UPTEC

Examensarbete 20 pOktober 2010

Multi-Body Unsteady Aerodynamics in 2D Applied to a Vertical-Axis Wind Turbine Using a Vortex Method

David Österberg

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Multi-Body Unsteady Aerodynamics in 2D Applied to aVertical-Axis Wind Turbine Using a Vortex Method

David Österberg

Vertical axis wind turbines (VAWT) have many advantages over traditional Horizontalaxis wind turbines (HAWT). One of the more severe problem of VAWTs are the complicated aerodynamicbehavior inherent in the concept. In contrast to HAWTs the blades experience varying angle of attack during its orbitalmotion. The unsteady flow leads to unsteady loads, and hence, to increased risk for problems with fatigue.

A tool for aerodynamic analysis of vertical axis wind turbines has been developed. The model, a Discrete vortex method, relies on conformal maps to simplify the taskto finding the flow around cylinders. After the simplified problem has been solved with Kutta condition,using the Fast Fourier transform, the solution is transformed back to the original geometry yielding the flowabout the turbine.

The program can be used for quick predictions of the aerodynamic blade loads for different turbines allowing the method to be validated by comparing the predictionsto experimental data from real vertical axis wind turbines. The agreement with experiment is good.

ISSN: 1401-5757, F10054Examinator: Tomas NybergÄmnesgranskare: Hans BernhoffHandledare: Paul Deglaire

Sammandrag

Vertikalaxlade vindkraftverk har manga fordelar over de traditionella horisontalaxlade. Ett stort problem arden komplicerade stromning som uppstar genom sadana vindkraftverk. Till skillnad fran horisontalaxladevindkraftverk ar aerodynamiken per automatik ostadig eftersom bladen upplever varierande anfallsvinkelunder sin cirkelbana. Det ostadiga flodet leder till problem med vaxlande laster vilket kan leda till problemmed utmattning.

Ett verktyg for simulering av flera viktiga aerodynamiska effecter hos vertikalaxlade vindkraftverk pre-senteras. Metoden, som tillhor klassen diskreta vortexmetoder, anvander conforma avbildningar for attforenkla problemet till flode runt cylindrar. Efter att det forenklade flodesproblemet losts med Kuttasvillkor, m.h.a. Fast Fourier transformen, erhalles flodet i den komplicerade geometrin genom en inversconform avbildning.

Programmet har t ex. kunnat anvandas for att snabbt forutsaga aerodynamiska krafter pa olika tur-biner. Modellen har darmed kunnat valideras genom jamforelse med experimentella data for vertikalaxladevindturbiner. Overensstammelsen anses vara god.

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Contents

1 Introduction 71.1 The Status of Vertical Axis Wind Turbine Technology . . . . . . . . . . . . . . . . . . . . 81.2 The Goal of This Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Previous Experimental Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Previous Aerodynamic Prediction Models . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Previous Multibody Potential Flow Solutions . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Aerodynamic Assumptions 122.1 Two-dimensional Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Inviscid Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Incompressible Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Boundary Layer Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Flow Around a Moving Rotor 163.1 Vortex Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Boundary conditions in Circle plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Construction of Complex Flow Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 Convection of Vortex Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.8 Routh’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.9 Convection in Circle Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Numerical Validation of the Model 264.1 Steady flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Blade Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Strickland/Klimas Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.4 Strickland/Oler Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Application to the Marsta 12kW Experimental Wind Turbine 31

6 Conclusions 36

A Method of Images 38A.1 The Circle Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38A.2 Image of a Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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Nomenclature

β Azimuthal angle of the rotor, as measured to a reference blade.

`, ` Tangent vector to ∂T . Function of ∂T

ω Vorticity ∇×v [s−1]

β Angular velocity of the rotor. β = 2λVD

∂T Boundary between the turbine and the fluid [m]

γ Circulation around a vortex [m2/s]

γk Vortex particle strength [m2/s]

sn Point in Ωs in the visinity of ste,n where vortices are shed

λ, TSR Tip-speed-ratio of the rotor. Defined as the ratio of the undisturbed wind speed and the speed ofthe fastest rotor blade.

T Stress tensor of the general Navier-Stokes equations

v Velocity field, often assumed two-dimensional [m/s]

xk, zk Vortex particle location [m]

x Radius vector, a point in the plane [m]

n, n The unit normal vector to ∂T . Function of ∂T

nn, nn The unit normal vector to ∂Tn. Function of ∂Tn

Ω Region of the plane containing the fluid [m2]

Ωs Area outside N disjoint discs in the plane

∂Ω The boundary of the fluid [m]

φ Velocity potential v =∇φ. [m2/s]

ψ The stream function, i.e., the harmonic conjugate of φ

ρ Mass density [kg/m3]

σ Solidity, σ = Nc/R

T Region in the plane occupied by the turbine [m2]

Tn Region in the plane occupied by a turbine part [m2]

Tsn Disc in the Circle plane corresponding to Tn in the Physical plane

z Unit normal vector to the plane of flow [m]

A Projected frontal area of the turbine

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bn Radius of disk Tsn

CP Power Coefficient, measure of aerodynamic efficiency CP = P12 ρv3A

cnk Coefficients of series describing the conformal map f

D Diameter of the rotor.

D/Dt Convective derivative, equivalent to ∂/∂t +v ·∇

f Conformal map f : Ωs→Ω

N Number of disjoint parts of the turbine section

N0 Number of terms kept in truncated series describing f

p Fluid pressure [N/m2]

R Characteristic core radius [m]

snk Center of disk Tsn

ste,n Point on ∂Tn where the tangential velocity is zero

V Undisturbed wind speed

vs Velocity in Circle plane, corresponding to v in the Physical plane

HAWT Horizontal Axis Wind Turbine

VAWT Vertical Axis Wind Turbine

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Chapter 1

Introduction

A wind turbine is a machine that converts the kinetic energy in the air into mechanical energy [1]. If themechanical energy gained results in a rotational movement around a vertical axis the turbine is called a Ver-tical axis wind turbine or a VAWT (see figure 1.1). There have been many, more or less successful attemptsat harnessing the power of the wind using the vertical axis concept. Today, however, the overwhelmingmajority of commercial turbines are of the competing Horizontal Axis Wind Turbine (HAWT) type.

