STUDY ON VERTICAL-AXIS WIND TURBINES USING STREAMTUBE AND DYNAMIC STALL MODELS DAVID VALLVERDÚ

Transcription

1 STUDY ON VERTICAL-AXIS WIND TURBINES USING STREAMTUBE AND DYNAMIC STALL MODELS BY DAVID VALLVERDÚ Treball Final de Grau submitted in partial fulfillment of the requirements for the degree of Enginyeria en Tecnologies Aeroespacials in Escola Tècnica Superior d Enginyeria Aeronàutica i Industrial de Terrassa Universitat Politècnica de Catalunya as a result of International Exchange program with Armour College of Engineering Illinois Institute of Technology Advisor Dietmar Rempfer Chicago, Illinois June 2014

2 c Copyright by DAVID VALLVERDÚ June 2014 ii

3 ACKNOWLEDGEMENT The production of this report wouldn t have been possible without the guidance and trust that my professor advisor, Dr. Dietmar Rempfer, deposited on me. Also, the invaluable dedication and help of my fellow office college and Master Graduate, Peter Kozak, has been decisive to my success and critical to overcome all obstacles and issues encountered. Thanks a lot to the rest of my office colleagues, who helped me and made my time working in IIT much more entertaining. Finally, I dedicate this report to all my old friends and family at home, specially my parents and sister for their support, and all the new friends I have made in this exciting stay in Chicago. iii

4 TABLE OF CONTENTS Page ACKNOWLEDGEMENT iii LIST OF TABLES vi LIST OF FIGURES viii LIST OF SYMBOLS ix LIST OF ABBREVIATIONS xii ABSTRACT xiii CHAPTER 1. INTRODUCTION Energy consumption Wind turbines Performance prediction of VAWTs STREAMTUBE MODEL Concepts Theory development Computational algorithm Results DYNAMIC STALL MODELS Concepts Theory development Computational algorithm modifications Results WAKE INTERACTION Concepts Theory development Algorithm modification Results CONCLUSION Summary Further work BIBLIOGRAPHY APPENDIX iv

5 A. SINGLE AND MULTIPLE STREAMTUBE MODELS A.1. Single streamtube model A.2. Multiple streamtube model B. OBTENTION OF THE LIFT AND DRAG CURVES FOR NACA C. DYNAMIC STALL NEEDED EXPRESSIONS C.1. Gormont method C.2. Berg modification C.3. Strickland adaptation C.4. Computational algorithm modifications v

6 LIST OF TABLES Table Page 2.1 Wind turbine characteristic values vi

7 LIST OF FIGURES Figure Page 1.1 Worldwide energy consumption in quadrillion of BTU. From EIA [8] Worldwide net electricity generated in quadrillion of BTU. From EIA [8] Classification according to rotation axis Characteristic power coefficient diagrams of wind turbines Drag (Savonius) and Lift (Darrieus) VAWTs H-rotor Darrieus VAWT Scheme of the flow velocities and forces from the blade point of view. τ is a unit vector tangent to the blade Scheme of the DMST discretization of a VAWT domain Flowchart of the calculation of λ 1 (θ (j) ) Flowchart of the calculation of λ 2 (θ (j) ) Flowchart of the calculation of C P C P against T SR Angle of attack for different dynamic stall models The two different values from which the C dyn L slope is calculated. In this example, the value used would be the slope of the dotted line, because it is lower Flowchart of the calculation of λ 1 (θ (j) ) for the dynamic stall modification Comparison of the power coefficient outputs for different dynamic stall models Angle of attack for different dynamic stall models Angle of attack for different dynamic stall models. Plain DMST α ref is the actual DMST α Blade torque coefficient for different dynamic stall models Blade lift coefficient for different dynamic stall models against angle of attack Effective angle of attack for plain DMST and Berg model. 45 vii

8 3.9 C P against T SR with Berg dynamic stall model Real and effective angle of attack for Berg model Visualization of the vorticity for T SR = 3 [9] Effective angle of attack comparison between Kozak and Vallverdú results for T SR = 3 [10] Flowchart of the calculation of C P for the dynamic stall and wake modification Effective angle of attack for DMST with Berg and wake modification C P against T SR with Berg dynamic stall and with/without wake interaction models Torque coefficient of one blade for T SR = 4 using Berg model C P against T SR with Berg dynamic stall and wake interaction models Former GUI used to interact with Vallverdú s code A.1 Scheme of the SST discretization of a VAWT domain A.2 Scheme of the MST discretization of a VAWT domain B.1 Lift and drag coefficients obtained from the Star-CCM+ simulation C.1 Flowchart of the calculation of λ 1 (θ (j) ) for the dynamic stall modification C.2 Flowchart of the calculation of C P for the dynamic stall modification viii

9 LIST OF SYMBOLS Symbol A A M c C D C F C L C Lα C P C T Definition Section normal to the flow, a.k.a swept area Berg parameter Blade chord length Drag coefficient Force coefficient Lift coefficient Linear lift coefficient slope Power coefficient Torque coefficient C Tb Blade torque coefficient D d d w ds dv F x i i j j K L Drag Unit vector with the direction of drag Wakes length Differential of surface Differential of volume Component of the force in the flow direction Natural index for multiple purposes, i ℵ Unit vector in the x axis direction Natural index for multiple purposes, j ℵ Unit vector in the y axis direction Delay parameter from Gormont s dynamic stall model Lift ix

10 l M n N b N th N w P R R e r w S T t t c T SR st U u x y Y + wall α α 0 Unit vector with the direction of lift Mach number Unit vector normal to δω Number of turbine blades Number of divisions of the azimuthal angle Number of wakes Power Turbine rotor radius Reynolds number Wakes influence ratio Non-dimensional rate of change of the angle of attack Torque Time Thickness Tip-speed ratio Streamtube subscript Free stream flow velocity Velocity of any perturbed flow Axis parallel to the wind, used in subscripts, too Axis normal to the wind Non-dimensional wall distance Blade angle of attack Angle of attack at zero lift x

11 α α eff α p α ref β t θ δθ δω θ θ λ ρ σ τ ϕ Ω Rate of change of the angle of attack Effective angle of attack Angle of attack modified with pitch Reference angle of attack Flight path angle with the instantaneous horizon tangent to the blade path Timestep between nodes Angular increment between nodes Angular width of any streamtube Boundary of the domain Ω Azimuthal angle Angular velocity Interference factor Flow density Solidity Unit vector tangent to the blade path Pitch Domain xi

12 LIST OF ABBREVIATIONS Abbreviation AC BEM BEMT BSD CFD COG DC DMST DNS EIA GUI HAWT IIT LES MST RANS SST UPC VAWT Meaning Alternating current Blade element method Blade element momentum theory Berkeley Software Distribution Computational fluid dynamics Center of gravity Direct current Double-multiple streamtube model Direct numerical simulation U.S. Energy Information Administration Graphical user interface Horizontal-axis wind turbine Illinois Institute of Technology Large eddy simulation Multiple streamtube model Reynolds-averaged Navier Stokes Single streamtube model Universitat Politècnica de Catalunya Vertical-axis wind turbine xii

13 ABSTRACT The expected exhaustion of fossil sources of energy requires the improvement of other renewable sources, such as wind turbines. Vertical-axis wind turbines present very attractive performance characteristics but extremely complicated aerodynamics. Therefore, research is still needed on this field. A code is developed in order to apply the blade element momentum theory to these wind turbines. BEMT is shown to give considerably reliable results, compared to its fast convergence. BEMT is improved with dynamic stall and wake interaction modifications, in order to include more physical effects. Modified BEMT models will help obtain fast and qualitatively reliable data on VAWT performance. Comparisons with other authors prove that the generated code provides data that are close to the real results. xiii

14 1 CHAPTER 1 INTRODUCTION 1.1 Energy consumption Our world needs new energy sources. This is a fact of our present time. From transportation to domestic life, including product manufacturing or any industrial procedure, all aspects of human life require the use of some source of energy. However, our main source, oil, will be exhausted in some decades. Moreover, its use is very harmful to all life species inhabiting this planet. We need to do something: we need to explore new ways to maintain our lifestyles relying on clean and renewable energies that are virtually inexhaustible. EIA [8] data show the worldwide energy consumption by type of energy source, see Figure 1.1. As the reader can see, most of the energy used nowadays comes from fossil origin coal, natural gas and most of the liquid sources are fossil energy sources. From all the energy used in electricity consumption, Figure 1.2 shows a classification by source. Note that still half of the electricity is generated from fossil origin sources. However, renewable energies still play a very insignificant role in the energy generation scene. Therefore, this report focuses on the analysis and performance prediction of a specific renewable energy generation device: the vertical-axis wind turbines. Thanks to our planet s self-rotation, the existence of atmosphere and the incidence on it of radiation energy coming form the sun, earth has a phenomena called wind. Wind consists of huge masses of air moving on the planets surface in different directions. Humans have used this as a source of energy since the first sailing boats were sent to explore new lands. Later on, when windmills were invented, we were able to use the energy of the wind, not only to propel boats, but also to perform tasks that required titanic forces, such as pumping water or smashing wheat [1]. Nowadays,

