DMST

STUDY ON VERTICAL-AXIS WIND TURBINES USING STREAMTUBE ANDDYNAMIC STALL MODELS BY ´ DAVID VALLVERDU Treball Final de Grau submitted in partial fulfillment of the requirements for the degree of Enginyeria en Tecnologies Aeroespacials in Escola T`ecnica Superior d’Enginyeria Aeron`autica i Industrial de Terrassa Universitat Polit`ecnica de Catalunya as a result of International Exchange program with Armour College of Engineering Illinois Institute of Technology Advisor Chicago, Illinois June 2014 Dietmar Rempfer c Copyright by ´ DAVID VALLVERDU June 2014 ii 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 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Energy consumption . . . . . . . . . . . . . . . . . . . 1.2. Wind turbines . . . . . . . . . . . . . . . . . . . . . . 1.3. Performance prediction of VAWTs . . . . . . . . . . . . 1 2 8 2. STREAMTUBE MODEL . . . . . . . . . . . . . . . . . . . . . . . 10 2.1. 2.2. 2.3. 2.4. . . . . 10 11 20 25 3. DYNAMIC STALL MODELS . . . . . . . . . . . . . . . . . . . . . 30 3.1. 3.2. 3.3. 3.4. Concepts . . . . . . . . Theory development . . Computational algorithm Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concepts . . . . . . . . . . . . . . . Theory development . . . . . . . . . Computational algorithm modifications Results . . . . . . . . . . . . . . . . 4. WAKE INTERACTION 4.1. 4.2. 4.3. 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 31 36 38 . . . . . . . . . . . . . . . . Concepts . . . . . . . Theory development . Algorithm modification Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 50 53 54 5. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Further work . . . . . . . . . . . . . . . . . . . . . . . 58 59 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 APPENDIX 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv . . . . . . SINGLE AND MULTIPLE STREAMTUBE MODELS . . .2. . . . . . . . . . A. Single streamtube model . . . . . . .A. . . . Multiple streamtube model . . C. . 72 73 74 74 74 . . . . . . . . . . . . .2. . . . . . . Computational algorithm modifications v . . Gormont method . . OBTENTION OF THE LIFT AND DRAG CURVES FOR NACA 0021 69 C.3. . . . . . Berg modification . . . . . . . . . . . . . . . . . . .4. . . . . . DYNAMIC STALL NEEDED EXPRESSIONS . . . . C. . . . . . . . . . .1. . . . Strickland adaptation . . . . A. . 64 65 67 B. . . . . . . . . . . C. C. . . . . . . . . .1. . . . . . . . . . . vi 25 . . .1 Page Wind turbine characteristic values . .LIST OF TABLES Table 2. . . . . . . . . . . . .7 Blade lift coefficient for different dynamic stall models against angle of attack. . .6 CP against T SR. . . . . . . . . . . . . .5 Drag (Savonius) and Lift (Darrieus) VAWTs. . . . . . . .7 Angle of attack for different dynamic stall models. From EIA [8]. . . . . . . .2 Scheme of the DMST discretization of a VAWT domain. . . 33 Flowchart of the calculation of λ1 (θ(j) ) for the dynamic stall modification. 27 3. . . . . . . . 44 Effective angle of attack for plain DMST and Berg model. . . . . . 23 2.3 Flowchart of the calculation of λ1 (θ(j) ). . . . . .5 Flowchart of the calculation of CP . . . . 7 1. . . . . . . . τ is a unit vector tangent to the blade. . .6 H-rotor Darrieus VAWT. . . .6 Blade torque coefficient for different dynamic stall models. . . 7 2. . . . . . . . . . . . . . . . . . . 42 3. . . . . . . 37 Comparison of the power coefficient outputs for different dynamic stall models. . . . 24 2. . . . . .2 3. . . .2 3. . . . . 16 2. . . 2 1. . . . . . . . .4 Flowchart of the calculation of λ2 (θ(j) ). . . . . . . . . . Plain DMST αref is the actual DMST α. . . . 26 2. . . . . . 4 1. .4 Characteristic power coefficient diagrams of wind turbines. . 22 2. . . . 40 3.LIST OF FIGURES Figure 1. because it is lower.1 Page Worldwide energy consumption in quadrillion of BTU. . . . 39 3. . . . . . . From EIA [8]. . . . . . . . . the value used would be the slope of the dotted line. . . . . . . . . . . In this example. . . . . . . . . . . . . .5 Angle of attack for different dynamic stall models. . 3 1. . . . .1 The two different values from which the CLdyn slope is calculated. . . .4 Angle of attack for different dynamic stall models. . . . . . . . . . . . . 45 1. . .1 Scheme of the flow velocities and forces from the blade point of view. . . . .3 Classification according to rotation axis. . . . . . . . . . . . . . . . 43 3. . . . . . . . . . . . . 13 2. .8 vii . . . . . . . . . . .3 3. . . . . . . . . . . . . 2 Worldwide net electricity generated in quadrillion of BTU. . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Scheme of the SST discretization of a VAWT domain. . .2 Effective angle of attack comparison between Kozak and Vallverd´ u results for T SR = 3 [10]. . . . .1 A. . . . 48 4. . . . . . . . . . . . . 76 viii . 53 Effective angle of attack for DMST with Berg and wake modification. . . .3. . . . 65 A. . . . . . . . . . . . . . . . . . . . . 71 C. . . . . 56 4. . . . . . .1 Lift and drag coefficients obtained from the Star-CCM+ simulation. . . . . . . . . . . . . . . . . . . . 67 B. . . . . . . . . . . . . . . . . . . . . . . . . .9 CP against T SR with Berg dynamic stall model. . . 46 3. . . . . . . 50 4.1 Visualization of the vorticity for T SR = 3 [9]. . . . . .5 5. . . . . .2 Scheme of the MST discretization of a VAWT domain.10 Real and effective angle of attack for Berg model. . . . . . . . . .3 4. . . . . . . . . . . . .1 Flowchart of the calculation of λ1 (θ(j) ) for the dynamic stall modification. . .2 Flowchart of the calculation of CP for the dynamic stall modification.6 Torque coefficient of one blade for T SR = 4 using Berg model. . . . . . 55 CP against T SR with Berg dynamic stall and with/without wake interaction models. 56 4. . . . . . . . . . 61 4. . . . . . . 75 C. . . 57 Former GUI used to interact with Vallverd´ u’s code. . . .7 CP against T SR with Berg dynamic stall and wake interaction models. . . . . 51 Flowchart of the calculation of CP for the dynamic stall and wake modification. . . . . . . . .4 4. . . a. j ∈ ℵ j Unit vector in the y axis direction K Delay parameter from Gormont’s dynamic stall model L Lift ix .a swept area Berg parameter Blade chord length CD Drag coefficient CF Force coefficient CL Lift coefficient CLα Linear lift coefficient slope CP Power coefficient CT Torque coefficient CTb Blade torque coefficient D Drag d Unit vector with the direction of drag dw Wakes length dS Differential of surface dV Differential of volume Fx Component of the force in the flow direction i Natural index for multiple purposes. i ∈ ℵ i Unit vector in the x axis direction j Natural index for multiple purposes.k.LIST OF SYMBOLS Symbol A AM c Definition Section normal to the flow. used in subscripts. too y Axis normal to the wind + Ywall Non-dimensional wall distance α Blade angle of attack α0 Angle of attack at zero lift x .l Unit vector with the direction of lift M Mach number n Unit vector normal to δΩ Nb Number of turbine blades Nth Number of divisions of the azimuthal angle Nw Number of wakes P Power R Turbine rotor radius Re Reynolds number rw Wakes influence ratio S Non-dimensional rate of change of the angle of attack T Torque t Time tc Thickness T SR st U∞ Tip-speed ratio Streamtube subscript Free stream flow velocity u Velocity of any perturbed flow x Axis parallel to the wind. α˙ αef f αp αref Rate of change of the angle of attack Effective angle of attack Angle of attack modified with pitch Reference angle of attack β Flight path angle with the instantaneous horizon tangent to the blade path ∆t Timestep between nodes ∆θ Angular increment between nodes δθ Angular width of any streamtube δΩ Boundary of the domain Ω θ Azimuthal angle θ˙ Angular velocity λ Interference factor ρ Flow density σ Solidity τ Unit vector tangent to the blade path ϕ Pitch Ω Domain xi . S.LIST OF ABBREVIATIONS Abbreviation AC BEM BEMT Meaning Alternating current Blade element method Blade element momentum theory BSD Berkeley Software Distribution CFD Computational fluid dynamics COG Center of gravity DC DMST Direct current Double-multiple streamtube model DNS Direct numerical simulation EIA U. Energy Information Administration GUI Graphical user interface HAWT Horizontal-axis wind turbine IIT Illinois Institute of Technology LES Large eddy simulation MST Multiple streamtube model RANS Reynolds-averaged Navier–Stokes SST Single streamtube model UPC Universitat Polit`ecnica de Catalunya VAWT Vertical-axis wind turbine xii . Therefore. compared to its fast convergence. BEMT is improved with dynamic stall and wake interaction modifications. such as wind turbines. A code is developed in order to apply the blade element momentum theory to these wind turbines. Vertical-axis wind turbines present very attractive performance characteristics but extremely complicated aerodynamics. in order to include more physical effects. BEMT is shown to give considerably reliable results. Comparisons with other authors prove that the generated code provides data that are close to the real results.ABSTRACT The expected exhaustion of fossil sources of energy requires the improvement of other renewable sources. research is still needed on this field. xiii . Modified BEMT models will help obtain fast and qualitatively reliable data on VAWT performance. Nowadays. Moreover. . will be exhausted in some decades.1 CHAPTER 1 INTRODUCTION 1. we were able to use the energy of the wind. including product manufacturing or any industrial procedure. most of the energy used nowadays comes from fossil origin – coal. Thanks to our planet’s self-rotation. the existence of atmosphere and the incidence on it of radiation energy coming form the sun. 