(a) Danish HAWT (b) Giromill VAWT (c) Darrieus VAWT

Figure 1.1: The three main types of wind turbines

However, the vertical axis principle often simplifies the turbines. In theory that could lead to bettereconomy. The advantages most commonly mentioned are 1) Generator can be placed on the ground wheremainainence is simpler and where weight and aerodynamic properties are irrelevant. 2) The turbine isinsensitive to the wind direction. Hence eliminating the need for a yawing mechanism. 3) If the blades arestraight as in figure 1.1(b) the blades experience the roughly the same type of flow throughout their span.Hence the blades can have a uniform blade profile, and 4) The primary fatigue loads are aerodynamic loadsinstead of gravitational loads. This is a strong argument for VAWTs for large scale machines.

In one sense, the price paid for structural simplicity is aerodynamic complexity. No simple theory existsthat can adequately account for the complicated interaction of the rotor and it’s wake or between the rotorand the tower. For straight bladed VAWTs of the type displayed in figure 1.1(b) one also have to investigatethe interaction between the rotor, the wake and the wing-tip vortices. These problems complicate the designprocess of VAWTs in comparison with HAWTs.

The power produced by a wind turbine is absorbed from the kinetic energy in the wind. It is; thus,proportional to the projected frontal area A of the turbine. Because the terminology is adopted from HAWTaerodynamics this is sometimes called the swept area of the turbine. It is obvious that a wind turbine cannotcapture all kinetic energy of the wind. If so the air would come to a standstill behind the turbine. Hencethere exists an upper limit for the aerodynamic efficiency. Often the so called Betz’ limit of (59.3%) ismentioned as this maximum. The technical name for aerodynamic efficiency of a wind turbine is the powercoefficient defined as

CP =P

12 ρV 3A

(1.1)

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where P is the absorbed mechanical power, ρ is the density of air (1.225 kg/m3, V is the wind speed and Ais the projected frontal area of the turbine.

Modern HAWTs has benefited much from research and experience; they have aerodynamic efficienciesclose to 50% where as VAWTs never exceeded 40%. Even so, there is no decisive argument showing thatVAWTs must have less efficiency than HAWTs. On the contrary, the VAWTs sweep the area twice, openinga slight possibility of reaching CPs exceeding the Betz limit. On the other hand, there are sections of the lapwhere the blades do not produce a positive torque. Furthermore the blades are obliged to move through theturbulent wake on the downwind part of a revolution.

Efficiency aside, equally important is to build turbines that can function properly for the intended life-time of the device. Specifically this means designing turbines that show less fluctuating loads on the bladesand shaft. For example the curved blades of Darrieus vertical axis wind turbines were known to fail fromfatigue as early as two years after construction. The emergence of modern materials has somewhat re-lieved this situation. Never-the-less decreasing variance in the loads leads to improved reliability and bettereconomy.

When designing VAWTs it is hence important to correctly understand the structure of the wake and howit corresponds to the blade loads.

(a) Giromill (b) Ropatec (c) Savonius

Figure 1.2: A few types of VAWT

1.1 The Status of Vertical Axis Wind Turbine TechnologyDuring the last thirty years both the horizontal-axis and the vertical-axis turbines have undergone extensivedevelopment. It should be noted that vertical axis machines have received much attention in the academiccommunity. The horizontal axis concept, on the other hand, has been successful in attracting commercialresearch and development.

The largest research effort to date was undertaken by Sandia National Laboratories in the USA [2].Sandia committed themselves to the curved bladed Darrieus VAWT concept (depicted in figure 1.1(c)) androutinely built and studied these turbines for over fifteen years. In parallel the National Research Councilof Canada were investigating a similar VAWT concept. The effort of Sandia and NRC resulted in a boomof interest in VAWTs in North America. Large commercial turbines were produced in by companies suchas FloWind, Vawtpower, Indal Tech., Lavalin and Adecon Inc.

When the VAWT trend reached Europe most companies opted for straight bladed giromill (depictedin figure 1.1(b)). Foremost by British company VAWT Ltd. and German company Heidelberg. Althoughthese companies discontinued their VAWT projects there has been recent development in connection tothe growing market for wind turbines due to the current climate concerns. Notably a new niche for smallscale VAWTs in city environment has emerged. For example, British company Quiet Revolution and dutchcompany Turby are prominent in this market. Another interesting new type of VAWT is the extreme highsolidity turbines introduced by Italian Ropatec that has a cross section as depicted in figure 1.2(b). Thesecompanies are emphasizing two properties of the VAWT. 1) The VAWTs are able to absorb power even inthe turbulent winds encountered in city landscapes. 2) The low tip-speed of the blades makes the VAWTsilent compared to HAWT.

Indeed this diploma work is a part of a current research project at Uppsala University that is tryingto reinvent the VAWT concept by emphasizing simplicity, reliability and especially longterm economicviability.

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1.2 The Goal of This ProjectIn this diploma work the goal is to to develop an aerodynamic model capable of predicting the behavior ofthe turbine wake and the instantaneous aerodynamic forces acting on the turbine parts. Furthermore, it isdesired that the model is flexible and fast enough to be used actively in the design process of VAWTs.

1.3 Previous Experimental WorkGiven the goals of this project the most relevant studies are those that can be used to validate the predictionsof the numerical model. Hence the most relevant experimental studies are based on setups able to:

1. Measure the pressure distribution over the blades.

2. Measure the force acting on the blades.

3. Measure the torque acting on the shaft.

4. Measure the power produced.

5. Visualize the structure of the wake.

Experiments of type 1-4 provides information of the same phenomena with decreasing level of detail (i.e.,the turbine performance). Experiments of type 5 provides a general insight into the aerodynamic behaviorand can be used when engineering wind farms.

Since the 1970s numerous vertical axis turbines have been built and tested in wind tunnels. However,there are large challenges in producing reliable and detailed performance data. To emulate free streamingconditions in a wind tunnel it is necessary that the model is kept small. In turn, this require that the angularfrequency of the model must be kept very high resulting in unwanted vibrations that simultaneously distortsthe flow and gives noise to the measurements. Further, it can be assumed that many of the results are keptas trade secrets of the companies who performed them.

