) vorticity in the flow field
The qualitative analysis of the fixed blade Darrieus turbine was performed using the pressure coefficient and vorticity distribution around the device for a tip speed ratio (λ) of 2 in a free stream velocity of 1 m/s. The pressure coefficient contour plot in Figure 5 shows the highly dynamic operation during the front half of the turbine as the flow is streaming from left to right, and illustrates the torque evolution of Figure 4. Also, the vortex shedding following the dynamic stall is evident by the distribution of negative pressure coefficient peaks that are convected downstream divergently. These observations are completed by the vorticity contour plot, in which the interaction between the blade and vortex shed from the preceding blade is obvious. More details about the dominant vorticity during the turbine’s rotation are available in the work of Gorle et al [5]. This study now moves forward to the investigation of blade pitching technique in order to prevent the vortex shedding from the blade without compromising on the improvement in the power characteristics.
IV – Blade pitching methodology
With the aim of mimicking a perfect fluid motion, and therefore constant circulation around the blades, the velocity condition over the profile is converted into control laws in terms of blade pitch in the context of potential flow physics. Indeed, Kelvin’s theorem requires that the fluctuations of circulation around the blade are compensated by opposing vortices which are then convected to the flow to form a trailing vortex. In order to prevent the formation of such vortex wakes downstream of the blades, the motion of the blade is controlled by a law which is derived from holomorphic functions [4], as shown in Figure 6,
assuming little mutual influence between blades. This section explains how the blade pitch control law is developed and its limitations.
Figure 6. Physical parameters of flow around the blade profile
III – 1 Mathematical background
Let the flat profile as shown in Figure 6 be set at an angle of $ %& , %& with respect to incoming flow. Considering the fluid velocity at infinity to be zero, the velocity field around the profile in terms of the axis system of the blade profile with the components ‘u’ and ‘v’ is given by,
(&)*+,- .%& /0& ... (5)
If l(t) and m(t) are the fluid velocity components with respect to profile’s orientation, then the speed of profile centre O in with respect to free-stream flow is
(&1 2 3 %& 4 3 0& ... (6)
For a constant circulation (Γ) around the profile, the flow potential around the flat plate of length 2l=4a after the transformation of the flow field around the flat plate can be given by
5 6, 3 7,8ln :;<= 2><;?4 3 @ ><;?A$B ... (7) Kutta-Joukowsky condition is satisfied when
Γ 4EF 4 3 F$B 3 ... (8)
From Figure 6, l=2a and the velocity components are defined by
4 3 sin θ sin @ I2 B ... (9a)
$B B ... (9b)
Substitution of (9) in (8) gives
Γ 2E2J sin θ sin 1 @ 2I 2
2 B K ... (10)
Defining the influencing parameter E as (1-2k)l/R and a constant circulation (Г) imposed on the profile as 2E2 / sin L where L is the equivalent profile incidence with respect to the relative velocity (V0+Rω), then the compatibility between the profile’s motion and the potential flow is expressed by the equation
MM 2
N 1 1 sin L @ 1 @2
N sin @ 2
N sin ... (11)
Although the domain of β in Equation 11 can theoretically be infinitely large, the practical application is limited to -50<β<100 for effective vorticity control. Outside this domain, dynamics of real fluid flow dominates the assumptions being made in developing the blade pitching control law and flow around the device would become erratic.
V – Results and discussion
V-I. Fixed blade pitching
For the chord length (2l) of 0.08m, turbine radius (R) of 0.3 m and free stream velocity of 1m/s, Figure 7 shows the blade orientation and corresponding variation of blade incidence (θ) at different azimuth angles (α) for relative blade incidence β of +100, +50 and -50 in the 1st and 2nd row respectively. It is noticed that, as the tip-speed ratio (λ) increases, the blade orientation tends to become that of classical Darrieus turbine blade. Further quantitative and qualitative analysis is based on the 3rd and 4th rows of Figure 7, which show the instantaneous torque (T) produced by a single blade as it moves through the azimuth, and vorticity distribution respectively. Effect of imposing constant circulation to the blades is strongly demonstrated by the smooth and even evolution of the instantaneous torque compared to the classical Darrieus case with fixed blades. It is important to notice that the blade provides a positive torque and hence becomes propulsive during the first half of the cycle for positive values of β, but loses power to the flow during the second half of the cycle as it operates at reverse camber and larger incidence. The reason why a constant circulation imposed to the blade induces cyclic positive to negative torque values is due to the fact that the blade’s angular motion is aligned with the free-stream direction during the front half of the cycle while it opposes in the rear half. It is therefore noticed that the power lost to the fluid flow during one complete rotation is either equal or more than the power extracted from the pitching blades with constant circulations which lead to negative performance coefficients that would not be practically feasible. This goes much worse as tip-speed ratio (λ) increases.