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. natural gas and most of the liquid sources are fossil energy sources.2 shows a classification by source. However. its use is very harmful to all life species inhabiting this planet. Later on. when windmills were invented. This is a fact of our present time. As the reader can see. From all the energy used in electricity consumption. such as pumping water or smashing wheat [1]. From transportation to domestic life. Therefore. all aspects of human life require the use of some source of energy. but also to perform tasks that required titanic forces. earth has a phenomena called wind. Humans have used this as a source of energy since the first sailing boats were sent to explore new lands.1. our main source. not only to propel boats.1 Energy consumption Our world needs new energy sources. EIA [8] data show the worldwide energy consumption by type of energy source. Note that still half of the electricity is generated from fossil origin sources. see Figure 1. However. Wind consists of huge masses of air moving on the planets surface in different directions. oil. Figure 1. renewable energies still play a very insignificant role in the energy generation scene. this report focuses on the analysis and performance prediction of a specific renewable energy generation device: the vertical-axis wind turbines. Figure 1. This rotor is connected – normally with some gearing system – to an . From EIA [8]. They transfer the momentum of the wind to a rotor through a set of blades attached to it.1: Worldwide energy consumption in quadrillion of BTU. From EIA [8]. 1.2 Wind turbines Wind turbines are electrical power generation devices whose energy source is the wind.2 wind can be used to produce electricity by using wind turbines. Figure 1.2: Worldwide net electricity generated in quadrillion of BTU. HAWTs and VAWTs respectively. one can find horizontal-axis and vertical-axis wind turbines. A very important non-dimensional parameter that defines the behavior of a wind turbine is the tip-speed ratio. both of them respectively to the ground. A good quick guide to wind turbines can be found in [29]. Therefore. (a) HAWT (b) VAWT Figure 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. U∞ (1. Then the output is fed into the general electrical grid. As the name suggests. One of the main classifications of turbines is made according to the axis around which the rotor spins. Some wind turbines incorporated in closed circuits with low power requirements might use a DC generator directly.3: Classification according to rotation axis. T SR is defined here because it will be used along the whole report. HAWTs twirl about a horizontal axis and VAWTs do so around a vertical one. .3 electrical AC generator. T SR = θ˙ R . 2) This result is reached by using continuity. it is probably wrong. the maximum power coefficient that any wind turbine can give is CP max = 1 2 16 Pmax = = 0. the ideal maximum is the Betz limit. Figure 1. it applies also for VAWTs. . 2 27 ρ A U∞ (1. Therefore.4 1. If a simulation or measurement gives a higher power extraction from the flow than the one specified by Betz limit. Figure 1.593 .2.1 Maximum power coefficient According to Betz. momentum and energy conservation in any type of turbine – see [4] for the whole deduction.4 is a diagram containing the characteristic power coefficient outputs of different wind turbines against their tip-speed ratios.4: Characteristic power coefficient diagrams of wind turbines. Another aspect that sets both types of turbine apart is the proximity of the blades to the supporting tower and the axis of rotation. HAWTs’ maximum CP is higher than that of VAWTs. from the aerodynamics point of view. This eliminates. This causes each blade to pass near the turbine tower periodically.a. Observe how. at least. are placed at a constant distance from the tower. producing a deep periodic noise that has proven harmful to the psychological human health [21]. To begin with. the towerblade interaction noises.2 Comparison between HAWTs and VAWTs Although both horizontal.3 . VAWT blades.4.5 1. Hence.k. HAWTs tend to behave in a much more steady way than VAWTs do.and vertical-axis turbines use the same principles to work.2. 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. this has made HAWTs’ performance characteristics much easier to predict. This makes HAWTs center of gravity. Historically. a. In contrast. VAWTs still have a considerable path ahead before being competitive [27]. much less effort has been invested in VAWTs – first significant efforts were made by Sandia National Laboratories during the 70’s [27]. in Figure 1.2. nowadays they are in a very advanced optimization state in design terms – almost the only improvement achievable is due to geographic position. 1 See Section 1. 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. or at least reduces dramatically. in general1 . This characteristic makes VAWTs more suitable candidates for inhabited regions. HAWT blades extent radially directly from the axis of rotation. they present very relevant behavioral differences due to their physical ones. This is because the HAWT’s plane of rotation is normal to the flow direction. greater than the HAWT blades. nowadays they are not as cost-effective as HAWTs. considerable effort has to be invested in making this components as lightweight as possible.4. though. Finally. Therefore. have simpler designs. instead. Therefore. Vertical axis configuration. bear in mind that their state of development is much lower. or T SR are higher than these of VAWTs. VAWTs are multi-directional. HAWTs optimal rotation speed regimes. if further investigation states that active blade pitch control is needed to optimize VAWTS. higher than that of VAWTs. Related to this. according to this section. However. research is still needed in the field to prove whether VAWTs are worth investing in or not. while HAWTs must be oriented against the wind at any time. VAWTs seem to be more advantageous than HAWTs. This reduces construction and design costs. Drag-based wind turbines. . Although. also. Hence. Only the latter type of VAWT is potentially competitive to the common HAWTs – see Figure 1. specially birds and other flying animals. This influences sound generation too: the slower the turbine rotates. 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.6 COG. the lower are the noise levels it reaches. Having the gear at ground level decreases maintainance costs. 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. 1. allows the VAWT heavy machinery to be placed at ground level. too. according to the main force they use to extract momentum from the wind.2. Hence.3 Vertical-axis wind turbines Vertical-axis wind turbines can be classified. Hence. simplicity in the design will be lost. VAWTs may be considered safer in case of breakdown and friendlier to the surrounding fauna. 2R (1. Another important parameter to characterize an H-rotor VAWT is the solidity.6 –. σ= Nb c .5: Drag (Savonius) and Lift (Darrieus) VAWTs. which consists of a set of vertical blades rotating around a vertical axis.7 A very common lift-based VAWT is the H-rotor type – see Figure 1. (a) Savonius rotor (b) Darrieus rotor Figure 1.3) which is the ratio between the length of the chord of each blade. c. Nb . The solidity is a representation of the . times the number of blades. 2R. The design is fairly simple and they can be simulated using 2D aerodynamics with less error than other less simple designs.6: H-rotor Darrieus VAWT. over the turbine diameter. Figure 1. 3 Performance prediction of VAWTs Performance prediction is critical to study the feasibility of vertical-axis wind turbines. Larsen [11] was the first to apply this type of method to VAWTs.8 blockage effect of the VAWT against the wind. they take into account the physics of the problem more faithfully than the other CFD models. but at a higher computational cost. Therefore. One can find experimental methods. • Wind tunnel experiments consist on observing the flow around a wind turbine . – Grid-centered methods discretize the Navier–Stokes equations in the volume domain of the VAWT. Different CFD models present different characteristics: – BEM: blade element methods are computationally inexpensive. computationally speaking. One of the most recent applications of grid-centered methods to VAWTs is the one made by Kozak [9]. and computational simulations. 1. 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. However they rely on critical empirical data that must be obtained by some other method. which predict the flow and turbine behavior. First applied on fluid dynamics by Glauert [5] and to VAWTs by Templin [28]. several methods exist to do so. Hence. However. He used finite volume methods to perform a 2D simulation. They are very fast and easy to apply – they are useful for preanalyses and qualitative interpretations. which observe reality. they are also the most expensive methods. They rely on stall models. – Vortex models are more precise than BEM. These are very varied. They require no previous models. if properly applied. other analyses are needed to finally be able to build a prototype with some guarantee of success. but the techniques used to obtain flow data are very complicated and expensive. These results will show that BEMT. This report focuses on developing a BEM and momentum theory code (BEMT) [30] with MATLAB [14] and interpreting the obtained results. However. even though being based on a fairly simplistic model.9 situated in a controlled environment. . • Real prototypes can be used to obtain performance data over a period of time in field situations. proves itself quite useful. a. and momentum theory. Therefore. Streamtube models are based on blade element method. This also means that the streamtube section expansion due to flow energy loss and continuity . the governing equations must be in integral form. 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. Hence. Streamtube model is a relatively simple method. there is no mass. Each streamtube connects both inlet and outlet boundaries and is parallel to the velocity of the flow at all points. As the name suggests.10 CHAPTER 2 STREAMTUBE MODEL 2.a. to achieve this. As said. The main advantage of this method is its ability to solve the domain without an internal volume mesh. a.k. especially for lower tip-speed ratios.k. This means that all components of the velocity perpendicular to the streamtubes are neglected. momentum nor energy – neglecting thermal effects – exchange between adjacent streamtubes. BEM. BEMT applications for VAWT cases give a fair approximation of the VAWT power output. specially in a CFD case. the turbine is being treated as a unidirectional flow problem only. Therefore. therefore. BEMT. However. compared to finite volume or vortex modeling. they undergo very important simplifications. BEMT applied to VAWTs is referred to as a streamtube model. 