The first wind tunnel measurements on Darrieus turbines were performed by the National ResearchCouncil (NRC) of Canada in 1972. NRC then proceeded to measure the performance of several turbines[3].

The experimental data that are most often cited were produced as a part of Sandia’s extensive windproject that over several years routinely built and tested Darrieus wind turbines. In particular, Strickland [4]performed the first (and possibly the only) published flow visualization of the developed wake.

Several authors have measured VAWT instantaneous blade forces. Tow tank studies done at Texas TechUniversity presented measurements of blade forces on a straight bladed giromill both during operatingconditions (tip-speed-ratio 5) and during dynamic stall condition (tip-speed-ratio 2.5) (described in [5]).The Texas Tech study also measured the velocity profile of the wake. Blade force measurements on astraight bladed VAWT operating in the dynamic stall regime were also produced by Vittecoq et al. [6]

British company VAWT Ltd produced several full scale H-rotors equipped with systems for instanta-neous blade force measurements [7]. These results are interesting because the size of the turbine reducesthe uncertainty in the force measurements. On the other hand the uncertainty in the flow conditions cannotbe controlled as in wind tunnel measurements.

An in-depth review of the experimental work to date can be found in Paraschivoiu[5].

1.4 Previous Aerodynamic Prediction ModelsThere has been, in the past, several attempts of modeling lift-driven VAWTs each with their own advantagesand disadvantages. The different methods can be classified in six groups:

1. Analytical maximum CP predictions

2. Fixed-wake vortex models

3. Streamtube models 1

1In the HAWT community these models are often called Blade Element Momentum (BEM)Methods

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4. Mesh-based CFD

5. Free-wake vortex models

The main fully analytical CP prediction model for vertical axis wind turbines is the double actuator discmodel. It models the flow as one steady streamtube that passes through the turbine. One can think of it astwo Betz turbines [8, 9] in tandem with a certain spacing in-between. The model was conceived by Templinin 1974 and was the first aerodynamic model of VAWTs [10]. Newman showed that the maximum CP,maxof such a double actuator-disc system is CP,max = 16/25 = 64% [11].

A semi-analytical attempt is the so called fixed-wake vortex model developed by Holme[12] and ex-tended by Fanucci et al. [13]. This model assumes an infinite number of wings but with solidity (σ = Nc/R)fixed to a certain value. At each azimuthal position there is a wing with a certain circulation around. Thiscirculation is calculated from the local angle of attack. Hence there is a vortex sheet bound to the turbine.There is also vortex sheets, which leaves the turbine due to the change in circulation between adjacentwings, and are modeled to be convected downstream at a uniform velocity. The forces on the blades can becalculated from Kutta-Joukowski principle and integrated to obtain the performance of the turbine. In mostfixed-wake analysis the maximum predicted CP is about 54% [14].

The most popular class of prediction models are streamtube models. There is a range of models thatvary in how detailed the analysis is. The most sophisticated streamtube model is the Double MultipleStreamtube model (DMST) first described by Paraschivoiu [5]. A simpler model is found in Wilson [15].In brief they model the flow as composed by a grid of linear streamtubes that partition the swept area. Foreach streamtube static airfoil data is used for calculation of the average blade forces from lift and drag usingthe relative velocity to calculate angle of attack and blade Reynolds number. This average force is then usedto calculate the loss of momentum and; thus, the slowdown of the wind. These models cannot predict thestructure of the wake but are on the other hand extremely fast and can with advantage be used for quickback-of-the-envelope calculations of blade forces and turbine performance. However, important effects likedynamic stall is not easily included. Also, because of the shortage of airfoil data for airfoils in flow withcurved streamlines it is not obvious how to study this effect.

In terms of thoroughness the streamtube models has its opposite in the method of direct numericalsimulation (DNS). In DNS the Navier-Stokes equations are solved numerically with some PDE-solvingmethod such as Finite Elements (FEM) or Finite Volumes (FVM) using a fine enough grid to capture allrelevant effects. To avoid simulating all fine scales an approach commonly used is the Reynolds AveragedNavier-Stokes Simulation (RANS) [5]. In theory this approach can be used for investigating all aspects ofthe turbine aerodynamics. However, the large computational cost associated with such methods limits theuse to specific details such as aeroelasticity but even these applications come with a large computationalcost. Various authors, notably Ferreira et al has used Large Eddy Simulations (LES) to study the effectof dynamic stall[16, 17]. The drawback is as with DNS that detailed simulations long enough for thewake to develop completely are very expensive. However, a recent article by Lida [18] shows promisingdevelopment in this direction.

The fifth group of models, which will be built upon herein, are discrete vortex methods (DVM). The liftforce of a blade in Darrieus motion is due to the circulation around the blade. However, because the liftand; thus, the circulation is changing during a revolution a continuous line of eddies are shed from eachwing in order to conserve the angular momentum of the air. Discrete Vortex Methods discretize this lineinto separate eddies and track the resulting eddies as they are convected downstream.

The first vortex simulations of Darrieus turbines in inviscid flow was performed by Strickland [4]. Itwas based on the principle of a lifting line approach using airfoil data sheets to calculate the circulation.Lifting line approaches has since then been the most popular of the vortex models. Another possibilityare panel methods, which can simulate the behavior of completely general airfoils as long as the flow isattached, without the need for experimental data. However, they have computational disadvantages becausethey require an inversion of a large full matrix and subsequent multiplication at each time-step.

An early model based on a conformal mapping technique was proposed by Wilson [14]. The idea wasto map an airfoil to a circle. The flow could then be found using the method of images. Conformal mappingtechniques share the principal advantages of the panel methods if an appropriate mapping can be found,at the same time conformal mappings are very efficient in terms of computational cost. An extension ofWilson’s conformal mapping approach was made by Deglaire [19] where a general manner of determininga suitable conformal map was found.