15 2 wind can be used to produce electricity by using wind turbines. Figure 1.1: Worldwide energy consumption in quadrillion of BTU. From EIA [8]. Figure 1.2: Worldwide net electricity generated in quadrillion of BTU. From EIA [8]. 1.2 Wind turbines Wind turbines are electrical power generation devices whose energy source is the wind. They transfer the momentum of the wind to a rotor through a set of blades attached to it. This rotor is connected normally with some gearing system to an

16 3 electrical AC generator. Then the output is fed into the general electrical grid. Some wind turbines incorporated in closed circuits with low power requirements might use a DC generator directly. One of the main classifications of turbines is made according to the axis around which the rotor spins. Therefore, one can find horizontal-axis and vertical-axis wind turbines, HAWTs and VAWTs respectively. As the name suggests, HAWTs twirl about a horizontal axis and VAWTs do so around a vertical one, both of them respectively to the ground. (a) HAWT (b) VAWT Figure 1.3: Classification according to rotation axis. A very important non-dimensional parameter that defines the behavior of a wind turbine is the tip-speed ratio, T SR = θ R U, (1.1) where θ is the rotational speed of the turbine, R is the tip radius of the blade and U is the velocity of the upwind free flow before it is influenced by the presence of the turbine. T SR is defined here because it will be used along the whole report. A good quick guide to wind turbines can be found in [29].

17 Maximum power coefficient According to Betz, the maximum power coefficient that any wind turbine can give is CP max = Pmax = = ρ A U (1.2) This result is reached by using continuity, momentum and energy conservation in any type of turbine see [4] for the whole deduction. Therefore, it applies also for VAWTs. If a simulation or measurement gives a higher power extraction from the flow than the one specified by Betz limit, it is probably wrong. Figure 1.4 is a diagram containing the characteristic power coefficient outputs of different wind turbines against their tip-speed ratios. the ideal maximum is the Betz limit. Figure 1.4: Characteristic power coefficient diagrams of wind turbines.

18 Comparison between HAWTs and VAWTs Although both horizontal- and vertical-axis turbines use the same principles to work, they present very relevant behavioral differences due to their physical ones. To begin with, HAWTs tend to behave in a much more steady way than VAWTs do, from the aerodynamics point of view. This is because the HAWT s plane of rotation is normal to the flow direction. Historically, this has made HAWTs performance characteristics much easier to predict. Hence, nowadays they are in a very advanced optimization state in design terms almost the only improvement achievable is due to geographic position. Observe how, in Figure 1.4, HAWTs maximum C P is higher than that of VAWTs. In contrast, much less effort has been invested in VAWTs first significant efforts were made by Sandia National Laboratories during the 70 s [27]. Because the prediction methods and technologies including simulation and empirical were too limited to cover such complicated aerodynamics with good enough precision at reasonable costs, VAWTs still have a considerable path ahead before being competitive [27]. Another aspect that sets both types of turbine apart is the proximity of the blades to the supporting tower and the axis of rotation. HAWT blades extent radially directly from the axis of rotation. This causes each blade to pass near the turbine tower periodically, producing a deep periodic noise that has proven harmful to the psychological human health [21]. VAWT blades, in general 1, are placed at a constant distance from the tower. This eliminates, or at least reduces dramatically, the towerblade interaction noises. This characteristic makes VAWTs more suitable candidates for inhabited regions. The fact that HAWTs rotate in a horizontal axis causes the heavy machinery transmission and generation elements of the rotor to be placed at a height, at least, greater than the HAWT blades. This makes HAWTs center of gravity, a.k.a. 1 See Section 1.2.3

19 6 COG, higher than that of VAWTs. Therefore, considerable effort has to be invested in making this components as lightweight as possible. Vertical axis configuration, instead, allows the VAWT heavy machinery to be placed at ground level. This reduces construction and design costs. Having the gear at ground level decreases maintainance costs, too. However, if further investigation states that active blade pitch control is needed to optimize VAWTS, simplicity in the design will be lost. Related to this, VAWTs are multi-directional, while HAWTs must be oriented against the wind at any time. Finally, HAWTs optimal rotation speed regimes, or T SR are higher than these of VAWTs. Therefore, VAWTs may be considered safer in case of breakdown and friendlier to the surrounding fauna, specially birds and other flying animals. This influences sound generation too: the slower the turbine rotates, the lower are the noise levels it reaches. Although, according to this section, VAWTs seem to be more advantageous than HAWTs, bear in mind that their state of development is much lower. Hence, nowadays they are not as cost-effective as HAWTs. Complete knowledge of all the parameters that affect VAWT performance might allow enough optimization so that VAWTs would match HAWTs when talking about power coefficient. Hence, research is still needed in the field to prove whether VAWTs are worth investing in or not Vertical-axis wind turbines Vertical-axis wind turbines can be classified, also, according to the main force they use to extract momentum from the wind. Hence, there are drag-based [20] if they use the component of force that the flow produces on the blade parallel to it and lift-based [12] if they use the normal component of the same force wind turbines. Only the latter type of VAWT is potentially competitive to the common HAWTs see Figure 1.4. Drag-based wind turbines, though, have simpler designs.

20 7 A very common lift-based VAWT is the H-rotor type see Figure 1.6, which consists of a set of vertical blades rotating around a vertical axis. The design is fairly simple and they can be simulated using 2D aerodynamics with less error than other less simple designs. (a) Savonius rotor (b) Darrieus rotor Figure 1.5: Drag (Savonius) and Lift (Darrieus) VAWTs. Figure 1.6: H-rotor Darrieus VAWT. Another important parameter to characterize an H-rotor VAWT is the solidity, σ = N b c 2R, (1.3) which is the ratio between the length of the chord of each blade, c, times the number of blades, N b, over the turbine diameter, 2R. The solidity is a representation of the

21 8 blockage effect of the VAWT against the wind. 1.3 Performance prediction of VAWTs Performance prediction is critical to study the feasibility of vertical-axis wind turbines. Hence, several methods exist to do so. These are very varied. One can find experimental methods, which observe reality, and computational simulations, which predict the flow and turbine behavior. A general classification would be the following [9] [27]: CFD: computational fluid dynamics encompasses all models that simulate flow physics and are applied using computer units. Different CFD models present different characteristics: BEM: blade element methods are computationally inexpensive. However they rely on critical empirical data that must be obtained by some other method. They are very fast and easy to apply they are useful for preanalyses and qualitative interpretations. First applied on fluid dynamics by Glauert [5] and to VAWTs by Templin [28]. Vortex models are more precise than BEM, but at a higher computational cost. They rely on stall models. Larsen [11] was the first to apply this type of method to VAWTs. Grid-centered methods discretize the Navier Stokes equations in the volume domain of the VAWT. Therefore, they take into account the physics of the problem more faithfully than the other CFD models. However, they are also the most expensive methods, computationally speaking. One of the most recent applications of grid-centered methods to VAWTs is the one made by Kozak [9]. He used finite volume methods to perform a 2D simulation. Wind tunnel experiments consist on observing the flow around a wind turbine

22 9 situated in a controlled environment. They require no previous models, but the techniques used to obtain flow data are very complicated and expensive. Real prototypes can be used to obtain performance data over a period of time in field situations. However, other analyses are needed to finally be able to build a prototype with some guarantee of success. This report focuses on developing a BEM and momentum theory code (BEMT) [30] with MATLAB [14] and interpreting the obtained results. These results will show that BEMT, if properly applied, proves itself quite useful, even though being based on a fairly simplistic model.