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. The combination is called blade element momentum theory. a.1 Concepts This chapter develops and explains the streamtube model applied to performance prediction of vertical-axis wind turbines. the governing equations. 11 is neglected2 . 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. 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. 2.2.1 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 conservation3 , respectively d dt I Z Ω d dt ρ u · n dS = 0 , ρ dV + ∂Ω I Z ρ u (u · n) dS = ρ u dV + Ω (2.1) X Fext (2.2) 1 2 ρ u (u · n) dS = −P , 2 (2.3) ∂Ω and d dt Z 2 I ρ u dV + Ω ∂Ω where Ω is the domain where these equations are applied and n is a unit vector normal P to ∂Ω and pointing outwards. Fext 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 13 2.2.2 Physical definitions One needs to define the geometry of a VAWT blade parametrically. Figure 2.1 is used for this purpose. L θ˙ R u ur −α u D β τ θ y 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. From it, the relative velocity (ur ) of the wind from the blade reference point of view, ur = U∞ s u U∞ 2 + T SR2 + 2 u T SR cos θ , U∞ (2.4) can be computed. Also, the flight path angle,  β = arctan T SR sinθ u/U∞ + T SR cos θ  , (2.5) and the angle of attack,  α = modulus π+β−θ 2π  −π , (2.6) 14 will be needed. Once the angle of attack is known, the lift, CL , and drag, CD , static coefficients of the VAWT blade profile can be obtained from experimental data4 , 1 ρ c u2r CL 2 (2.7) 1 ρ c u2r CD . 2 (2.8) L= and D= 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, 1 P = 2π Z 2π θ˙ T dθ , (2.11) 0 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. 5. The flow is not perturbed by the turbine. the contribution of all front and rear half-cycle streamtubes is added together. so a different velocity of the flow can be calculated at any position of the blade. state e is too close to both states 1 and 2 to allow this hypothesis. State w: this is the wake state. in turn. the flow is considered to be far enough from both disks. It acts as the input state for Disk 1. The flow is perturbed by both Disks 1 and 2. .15 2. Figure 2.2 illustrates the DMST scheme. However.3 Double-multiple streamtube model The most precise streamtube model is DMST. This is an approximation. 4. 3. at the same time. State e acts as the output state for Disk 1 and as the input state for Disk 2. In reality. State e: this is the equilibrium state. State 1: this is the state of the flow when interacting with the upwind actuator disk. Moreover. Then. State ∞: this is the free stream state. the turbine domain is discretized in multiple streamtubes parallel to the flow. hence it is steady. Observing Figure 2. as it is a method that seeks very fast convergence through simple modeling. State 2: this is the state of the flow when interacting with the downwind actuator disk.2.2. The flow is considered to travel through two consecutive actuator disks which. 2. DMST uses this assumption anyway. because it allows to consider the energy losses of the flow separately for the front and rear half of the VAWT. extract energy from it. also called the rear half-cycle of the turbine. one can distinguish five different states of the flow: 1. In this state. also called the front half-cycle of the turbine. It acts as the output state for Disk 2. 2π) .2: Scheme of the DMST discretization of a VAWT domain. x u1 U∞ θ˙ R Ω1. π) . st y One streamtube x Figure 2. (2. Equation 2.16 Disk 2 Disk 1 u1 U∞ ue Only 1D Upwind zone F1. conservation equations can be applied. st A2. 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 A1. A2.12b) where θst is the value of θ the blade has when it is going through the middle point of the given streamtube. Now. (2.st = R δθ (− sin θst ) .x u2 uw θ A1. st u2 δθ ue uw Downwind zone F 2.12a) ∀ θst ∈ (π.1 mathematically expresses the fact that there is no mass exchange between streamtubes: each streamtube .2. st Ω2. ∀ θst ∈ (0. According to Figure 2.st = R δθ sin θst . the velocities6 of the equilibrium and wake states.2.17) and that of the rear half-cycle. (2. for each streamtube with section Ai. 5 Note that. (2. as noted previously in this chapter.st = ρ u2 A2.14 and 2. (2.17 mass flow (m ˙ st ) is constant.13) Equation 2.14b) Similarly.16b) u2 = (2 λ1 − 1) λ2 . once evaluated in one front and one back streamtube in the direction parallel to the flow.13). . gives the expressions5 m ˙ (ue − U∞ ) = −F1. 2 (2.x . from Equation 2.15a) (2.15b) Using the interference factor: λ1 = u1 /U∞ and λ2 = u2 /ue . flow velocity should be constant. as they all refer only to a single streamtube 6 Note that flow velocity expressions violate continuity (Equation 2. (2. (2.x . from now on.st = ct.15. In an incompressible flow. one obtains 1 2 m ˙ (u2e − U∞ ) = −F1. are found.x u2 .16a) uw = ue (2 λ2 − 1) . ue = U∞ (2 λ1 − 1) . and combining Equations 2. Therefore. 2 1 m ˙ (u2w − u2e ) = −F2.st : m ˙ = ρ u1 A1. the subscript st is omitted on all forces and velocities. by considering that there is no streamtube expansion and no spanwise components of the velocity.3.x u1 . U∞ (2.14a) m ˙ (uw − ue ) = −F2. x (θ) dθ = θi.x = F2.x = 4 λi (1 − λi ) .st +δθ/2 Fi. 2 .    4 λi (1 − λi ) : λi ≥ 43/60 (2. the other expression for the thrust coefficient at each streamtube. 2 .12.r (CD cos β − CL sin β) .     1849/900 − 26 λi /15 : λi < 43/60 CFi. the thrust coefficient expression from the conservation analysis. Equation 2. (2.r (CD cos β − CL sin β) .20 and 2. together with Equations 2.18 2 Using the thrust coefficient expression – CF1. and CF2.st U∞ ). 2 .20) is needed. This expression is obtained from load analysis.x = .19) according to experimental data. 2 ∀i = 1.x Nb = 2π Z θi.16. ∀i = 1. As shown in Equation 2. Glauert [5] proposed a linear modification to this result.x /(1/2 ρ A1. for the front half-cycle. 2.x /(1/2 ρ A2. 2 . 2 .st ) . To be able to solve the interference factor.22b) . 2π ∀i = 1.21.x = CF2. However. (2. 2 (2 λ − 1) π sin θ U∞ 1 CF1. Fi.6. 2 π sin θ U∞ u22.st u2e ).22a) ∀i = 1.18) is obtained.x (θi.x are obtained. (2. ∀i = 1. (2.r −σ = (CD cos β − CL sin β) .14 and 2. ∀i = 1.9. σ u21. Then.x (θ) = 1 ρ c u2i. due to the inadequacy of the unidirectional flow assumption below λ ≈ 0. using the thrust coefficient expressions.13. 2. CFi. This results in Fi.21) The reader will note that this integral is solved with a first order approximation.st −δθ/2 Nb δθ Fi. another expression for the thrust at any given azimuthal angle. Then. (2. lift and drag coefficients can be used to obtain the instantaneous thrust each blade receives. 2 . for the rear one – and combining Equations 2.20 is time-averaged along one period in one single streamtube of width δθ for all the Nb blades of the turbine.x = F1. From the interference factor. etc. Also see that this system can be solved for any value of θ – that is why the interference factors are using dependence notation (θ). CP = P/(1/2 ρ A U∞ ).10.22b is λ1 (2π − θ) . Nb ]. ∀i ∈ [1. so there are an infinite number of streamtubes. bi . the power coefficient that all blades produce. CT (θ) = T (θ)/ (1/2 ρ A U∞ Note that the swept area is A = 2R because this is a 2D case. note that the value of λ1 inserted in Equation 2.1 + CP. Then. 7 (2.2. 3 Finally. Then. λ1 (θ) and λ2 (θ) can both be computed – note that λ1 7 participates in Equation 2. b1 . one should add the instantaneous torque of each blade. Remember that δθ → 0. velocities. To obtain the whole instantaneous torque coefficient using turbine parameters.22.) can be obtained using the equations shown in Section 2. distributed equidistantly. the first order approximation introduced in the integral of Equation 2.25) For θ > π. it will be limited by the algorithms used to solve this system and to integrate the torque along the blade path. all variables (β(θ).24) CTbi θ + Nb i=1 3 Note that for any Nb number of blades. 2 R 2 1/2 ρ c U∞ U∞ (2. is CP = CP.2 . force coefficients. so the flow going through the front half-cycle must be solved before it can be solved in the rear half-cycle.19 and 2. α(θ). Also.22b.23) according to Equation 2.   Nb σ X i−1 CT (θ) = 2π . the whole torque coefficient has a periodicity of 2π/Nb . would be subject to the total torque coefficient. needed to average the output power of the turbine.19 By using Equations 2. each blade non-dimensional torque at any given azimuthal angle is CTb (θ) = u2r Tb (θ) = (CL l + CD d) · τ . Then a turbine with Nb equidistant blades. too. 2 R). please notice that θ is the azimuthal angle of the first blade. (2.2. Instead.21 does not limit the order of precision of the whole problem. This method. it neglects several effects that gain great relevance in the case of wind turbines.27) . Therefore. a discretization in the θ domain. The results described in this report have been obtained by using a second-order trapezoidal integration.20 where CP. the 1D simplification introduces some error by neglecting span-wise velocity and 3D effects on the blade.back = 2π π (2.26b) are obtained by averaging the instantaneous power in each half-cycle.2 ≡ CP. Also.22 is nonlinear. 2π 0 Z σ T SR 2π CTb dθ . according to Equation 2. θ = 2       (j) (1)  θ = θ + (j − 1) ∆θ  (2.26a) (2.11. another numerical procedure is required to solve it. numerical integration is necessary to obtain the performance characteristics of the VAWT. although requiring some numerical calculations to solve the interference factor equation and to numerically integrate the torque to obtain the power. Z σ T SR π CP.front = CTb dθ .1 . is very straightforward and easy to implement. CP. 2. so no analytical integration of the torque – in order to get the power – can be computed either. 