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1.5 Previous Multibody Potential Flow SolutionsMilne-Thomson first stated and proved the circle theorem [20] that allows the introduction of a cylinder inany known potential flow. Williams [21] used the circle theorem iteratively to derive a solution to the flowaround two circles. Resent work by Crowdy [22] demonstrated an analytical expression for potential flowaround multiple cylinders. In this diploma work, however, an iterative method will be used.

Ives [23] used Laurent-series, which were obtained by the discrete Fourier transform, to find the poten-tial flow around a wide class of two dimensional shapes. Deglaire[19] also used discrete Fourier transformwhen extending Williams work to a flow around an arbitrary number of circles. Deglaire suggest the use ofhis technique for application to vertical axis wind turbines. Because most VAWTs rotate as solid bodies itis not necessary to remap at each time-step.

Another approach was described in a paper by Zannetti et al. [24]. They used sophisticated mathemat-ical techniques to map two specific wings conformally to a simpler geometry. However, their approach islimited to two blades. Also they appear to have no way to try different blade sections in a general manner.

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Chapter 2

Aerodynamic Assumptions

In general air flow is governed by the Navier-Stokes equation.

ρDvDt

=−∇p+∇ ·T+ f (2.1)

where D/Dt ≡ ∂/∂t +v ·∇, T is a tensor describing the viscous properties of the fluid; v the velocity; p thepressure and ρ is the air density (1.225 kg/m3).

There is little hope of finding analytical solutions to (2.1) in all but the simplest of geometries. Evenfor cases where an analytical solution exist they inevitably break down as the speed of the flow increasesbecause the equations destabilize. In the case of a vertical axis wind turbine the complexity of the geometryand motion forces us to resort to numerical methods. However, solving the Navier-Stokes equations makes,even then, a very challenging task because of the high sensitivity of the solutions on the various simulationparameters such as step-length in the time and space discretization. With today’s computer resources, nu-merical solution in full 3D of (2.1) for a VAWT-geometry is unfeasible, even when using the most powerfulsupercomputers. In applications it is; thus, necessary to make approximations and simplifications and inorder to do that, avaliable a priori knowledge about both the flow must be described and taken into account.

Wind turbines are operating within the earth surface boundary layer. Therefore the wind speed is sig-nificantly dependent of the distance to the ground i.e., wind shear. Further, depending on the roughness ofthe earth surface the flow will be more or less turbulent. Moreover still, some aspects of the flow aroundstraight bladed giromills are known. A notable effect is the downwash effect at the tip of each blade. Theformation of tip-vortices which decrease the lift and increase the drag. We thus must ask: Can we simplify,model or neglect:

1. Height dependent wind speed differences

2. Turbulence in the incoming flow?

3. Tip effects from the rotor blades?

4. Viscous effects such as skin friction drag on the struts and blades?

The average wind speed is increasing with height over ground. Because a wind turbine tower is typically1-100 meter high, the velocity gradient at these heights indicate that there is a velocity difference betweenthe bottom and the top parts of the rotor. Further, this difference is smaller the further from the groundthe turbine is mounted. In the case of a very small turbine with a 10 meter tower and 5 meter blades thedifference in undisturbed wind velocity might be as much as 20% between the bottom and top of the partsof the rotor. An important implication of this effect is that the tip-speed-ratio varies throughout the heightof the turbine.

Although the average wind speed is indeed increasing with height over ground. It seems like the shorttime variance in wind speed, however, is roughly constant. This means that the impact of natural turbulencewill be less on larger turbines [9]. That is not to say it is without importance.

Moreover, while the turbine operates it is absorbing kinetic energy from the air. Hence the air speeddownwind from the rotor (the wake) must be lower than the undisturbed wind speed. Because the air is freestreaming it can be assumed that a part of the air that was flowing in the projected frontal area of the rotor

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in the end is flowing around the rotor where it is not producing power. See figure 2.1 for an illustration ofthe general properties of the expanding flow.

Figure 2.1: The behavior of the air streams around an H-rotor type wind turbine

The lift of an airfoil is due to the pressure difference between its two sides. Air flow follows pressuregradients; the tip of each blade thus provides the air opportunity to move from the lower to the upper sidehence evening out this pressure somewhat. This results in two things. 1) There is a reduction of lift due tothe lost pressure. 2) The air flowing over the tip of the blade results in a vortex line emanating from eachblade-tip. These vortex lines induces an increase in drag which is related to the lift force.

Further, it is known that flow around a cylinder exhibits three dimensional eddies at Reynolds numbersgreater than 104 and fully turbulent (a large amount of eddies in various sizes) at Reynolds numbers greaterthan 105. It is thus resonable to assume that the flow is three-dimensional for airfoils in Darrieus motion inoperating conditions (typically the blade Reynolds number is exceeding Re > 105).

2.1 Two-dimensional ApproximationThere are indeed complicated three-dimensional effects present in the flow. A treatment of these effects arenecessary in order not to overestimate the performances of a given turbine.

It is possible to estimate the losses due to the finite span of the blades through the lifting line theoryof Prandtl [25]. But there is an important point to make before the standard corrections for finite span areapplied to the results. For steady conditions the starting vortex of a wing is too far away to influence theflow. Thus there is no downwash from it felt by the wing and hence no drag due to lift. Thus the drag dueto lift is completely dominated by the downwash from the wingtip vortices. Hence, only 3D wings havedrag due to lift in steady flight. On the other hand, the influence from the starting vortex dominates thedownwash for a wing that is in the process of building up its lift. And the downwash from a 2D startingvortex is stronger than for a finite length 3D starting vortex.

Given that the blade-tip losses can be estimated from knowledge of the performance of infinite spanblades it makes sense to study the simplified case of infinite aspect ratio. Furthermore if the effect ofthe wind shear is to be studied a rudimentary 3D model can be accomplished by running multiple simu-lations for different height intervals. In addition, small three dimensional eddies can be modelled usingKolmogorov’s theory or other turbulence models [26]. Hence it will be assumed that the flow is two-dimensional; neglecting the influence of the three dimensional turbulence effects.

We thus want to find the two-dimensional flow which satisfies the boundary conditions of the undis-turbed wind velocity far up-stream, and no-through and no-slip conditions at the rotor boundaries.