23 10 CHAPTER 2 STREAMTUBE MODEL 2.1 Concepts This chapter develops and explains the streamtube model applied to performance prediction of vertical-axis wind turbines. Streamtube models are based on blade element method, a.k.a. BEM, and momentum theory. The combination is called blade element momentum theory, a.k.a. BEMT. Streamtube model is a relatively simple method, compared to finite volume or vortex modeling. The main advantage of this method is its ability to solve the domain without an internal volume mesh. However, to achieve this, the governing equations must be in integral form. Therefore, specially in a CFD case, they undergo very important simplifications. The result is a very fast computational method with significant error in the final result the amount of error is very dependent on the nature of the problem studied and, therefore, the governing equations. BEMT applications for VAWT cases give a fair approximation of the VAWT power output, especially for lower tip-speed ratios. As said, BEMT applied to VAWTs is referred to as a streamtube model. As the name suggests, all streamtube models consist of applying each governing equation in its integral form to every one and each of the domains defined by the streamtubes. Each streamtube connects both inlet and outlet boundaries and is parallel to the velocity of the flow at all points. Therefore, there is no mass, momentum nor energy neglecting thermal effects exchange between adjacent streamtubes. This means that all components of the velocity perpendicular to the streamtubes are neglected. Hence, the turbine is being treated as a unidirectional flow problem only. This also means that the streamtube section expansion due to flow energy loss and continuity

24 11 is neglected 2. Applying governing equations in their integral form entails the need of empirical data as an input to solve a turbine. Specifically, streamtube methods need the characteristic curves for lift and drag at any given angle of attack. Also, this method is unable to take into account turbulent effects such as wake interaction, very important at high T SR, or the influence of dynamic stall, dominant when the VAWT is working at low T SR and related to the empirical dependency. The following chapters discuss these issues. 2.2 Theory development There are several versions of the streamtube model, each of them using a different boundary discretization. These variations are the single, multiple and doublemultiple streamtube models, a.k.a. SST, MST and DMST, respectively. STT, devised by Templin [28], considers the whole turbine as an actuator disk placed inside a single streamtube. MST, introduced by Strickland [24], improves SST by dividing the former single streamtube into multiple discrete streamtubes. Theoretical developments for both models can be found in Appendix A. Finally DMST, proposed by Paraschivoiu [15] [16], improves MST by considering the turbine as two separate actuator disks one for the upwind or front half-cycle and a second one for the downwind or rear half-cycle. All streamtube models (BEMT) follow the same principle. First, they use the governing conservation equations to calculate the thrust force on each streamtube the force parallel to it. Then, the same force is calculated from a load analysis. Both expressions depend on the flow velocity. Hence, they can be coupled to obtain 2 Paraschivoiu [18] proved that neglecting the streamtube section expansion, which induces spanwise velocity, does not introduce significant error relatively to other omitted effects in BEMT application to VAWTs.

25 12 a system of equations to solve the flow velocity field in the streamtube: the load analysis result is introduced into the momentum loss result. From the velocity field, all performance dependent variables of the turbine can be computed Governing equations First of all, the three governing equations applied to any streamtube must be defined in integral formulation. These are mass, momentum and energy conservation 3, respectively and d dt Ω d dt Ω d dt Ω ρ dv + ρ u dv + Ω ρ u 2 dv + Ω Ω ρ u n ds = 0, (2.1) ρ u (u n) ds = F ext (2.2) 1 2 ρ u2 (u n) ds = P, (2.3) where Ω is the domain where these equations are applied and n is a unit vector normal to Ω and pointing outwards. F ext is the sum of the forces that the flow receives and P is the power output of the portion of the turbine inside the domain. Hence, P > 0 when the turbine is extracting energy from the flow. Bold symbols are vectors. Otherwise, they are scalars. BEMT treats the problem as a steady case, so all time derivate components from momentum and energy equations can be eliminated. 3 Note that dynamic and other magnitudes that include dimension units surface, volume, force, torque, power have one less dimension than usual, because all the development is using 2D approach. For example, for a hypothetical force, F, the units are [F ] = N/m

26 Physical definitions One needs to define the geometry of a VAWT blade parametrically. Figure 2.1 is used for this purpose. L u θ R α D β u u r τ y θ θ x Figure 2.1: Scheme of the flow velocities and forces from the blade point of view. τ is a unit vector tangent to the blade. of view, From it, the relative velocity (u r ) of the wind from the blade reference point u r U = ( u U can be computed. Also, the flight path angle, ) 2 + T SR u U T SR cos θ, (2.4) ( ) T SR sinθ β = arctan u/u + T SR cos θ, (2.5) and the angle of attack, ( ) π + β θ α = modulus π, (2.6) 2π

27 14 will be needed. Once the angle of attack is known, the lift, C L, and drag, C D, static coefficients of the VAWT blade profile can be obtained from experimental data 4, L = 1 2 ρ c u2 r C L (2.7) and D = 1 2 ρ c u2 r C D. (2.8) Note that the units of these forces are [N/m]: as said previously in this chapter, the turbine is approached as a 2D problem. Hence, all dynamic magnitudes lack one order of distance unit. The whole force that the blade receives from the wind expressed as a function of lift and drag components, F = [D cos β L sin β] i + [D sin β + L cos β] j, (2.9) is used to calculate the torque that the turbine receives from each blade, T = R F t = R (L l + D d) τ, (2.10) where l = sin β i+cos β j, d = cos β i+sin β j and τ = cos θ i sin θ j. Therefore, the power coefficient obtained from one blade, P = 1 2π 2π 0 θ T dθ, (2.11) is the result of averaging the instantaneous torque times the angular velocity along a period of azimuthal angle. 4 Appendix B develops how these data sere obtained.

28 Double-multiple streamtube model The most precise streamtube model is DMST, because it allows to consider the energy losses of the flow separately for the front and rear half of the VAWT. The flow is considered to travel through two consecutive actuator disks which, in turn, extract energy from it. Moreover, at the same time, the turbine domain is discretized in multiple streamtubes parallel to the flow, so a different velocity of the flow can be calculated at any position of the blade. Then, the contribution of all front and rear half-cycle streamtubes is added together. Figure 2.2 illustrates the DMST scheme. Observing Figure 2.2, one can distinguish five different states of the flow: 1. State : this is the free stream state. The flow is not perturbed by the turbine. It acts as the input state for Disk State 1: this is the state of the flow when interacting with the upwind actuator disk, also called the front half-cycle of the turbine. 3. State e: this is the equilibrium state. In this state, the flow is considered to be far enough from both disks, hence it is steady. This is an approximation. In reality, state e is too close to both states 1 and 2 to allow this hypothesis. However, DMST uses this assumption anyway, as it is a method that seeks very fast convergence through simple modeling. State e acts as the output state for Disk 1 and as the input state for Disk State 2: this is the state of the flow when interacting with the downwind actuator disk, also called the rear half-cycle of the turbine. 5. State w: this is the wake state. The flow is perturbed by both Disks 1 and 2. It acts as the output state for Disk 2.

29 16 Disk 2 Disk 1 U u 1 u e u 2 u w Upwind zone Only 1D Downwind zone F 1, x u 1 u e F 2,x u 2 u w U θ R δθ θ Ω 1, st A 1, st A 2, st Ω 2, st y x One streamtube Figure 2.2: Scheme of the DMST discretization of a VAWT domain. According to Figure 2.2, if one considers each streamtube of angular width δθ as being infinitesimally thin which means that δθ 0 its swept area can be computed as A 1,st = R δθ sin θ st, θ st (0, π), (2.12a) A 2,st = R δθ ( sin θ st ), θ st (π, 2π), (2.12b) where θ st is the value of θ the blade has when it is going through the middle point of the given streamtube. Now, conservation equations can be applied. Equation 2.1 mathematically expresses the fact that there is no mass exchange between streamtubes: each streamtube

30 17 mass flow (ṁ st ) is constant. Therefore, for each streamtube with section A i,st : ṁ = ρ u 1 A 1,st = ρ u 2 A 2,st = ct. (2.13) Equation 2.2, once evaluated in one front and one back streamtube in the direction parallel to the flow, gives the expressions 5 ṁ (u e U ) = F 1,x, ṁ (u w u e ) = F 2,x. (2.14a) (2.14b) Similarly, from Equation 2.3, one obtains 1 2 ṁ (u2 e U ) 2 = F 1,x u 1, (2.15a) 1 2 ṁ (u2 w u 2 e) = F 2,x u 2. (2.15b) Using the interference factor: λ 1 = u 1 /U and λ 2 = u 2 /u e, and combining Equations 2.14 and 2.15, the velocities 6 of the equilibrium and wake states, u e = U (2 λ 1 1), u w = u e (2 λ 2 1), (2.16a) (2.16b) and that of the rear half-cycle, u 2 U = (2 λ 1 1) λ 2, (2.17) are found. 5 Note that, from now on, the subscript st is omitted on all forces and velocities, as they all refer only to a single streamtube 6 Note that flow velocity expressions violate continuity (Equation 2.13). In an incompressible flow, by considering that there is no streamtube expansion and no spanwise components of the velocity, flow velocity should be constant, as noted previously in this chapter.