For β equal to -50, power is lost for almost entire cycle with a major portion during first half but close to zero during the second half.
The vorticity contours in Figure 7 correspond to the tip-speed ratio (λ) of 2 and similar trends were observed for other values of λ for a given circulation (or β). When β=+5°, even for large values of α, the angle of incidence is comparatively lower and therefore the flow is completely attached to the blade and the flow fields remain vorticity free. On the other hand, the limiting cases of β equal to 10° and -5° slightly missed the credibility of completely arresting the vortex shedding from the blades. Particularly, turbine operating with a constant blade circulation with β equal to -5° produces comparatively stronger vortices shed from each blade as they follow a streamwise motion. In this case, the vortex zone begins to appear for α approaching 1500 and shedding occurs around α=200°. It is also observed that the extension of the wake zone strongly depends on the angle of incidence of the blade. Interestingly, there is no noticeable interaction between the blades and the vortices shed by preceding blades in any of these 3 cases.
(a) (b) (c)
Blade orientation with the free-stream coming from left to right for different pitch-control regimes in comparison with classical Darrieus blade (outermost, black coloured profiles)
(a) (b) (c)
Local incidence of the blade (θ) as a function of its azimuthal position (α)
(a) (b) (c)
Torque evolution from one blade over one cycle
(a) (b) (c)
Vortex shedding from the blades with controlled blade pitching
Figure 7. Comparison of different blade pitch regimes in terms of blade orientation (top row), blade incidence (2nd row), calculated torque for one cycle (3rd row), and vorticity distribution
(last row). (a) β = +100 (b) β = +50 and (c) β = -50
Propulsive
Power loss Power loss
Propulsive
Power loss
(a) (b) (c)
Blade orientation with the free-stream coming from left to right for different pitch-control regimes.
(a) (b) (c)
Local incidence of the blade (θ) as a function of its azimuthal position (α)
(a) (b) (c)
Torque evolution from one blade over one cycle
Figure 8. Comparison of different blade pitch regimes in terms of blade orientation (top), blade incidence (middle) and calculated torque for one cycle (bottom)
(a) β = +100 → -50 (b) β = +50 → -50 and (c) β = +50 → -50 with transition points of β = +100 → -50
V.II Variable blade pitching
As understood so far, the success of controlling the vortex shedding from the Darrieus water turbine blades at any given tip-speed ratio (λ) is largely penalized by power lost to the fluid flow due to the blades subject to large negative incidences for more than half of the operation cycle. One way to improve the propulsive characteristics of the device without losing the emphasis on the vorticity distribution is to apply variable blade pitching that imposes constant circulation to the blade. As noticed in Figure 7, the front half of the cycle is
Propulsive Propulsive Propulsive
mostly propulsive while the rear half is equally losing the power for β equal to +50 and +100. On the other hand, rear half of the cycle for β equal to -50 is losing lesser power when compared to positive values of β. An efficient merge of the blade’s incidence correspond to β equal to +100 or +50 for front half of the cycle with that correspond to -50 for rear half of the cycle can greatly reduce the power losing to flow. The key issues here are,
• The optimal location of the transition points on the θ-α profile, in order to maximise the power output. These points were identified by careful examination of instantaneous torque plots in Figure 7. The transition is needed between the point where the torque profile for positive β becomes negative and for negative β becomes positive (or better). And, this continues also when the torque profile for negative β becomes positive and for positive β becomes negative. Therefore, each cycle is normally subject to appropriate transition twice.
• Transition scheme not only to preserve the numerical solver’s stability but also to avoid the secondary vortex shedding from the blade due to transition. For a smooth transition, polynomial interpolations fitted on the slopes of θ-α profiles corresponding to positive and negative β between the chosen transition points are used.