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Although a three dimensional simulation would be preferable, there has not been time to develop andtest such a tool. Also, the computational cost would be substantially higher.

2.2 Inviscid ApproximationLudwig Prandtl discovered that given a Reynolds number sufficiently large, the flow over a surface canbe divided into two distinct regions. There is one region away from the boundary surface where the fluidbehaves as an ideal fluid. In the other region, which surrounds the boundary surface, viscosity is important.This other region is called the boundary layer. Prandtl also pointed out that the thickness of the boundarylayer tends to zero as the Reynolds number increases (see [27] for an interesting article explaining theboundary layer concept; its history and significance). For a VAWT the Reynolds number is relatively high(typically exceeding 105). It is thus reasonable to make the assumption that the boundary-layer thicknesscan be approximated to zero. This simplifies the problem at hand to solving the Euler equation with no-through boundary condition.

A problem with inviscid flow around a wing is that the flow is not uniquely defined. There can bean arbitrary circulating flow around the wing. In fact, this circulation determines the placement of thestagnation points in the case of steady flow.

It is further assumed that all shear in the flow takes place in the boundary layer or in a thin wake behindthe lifting bodies. Thus the flow outside the boundary layer and not in the wake is irrotational.

2.3 Incompressible ApproximationA rule of thumb is that a fluid behaves in an incompressible manner when the Mach number is smaller than0.3 [28]. This means that air-flow can be considered incompressible for speeds lower than 100 m/s. As itturns out this approximation applies to a majority of VAWTs in operating conditions.

In an incompressible inviscid fluid, disturbances travel at a very high speed (if the fluid is perfectlyincompressible the speed of sound is infinite). Thus, for the flow problem to be well posed it is necessaryto specify the flow at some initial time t = 0 as well as on the boundaries of the domain at all times.

Under these assumptions the fluid flow is governed by the Euler equation and the Continuity equation

DvDt

=−1ρ∇p (Euler equation) (2.2)

∇ ·v = 0 (Continuity equation) (2.3)

where v is the velocity field; p the pressure field; ρ the density and D/Dt = ∂/∂t + v ·∇ the convectivederivative.

2.4 Boundary Layer ConsiderationsThe Euler equations poses no restrictions on the flow regarding the continuity of the velocity field. Suchsolutions are however unphysical because all real fluids have a certain viscosity although it might be small.In particular at the sharp trailing edge of a moving wing the flow comes together from the upper and lowerside of the wing. If the stagnation point deviates from the trailing edge the resulting sharp velocity sheargives rise to the formation of a vortex inside the boundary layer. Indeed as the rotor starts up a continuoussheet of vorticity is shed from the trailing edge of each wing. Figure 2.2 and figure 2.3 illustrates the effect.

The angular momentum of the air is conserved through time. The reaction of the angular momentumof the vortex sheet must thus be contained in a counter-circulation around each wing. This circulation isindeed giving rise to the lift force. The variation of angle of attack throughout the Darrieus motion impliesa corresponding variation in circulation and that a turbine blade leaves a continuous line of vorticity behindits trailing edge. This line is called the wake of the wing.

The demand continuous velocity around the trailing edge is called the Kutta condition after the Germanmathematician and aerodynamicist Martin Wilhelm Kutta [29]. Herein we will call any point that is requireda priori to be a stagnation point a point with Kutta condition.

14

(a) zero circulation (b) zero circulation

(c) Kutta condition (d) Kutta condition

Figure 2.2: Streamlines and Velocity field at the trailing edge of a 2D wing at an angle of attack.

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0.6

0.65

0.7

Figure 2.3: Simulated vortex sheet with starting vortex. The upper image depicts a complete boundary layer vortexsimulation at Re=8000 (by courtesy of Paul Deglaire) whereas the lower image shows the approximation of an infinitlythin boundary layer (Re→ ∞) used in the present model.

15

Chapter 3

Flow Around a Moving Rotor

A shear layer is is formed from the trailing edge of each wing, as explained in section 2.4. This sheet will bemodeled as a discrete chain of distinct eddies (called vortices, sing. vortex). The dynamics of such vorticescan be determined from the principal equations of fluid dynamics.

3.1 Vortex DynamicsIn nature a two-dimensional vortex consists of two regions that exhibits different behavior. In the vortexcenter there is a rotational core that rotates as a solid body. Outside the core the flow is almost irrotational.The velocity field from a single vortex placed at the origin can; accordingly, be accurately modeled [26] as

v(x) =

γ

2πz×x if |x| ≤ R

γ

2π

z×x|x|2 if |x|> R.

(3.1)

where R is the core radius and γ is the circulation around the vortex. This type of vortex is often called aRankine vortex. In the special case of R = 0 the vortex shall be called a point vortex (in 3D it is called a linevortex). A smoother vortex model is the so called Lamb-Oseen vortex, which is modeled as

v(x) =γ

2π

(1− e−(

|x|R )2) z×x|x|2

=γ

2π

(1− e−(

|x|R )2)∇log(|z×x|) (3.2)

It is important to note that v(x) is essentially conservative outside the vortex core (i.e when |x| R). Plotsof the velocity profile of the different vortex modeles are displayed in figure 3.1

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

Radial distance to the vortex center

Vel

ocity

Rankine vortexOseen vortexPoint vortex

Figure 3.1: Velocity profiles of the most common vortex models

It is possible to determine rules for the kinematics of vortices from Euler’s equation

DvDt

=−1ρ∇p (3.3)

16

where v is the velocity; p the pressure and ρ is the density.Taking the curl of (3.3)

D(∇×v)Dt

= 0

Vorticity is defined as ω =∇×v, which gives the vorticity equation of incompressible flow

DωDt

= 0 (3.4)

Given the vorticity field (i.e., a solution to (3.4)) it is possible to express the velocity field by use of theBiot-Savart relation, in our case for two dimensions [30]

v(x) =∇φ′− 1

2π

∫∫Ω

ω(x′)× (x′−x)|x−x′|2

dΩ (3.5)

where φ′ is the irrotational velocity potential generated by the boundary conditions.The trick of vortex particle methods is to temporarily assume that all vorticity is concentrated around a

countable set of points xk called vortex particles. The vorticity field can then be represented as