31 18 Using the thrust coefficient expression C F1,x = F 1,x /(1/2 ρ A 1,st U 2 ), for the front half-cycle, and C F2,x = F 2,x /(1/2 ρ A 2,st u 2 e), for the rear one and combining Equations 2.13, 2.14 and 2.16, the thrust coefficient expression from the conservation analysis, C Fi,x = 4 λ i (1 λ i ), i = 1, 2, (2.18) is obtained. However, due to the inadequacy of the unidirectional flow assumption below λ 0.6, Glauert [5] proposed a linear modification to this result, 1849/ λ i /15 : λ i < 43/60 C Fi,x = 4 λ i (1 λ i ) : λ i 43/60 according to experimental data., i = 1, 2, (2.19) To be able to solve the interference factor, another expression for the thrust at any given azimuthal angle, F i,x (θ) = 1 2 ρ c u2 i,r (C D cos β C L sin β), i = 1, 2, (2.20) is needed. This expression is obtained from load analysis. As shown in Equation 2.9, lift and drag coefficients can be used to obtain the instantaneous thrust each blade receives. Then, Equation 2.20 is time-averaged along one period in one single streamtube of width δθ for all the N b blades of the turbine. This results in F i,x = N b 2π θi,st +δθ/2 θ i,st δθ/2 F i,x (θ) dθ = N b δθ 2π F i,x(θ i,st ), i = 1, 2. (2.21) The reader will note that this integral is solved with a first order approximation. Then, using the thrust coefficient expressions, together with Equations 2.12, 2.20 and 2.21, the other expression for the thrust coefficient at each streamtube, C F2,x = are obtained. C F1,x = σ π sin θ u 2 1,r U 2 σ (C D cos β C L sin β), i = 1, 2, (2.22a) π sin θ u 2 2,r U 2 (2 λ 1 1) (C D cos β C L sin β), i = 1, 2, (2.22b)

32 19 By using Equations 2.19 and 2.22, λ 1 (θ) and λ 2 (θ) can both be computed note that λ 7 1 participates in Equation 2.22b, so the flow going through the front half-cycle must be solved before it can be solved in the rear half-cycle, too. Also see that this system can be solved for any value of θ that is why the interference factors are using dependence notation (θ). Remember that δθ 0, so there are an infinite number of streamtubes. Then, the first order approximation introduced in the integral of Equation 2.21 does not limit the order of precision of the whole problem. Instead, it will be limited by the algorithms used to solve this system and to integrate the torque along the blade path, needed to average the output power of the turbine. From the interference factor, all variables (β(θ), α(θ), velocities, force coefficients, etc.) can be obtained using the equations shown in Section Then, each blade non-dimensional torque at any given azimuthal angle is C Tb (θ) = according to Equation T b (θ) 1/2 ρ c U 2 R = u2 r (C U 2 L l + C D d) τ, (2.23) To obtain the whole instantaneous torque coefficient using turbine parameters, C T (θ) = T (θ)/ (1/2 ρ A U 2 R), one should add the instantaneous torque of each blade. Note that the swept area is A = 2R because this is a 2D case. Then a turbine with N b equidistant blades, b i, i [1, N b ], would be subject to the total torque coefficient, C T (θ) = σ N b N b i=1 ( C Tbi θ + i 1 ) 2π 3. (2.24) Note that for any N b number of blades, distributed equidistantly, the whole torque coefficient has a periodicity of 2π/N b. Also, please notice that θ is the azimuthal angle of the first blade, b 1. is Finally, the power coefficient that all blades produce, C P = P/(1/2 ρ A U 3 ), C P = C P,1 + C P,2, (2.25) 7 For θ > π, note that the value of λ 1 inserted in Equation 2.22b is λ 1 (2π θ)

33 20 where C P,1 and C P,1, C P,1 C P,front = σ T SR 2π C P,2 C P,back = σ T SR 2π π 0 2π π C Tb dθ, C Tb dθ, (2.26a) (2.26b) are obtained by averaging the instantaneous power in each half-cycle, according to Equation This method, although requiring some numerical calculations to solve the interference factor equation and to numerically integrate the torque to obtain the power, is very straightforward and easy to implement. However, it neglects several effects that gain great relevance in the case of wind turbines. The most significant are the effect of the blades working in angles of attack beyond the static stall angle, the unsteadiness nature of the problem (which includes the effects of dynamic stall) and the interaction of the blades and the wakes that detach from themselves. Also, the 1D simplification introduces some error by neglecting span-wise velocity and 3D effects on the blade. 2.3 Computational algorithm No analytical solution of λ(θ) exists, so no analytical integration of the torque in order to get the power can be computed either. Therefore, numerical integration is necessary to obtain the performance characteristics of the VAWT. Also, as the resulting equation from combining Equations 2.19 and 2.22 is nonlinear, another numerical procedure is required to solve it. The results described in this report have been obtained by using a second-order trapezoidal integration. Therefore, a discretization in the θ domain, θ = π N th θ (1) = θ 2 j [1, 2N th ], (2.27) θ (j) = θ (1) + (j 1) θ

34 21 was required. It consists in N th volume-centered θ (j) smaller domains for each half of the turbine in other words, j [1, 2N th ]. Note that doubling N th means quadrupling the number of discretization nodes θ (j). Interference factor λ is calculated individually for each value of j. To do so, interpolation or bisection methods are used, depending on each iteration outputs. Afterwards, the trapezoidal integration is applied in the whole discretized domain. Figures 2.3, 2.4 and 2.5 show the algorithm used. MATLAB [14] is the tool that has been used to implement this algorithm and to plot all the results. The produced code is object-oriented to ease the addition of features in a modular way. Also, it is parametrized, so it can be used to obtain results for different input values and turbines swiftly. The code is published in Mathworks File Exchange website [30] with a BSD license. It also includes an example file to show how to use it. This code used Subashki et al. [26] MATHEMATICA [13] code as a base.

35 22 Start T SR, θ (j) Start with j = 1 Suppose λ 1 Compute β 1, u 1,r Compute α 1 Compute C L,1, C D,1 Compute C F1,x from Eq. 2.22a Compute λ 1, from Eq Convergence criteria met? no Apply bisection or interpolation yes no j = N th? j = j + 1 yes λ 1 (θ (j) ) Equilibrium Figure 2.3: Flowchart of the calculation of λ 1 (θ (j) ).

36 23 Equilibrium T SR, θ (j), λ 1 (θ (j) ) Start with j = N th + 1 Suppose λ 2 Compute β 2, u 2,r Compute α 2 Compute C L,2, C D,2 Compute C F2,x from Eq. 2.22a Compute λ 2, from Eq Convergence criteria met? no Apply bisection or interpolation yes no j = 2N th? j = j + 1 yes λ 2 (θ (j) ) Postprocess Figure 2.4: Flowchart of the calculation of λ 2 (θ (j) ).

37 24 Postprocess T SR, θ (j), λ 1 (θ (j) ), λ 2 (θ (j) ) Compute β(θ (j) ), u r (θ (j) ) Compute α(θ (j) ) Compute C L (θ (j) ), C D (θ (j) ) Compute C Tb (θ (j) ) Compute C P 1, C P 2 Compute C P using 2 nd order trapezoidal integration C P Stop Figure 2.5: Flowchart of the calculation of C P.

38 Results This section discusses the results obtained from applying DMST to a Darrieus H-rotor wind turbine with the following characteristics: Table 2.1: Wind turbine characteristic values VAWT characteristics N b 3 R e (blade) Blades profile NACA 0021 R 515 mm c 85.8 mm σ This case is chosen because there are enough data in the literature with which to compare results [9]. The characteristic static lift and drag curves for the profile NACA 0021 at the specified Reynolds number have been obtained according to Appendix B. The discretization parameter is N th = 51, the one that proved to give enough precision at lowest computational cost. The curve of power coefficient against tip-speed ratio obtained for the presented case is shown in Figure 2.6. Note that at low tip-speed ratios both front and back half-cycles of the turbine extract a similar amount of energy from the flow. However as the T SR increases, the front half of the turbine extracts more energy from the flow. Therefore, when the same flow reaches the back half, it carries much less energy, so the back half of the turbine cannot extract as much energy Furthermore, above values of T SR a bit higher than 3, the front half-cycle extracts so much energy than the rear half-cycle is actually giving some of it back to the flow, instead of extracting it 8. Hence, there is a value of T SR, about 2.7, where the balance front/back ex- 8 This phenomena becomes very relevant when adding wake interaction modifi-

39 26 traction gives the optimum power coefficient. Although DMST is a reasonably crude model, this characteristic is represented well enough. 0.5 C P against TSR C P C P C P,front C P,back TSR Figure 2.6: C P against T SR Effect of solidity To appreciate the effect of solidity on the output power, the DMST simulation has been run with the same turbine only changing the solidity value, by modifying the diameter if the chord of each blade is changed, the Reynolds number of the blade would change too. However, the purpose of this subsection is to see the effect of solidity only. The results can be seen in Figure 2.7. According to the DMST model, the bigger the solidity is, the lower is the optimal tip-speed ratio. The maximum achievable power coefficient seems to be maximum for the initial case, decreasing when solidity either increases or decreases. cations to DMST. Therefore, it is discussed further in Chapter 4, where wake interactions are covered.