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. However.3 Computational algorithm No analytical solution of λ(θ) exists.1 and CP. Also. Therefore. as the resulting equation from combining Equations 2.1 ≡ CP. 2Nth ] .      ∆θ = Nπth     ∆θ (1) ∀j ∈ [1.19 and 2. The most significant are the effect of the blades working in angles of attack beyond the static stall angle. 2. Afterwards. The produced code is object-oriented to ease the addition of features in a modular way. It also includes an example file to show how to use it. To do so. Also. This code used Subashki et al. Figures 2. 2Nth ]. MATLAB [14] is the tool that has been used to implement this algorithm and to plot all the results. interpolation or bisection methods are used. the trapezoidal integration is applied in the whole discretized domain. it is parametrized. so it can be used to obtain results for different input values and turbines swiftly. It consists in Nth volume-centered θ(j) smaller domains for each half of the turbine – in other words. Interference factor λ is calculated individually for each value of j.21 was required. Note that doubling Nth means quadrupling the number of discretization nodes θ(j) . depending on each iteration outputs.3.5 show the algorithm used. The code is published in Mathworks File Exchange website [30] with a BSD license. [26] MATHEMATICA [13] code as a base.4 and 2. . j ∈ [1. θ(j) Start with j = 1 Suppose λ1 Compute β1 . u1.22 Start T SR. from Eq.22a Compute λ1 . 2. CD.19 Convergence criteria no Apply bisection or interpolation met? yes j = Nth ? no j = j+1 yes λ1 (θ(j) ) Equilibrium Figure 2.3: Flowchart of the calculation of λ1 (θ(j) ). . 2.r Compute α1 Compute CL.x from Eq.1 .1 Compute CF1. θ(j) . 2. u2. .x from Eq.r Compute α2 Compute CL. CD.2 . λ1 (θ(j) ) Start with j = Nth + 1 Suppose λ2 Compute β2 . from Eq.22a Compute λ2 .19 Convergence criteria no Apply bisection or interpolation met? yes j = 2Nth ? no j = j+1 yes λ2 (θ(j) ) Postprocess Figure 2. 2.2 Compute CF2.4: Flowchart of the calculation of λ2 (θ(j) ).23 Equilibrium T SR. CD (θ(j) ) Compute CTb (θ(j) ) Compute CP 1 . λ2 (θ(j) ) Compute β(θ(j) ). . CP 2 Compute CP using 2nd order trapezoidal integration CP Stop Figure 2. θ(j) . ur (θ(j) ) Compute α(θ(j) ) Compute CL (θ(j) ). λ1 (θ(j) ).5: Flowchart of the calculation of CP .24 Postprocess T SR. when the same flow reaches the back half.7. The characteristic static lift and drag curves for the profile NACA 0021 at the specified Reynolds number have been obtained according to Appendix B.1: Wind turbine characteristic values VAWT characteristics Nb Re (blade) Blades profile 3 3 · 105 NACA 0021 R 515 mm c 85.6. The discretization parameter is Nth = 51. where the balance front/back ex8 This phenomena becomes very relevant when adding wake interaction modifi- . about 2. there is a value of T SR. The curve of power coefficient against tip-speed ratio obtained for the presented case is shown in Figure 2.4 Results This section discusses the results obtained from applying DMST to a Darrieus H-rotor wind turbine with the following characteristics: Table 2.25 2. Hence. the front half-cycle extracts so much energy than the rear half-cycle is actually giving some of it back to the flow. the one that proved to give enough precision at lowest computational cost. it carries much less energy. above values of T SR a bit higher than 3. the front half of the turbine extracts more energy from the flow.2499 This case is chosen because there are enough data in the literature with which to compare results [9]. instead of extracting it8 . However as the T SR increases. 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. Therefore.8 mm σ 0. so the back half of the turbine cannot extract as much energy Furthermore. the bigger the solidity is.2 0. Although DMST is a reasonably crude model. decreasing when solidity either increases or decreases.3 1. 2. it is discussed further in Chapter 4. Therefore. The maximum achievable power coefficient seems to be maximum for the initial case.1 0 −0. this characteristic is represented well enough. cations to DMST. the lower is the optimal tip-speed ratio. However. where wake interactions are covered.5 3 3.5 4 4. the purpose of this subsection is to see the effect of solidity only.3 CP 0.7.4.5 5 TSR Figure 2.1 C P −0.5 0. C against TSR P 0. The results can be seen in Figure 2. .front CP.26 traction gives the optimum power coefficient.1 Effect of solidity To appreciate the effect of solidity on the output power. the Reynolds number of the blade would change too.5 2 2. According to the DMST model.back −0.2 C P.4 0. the DMST simulation has been run with the same turbine only changing the solidity value.6: CP against T SR. by modifying the diameter – if the chord of each blade is changed. 2499 σ = 0.5 4 4. However.1 −0.2 0. as σ increases.5 3 3.5 5 TSR (b) Power coefficient on the front half- (c) Power coefficient on the back half- cycle. Also note that.1 −0.1 σ = 0.5 0.3 0.5 2 2.5 2 2.05 1.5 4 4. Figure 2.7: Angle of attack for different dynamic stall models. Therefore.5 1.1 1.35 σ = 0.1 0.2 −0.5 5 TSR 2 2.4 0 −0.05 σ = 0.5 4 4.3 0.27 CP against TSR 0.05 −0.6 0.2499 σ = 0. which is an effect that gains relevance with solidity growth.25 0 0.5 3 3.05 σ = 0.3 CPback CPfront 0. the front part of the turbine produces much . CPfront against TSR CPback against TSR 0.45 0.4 0. remember that DMST neglects wake interactions.4 CP 0.4 0 −0.15 0.5 3 3. cycle.1 σ = 0.5 −0.2 0. it might be that the optimal solidity was lower than the one suggested by DMST.4 0.5 5 TSR (a) Power coefficient on the whole turbine.2 0.1 σ = 0. there is not much difference. From the computational point of view. note that according to DMST. Moreover. which is produced. 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. by wake generation. Multiple and Double-Multiple Streamtube models are compared. 2. It is true that DMST must solve two different values of λ for each one of MST.3 Comparison with other Streamtube models As said before. the more energy they can extract. for the values of T SR that the back half of the turbine produces negative power. However. The main difference. is that DMST is able to distinguish the different behavior of the flow in the front and the rear half of the turbine.4. it has less energy to be extracted. 2. DMST performs even better than MST in general because the latter has . However. Hence. because λ2 depends of λ1 .4.2 Effect of the number of blades The effect of the number of blades only means a change in solidity. because both models solve the interference factor at many different values of θ. it loses more energy if the solidity is higher. because the larger the blades are respectively to the domain. two turbines with different number of blades but same solidity and blade Reynolds number would behave identically. Hence. when it reaches the rear part. the T SR for which the rear half-cycle produces zero power keeps decreasing as solidity increases. an increase of solidity – which means bigger blades relatively to the diameter – implies that more energy is given back to the flow. of course. First. in fact. This is logical. When the flow first meets the turbine. This is noticed by observing the instantaneous torque for each model at any azimuthal angle. This is because DMST is not taking into account interaction between blades.28 more power and the rear much less. DMST is the most precise version fo the three Streamtube models. However. all three have been tested. because the integration steps level the difference in torque. the output torque is not only mirrored from front to back halve. 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. DMST is worthier using because it overtakes MST both in precision and convergence speed. Hence. In this case. 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. . centered at 90◦ and 270◦ . globally. When testing SST. Therefore. the precision of this method is even worse. because the interference factor is solved only once in the whole turbine. respectively. Nonetheless. However. Morover. as said before. SST is only worth it using if only very low-end computational resources are available. the generation of the code of both models is very similar. It can be useful as a very preliminary tool to get an idea of the turbine performance. convergence is reached much faster than both DMST or MST.29 more problems to reach convergence. DMST is not that complicated and slow. but it is also symmetric in each half. The final power output is less different than that of the torque. as expected. In this latter case. Therefore. static lift and drag curves have been used. presents a hysteresis behavior. In other words. 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. so that it includes these effects. the induced velocity. angles of attack are lower.1. which leads to smaller oscillations of the blade respectively to the relative movement of the flow. Such an effect is only dominant at low T SR. Therefore. applying a dynamic stall model to the DMST method means computing some additional steps interlaced with the main algorithm. a lot of effort has been invested into developing modifications to the original DMST. Dynamic stall. Therefore.1 Concepts As mentioned in the previous chapter. However. component of the relative velocity.30 CHAPTER 3 DYNAMIC STALL MODELS 3. . opposite to the usual static stall observed in wind tunnels – and in fixed blades simulations –. according to Equation 2.6. 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. is much higher than the wind velocity component. if a blade changes its position. but it adapts gradually to the new state within a definite interval of time. ur . U∞ – see Figure 2. Most of the dynamic stall models. the flow does not react immediately. dynamic stall plays a very relevant role in the VAWT problem. In high T SR cases this effect ceases to be determinant – wake interactions gain strength and become the major factor. according to [18]. θ˙ R. Obviously there is significant amount of error introduced by this fact. 31 3. so it is negative when the angle of attack approaches 0 and positive when it moves away from 0. so the upwind flow changes its state more easily. but it has been used as a very strong base for later adaptations.5 : α˙ < 0 is a parameter defined according to Gormont’s empirical observations. These are: 3. several dynamic stall models are explained and compared. This effect generates a discontinuity that can induce severe instabilities to the flow surrounding the points where α˙ changes sign9 . the flow near the leading edge of the blade is pushed to the surface of it. because they operate at much higher angles of attack.2.