ω(x) = ∑k

γkδ(x−xk)z (3.6)

where δ(x) is the Dirac distribution. The velocity field can be calculated from (3.5) and (3.6) to be

v(x) =∇φ′(x)−∑

k

γk

2π∇log |z× (x−xk)|. (3.7)

To avoid the singularity and obtain a more physical velocity field the point vortices are now replaced withLamb-Oseen vortices with a small core radius R. Thus, we have

v(x) =∇φ′(x)−∑

k

(1− e−

( |x−xk |R

)2)

γk

2π∇log |z× (x−xk)| (3.8)

Clearly (3.8) and (3.7) coincide almost identically outside the vortex cores.Now, Helmholtz proved (in a seminal 1858 paper [31]) that a fluid element initially located at the center

of a vortex will continue to stay at the center for all time. Another way of saying that is that a vortex movewith the same velocity as the fluid does.

This means that the movement of vorticity is governed by a system of ordinary differential equations

dxk

dt=∇φ

′(x)∣∣∣xk−∑

j 6=k

1− e−(|xk−xj|

R

)2 γ j

2π∇log |z× (x−xj)|

∣∣∣xk

(3.9)

3.2 Potential FlowOutside the cores of the vortices it holds

v(x) =∇φ′(x)−∑

k

γk

2π∇log |z× (x−xk)|.

Factoring out the ∇ from the previous expression

v(x) =∇(

φ′(x)−∑

k

γk

2πlog |z× (x−xk)|

)def≡ ∇φ

The continuity equation (2.3) for incompressible fluids reads

∇ ·v = 0

17

This implies that ∇φ(x) satisfies the Laplace equation

∇ ·∇φ = ∆φ = 0 (3.10)

i.e φ is a harmonic function.It is beneficent to construct an analytic function from φ(x,y) and its harmonic conjugate ψ(x,y). The

well known Complex Flow Potential is obtained, which in this case is induced by the boundaries of Ω.

w(z) = w(x+ iy) = φ(x,y)+ iψ(x,y) (3.11)

ψ is called the Stream function and is of great importance for fluid dynamics [20]. Physically ψ(z) mea-sures the fluid flux through the line connecting z and a reference point z0. In the following, the commonassociation between a vector x ∈ R2 and the complex number z≡ x+ iy ∈ C will be used. Then (3.11) canbe written

w(z) = wext(z)−∑k

γk

2πlog(z− zk)

Equations (3.8) and(3.9) can now be written concisely as

v(z) =dwdz

(3.12)

dzk

dt=

dwdz

∣∣∣zk

(3.13)

3.3 Boundary ConditionsThe potential flow can be described as superposition of four types of flow. The first part is the wind; thesecond is the headwind due to the rotation of the rotor; the third is the flow due to free vortices in the flow;and the fourth and last is the action from the boundaries of the turbine.

It is assumed that there exists a complex flow potential that governs the flow as described in section 3.2

w(z) = w∞(z)+wγ(z)+ wγ(z)+η(z). (3.14)

Here w∞(z) denotes the flow of the undisturbed wind; wγ(z) denotes the flow induced by free vortices; andη(z) denotes the disturbance induced by the turbine. The function wγ is the complex flow potential of thespecial vortices that are currently forming in the imagined boundary layers of the blades as described insection 2.4.

The complex potential η(s) in turn, can be written as

η(z) =N

∑n=1

ηn(z).

where ηn(z) denotes the disturbance of each turbine part ∂Tn. Equipped with this notation the boundaryconditions can be determined for the parts individually. The far field is only effected by the wind. Hencethe boundary conditions become

dw∞

dz→V

dwγ

dz→ 0

dwdz→ 0

dη

dz→ 0 as |z| → ∞ (3.15)

The physical reality is that no fluid can penetrate the surface of the rotor. I.e., the no-through boundarycondition. This means that the wings of the turbine must be streamlines. When the turbine is in operationthe rotor moves in rigid body rotation. Assuming that β denotes the azimuthal angle measured to a referenceblade of the turbine then the angular velocity of the rotor is defined by

β =2λV

D

where λ is the tip-speed-ratio and D the turbine diameter.

18

Accordingly, the turbine wings must be streamlines in the headwind−iβz. This corresponds to a stream-function β|z|2/2. The potentials w∞(z) and wγ(z) are induced by the wind and the free vortices; they arefree to take any value on ∂T . It is; therefore, necessary that the complex flow potential η(z) satisfy

ℑ(η(z)+w∞(z)+wγ(z)+ wγ(z))+12

β|z|2 =Cn z ∈ ∂Tn (3.16)

where Cn is a real constant that may be different on different wings. The notation ∂Tn has been introducedto mean the boundary of blade n. Similarly ∂T will denote the boundary of the whole rotor i.e., the unionof the blades and Ω denotes the fluid region.

In applications it is known that certain points zte,k on the boundary are stagnation points (in particularthe trailing edges of the blades). These relations will be called Kutta conditions even though they mean aslightly more general thing than what is normally called Kutta condition.

v(zte,k) =dwdz

(zte,k) = iβzte,k zte,k ∈ ∂T (3.17)

3.4 Conformal MappingIt is a property of certain functions called conformal maps that a solution φ(x,y) to the Laplace equation(3.10) can be transformed to a different geometry, using the conformal map, yet still solve the Laplace’sequation in the new coordinates.

This is of interest if a conformal map between the area outside the rotor and a simpler geometry can befound.

Specifically, if a function φ(x,y) that solves Laplace equation is transformed with a conformal map

f (u+ iv) = x(u,v)+ iy(u,v) to the new function φ′(u,v)def≡ φ(x(u,v),y(u,v)) then

∂2φ′

∂u2 +∂2φ′

∂v2 = 0

In a recent paper Deglaire proposed a method for mapping the area outside N distinct simply connectedsubsets of C to the area outside N distinct discs conformally [19].