40 CP against TSR CP σ = σ = 0.05 σ = σ = TSR (a) Power coefficient on the whole turbine CP front against TSR 0.2 CP back against TSR CP front σ = 0.1 σ = 0.05 σ = σ = 0.4 CP back TSR (b) Power coefficient on the front halfcycle TSR (c) Power coefficient on the back halfcycle. Figure 2.7: Angle of attack for different dynamic stall models. However, remember that DMST neglects wake interactions, which is an effect that gains relevance with solidity growth. Therefore, it might be that the optimal solidity was lower than the one suggested by DMST. Also note that, as σ increases, the front part of the turbine produces much

41 28 more power and the rear much less. Hence, the T SR for which the rear half-cycle produces zero power keeps decreasing as solidity increases. This is logical, because the larger the blades are respectively to the domain, the more energy they can extract. When the flow first meets the turbine, it loses more energy if the solidity is higher. Hence, when it reaches the rear part, it has less energy to be extracted. Moreover, for the values of T SR that the back half of the turbine produces negative power, an increase of solidity which means bigger blades relatively to the diameter implies that more energy is given back to the flow Effect of the number of blades The effect of the number of blades only means a change in solidity. However, note that according to DMST, two turbines with different number of blades but same solidity and blade Reynolds number would behave identically. This is because DMST is not taking into account interaction between blades, which is produced, in fact, by wake generation Comparison with other Streamtube models As said before, DMST is the most precise version fo the three Streamtube models. However, all three have been tested. First, Multiple and Double-Multiple Streamtube models are compared. The main difference, of course, is that DMST is able to distinguish the different behavior of the flow in the front and the rear half of the turbine. This is noticed by observing the instantaneous torque for each model at any azimuthal angle. While MST shows a symmetric distribution centered on the 180, DMST clearly reports a different distribution between front and back halves although the shape of the rear part is similar to the one of the front part, because λ 2 depends of λ 1. From the computational point of view, there is not much difference, because both models solve the interference factor at many different values of θ. It is true that DMST must solve two different values of λ for each one of MST. However, DMST performs even better than MST in general because the latter has

42 29 more problems to reach convergence, probably due to the fact that the λ equation is less smooth it has to take into account both front and back halves in one only step. The final power output is less different than that of the torque, because the integration steps level the difference in torque. However, as said before, DMST is worthier using because it overtakes MST both in precision and convergence speed. When testing SST, as expected, convergence is reached much faster than both DMST or MST, because the interference factor is solved only once in the whole turbine. In this case, the output torque is not only mirrored from front to back halve, but it is also symmetric in each half, centered at 90 and 270, respectively. Hence, the precision of this method is even worse. It can be useful as a very preliminary tool to get an idea of the turbine performance. Nonetheless, DMST is not that complicated and slow. Morover, the generation of the code of both models is very similar. Therefore, globally, SST is only worth it using if only very low-end computational resources are available. General-use computers nowadays have no problems with DMST and can solve a whole turbine for a reasonable range of T SR a few minutes.

43 30 CHAPTER 3 DYNAMIC STALL MODELS 3.1 Concepts As mentioned in the previous chapter, dynamic stall plays a very relevant role in the VAWT problem. However, static lift and drag curves have been used. Obviously there is significant amount of error introduced by this fact. Therefore, a lot of effort has been invested into developing modifications to the original DMST, so that it includes these effects. Dynamic stall, opposite to the usual static stall observed in wind tunnels and in fixed blades simulations, presents a hysteresis behavior. In other words, if a blade changes its position, the flow does not react immediately, but it adapts gradually to the new state within a definite interval of time. Such an effect is only dominant at low T SR, when the turbine rotates slowly enough to allow the development of this hysteresis cycle the magnitude of the blade movement is similar or smaller than that of the flow movement. In high T SR cases this effect ceases to be determinant wake interactions gain strength and become the major factor. In this latter case, the induced velocity, θ R, component of the relative velocity, u r, is much higher than the wind velocity component, U see Figure 2.1. Therefore, angles of attack are lower, according to Equation 2.6, which leads to smaller oscillations of the blade respectively to the relative movement of the flow. Most of the dynamic stall models, according to [18], applied to DMST consist of a series of semi-empirical procedures applied in the calculation of the lift and drag coefficients of the VAWT blade. Therefore, applying a dynamic stall model to the DMST method means computing some additional steps interlaced with the main algorithm.

44 Theory development These are: In this section, several dynamic stall models are explained and compared Gormont model Gormont [6] proposed this method to take into account dynamic stall in helicopter blades. It is not optimal for VAWTs, because they operate at much higher angles of attack, but it has been used as a very strong base for later adaptations. The Gormont method consists of applying a certain delay,δα, on the angle of attack, based on hysteresis behavior. After applying this delay, the resulting reference angle of attack is α ref = α K δα, (3.1) where K, 1 : α 0 K = 0.5 : α < 0, (3.2) is a parameter defined according to Gormont s empirical observations. Therefore, the delay experienced when the absolute angle of attack is increasing is twice the one observed when it is decreasing. One possible qualitative explanation would be the following. When the angle of attack is increasing, the flow near the leading edge of the blade is pushed to the surface of it, making it more difficult for the upwind flow to change its state. However, when the blade angle of attack is decreasing, this same flow is left behind, so the upwind flow changes its state more easily. This effect generates a discontinuity that can induce severe instabilities to the flow surrounding the points where α changes sign 9. 9 α is the rate of change of the absolute value of α, so it is negative when the angle of attack approaches 0 and positive when it moves away from 0.

45 32 The term δα is proportional to the non-dimensional rate parameter S, c α S = 2u r, (3.3) which can also be interpreted as the square root of the magnitude of the leading edge velocity of a blade rotating around its half chord point over the magnitude of the relative wind velocity. In other words, it compares the blade movement to the flow relative movement around it. The higher S is, the greater the applied delay. This is coherent with what was explained in Section 3.1 dynamic stall is important for low tip-speed ratios, when α oscillations are bigger. The slope of the line δα(s) decreases both with thickness over chord ratio and Mach number of the flow relative to the blade growth. Then, the thicker the blade is and the less compressible the flow is, the smaller is the delay on the angle of attack. The specific expression for δα can be found Appendix C. Once the delayed angle of attack, α ref, is found, the modified lift and drag coefficients can be calculated. The lift coefficient, C dyn L = C L,0 (α 0 ) + m (α α 0 ), (3.4) is calculated by using the potential equation with a modified slope, m, the minimum value of either the slope of the linear part of C L or the value obtained with the following expression: (C L (α ref ) C L (α 0 ))/(α ref α 0 ). Figure 3.1 shows both mentioned slopes for an example value of α ref. Also, the mathematical expression for m can be found in Appendix C. The symbol α 0 refers to the angle of attack at zero lift. The drag coefficient, C dyn D = C D(α ref ), (3.5) is obtained simply by using α ref to extract it from the available static data.

46 C L linear C L alternate CL C L α ref = 17.5 o α (deg) Figure 3.1: The two different values from which the C dyn L slope is calculated. In this example, the value used would be the slope of the dotted line, because it is lower Berg modification Berg [3] considered that the α range at which helicopter blades work is much lower than that of a VAWT, which can perfectly go beyond the stall angle of attack (α ss ). Therefore, Berg theorized that the pure Gormont method might over predict the lift and drag for high angles of attack. He proposed the following modification for both lift, CL mod = [ C L + ] A M α ss α A M α ss α ss (C dyn L C L ) : α A M α ss, (3.6) C L : α > A M α ss and drag 10, so that both coefficients tend to the static values for very high values of α, deep into the non-linear regime. The parameter A M is set by the user of the method. According to [3] the best option is A M = 6. However, according to [18], some VAWTs show a better response when using A M. Note that this is the same as using the pure Gormont method with no modifications. The results shown 10 Only the lift modification is shown. The drag modification is the same expression but changing all L subscripts for D. See Appendix C for the drag expression.