The three transition laws analyzed in this study are,
i. β equal to +100 for the front half of the cycle followed by β equal to -50 for the rear half with a set of transition points P1 for λ =[1.5, 3.5]
ii. β equal to +50 for the front half of the cycle followed by β equal to -50 for the rear half with a set of transition points P2 for λ =[1.5, 3.5]
iii. β equal to +50 for the front half of the cycle followed by β equal to -50 for the rear half with a set of transition points P1 for λ =[1.5, 3.5]
The blade orientation, corresponding plot of blade incidence (θ) against azimuthal position (α) highlighting the transition part, and instantaneous torque measurement using CFD analysis is furnished in Figure 8. It is clear from the torque evolution plots that the variable blade pitching has successfully improved the overall power output by eliminating the blade stall in the rear part of the cycle there by reducing the amount of power lost to the fluid flow.
The In addition, the second transition also has a small adverse affect on the front part of the cycle where the peak of propulsive characteristic has slightly fallen down compared to that of classical Darrieus turbine blade with fixed blades.
(a) (b) (c)
Figure 9. Comparison of coefficients of performance (COP) for different blade pitch regimes (a) β = +100 → -50 (b) β = +50 → -50 and (c) β = +50 → -50 with transition points of β = +100 → -50
In order to assess the overall affect of variable blade pitching, the evaluation of performance coefficient is necessary which is shown in Figure 9. Compared to classical Darrieus turbine with fixed blades, more than 100% gain in the performance for a tip-speed ratio (λ) of 1.5 and 2 is evident with the transition from β equal to +100 to -50. However, the coefficient of performance for the same transition is slightly lesser than that of the fixed blades. This can be overcome by identifying more appropriate transition points at precise locations on the θ-α plot. Another important conclusion is that the optimal tip-speed ratio with this transition is 3 while the classical case provides 2.5.
(a) (b)
Figure 10. Vorticity fields of the turbine with variable blade pitching.
(a) β = +100 → -50 (b) β = +50 → -50
With the transition from β equal to +50 to -50, performance coefficient is increased for more than 2.5 times compared to the fixed blade design for a tip speed ratio (λ) of 1.5.
Thereafter, the both the designs follow the same trend until the optimum tip speed ratio of 2.5 is reached. The performance of this transition for λ beyond the optimum value is inferior to the classical case with fixed blades. The transition case of β from +50 to -50 applied at the points same as the transition of β from +100 to -50, has expressed lower performance quality and is of no interest. Finally, the effectiveness of first and second transition laws is compared through the vorticity contours on the same scale in Figure 10 for a tip-speed ratio (λ) of 2 and free-stream velocity (V0) of 1 m/s in suppressing the severe vortex shedding from the blades and thus avoiding possible blade-vortex interaction. Although the transition of β from +100 to -50 is a comparatively better than that from +50 to -50, both laws have performed tremendously well in controlling the drastic sheds of vortex from the fixed blades of classical turbine. Although little spots of vorticity reveal localized vortex shedding from the variable pitching blades due to the transition from one regime to the other are observed, there is no interaction between the blades and vortex.
V – Conclusion
As a part of ongoing research, the present study analyzed the performance of a 4-bladed vertical axis turbine with fixed and variable blade pitching. Prospective potential flow application on the basis of Couchet theory is involved in the development of a control law that decides blade pitching. Along this article, an analytical construction and corresponding numerical methodology applied to CFD calculations is presented. This study shows the complexity in handling both the constant circulation paradigm and punctual circulation changes for better performance. The use of SST k-ω model for performing numerical analysis of pitching blade performance after validating against classical Darrieus case yielded
satisfactory results. Beyond the simple dynamic laws suggested by various scholars, the development of more sophisticated, physics-based, control laws were analyzed using CFD. A precise characterization of the circulation to be applied to the blades not only to prevent the vortex shedding but also to improve the propulsive features has been undertaken in order to identify the most appropriate pitch regimes. Considering the real conditions of fluid flow and assumptions in replicating the practical device operation in 2D CFD computation, numerical solver has produced justified results for fixed and variable pitching blades within feasible range of relative blade incidence. In addition to the numerical investigation, further validation of the developed control laws in the experimental facilities of PPRIME laboratory [5] is planned.
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