Figure 3.2: The idea of using conformal mappings to simplify the problem

Deglaire suggests that the conformal mapping can be written as a kind of Laurent-series

z = f (s) = s+N

∑n=1

∞

∑k=1

cnkbkn

(s− scn)k ≈ s+N

∑n=1

N0

∑k=1

cnkbkn

(s− scn)k . (3.18)

where scn are the circle centers and bn the circle radii. He then develops an algorithm to determine thecomplex coefficients cnk. This technique provides a straight forward manner to find maps as in figure 3.2for an extended family of rotors. The task is; thus, changed to the simpler problem of finding φ for the flowaround N cylinders.

Accordingly, let z= f (s) be a function that maps conformally, the area outside N circles s : |s−scn|= bn,in what shall be denoted the circle plane, to the area outside N airfoils, denoted the physical plane.

19

To obtain the velocity in the physical plane the fact that the complex flow potential is invariant underconformal transformations can be used (see for example [32]). Thus, the velocity carries over to the physicalplane by the simple formula

v(s) = vs(s)1

f ′(s). (3.19)

here a note of caution is necessary. Strictly speaking (3.19) is only valid if vs(s) is irrotational at s. Strictlythis is not the case when there are vortices in the flow with core radius R > 0. Nonetheless (3.19) will beused with the restriction that the core radius must be kept small.

Because the two functions φ(x,y) and ψ(x,y) both solves Laplace’s equation it is easy to varify that thecomplex flow potential w also is invariant under a conformal transformation. From equation (3.14):

w(s) = w∞(s)+wγ(s)+ wγ(s)+η(s). (3.20)

3.5 Boundary conditions in Circle planeWe now want to express the boundary conditions (3.15) and (3.16) in the simpler circle plane. The windcorrespond to a boundary condition at infinity. The conformal map f , (3.18), which relate the physicalplane to the circle plane, has the property

f ′(s)→ 1, as |s| → ∞

This implies that the boundary condition (3.15) — which describe the behaviour as |s| → ∞ — can beexpressed independently of f as

dw∞

ds→V

dwγ

ds→ 0

dwγ

ds→ 0

dη

ds→ 0 as |s| → ∞ (3.21)

The boundary condition at the turbine surface is slightly more complicated. That is due to the rotationof the rotor. This means that the shape of the wings matter because the distance to the center of rotationdefines the boundary condition of a point. The boundary condition (3.16) takes the form

ℑ(η(s)+w∞(s)+wγ(s)+ wγ(s))+12

β| f (s)|2 =Cn s ∈ ∂Tsn (3.22)

In the circle plane the Kutta condition can be written as

vs(ste,k) = iβ f (ste,k) f ′(s)

Because the velocity is otherwise bound to follow the circle it is enough to require

vs(ste,k)− iβ f (ste,k) f ′(s)⊥ `(ste,k) (3.23)

where ste,k denotes a point on one of the circles zte,k = f (ste,k) and `(s) denotes a tangent vector to the circle.

3.6 Construction of Complex Flow PotentialFrom the boundary condition (3.21) and section 3.1 the following forms of the partial complex potentialsof w in equation 3.20 are proposed

w∞(s) =V s

wγ(s) = ∑k

iγk

2πlog(s− svk)

wγ(s) = ∑n

iγn

2πlog(s− sn)

η(s) =N

∑n=1

ηn(s)

20

the wind velocity V ; the vortices (γk,svk) and the kutta vortex release positions sn are all assumed tobe known. The goal of this section is to present an algorithm to determine η(s) and γn such that theno-through and the Kutta conditions are satified. In order to describe the algorithm a notation for partialresults under iteration step j is used as in the following

w( j)(s)γ =iγ( j)

n

2πlog(s− sn)

η( j)n (s) =− iγ( j)

n

2πlog(s−∆n(sn))

ηn(s) = ∑j=1

(η( j)n (s)+ η

( j)n (s)

)where a symbol ∆n(s) =

b2n

s−scn+ scn for the point symmetric with respect to circle n has been introduced.

The condition (3.22) can be reformulated as

ℑ(ηn(s)) =−ℑ(w∞(s)+wγ(s)+ wγ(s)+ ∑m 6=n

ηm(s))−12

β| f (s)|2 +Cn s ∈ ∂Tn

The approach will be to first find the potential as if the blades has no influence on each other (i.e.,∑m6=n ηm(s) is neglected for the moment). Also, for the moment the Kutta condition is ignored. Hence

determine η(0)n (s) such that

ℑ(η(0)n (s)) =−ℑ(w∞(s)+wγ(s))−

12

β| f (s)|2 +Cn s ∈ ∂Tn.

This is possible because all entities are known. Then a first approximation γ0n can be determined such that

dw(0)γ

ds

∣∣∣ste,n

+dη

(0)n

ds

∣∣∣ste,n≡ iγ(0)n

2π

[1

ste,n− sn− 1

ste,n−∆n(sn)

]

=−dη(0)

ds

∣∣∣ste,n− dw∞

ds

∣∣∣ste,n

+dwγ

ds

∣∣∣ste,n

+ iβ f (ste,n) f ′(s)∣∣∣ste,n

The iteration step is to determine η( j+1)n (s) such that

ℑ(η( j+1)n (s)) =−ℑ( ∑

m6=n

(η( j)m (s)+ η

( j)m (s)

)+ w( j)

γ (s))+Cn

and γ j+1n such that

dw( j+1)γ

ds

∣∣∣ste,n

+dη

( j+1)n

ds

∣∣∣ste,n≡ iγ( j+1)

n

2π

[1

ste,n− sn− 1

ste,n−∆n(sn)

]=− ∑

m 6=n

dη( j+1)m

ds

∣∣∣ste,n

It shall be assumed that ηn(s) can be written as a series

ηn(s) =∞

∑k=1

ankbkn

(s− scn)k

furthermore, is is also assumed that the functions η( j)n (s) have the same form

η( j)n (s) =

∞

∑k=1

a( j)nk bk

n

(s− scn)k

Under this assumption η is analytic on Ω and the boundary condition (3.21) is automatically fullfilled. Tosatisfy (3.22) ∂Tsn is parameterized by scn +bneiθ. On ∂Tsn it; thus, holds that