47 34 on this paper are computed using Berg s suggestion Strickland et al. adaptation Strickland et al. [25] considered that Gormont method was developed for very thin blades, as the typical helicopter blade. However, VAWTs typically use blades with higher thickness over chord ratio. That is why they modified the delay so S c = 0. See Equation C.2 in Appendix C. Also, as the typical rotation speed of a VAWT is much lower than that of a helicopter, the flow was considered to be incompressible the Mach number is not used as an input parameter of the model. Finally, Strickland et al. supposed that dynamic stall effects only occur after the angle of attack is higher than the stall angle of attack, α α ss. Then, the resulting reference angle of attack is: α ref = α γ K S S α, (3.7) where K and S are described in Equations 3.2 and 3.3, respectively. S α is the sign of α as is described in Section Expression for γ can be found in Appendix C. From Strickland s angle of attack, lift, CL str = C L (α) ( ) α α ref α 0 : α < α ss C L (α ref ) : α α ss, (3.8) and drag, CD str = C D (α) : α < α ss C D (α ref ) : α α ss, (3.9) coefficients are obtained Paraschivoiu adaptation Paraschivoiu [17], after some experimental observation, concluded that the part of the blade path comprised in θ [105, 225 ] is mainly dominated by turbulent aerodynamics and wake interactions. Therefore, they considered that in this region

48 35 dynamic stall can be neglected. Therefore, their modification basically consists of applying Strickland adaptation only outside this azimuthal angle range. This adaptation might be a bit crude, because it is based on experimental data from one specific turbine under some very particular conditions Indicial method and adaptations Another different approach to dynamic stall modeling is the indicial method. It was first introduced by Beddoes and Leishman [2] and first applied to VAWTs by Paraschivoiu et al. [19]. This method consists of accepting that the total effect of lift and drag modification due to dynamic stall results of the individual addition of different effects. The first step consists on calculating the attached flow unsteady behavior by using potential flow theory and the addition of unsteady empirical correlations. Second, separation in the leading and trailing edge are taken into account. Finally, an optional third phase introduces some corrections of the separation for deep stall, when the angle of attack is far beyond the stall angle of attack. All these effects are supposed to have indicial behavior. This means that they grow or decrease with time according to lag and impulsive functions. These indicial functions are of the form of summations of exponential functions with different coefficients and characteristic time parameters. All modifications depend also on the rate of change over time of the angle of attack, as did the Gormont based models. This method, however, depends on several coefficients mainly the lag and impulsive parameters that must be determined empirically and/or by potential flow approximations. Also, it considerably increases the computational requirements, because the introduced modifications must be solved by iterative methods, which DMST must apply at every iteration of the interference factor calculation. According to [18], the indicial method gives an overall better approximation of the dynamic stall models than the Gormont based ones. However, for qualitative purposes the difference is not very high compared to Berg adaptation, for example. For all these reasons,

49 36 the indicial method is not explained nor was developed during the redaction of this report. More information can be found in [18]. Further development on streamtube models must include the development of DMST code with Indicial dynamic model. Several adaptations of the original Beddoes and Leishman model have been applied to wind turbines in general, such as Hansen [7]. 3.3 Computational algorithm modifications In order to insert any of the described dynamic stall models into the DMST algorithm proposed in Section 2.3, three additional steps must be introduced. Basically, Compute C L, C D box of each Figure 2.3, 2.4 and 2.5 must be substituted by the specific calculation of the modified lift and drag coefficients. The flowchart for the modified DMST of the front half is shown in Figure 3.2. The rest of the algorithm can be found in Appendix C. In the modified flowcharts, t is the time that the blade needs to travel an arch of amplitude θ. Hence, it is t = θ θ. (3.10) Also, note that the superscript dyn refers to any dynamic stall model, not only Gormont s, as suggested in the previous section. As seen in Figure 3.2, the rate of change of the angle of attack, α, and lift and drag coefficients are calculated using the absolute values of the angle of attack. This is because all Gormont based models are described for positive angles of attack. When working with α < 0, all signs should be switched. Instead, observe that this algorithm works with the angle of attack in absolute value and, afterwards, it changes the sign of the lift if the blade is working at α < 0 note that S α is the sign of α. Note that this produces the same result and is much easier to implement in a computational code. Also note that α is calculated with first order approximation, using the value of α of the previous azimuthal angle. In θ (1) and θ (N th+1), α(θ (j 1) ) is substituted by

50 37 Start T SR, θ (j) Start with j = 1 Suppose λ 1 Compute β 1, u 1,r Compute α 1, α 1 = [ α1 (θ (j) ) α1 (θ (j 1) ) ] / t Compute C dyn L,1 = S α C dyn L,1 ( α 1, α 1 ), C dyn L,1 ( α 1, α 1 ) Compute C F1,x from Eq. 2.22a Compute λ 1, from Eq Convergence criteria met? no Apply bisection or interpolation yes no j = N th? j = j + 1 yes λ 1 (θ (j) ) Equilibrium Figure 3.2: Flowchart of the calculation of λ 1 (θ (j) ) for the dynamic stall modification.

51 38 α(θ (j) ). This assumption forces the angle of attack at θ = 0, 180 to be zero, so that Equation 2.6 is not violated. Dynamic stall modification is included in Vallverdú s code [30]. 3.4 Results First, all models will be compared. Afterwards, once the best dynamic stall model is chosen, the stall effect itself will be studied. To do so, a new variable has been introduced, which is the effective angle of attack, α eff, α eff = C L C Lα, (3.11) where C Lα is the slope of the linear part of the lift coefficient curve for the NACA 0021 profile. It can be found in Equation B.1, in Appendix B. α eff is, basically, a normalization of the lift. However, it is useful to identify non-linear zones of the blade path by comparing it with the actual α if both angles are equal, it means that the blade is working in the linear regime at that point of its path Comparison of different models As said, this section is mainly aimed at comparing the different stall models with each other and with the plain DMST model. Lets begin with the power coefficient output for the initial case proposed in Table 2.1. The result can be seen in Figure 3.3. The reader can see that, as explained in the theory, Gormont over-predicts performance relatively to Berg. Also, observe that for T SR lower than 3 all stall models predict better performance than the original DMST. This is due to the stall happening slower in a dynamic regime than a static one, thanks to the delay introduced by the hysteresis effect. Therefore, dynamic stall models predict that the time that the blade is experiencing stall is less than what a quasi-static approach gives. Next, the lift characteristic reference angle of attack and blade torque coeffi-

52 C P against TSR 0.4 C P plain DMST Gormont Berg A M = 6 Strickland Paraschivoiu TSR Figure 3.3: Comparison of the power coefficient outputs for different dynamic stall models. cient at each azimuthal angle have been obtained see Figures 3.5 and 3.6. These two are the representation of the dynamic stall modifications. Remember that all proposed adaptations consist of two steps: first they modify the angle of attack and then the lift and drag coefficients from this angle of attack. Hence, Figure 3.5 shows the first modification and 3.6 is a way to see the second one, because the torque coefficient directly depends on the lift and drag coefficients, according to the DMST theory. The lift characteristic reference angle of attack for all models curves show how the angle of attack is delayed to consider dynamic stall. Also, it shows very well how this delay is greater when the angle of attack is increasing than when it is decreasing. Also, specially in Figure 3.5c, note that Strickland-based models predict

53 40 15 α (deg) for TSR = plain DMST Gormont Berg A M = 6 Strickland Paraschivoiu θ (deg) (a) T SR = α (deg) for TSR = 2 10 α (deg) for TSR = θ (deg) θ (deg) (b) T SR = 2 (c) T SR = 3 Figure 3.4: Angle of attack for different dynamic stall models. a less radical delay than Gormont/Berg models 11. This is observed more closely in the torque curves, where the changes of sign of α affect the Gormont/Berg curves to a higher extent than the Strickland based ones. The loss of torque according to same. 11 Remember from the theory development that Berg and Gormont α ref are the