δα. the resulting reference angle of attack is αref = α − K δα . One possible qualitative explanation would be the following. making it more difficult for the upwind flow to change its state. It is not optimal for VAWTs. Therefore. when the blade angle of attack is decreasing.2)    −0. The Gormont method consists of applying a certain delay. on the angle of attack.2 Theory development In this section. K=     1 : α˙ ≥ 0 . After applying this delay. (3.1 Gormont model Gormont [6] proposed this method to take into account dynamic stall in helicopter blades. However. When the angle of attack is increasing. . the delay experienced when the absolute angle of attack is increasing is twice the one observed when it is decreasing. (3. based on hysteresis behavior.1) where K. this same flow is left behind. 9 α˙ is the rate of change of the absolute value of α. s.32 The term δα is proportional to the non-dimensional rate parameter S. . . c α˙ . . . . S= . 3) 2ur . (3. 5) . The symbol α0 refers to the angle of attack at zero lift. the mathematical expression for m can be found in Appendix C. is obtained simply by using αref to extract it from the available static data. it compares the blade movement to the flow relative movement around it. The higher S is. CLdyn = CL. m. Also. 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. The drag coefficient.1 – dynamic stall is important for low tip-speed ratios. dyn CD = CD (αref ) . the modified lift and drag coefficients can be calculated. Figure 3.0 (α0 ) + m (α − α0 ) . Then. The specific expression for δα can be found Appendix C. when α oscillations are bigger. (3.4) is calculated by using the potential equation with a modified slope. αref . is found. the smaller is the delay on the angle of attack. the minimum value of either the slope of the linear part of CL or the value obtained with the following expression: (CL (αref ) − CL (α0 ))/(αref − α0 ). the thicker the blade is and the less compressible the flow is. the greater the applied delay. Once the delayed angle of attack. (3. 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.1 shows both mentioned slopes for an example value of αref . This is coherent with what was explained in Section 3. In other words. The lift coefficient. He proposed the following modification for both lift. some VAWTs show a better response when using AM → ∞.6) : α > AM αss and drag10 . The results shown 10 Only the lift modification is shown.1: The two different values from which the CLdyn slope is calculated. so that both coefficients tend to the static values for very high values of α. Note that this is the same as using the pure Gormont method with no modifications. CLmod =  i h  −α   CL + AAMααss−α (CLdyn − CL ) : α ≤ AM αss ss M ss    CL . . See Appendix C for the drag expression.2 0 αref = 17. deep into the non-linear regime. The parameter AM is set by the user of the method. In this example.4 0. The drag modification is the same expression but changing all L subscripts for D . Therefore. because it is lower.33 CL 1. According to [3] the best option is AM = 6. according to [18]. Berg theorized that the pure Gormont method might over predict the lift and drag for high angles of attack.2 alternate CL 1 CL 0.6 0.5o 0 5 10 15 α (deg) 20 25 30 Figure 3. the value used would be the slope of the dotted line.4 linear CL 1. However. 3.8 0.2 Berg modification Berg [3] considered that the α range at which helicopter blades work is much lower than that of a VAWT.2. which can perfectly go beyond the stall angle of attack (αss ). (3. they considered that in this region .3. adaptation Strickland et al. Also.7) where K and S are described in Equations 3.2 in Appendix C.1.8) and drag. Then. 3. (3. Expression for γ can be found in Appendix C. the resulting reference angle of attack is: αref = α − γ K S Sα˙ .     CL (α) : α < αss str CL =     α  CL (αref ) : α ≥ αss αref −α0 .2. 225◦ ] is mainly dominated by turbulent aerodynamics and wake interactions. 3. Therefore. However. str CD =     CD (α) : α < αss .9)    CD (αref ) : α ≥ αss coefficients are obtained.4 Paraschivoiu adaptation Paraschivoiu [17].2.2 and 3. lift. That is why they modified the delay so Sc = 0. concluded that the part of the blade path comprised in θ ∈ [105◦ . See Equation C. α ≥ αss . From Strickland’s angle of attack. VAWTs typically use blades with higher thickness over chord ratio. after some experimental observation. as the typical helicopter blade. Finally.34 on this paper are computed using Berg’s suggestion. (3. (3. Sα˙ is the sign of α˙ as is described in Section 3. as the typical rotation speed of a VAWT is much lower than that of a helicopter. [25] considered that Gormont method was developed for very thin blades.3 Strickland et al. the flow was considered to be incompressible – the Mach number is not used as an input parameter of the model. Strickland et al. respectively.2. supposed that dynamic stall effects only occur after the angle of attack is higher than the stall angle of attack. This method.2. Second. for qualitative purposes the difference is not very high compared to Berg adaptation. for example. Also. [19]. 3. This adaptation might be a bit crude. when the angle of attack is far beyond the stall angle of attack. The first step consists on calculating the attached flow unsteady behavior by using potential flow theory and the addition of unsteady empirical correlations. their modification basically consists of applying Strickland adaptation only outside this azimuthal angle range. These indicial functions are of the form of summations of exponential functions with different coefficients and characteristic time parameters. . It was first introduced by Beddoes and Leishman [2] and first applied to VAWTs by Paraschivoiu et al. however.35 dynamic stall can be neglected.5 Indicial method and adaptations Another different approach to dynamic stall modeling is the indicial method. According to [18]. the indicial method gives an overall better approximation of the dynamic stall models than the Gormont based ones. an optional third phase introduces some corrections of the separation for deep stall. Finally. it considerably increases the computational requirements. All these effects are supposed to have indicial behavior. All modifications depend also on the rate of change over time of the angle of attack. Therefore. However. depends on several coefficients – mainly the lag and impulsive parameters – that must be determined empirically and/or by potential flow approximations. because it is based on experimental data from one specific turbine under some very particular conditions. 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. which DMST must apply at every iteration of the interference factor calculation. For all these reasons. This means that they grow or decrease with time according to lag and impulsive functions. as did the Gormont based models. because the introduced modifications must be solved by iterative methods. separation in the leading and trailing edge are taken into account. as suggested in the previous section. α(θ(j−1) ) is substituted by . observe that this algorithm works with the angle of attack in absolute value and.2. Also note that α˙ is calculated with first order approximation. such as Hansen [7]. ˙ 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.2.3.5 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. α.4 and 2. 2. Note that this produces the same result and is much easier to implement in a computational code.3 Computational algorithm modifications In order to insert any of the described dynamic stall models into the DMST algorithm proposed in Section 2. it is ∆t = Also. CD box of each Figure 2.3. three additional steps must be introduced. the rate of change of the angle of attack. Basically. afterwards. using the value of α of the previous azimuthal angle. Hence. Compute CL .36 the indicial method is not explained nor was developed during the redaction of this report. Instead. Further development on streamtube models must include the development of DMST code with Indicial dynamic model. As seen in Figure 3. In θ(1) and θ(Nth +1) . θ˙ (3. 3. not only Gor- mont’s. When working with α < 0. The rest of the algorithm can be found in Appendix C. More information can be found in [18]. Several adaptations of the original Beddoes and Leishman model have been applied to wind turbines in general.10) refers to any dynamic stall model. note that the superscript dyn ∆θ . it changes the sign of the lift if the blade is working at α < 0 – note that Sα is the sign of α. all signs should be switched. In the modified flowcharts. ∆t is the time that the blade needs to travel an arch of amplitude ∆θ. θ(j) Start with j = 1 Suppose λ1 Compute β1 .37 Start T SR. u1.r . . .  . α˙ 1 = . Compute α1 . α1 (θ(j) ). − . α1 (θ(j−1) ). 22a Compute λ1 .2: Flowchart of the calculation of λ1 (θ(j) ) for the dynamic stall modification.1 (|α1 | .1 = Sα CL. CL. α˙ 1 ) Compute CF1. from Eq. α˙ 1 ).x from Eq.19 Convergence criteria no Apply bisection or interpolation met? yes j = Nth ? no j = j+1 yes λ1 (θ(j) ) Equilibrium Figure 3. . 2.1 (|α1 | . /∆t dyn dyn dyn Compute CL. 2. basically. However. The result can be seen in Figure 3.4 Results First. dynamic stall models predict that the time that the blade is experiencing stall is less than what a quasi-static approach gives.4. it is useful to identify non-linear zones of the blade path by comparing it with the actual α – if both angles are equal. Dynamic stall modification is included in Vallverd´ u’s code [30]. all models will be compared. 