54 41 Gormont/Berg curves in azimuthal angle around θ = 90 might be caused by the sudden update of the flow state due to the change of sign of α all the accumulated dynamic load is released when the blade starts exposing its suction surface, causing a sudden increase of drag that brings torque down. Torque curves are also very suitable to appreciate how Berg method is meant to soften Gormont s over-prediction at high angles of attack. Figure 3.7 shows the lift coefficient at any angle of attack. It is included, mainly, to show the hysteresis effect explained in the previous Section. Please observe how all dynamic stall models represent the dynamic delays by giving different lift at a same angle of attack. When its absolute value is increasing, the dynamic lift does not sense stall until a higher angle of attack than the static angle of attack, α ss 10. However, when it is decreasing, the dynamic lift adapts faster to the static value. This is due to Gormont s observation explained in Section 3.2.1: the flow adapts swifter when the blade angle of attack is decreasing than when it is increasing. Also see that Paraschivoiu adaptation result does not present hysteresis in some regions in general, all other figures show the same behavior of the Paraschivoiu curves. These are the regions where turbulence was considered dominant and, therefore, no dynamic model was applied. Because the Strickland model was the first and most crude adaptation to VAWTs and the Gormont model was developed mainly for helicopters, the Berg model is the one chosen to treat dynamic stall in this text. Also, it presents the advantage of being parameterizable by varying A M. Finally, Berg results are more similar to experimental measurements [18] and finite volume CFD results [9] than the other dynamic stall models. A said in the previous section, indicial mehtods seem to be very good at dynamic stall simulation. However, they add computational difficulties much higher than those inherent to Berg adaptation. Streamtube models are aimed at simple and quick qualitative studies, so it was considered that the overall computational efficiency of the Berg model is more suitable for a DMST model than

55 42 15 α ref (deg) for TSR = 2.5 for lift characteristic plain DMST Gormont Berg A M = 6 20 Strickland Paraschivoiu θ (deg) (a) T SR = α ref (deg) for TSR = 2 for lift characteristic 10 α ref (deg) for TSR = 3 for lift characteristic θ (deg) θ (deg) (b) T SR = 2 (c) T SR = 3 Figure 3.5: Angle of attack for different dynamic stall models. Plain DMST α ref is the actual DMST α. the one of the indicial models. Finally, this chapter contrasts Berg results with the plain DMST results. This is done by using the effective angle of attack from Figure 3.8. According to these

56 43 C Tb for TSR = plain DMST Gormont Berg A M = 6 Strickland Paraschivoiu θ (deg) (a) T SR = C Tb for TSR = C Tb for TSR = θ (deg) θ (deg) (b) T SR = 2 (c) T SR = 3 Figure 3.6: Blade torque coefficient for different dynamic stall models. curves, the dynamic stall model delays the overall effective performance of the blade by smoothening the sudden changes of the quasi-static approach. Specially the front part of the turbine shows very well this inertia effect that tends to treat the state of operation of the blade in a more conservative way.

57 C L for TSR = C L plain DMST Gormont Berg A M = 6 Strickland Paraschivoiu α (a) T SR = C L for TSR = C L for TSR = C L 0 C L α α (b) T SR = 2.5 (c) T SR = 3 Figure 3.7: Blade lift coefficient for different dynamic stall models against angle of attack. When comparing the power between plain DMST, Figure 2.6 and Berg modification, Figure 3.9, one sees that differences in the back half performance are very little. The front half, instead, a presents a noticeable improvement in performance for T SR lower than 3 and a less detectable exacerbation of the power output for higher

58 45 15 α eff (deg) for TSR = plain DMST Berg A M = θ (deg) (a) T SR = α eff (deg) for TSR = 2 10 α eff (deg) for TSR = θ (deg) θ (deg) (b) T SR = 2 (c) T SR = 3 Figure 3.8: Effective angle of attack for plain DMST and Berg model. values. On the whole, this means that most of the dynamic stall occurs in the front half and that it tends to improve the overall performance of the turbine, because it predicts that stall comes later than in a quasi-static evolution, but it disappears faster than it appears. The lower power predicted for high values of T SR can be neglected and will not be commented further, because when the turbine is working at this range

59 46 of T SR, dynamic stall looses dominance to give it to wake interactions [9]. 0.5 C P against TSR C P C P C P,front C P,back TSR Figure 3.9: C P against T SR with Berg dynamic stall model.

60 Stall study using DMST with Berg adaptation Figure 3.10 is used, mainly, to detect the regions where the blade is working under stall 12 regime. See that for low tip-speed ratios, most of the stall occurs evenly both in the rear and front halves of the turbine. Both parts, according to Figure 3.9 are extracting a similar amount of energy from the flow, which means that the interference factor of both halves is high enough. Therefore, the flow component of the relative velocity of the blade is important enough so that both halves work at high angles of attack for an extended period of time. However, as T SR grows, the front half starts to extract all the energy from the flow, so it reaches the rear half with very slow velocities, which produces a huge fall in the angle of attack. Hence, the blade in the rear half does not reach stall regime. The reference angle of attack has also been included in Figure 3.10 to give an idea of the delay introduced by the constant motion of the blade. 12 This section only works with Berg addition to DMST, so stall always refer to dynamic stall, the proper type of stall of oscillating aerodynamic problems.

61 48 15 α (deg) for TSR = 2.5, Berg model α α eff α ref θ (deg) (a) T SR = α (deg) for TSR = 2, Berg model 10 α (deg) for TSR = 3, Berg model θ (deg) θ (deg) (b) T SR = 2 (c) T SR = 3 Figure 3.10: Real and effective angle of attack for Berg model.

62 49 CHAPTER 4 WAKE INTERACTION 4.1 Concepts Wake interaction is the effect of the turbulent wakes generated by the blades on the front half received on the back half. However, BEMT is unable to model turbulence, so wake interaction is neglected in streamtube models. Also, modeling this effect by using the fundamental physics that describe turbulence or some turbulence model and applying it directly to streamtube models would be very detrimental to the computational efficiency of BEMT in fact, the problem would become a mixture of BEMT and a grid-centered algorithm. BEMT s main advantage, actually, is computational efficiency. Therefore, a workaround was needed. Kozak [9] studied the turbine described on Table 2.1 with a grid-centered method. He used STAR- CCM+ [23] to apply 2D finite volume discretization to the Navier Stokes equations. Kozak and Vallverdú et al. [10] compared the effective angle of attack obtained from Vallverdú s code [30] with dynamic stall the green curve in Figure 3.8 with Kozak s own results shown on [9]. From this comparison, they were able to develop a much simpler and effective wake interaction model for DMST that was shown to correlate well with finite volume simulation.

63 Theory development The Kozak et al. wake model follows the subsequent observation. By studying graphic visualizations of the vorticity generated by the blades [9] see Figure 4.1, they realized that the number of wakes, N w = 0.85 N b T SR, (4.1) that exist in the rear half of the turbine at a given moment is proportional to the tip speed ratio and the number of blades. Figure 4.1: Visualization of the vorticity for T SR = 3 [9]. Also, they observed that each blade hits each wake twice. They considered that the width of a wake is equal to the thickness of a blade. This hypothesis neglects the wake diffusion, but it is accurate enough for streamtube application. Therefore, a distance value, d w = 2 t N w, (4.2) can be obtained. This distance, d w, represents the sum of all the parts of the blade path that crosses a wake. However, this value is more useful if used as a ratio, r w = d w π R, (4.3) where the denominator is the distance covered by a blade in the rear half-cycle of the turbine. Note that r w does not depend anymore on the number of blades of

64 51 the turbine, because both numerator and denominator include the number of blades, hence it disappears. It represents the proportion of the blade path in the back that cuts through a wake. Figure 4.2: Effective angle of attack comparison between Kozak and Vallverdú results for T SR = 3 [10]. Kozak and Vallverdú et al. assumed that the flow inside the wake travels at the same speed and with the same direction than the blade that generated it. Therefore, according to that assumption, when a blade hits a wake in the rear part, its relative velocity is diminished a lot, and so does its instantaneous angle of attack. In other words, when a blade hits a wake, its lift and drag coefficients decrease dramatically. This hypothesis may seem to be very crude, because it neglects several diffusion and transport effects. Also, the blade and the wake do not hit in parallel directions. However, following this line of thought, Kozak and Vallverdú et al. multiplied Vallverdú s results of the rear half-cycle effective angle of attack by the (1 r w ) term they dropped lift and drag down to zero whenever a blade hits a wake. Afterwards, they compared the output with Kozak s [9] effective angle of attack: the result was very satisfactory. Both curves, in the range of θ [π, 2π] fit almost perfectly see Figure 4.2 for the T SR = 3 example 13. Hence, the hypothesis taken related to the flow motion in the wake proves not to be so wrong. The effective angle of attack, as said before, is basically a representation of 13 Curves for other values of T SR can be found in [10].