3. as explained in the theory. Gormont over-predicts performance relatively to Berg. thanks to the delay introduced by the hysteresis effect. it means that the blade is working in the linear regime at that point of its path. This is due to the stall happening slower in a dynamic regime than a static one. observe that for T SR lower than 3 all stall models predict better performance than the original DMST.6 is not violated.1. The reader can see that. Next. Lets begin with the power coefficient output for the initial case proposed in Table 2. which is the effective angle of attack.1. the lift characteristic reference angle of attack and blade torque coeffi- . in Appendix B.11) where CLα is the slope of the linear part of the lift coefficient curve for the NACA 0021 profile.38 −α(θ(j) ). so that Equation 2. To do so.1 Comparison of different models As said. It can be found in Equation B. 180◦ to be zero. Therefore. once the best dynamic stall model is chosen. the stall effect itself will be studied. 3. αef f . this section is mainly aimed at comparing the different stall models with each other and with the plain DMST model. Afterwards. Also. αef f is.3. CLα (3. a normalization of the lift. a new variable has been introduced. This assumption forces the angle of attack at θ = 0◦ . αef f = CL . note that Strickland-based models predict .4 0. because the torque coefficient directly depends on the lift and drag coefficients.35 plain DMST Gormont Berg AM = 6 0. The lift characteristic reference angle of attack for all models curves show how the angle of attack is delayed to consider dynamic stall.3: Comparison of the power coefficient outputs for different dynamic stall models.5 TSR Figure 3. Also. Hence.6 is a way to see the second one.5c. according to the DMST theory. These two are the representation of the dynamic stall modifications.5 and 3.5 3 3. Figure 3. it shows very well how this delay is greater when the angle of attack is increasing than when it is decreasing.39 C against TSR P 0. specially in Figure 3.45 CP 0.5 shows the first modification and 3. Also. cient at each azimuthal angle have been obtained – see Figures 3.3 Strickland Paraschivoiu 0.25 2 2.6. 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. 40 α (deg) for TSR = 2. This is observed more closely in the torque curves.5 15 10 5 0 −5 plain DMST Gormont Berg A = 6 −10 M Strickland Paraschivoiu −15 −20 0 50 100 150 200 θ (deg) 250 300 350 400 (a) T SR = 2. Remember from the theory development that Berg and Gormont αref are the . a less radical delay than Gormont/Berg models11 . The loss of torque according to 11 same.5 α (deg) for TSR = 2 α (deg) for TSR = 3 30 10 20 5 10 0 0 −5 −10 −10 −20 −30 0 50 100 150 200 θ (deg) 250 (b) T SR = 2 300 350 400 −15 0 50 100 150 200 θ (deg) 250 300 350 400 (c) T SR = 3 Figure 3. where the changes of sign of α˙ affect the Gormont/Berg curves to a higher extent than the Strickland based ones.4: Angle of attack for different dynamic stall models. Because the Strickland model was the first and most crude adaptation to VAWTs and the Gormont model was developed mainly for helicopters. This is due to Gormont’s observation explained in Section 3. It is included. 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. Please observe how all dynamic stall models represent the dynamic delays by giving different lift at a same angle of attack.7 shows the lift coefficient at any angle of attack. no dynamic model was applied. therefore. causing a sudden increase of drag that brings torque down. all other figures show the same behavior of the Paraschivoiu curves. to show the hysteresis effect explained in the previous Section. However.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. When its absolute value is increasing. Also see that Paraschivoiu adaptation result does not present hysteresis in some regions – in general. it presents the advantage of being parameterizable by varying AM . Streamtube models are aimed at simple and quick qualitative studies. indicial mehtods seem to be very good at dynamic stall simulation.2. when it is decreasing. they add computational difficulties much higher than those inherent to Berg adaptation. Finally. the dynamic lift adapts faster to the static value. αss ≈ 10◦ .1: the flow adapts swifter when the blade angle of attack is decreasing than when it is increasing. A said in the previous section. so it was considered that the overall computational efficiency of the Berg model is more suitable for a DMST model than . the dynamic lift does not sense stall until a higher angle of attack than the static angle of attack. Also. Berg results are more similar to experimental measurements [18] and finite volume CFD results [9] than the other dynamic stall models. mainly. However. These are the regions where turbulence was considered dominant and. the Berg model is the one chosen to treat dynamic stall in this text. 42 αref (deg) for TSR = 2. Plain DMST αref is the actual DMST α.5 αref (deg) for TSR = 2 for lift characteristic αref (deg) for TSR = 3 for lift characteristic 30 10 20 5 10 0 0 −5 −10 −10 −20 −30 0 50 100 150 200 θ (deg) 250 300 (b) T SR = 2 350 400 −15 0 50 100 150 200 θ (deg) 250 300 350 400 (c) T SR = 3 Figure 3. According to these .8.5 for lift characteristic 15 10 5 0 −5 −10 plain DMST Gormont Berg A = 6 −15 M −20 Strickland Paraschivoiu −25 0 50 100 150 200 θ (deg) 250 300 350 400 (a) T SR = 2. the one of the indicial models. This is done by using the effective angle of attack from Figure 3.5: Angle of attack for different dynamic stall models. Finally. this chapter contrasts Berg results with the plain DMST results. 5 b 2.5 0 0 −0.5 Strickland Paraschivoiu 1 0.43 CT for TSR = 2.5 0.6: Blade torque coefficient for different dynamic stall models.5 1.5 0 50 100 150 200 θ (deg) 250 300 350 400 (a) T SR = 2. .5 2 plain DMST Gormont Berg AM = 6 1. the dynamic stall model delays the overall effective performance of the blade by smoothening the sudden changes of the quasi-static approach.5 2.5 0 50 (b) T SR = 2 100 150 200 θ (deg) 250 300 350 400 (c) T SR = 3 Figure 3. curves. 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.5 1 1 0.5 0 −0.5 2 2 1.5 CT for TSR = 2 CT for TSR = 3 b b 2.5 −1 0 50 100 150 200 θ (deg) 250 300 350 400 −0. a presents a noticeable improvement in performance for T SR lower than 3 and a less detectable exacerbation of the power output for higher .5 0.8 0.5 CL for TSR = 3 1. The front half.5 5 10 15 −1.9.5 −30 −20 −10 0 α 10 20 30 (a) T SR = 2 CL for TSR = 2.5 CL CL 0 0 −0.4 −0.4 0. When comparing the power between plain DMST. instead.5 0 −0.6 −0.5 −0. Figure 2.5 1 CL 0.2 −0.2 0.6 and Berg modification.7: Blade lift coefficient for different dynamic stall models against angle of attack.2 −15 −10 −5 0 5 10 α (c) T SR = 3 Figure 3.5 plain DMST Gormont Berg AM = 6 −1 Strickland Paraschivoiu −1.6 1 0.44 C for TSR = 2 L 1. one sees that differences in the back half performance are very little.5 −20 −15 −10 −5 0 α (b) T SR = 2.8 −1 −1 −1. Figure 3. because it predicts that stall comes later than in a quasi-static evolution.8: Effective angle of attack for plain DMST and Berg model.5 αeff (deg) for TSR = 2 αeff (deg) for TSR = 3 15 10 10 5 5 0 0 −5 −5 −10 −10 −15 0 50 100 150 200 θ (deg) 250 (b) T SR = 2 300 350 400 −15 0 50 100 150 200 θ (deg) 250 300 350 400 (c) T SR = 3 Figure 3. because when the turbine is working at this range . but it disappears faster than it appears. values.5 15 10 5 0 plain DMST Berg AM = 6 −5 −10 −15 0 50 100 150 200 θ (deg) 250 300 350 400 (a) T SR = 2. On the whole. The lower power predicted for high values of T SR can be neglected and will not be commented further.45 αeff (deg) for TSR = 2. 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. 2 CP 0.back 2 2.5 0.5 3 3.4 0.2 −0.4 1.front CP.5 TSR Figure 3.9: CP against T SR with Berg dynamic stall model.3 −0.3 0.5 CP CP. dynamic stall looses dominance to give it to wake interactions [9].1 −0.46 of T SR. 5 .1 0 −0.5 4 4. C against TSR P 0. The reference angle of attack has also been included in Figure 3.2 Stall study using DMST with Berg adaptation Figure 3.4. See that for low tip-speed ratios. according to Figure 3.9 are extracting a similar amount of energy from the flow. most of the stall occurs evenly both in the rear and front halves of the turbine. 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. the front half starts to extract all the energy from the flow. However. so it reaches the rear half with very slow velocities. .10 to give an idea of the delay introduced by the constant motion of the blade. as T SR grows. so stall always refer to dynamic stall. Hence. the blade in the rear half does not reach stall regime. which means that the interference factor of both halves is high enough.10 is used.47 3. mainly. 12 This section only works with Berg addition to DMST. Both parts. the proper type of stall of oscillating aerodynamic problems. Therefore. to detect the regions where the blade is working under stall12 regime. which produces a huge fall in the angle of attack. 10: Real and effective angle of attack for Berg model.5 α (deg) for TSR = 2. Berg model 30 10 20 5 10 0 0 −5 −10 −10 −20 −30 0 50 100 150 200 θ (deg) 250 (b) T SR = 2 300 350 400 −15 0 50 100 150 200 θ (deg) 250 300 350 (c) T SR = 3 Figure 3. Berg model 15 10 5 0 −5 −10 −15 α αeff −20 α ref −25 0 50 100 150 200 θ (deg) 250 300 350 400 (a) T SR = 2. Berg model α (deg) for TSR = 3.5. 400 .48 α (deg) for TSR = 2. 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. Kozak and Vallverd´ u et al.8 – with Kozak’s own results shown on [9]. From this comparison. BEMT’s main advantage. 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. [10] compared the effective angle of attack obtained from Vallverd´ u’s code [30] with dynamic stall – the green curve in Figure 3. . Kozak [9] studied the turbine described on Table 2.1 Concepts Wake interaction is the effect of the turbulent wakes generated by the blades on the front half received on the back half. so wake interaction is neglected in streamtube models. However. is computational efficiency. BEMT is unable to model turbulence. actually. the problem would become a mixture of BEMT and a grid-centered algorithm.49 CHAPTER 4 WAKE INTERACTION 4. a workaround was needed.1 with a grid-centered method. He used STARCCM+ [23] to apply 2D finite volume discretization to the Navier–Stokes equations. Therefore. 