65 52 the lift coefficient. Therefore, both Kozak and Vallverdú et al. [10] concluded that the wake effect could be included in a simple way in streamtube model by reducing both final lift and drag coefficients by r w, CL wake = CD wake = C dyn L : θ (0, π) C dyn L (1 r w) : θ (π, 2π) C dyn L : θ (0, π) C dyn L (1 r w) : θ (π, 2π), (4.4a). (4.4b) This reduction should be applied after the determination of the interference factor but before the postprocessing in other words, before the calculation of torque and power coefficient. Note that the wake interaction is approached as an average effect on the whole rear half-cycle of the turbine. However, the real wake interaction is a very discrete effect [9].

66 Algorithm modification The wake model applied to DMST with dynamic stall would only change, therefore, in the postprocessing part. See Figure 4.3. Postprocess T SR, θ (j), λ 1 (θ (j) ), λ 2 (θ (j) ) Compute β(θ (j) ), u r (θ (j) ) Compute α(θ (j) ), α(θ (j) ) as in Fig. 3.2 and C.1 Compute C dyn L = S α C dyn ( α, α), Cdyn( α, α) L D Compute CL wake, CD wake from Eq. 4.4 Compute C Tb (θ (j) ) Compute C P 1, C P 2 Compute C P using 2 nd order trapezoidal integration C P Stop Figure 4.3: Flowchart of the calculation of C P for the dynamic stall and wake modification. Wake interaction modification is included in Vallverdú s code [30].

67 Results Comparison between DMST with Berg modification and with/without wake interaction model The effective angle of attack obtained with the new wake interaction modification applied to Berg DMST can be seen in Figure 4.4. It clearly shows that the model that includes wake interaction presents a slightly reduced effective angle of attack respectively to the non wake model, only in the back half of the turbine. Of course, a real case would show reductions only at discrete regions, not a general decrement, as explained in the last paragraph of Section 4.2. However, BEMT is treating the problem as steady, so the average reduction is applied to the whole back half. Figure 4.5 shows the power coefficient output for DMST modified with Berg and wake interaction compared to only Berg. As can be seen, for low T SR the wake interaction model predicts lower power output. However, it reports better performance for high T SR. Low T SR have higher angles of attack: high α C L C D. Therefore, both coefficients are reduced a similar amount and the resulting torque gets diminished 14. However, as T SR keeps increasing, the front half-cycle of the turbine extracts more and more energy from the flow. Hence, the back half-cycle gradually extracts less and less energy. As mentioned in Chapter 2, at some T SR slightly over 3, the back half stops producing power. From this value and above, it becomes drag dominant and it even gives energy back to the flow see how, in Figure 4.6, the torque is negative in the back half-cycle. Therefore, if effective angle of attack is reduced, the drag force that dominates the rear half-cycle torque decreases. Then, the absolute value of the torque decreases, too: less opposing torque means less energy back to the flow, which means more energy extracted from the flow on the whole. Also, according to Figure 4.5, note that the wake interaction effect increases 14 High angles of attack registered at low tip speed ratios are those when C L C D, according to Figure B.1 on Appendix B

68 55 linearly with T SR, according to the theory. 15 α eff (deg) for TSR = Berg A M = 6 Berg A M = 6 with wake θ (deg) (a) T SR = α eff (deg) for TSR = 2 10 α eff (deg) for TSR = θ (deg) θ (deg) (b) T SR = 2 (c) T SR = 3 Figure 4.4: Effective angle of attack for DMST with Berg and wake modification. Computationally, this wake interaction model is so simple that it virtually does not change the code performance. Therefore, as it shows better agreement

69 C P against TSR C P Berg A M = 6 Berg A M = 6 with wake TSR Figure 4.5: C P against T SR with Berg dynamic stall and with/without wake interaction models One blade torque coefficient Upwind Downwind θ Figure 4.6: Torque coefficient of one blade for T SR = 4 using Berg model. with Kozak s results for the same turbine [9], it will be added permanently to the calculation of the turbine performance. Note that the rule stated in Equation 4.1 might vary according to different solidity values, type of blade, number of blades or

70 57 other parameters. Therefore, further experiments or simulations should be performed in order to determine if these dependencies exist and, in case they do, how they behave. On the whole, the final performance result can be seen in Figure 4.7. The optimal T SR of this turbine, according to the proposed model DMST with Berg modification for dynamic stall and Kozak and Vallverdú variation for wake interaction is around However, the curve shows that the performance is fairly acceptable in the range T SR [2.3, 3.2], approximately. 0.5 C P against TSR C P C P C P,front C P,back TSR Figure 4.7: C P against T SR with Berg dynamic stall and wake interaction models.

71 58 CHAPTER 5 CONCLUSION 5.1 Summary Streamtube models have been implemented in MATLAB [14] in order to obtain a simple and relatively computationally inexpensive method to predict vertical-axis wind turbine performance. With this method, data about power, torque, angle of attack and flow velocities can be calculated, amongst other variables, for different parametric values of tip speed ratio, solidity, chord length, free stream velocity and others. Also, performance for different airfoil blade profiles can be computed by using different lift and drag characteristic curves. Three basic streamtube models are available: single, multiple and doublemultiple. DMST has been chosen due to its higher computational efficiency relative to precision over convergence behavior. Afterwards, a dynamic stall model has been applied to include dynamic stall effects in the original model. The Berg adaptation of the Gormont method has been chosen, again because it gives the best balance of precision over convergence. Finally, a simplified wake interaction model developed by Kozak has been added too. The results obtained show physical consistency to reality to some extent, and are very useful to obtain fast qualitative data about any VAWT. However, they are not suitable for precise design or flow observation. Finite volume methods using RANS, LES or DNS with Navier Stokes equations are needed to obtain precise data. Nonetheless, BEMT is useful. Thanks to its ease of use and fast convergence, it can help detecting major problems on more complicated simulations. Likewise, it can serve to provide initial data for problems with poorer convergence or to orient some

72 59 turbine design project at its initial to intermediate stages. 5.2 Further work To improve the streamtube model and make it more competitive, some more work should be done. First, the Beoddes-Leishman indicial method should be implemented and tested. This would help to improve the dynamic-stall prediction. Further investigation should be done to improve and generalize the already applied wake interaction method. A discretization of streamtubes in the z direction, such as the one proposed in [18], might be applied to include 3D effects of the flow, such as the boundary layer that the ground induces to the wind. Also a tower interaction modification might be included. Finally, some corrections to the model could be performed to consider the spanwise components of the velocity. However, some convergence evaluation should be performed to state if the method is still fast enough so these modifications are worth applying. More comparisons with real cases or other simulations should be done, for different input parameters and turbines. This would help validating or limiting to a certain range of some parameter the code. Finally, one could also include design optimization procedures to find better work regimes of the turbine. Some of them include finding the optimum variable blade pitch, free stream velocity or variable local blade tip speed ratio. Also, nonsymmetric airfoils might be tested to see if there is any improvement on the turbine performance. VAWTs still have a long optimization path ahead, so BEMT fast convergence characteristic will still be useful to guide and help more complicated methods and make them more efficient.

73 Variable Pitch Although not finished, some work has been done already with the variable pitch optimization issue. H-rotor VAWT design allows assembling the blades in a way that their pitch angle, relative to their supporting arm, is variable. With this modification, the blade orientation can be controlled to modify the angle of attack. Therefore, some optimization is needed to determine the best variable pitch values at any azimuthal angle. The angle of attack has been redefined as α p, α p = α + ϕ, (5.1) where ϕ is the added pitch to the blade, positive when the blade is facing outwards its path, and α is calculated according to Equation 2.6. The optimization algorithm tries to find the best ϕ(θ) by imposing a constant effective angle of attack at both halves of the turbine [9]. Remember that the effective angle of attack is a representation of the lift. Therefore, the imposed value must be that of maximum lift without stall, α ss. However, further work must be done to implement this modification to the stable version of the code published on Mathworks [30] Graphical User Interface A GUI was developed in the first stages of the project, in order to make the code more user friendly. However, it slowed down the development of the project and limited its versatility, because the GUI had to be constantly kept updated with it. Therefore, the GUI branch was temporally abandoned. However, a not so relevant future task that might be carried is the update and maintenance of this GUI. Figure 5.1 shows the aspect of the abandoned GUI.

74 Figure 5.1: Former GUI used to interact with Vallverdú s code. 61