50 4. this value is more useful if used as a ratio.2) can be obtained. they realized that the number of wakes. wake model follows the subsequent observation. dw = 2 t Nw .2 Theory development The Kozak et al.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. represents the sum of all the parts of the blade path that crosses a wake. However.85 Nb T SR . πR (4. (4. (4. They considered that the width of a wake is equal to the thickness of a blade. Figure 4. rw = dw . Also.3) where the denominator is the distance covered by a blade in the rear half-cycle of the turbine. but it is accurate enough for streamtube application. Nw = 0. This distance. This hypothesis neglects the wake diffusion. Note that rw does not depend anymore on the number of blades of . Therefore. a distance value. By studying graphic visualizations of the vorticity generated by the blades [9] – see Figure 4.1 –. they observed that each blade hits each wake twice.1: Visualization of the vorticity for T SR = 3 [9]. dw . as said before. However. when a blade hits a wake. because both numerator and denominator include the number of blades. Afterwards. This hypothesis may seem to be very crude. when a blade hits a wake in the rear part. assumed that the flow inside the wake travels at the same speed and with the same direction than the blade that generated it. Kozak and Vallverd´ u et al. . Therefore.51 the turbine. Both curves. the hypothesis taken related to the flow motion in the wake proves not to be so wrong. It represents the proportion of the blade path in the back that cuts through a wake. because it neglects several diffusion and transport effects. Hence. its lift and drag coefficients decrease dramatically. its relative velocity is diminished a lot. the blade and the wake do not hit in parallel directions. following this line of thought. is basically a representation of 13 Curves for other values of T SR can be found in [10]. according to that assumption. and so does its instantaneous angle of attack.2 for the T SR = 3 example13 . hence it disappears. Figure 4. 2π] fit almost perfectly – see Figure 4. In other words. Also.2: Effective angle of attack comparison between Kozak and Vallverd´ u results for T SR = 3 [10]. they compared the output with Kozak’s [9] effective angle of attack: the result was very satisfactory. multiplied Vallverd´ u’s results of the rear half-cycle effective angle of attack by the (1 − rw ) term – they dropped lift and drag down to zero whenever a blade hits a wake. The effective angle of attack. in the range of θ ∈ [π. Kozak and Vallverd´ u et al. 2π) L     CLdyn : θ ∈ (0. both Kozak and Vallverd´ u et al. the real wake interaction is a very discrete effect [9]. 2π) . . π) wake CD =    C dyn (1 − rw ) : θ ∈ (π. [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 rw . Therefore.52 the lift coefficient.     CLdyn : θ ∈ (0.4b) L This reduction should be applied after the determination of the interference factor but before the postprocessing – in other words. (4. π) wake CL =    C dyn (1 − rw ) : θ ∈ (π. before the calculation of torque and power coefficient. However. Note that the wake interaction is approached as an average effect on the whole rear half-cycle of the turbine.4a) . (4. 1 dyn Compute CLdyn = Sα CLdyn (|α| .2 and C. ur (θ(j) ) Compute α(θ(j) ). Wake interaction modification is included in Vallverd´ u’s code [30].3 Algorithm modification The wake model applied to DMST with dynamic stall would only change. CD from Eq. CP 2 Compute CP using 2nd order trapezoidal integration CP Stop Figure 4. in the postprocessing part. Postprocess T SR. 4. α(θ ˙ (j) ) as in Fig. See Figure 4. θ(j) . . λ2 (θ(j) ) Compute β(θ(j) ).3: Flowchart of the calculation of CP for the dynamic stall and wake modification. therefore.4 Compute CTb (θ(j) ) Compute CP 1 . ˙ CD (|α| .3. α). λ1 (θ(j) ). 3.53 4. α) ˙ wake Compute CLwake . However. Also. it reports better performance for high T SR. As can be seen. not a general decrement.5 shows the power coefficient output for DMST modified with Berg and wake interaction compared to only Berg. Figure 4. BEMT is treating the problem as steady. according to Figure 4. the back half-cycle gradually extracts less and less energy. the torque is negative in the back half-cycle. as T SR keeps increasing. in Figure 4. for low T SR the wake interaction model predicts lower power output.6.5. only in the back half of the turbine.2. From this value and above.54 4.1 on Appendix B 14 . note that the wake interaction effect increases High angles of attack registered at low tip speed ratios are those when CL ∼ CD . according to Figure B.4 Results 4. As mentioned in Chapter 2.4. However. both coefficients are reduced a similar amount and the resulting torque gets diminished14 . Therefore. Low T SR have higher angles of attack: high α ⇒ CL ∼ CD . the back half stops producing power. Therefore. the front half-cycle of the turbine extracts more and more energy from the flow. It clearly shows that the model that includes wake interaction presents a slightly reduced effective angle of attack respectively to the non wake model. too: less opposing torque means less energy back to the flow. if effective angle of attack is reduced. which means more energy extracted from the flow on the whole. as explained in the last paragraph of Section 4.1 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. Of course. it becomes drag dominant and it even gives energy back to the flow – see how.4. However. a real case would show reductions only at discrete regions. so the average reduction is applied to the whole back half. Then. at some T SR slightly over 3. Hence. the absolute value of the torque decreases. the drag force that dominates the rear half-cycle torque decreases. 5 αeff (deg) for TSR = 2 αeff (deg) for TSR = 3 15 10 10 5 5 0 0 −5 −5 −10 −10 −15 0 50 100 150 200 θ (deg) 250 (b) T SR = 2 300 350 400 −15 0 50 100 150 200 θ (deg) 250 300 350 400 (c) T SR = 3 Figure 4. αeff (deg) for TSR = 2. as it shows better agreement . Therefore.5 15 10 5 0 Berg AM = 6 Berg A = 6 with wake −5 M −10 −15 0 50 100 150 200 θ (deg) 250 300 350 400 (a) T SR = 2. this wake interaction model is so simple that it virtually does not change the code performance.55 linearly with T SR.4: Effective angle of attack for DMST with Berg and wake modification. according to the theory. Computationally. 1 0 Berg AM = 6 Berg AM = 6 with wake −0.5 2 2.5 0.1 might vary according to different solidity values.5 4 4.6: Torque coefficient of one blade for T SR = 4 using Berg model. with Kozak’s results for the same turbine [9].5 3 3. One blade torque coefficient 2 1.5 −1 0 100 200 θ 300 400 Figure 4.5 Upwind Downwind 1 0. it will be added permanently to the calculation of the turbine performance.2 0.5 0 −0.1 −0. number of blades or .56 C against TSR P 0. Note that the rule stated in Equation 4.4 0.2 1.3 CP 0.5 5 TSR Figure 4.5: CP against T SR with Berg dynamic stall and with/without wake interaction models. type of blade. 4 0.7. Therefore.5 2 2.2 0.front CP.3 1. the final performance result can be seen in Figure 4. 3.1 0 −0. However.2].5 5 TSR Figure 4. The optimal T SR of this turbine.5 4 4.5 0.3. in case they do.5 3 3. according to the proposed model – DMST with Berg modification for dynamic stall and Kozak and Vallverd´ u variation for wake interaction – is around 2. further experiments or simulations should be performed in order to determine if these dependencies exist and.57 other parameters.1 C P −0.3 CP 0. C against TSR P 0.2 C P.75.back −0. the curve shows that the performance is fairly acceptable in the range T SR ∈ [2.7: CP against T SR with Berg dynamic stall and wake interaction models. how they behave. On the whole. . approximately. The results obtained show physical consistency to reality to some extent.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. torque. With this method. DMST has been chosen due to its higher computational efficiency relative to precision over convergence behavior. Thanks to its ease of use and fast convergence. LES or DNS with Navier–Stokes equations are needed to obtain precise data. chord length. solidity. they are not suitable for precise design or flow observation. Three basic streamtube models are available: single. again because it gives the best balance of precision over convergence.58 CHAPTER 5 CONCLUSION 5. Finally. Likewise. Also. performance for different airfoil blade profiles can be computed by using different lift and drag characteristic curves. However. free stream velocity and others. The Berg adaptation of the Gormont method has been chosen. for different parametric values of tip speed ratio. it can help detecting major problems on more complicated simulations. it can serve to provide initial data for problems with poorer convergence or to orient some . a dynamic stall model has been applied to include dynamic stall effects in the original model. Nonetheless. multiple and doublemultiple. amongst other variables. Finite volume methods using RANS. data about power. angle of attack and flow velocities can be calculated. BEMT is useful. a simplified wake interaction model developed by Kozak has been added too. Afterwards. and are very useful to obtain fast qualitative data about any VAWT. . 5.59 turbine design project at its initial to intermediate stages. More comparisons with real cases or other simulations should be done. some corrections to the model could be performed to consider the spanwise components of the velocity. Some of them include finding the optimum variable blade pitch. This would help validating – or limiting to a certain range of some parameter – the code. might be applied to include 3D effects of the flow. such as the one proposed in [18]. so BEMT fast convergence characteristic will still be useful to guide and help more complicated methods and make them more efficient. such as the boundary layer that the ground induces to the wind. First. Finally. free stream velocity or variable local blade tip speed ratio. some convergence evaluation should be performed to state if the method is still fast enough so these modifications are worth applying. one could also include design optimization procedures to find better work regimes of the turbine. 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. nonsymmetric airfoils might be tested to see if there is any improvement on the turbine performance. VAWTs still have a long optimization path ahead. for different input parameters and turbines. Also. Also a tower interaction modification might be included. the Beoddes-Leishman indicial method should be implemented and tested. However.2 Further work To improve the streamtube model and make it more competitive. some more work should be done. Finally. However.1 shows the aspect of the abandoned GUI. H-rotor VAWT design allows assembling the blades in a way that their pitch angle. some work has been done already with the variable pitch optimization issue. is variable. 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]. However. (5.2. However.1 Variable Pitch Although not finished. some optimization is needed to determine the best variable pitch values at any azimuthal angle. Remember that the effective angle of attack is a representation of the lift. relative to their supporting arm. With this modification. αss . Therefore. 5.2. . Therefore. the blade orientation can be controlled to modify the angle of attack. αp = α + ϕ . it slowed down the development of the project and limited its versatility. because the GUI had to be constantly kept updated with it. The angle of attack has been redefined as αp .2 Graphical User Interface A GUI was developed in the first stages of the project. a not so relevant future task that might be carried is the update and maintenance of this GUI. further work must be done to implement this modification to the stable version of the code published on Mathworks [30]. in order to make the code more user friendly.6. positive when the blade is facing outwards its path. Figure 5.60 5. Therefore. the GUI branch was temporally abandoned. and α is calculated according to Equation 2.1) where ϕ is the added pitch to the blade. the imposed value must be that of maximum lift without stall. 61 Figure 5.1: Former GUI used to interact with Vallverd´ u’s code. . [6] R. 1935. Roskilde. Berg. The MathWorks Inc. Kozak. [10] P. Airplane Propellers. 2014. Double-Multiple Streamtube Model for Darrieus Wind Turbines. Larsen.G. volume 8. Betz. Version 9. [11] H. Finite Volume Approach. Version 8. Second DOE/NASA Wind Turbines Dynamics Workshop. H. Gormont and Boeing Vertol Company. C. 34(3):3–17. 1975. Pa. NASA CP-2185. http://www. Natick. Das Maximum der theoretisch m¨oglichen Ausnutzung des Windes durch Windmotoren. [12] D. Colorado State University.835. 1920. pages 1–3. [4] A. 2013. Kozak. July 2013. E. Accessed on June 15th 2014. [14] MATLAB. Minneapolis. Wolfram Research. D. Beddoes and J. Illinois USA.62 BIBLIOGRAPHY [1] Wind Energy Foundation. Springer Berlin Heidelberg. [5] H. A. [9] P. 1983. Risøo National Laboratory. Turbine having its rotating shaft transverse to the flow of the current. A.018. December 8 1931. Vallverd´ u. M. US Patent 1. Journal of the American Helicopter Society. Energy Information Administration. Paraschivoiu. pages 169–360.windenergyfoundation. July 1989. 6:406–412. Summary of a Vortex Theory for Cyclogyro. Army Air Mobility Research and Development Laboratory.cfm. Leishman.1.701 (R2013b). Solar Energy Soc. [13] MATHEMATICA. 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January 1992.007 (9). 2014. [27] H. 1988.63 [17] I. http://people. A one-equation turbulence model for aerodynamic flows. Sutherland. [24] J. Nguyen. David. Reproduced by National Technical Information Service. Shepherd. Rempfer. [28] R. Dirks. Kozak. Paraschivoiu. [23] STAR-CCM+. [21] Daniel. http://www. Indicial Method calculating Dynamic Stall on a Vertical-Axis Wind Turbine. 2002. Sandia Laboratories. Journal of Propulsion and Power. 1974. and P. Hill. Paraschivoiu and S. Noise and Health. D. US Patent 1. Accessed on June 17th 2014. and Ashwill T. Advanced Energy Projects Dept. Berg. June 1930. Technical Report SAND2012-0304. 64 APPENDIX A SINGLE AND MULTIPLE STREAMTUBE MODELS . (A. m ˙ = ρ u A = ct. For physical understanding and nomenclature conventions refer to Chapter 2.1) . both models are presented with brief explanation.1 Single streamtube model Disk u U∞ Only 1D Upwind zone Fx U∞ uw θ˙ R Downwind zone Fx u θ u uw A Ω y x Single streamtube Figure A.1: Scheme of the SST discretization of a VAWT domain. In this appendix.65 Single and multiple streamtube models are similar to DMST. but more simplified. A. Note that A = 2R. 3) Interference factor: λ = u/U∞ .66 m ˙ (uw − U∞ ) = −Fx (A.11: σ T SR CP = 2π Z 2π CTb dθ 0 (A.8) From Equation 2.2) 1 2 ) = −Fx u m ˙ (u2w − U∞ 2 (A.5) : λ ≥ 43/60 From Equation 2.9: Nb Fx = 2π CFx Z 2π 0 σ = 2π Z 1 ρ c u2r (CD cos β − CL sin β) dθ 2 2π 0 ur (CD cos β − CL sin β) dθ U∞ (A.7) Using Equations A. λ is obtained.5 and A. u = U∞ (2 λ − 1) (A.10: CTb (θ) = u2r (CL l + CD d) · τ 2 U∞ (A. Thrust coefficient: CFx = Fx /(1/2 ρ A U∞ CFx =     1849/900 − 26 λ/15 : λ < 43/60    4 λ(1 − λ) (A.7.4) 2 ). From Equation 2.6) (A.9) . Ast =     R δθ sin θst : θst ∈ (0.12) . (A.10)    R δθ (− sin θst ) : θst ∈ (π. π) ⇒ Ast (θst ) = Ast (2π − θst ) (A.2: Scheme of the MST discretization of a VAWT domain.2 Multiple streamtube model Disk 1 u U∞ uw Only 1D Fx U∞ θ˙ R Ω st u δθ Fx u uw θ A st y x One streamtube Figure A. 2π) m ˙ = ρ u A = ct.11) m ˙ (uw − U∞ ) = −Fx (A.67 A. 16) Note that Fx (θ) = Fx (2π − θ).18) From Equation 2.19) . CFx =     1849/900 − 26 λ/15 : λ < 43/60    4 λ(1 − λ) (A. λ is obtained.14) 2 Thrust coefficient: CFx = Fx /(1/2 ρ Ast U∞ ).x (θ) = 1 ρ c u2i.13) Interference factor: λ = u/U∞ .r (CD cos β − CL sin β) 2 (A. u = U∞ (2 λ − 1) (A.15) : λ ≥ 43/60 From Equation 2.17) Using Equations A. π] (A. CFx (θ) = 2 σ u2r (CD cos β − CL sin β) 2 π sin θ U∞ ∀θ ∈ [0. π] (A.17.15 and A.9: Fi.68 1 2 m ˙ (u2w − U∞ ) = −Fx u 2 (A.10: CTb (θ) = u2r (CL l + CD d) · τ 2 U∞ ∀θ ∈ [0. From Equation 2.11: σ T SR CP = 2π Z π CTb dθ 0 (A. 69 APPENDIX B OBTENTION OF THE LIFT AND DRAG CURVES FOR NACA 0021 . the problem was considered to be implicit unsteady. In fact. The software used was STAR-CCM+ [23]. the data for α > 30◦ is extrapolated. Therefore. The slope of the linear part of the lift. some points had to be neglected because the Ywall < 1. Also.5 principle was violated. The case that is presented in this article works at a Reynolds number of around 3 · 105 . The high angle of attack cases presented a periodic unsteady behavior. but it is not decisive to the obtained results. + Also.1) .70 As explained in Chapter 2. is: CLα = 5. However. so the resulting forces were obtained by averaging the last period of the output report. A series of finite volume numeric simulations were performed in order to obtain the static lift and drag forces on the Naca 0021 profile at different angles of attack. due to Ywall being over 1.1.5.1. according to this data. ideal gas and separated flow.2261 rad−1 . A very fine overset mesh was used – similar to the finite volume method explained in this article. turbulent – Spalart-Allmaras [22] model was used –. The specifications of the simulation are the ones found in Kozak [9] thesis. The resulting lift and drag curves can be seen in Figure B. note that the cases studied in this article rarely present angles of attack over 20◦ . streamtube methods rely very strongly on experimental or known data of the lift and drag coefficients at each angle of attack for the profile that is being studied. this extrapolation main purpose is to ensure convergence. (B. 4 0.8 4 C CL D 1.4 2 0.1: Lift and drag coefficients obtained from the Star-CCM+ simulation.6 3 0.2 6 1 5 0.2 1 0 0 20 40 α (deg) 60 0 0 20 40 60 α (deg) Figure B.71 C (α) CL(α) D 7 1. . 72 APPENDIX C DYNAMIC STALL NEEDED EXPRESSIONS . min 1.0 + 2.4 + 5.1)    γ1 Sc + γ2 (S − Sc ) : S > Sc   tc Sc = 0.9 + 2.5)    1.8) .73 This Appendix contains the needed expressions for the stall models that are note physically interpretable and hence not suitable for the main report.5 (0. C.0 (0. αref − α0 αss − α0  (C.006 − c γ1 =     γ2 2    0 γ2 = γmax γmax = (C.4) : for lift characteristic (C.06 − tcc ) : for lift characteristic (C.7)    0.006 + 1.3) : for drag characteristic    M − M2 max 0.6) : for drag characteristic : for lift characteristic (C.5 (0.06 − tcc )    0.4 + 6.2) : for lift characteristic (C.0 (0.06 − tc ) : for drag characteristic c M1 =     0. M1 − M2     1.1 Gormont method δα =     γ1 S : S ≤ Sc (C.2 M2 =     0.5 0.7 + 2.06 − tc ) : for drag characteristic c  CL (αref ) − CL (α0 ) CL (αss ) − CL (α0 ) m = min .06 − tcc ) (C.5 (0. 06 − tc ) : for drag characteristic c C. (C.5 (0.2 Berg modification mod CD =  i h  dyn −α   CD + AAMααss−α (CD − CD ) : α ≤ AM αss ss M ss    CD (C.06 − tcc ) : for lift characteristic    1.10) .1 and C.0 − 2.0 (0.4 − 6.74 C.3 Strickland adaptation γ=     1.9) : α > AM αss C.2.4 Computational algorithm modifications See Figures C. θ(j) . u2.75 Equilibrium T SR.r . λ1 (θ(j) ) Start with j = Nth + 1 Suppose λ2 Compute β2 . . .  . Compute α2 . α˙ 2 = . α2 (θ(j) ). − . α2 (θ(j−1) ). /∆t dyn dyn dyn Compute CL.1: Flowchart of the calculation of λ1 (θ(j) ) for the dynamic stall modification. from Eq.2 = Sα CL.2 (|α2 | .2 (|α2 | .22b Compute λ2 . 2. 2.x from Eq.19 Convergence criteria no Apply bisection or interpolation met? yes j = 2Nth ? no j = j+1 yes λ2 (θ(j) ) Postprocess Figure C. α˙ 2 ) Compute CF2. . α˙ 2 ). CD. ˙ CD (|α| . θ(j) .2: Flowchart of the calculation of CP for the dynamic stall modification. λ2 (θ(j) ) Compute β(θ(j) ). ur (θ(j) ) Compute α(θ(j) ). 3. α) ˙ Compute CTb (θ(j) ) Compute CP1 . . α).2 and C. α(θ ˙ (j) ) as in Fig. λ1 (θ(j) ).1 dyn Compute CLdyn = Sα CLdyn (|α| . CP2 Compute CP using 2nd order trapezoidal integration CP Stop Figure C.76 Postprocess T SR.