η( j)n (scn +bneiθ) =

∞

∑k=1

a( j)nk e−ikθ

21

This expression can be identified as negative-frequency part of a Fourier series. If the series is truncatedafter Nco terms and then evaluated at 2Nco equally spaced points on the circles. Then the coefficients can beobtained by use of the inverse Discrete Fourier Transform (F −1)

a(0)nk k = F −1−iℑ(w∞(snp)+wγ(snp))−i2

β| f (snp)|2p

a( j+1)nk k = F −1−iℑ( ∑

m 6=n

(η( j)m (snp)+ η

( j)m (snp)

)+ w( j)

γ (snp)p

where snp = scn +bneiπp/Nco

In practice, however, if there are vortices very close to circle n the number of terms Nco becomes veryhigh. The performance of the method is improved if the close vortices is handled separatelly using themethod of images.

For each disc, one can divide wγ(s) in one part coming from vortices close to the disc and anothercoming from vortices far from the disc. Empirically, the best cut-off distance has been determined to be0.1bn from the boundary of circle n [19].

wγ(s) = ∑k:

svk∈Nn

iγk

2πlog(s− svk)+ ∑

k:svk /∈Nn

iγk

2πlog(s− svk) Nn

def≡ s : |scn− s| ≤ 1.1bn

now, η(0)n (s) is written

η(0)n (scn +bneiθ) =

∞

∑k=1

a(0)′

nk e−ikθ + ∑k:

svk∈Nn

iγk

2πlog(s−∆n(svk))

where the coefficients a(0)′

nk k are calculated by

a(0)′

nk k = F −1−iℑ(w∞(snp)+ ∑k:

svk /∈Nn

iγk

2πlog(snp− svk)+

i2

β| f (snp)|2)p

Figure 3.3 and 3.4 displays the convergence of the algorithm for the case of steady flow around a Ropatecwind turbine.

Figure 3.3: Streamlines of steady flow around a standing Ropatec rotor in the transformed and physical plane respec-tively. Note there is no Kutta condition applied to the central body.

22

0 2 4 6 8 1010−8

10−6

10−4

10−2

100

Iterations (M)

Err

or (m

ax fl

ux tr

ough

bou

ndar

y)

Figure 3.4: The convergence of the algorithm described in section 3.6

3.7 Convection of Vortex ParticlesThe movement of the vortices are governed by equation (3.13) which reads

dzk

dt=

dwdz

∣∣∣zk

(3.24)

with the current notation this can be written

dzk

dt=V +

dη

dz

∣∣∣zk+ ∑

j 6=k

1− e−(|zk−z j |

R

)2 iγk

2π

1zk− z j

(3.25)

which is a system of Nv ordinary differential equations. These equations give the velocity of each vortexparticle and; accodingly, contain the information needed to advance the vortices. Simply summing (3.24)for each vortex—called Direct Summation or sometimes Naıve Summation—of course yield the velocities.It is called naıve because the number of basic operations in the computation scales as O(N2

v ). A respectablework given that it must be computed in every time-step and for perhaps 100000 vortices. More efficientmethods was developed in the eighties. These exploit the fact that a cluster of vortices itself behave as alarge vortex placed at the center of vorticity (defined analogously to the center-of-mass concept of particlemechanics). A cluster of vortices; thus, moves around this point while at the same time moving somewhatas shown in figure 3.5.

This self-similar property can be efficiently exploited if recursively applied on single vortices, clustersof vortices and clusters of clusters of vortices and so fourth. During the eighties this approach spawneda number of efficient O(Nv logNv) divide-and-conquer algorithms that has been applied successfully. Theapproaches mainly differ in how they determine the clusters, a common method beeing the Particle-in-Cellalgorithm [30], which relies on superimposion of a grid over the plane. In this method a cell of the gridis thought of as defining a cluster. Another method was proposed by Barnes and Hut [33] in 1989. Intheir method the box containing all vortices is recursively divided until each box only contains one or zerovortices. In this manner a tree is built where each branch is defining a cluster. The idea is that the branchesfar from each other only interact through their centers of vorticity. The most sophisticated algorithm todate is the Fast Multipole Method (FMM) of Greengard and Rokhlin [34]. FMM does not simply replacea cluster far away with a replacement vortex at the center of vorticity. The cluster is instead represented

23

Figure 3.5: Qualitative picture of the hierarchical motion of vortex clusters. The center-of-mass symbol represents thecenter of vorticity.

by a truncated Taylor series. FMM has the advantage that the error can be kept arbitrarily small. However,FMM relies on the potential field representation of the velocity. The error from this is however not believedto be large.

In this project the fast multipole method has been used for finding the velocities of the vortices. Theauthor of this report has had access to a very efficient implementation by Stefan Engblom of the Divisionof Scientific Computing Department at Uppsala University. Further explanation can be found in [35].

Once the velocities of the vortices are known; they can be advanced in accordance with some numericalintegration method. In the continuous case ∆t→ 0 it is impossible for a vortex to be convected into a solidregion. This is however not true when ∆t is discrete. It is therefore necessary to perform a check at eachtime-step whether a vortex has moved into a solid body. Two actions are possible when a vortex is foundin a forbidden zone. 1.) The vortex could be removed from the flow or 2.) the vortex can be moved to theclosest boundary. In the case of the former alternative the boundary layer can be said to have absorbed thevortex. The reaction must; accordingly, be a change in circulation around the corresponding wing. In thisproject the latter option has been chosen for the simplicity of implementation.

3.8 Routh’s RuleThe convection of vortex particles is complicated by the use of conformal mappings to transform the do-main. A condition for the relation between velocities in the two planes (3.19) to be valid is that vs isirrotational at the point s. This condition is satisfied everywhere except on the vortex particles themselves.

The velocity of a vortex is determined by the velocity induced at it’s center by all other causes exceptthe vortex itself. Thus, to derive the velocity in the physical plane it is logical to start with the complexflow potential in the physical plane. It is; hence, assumed that a vortex is located at zv = f (sv). Then theinvariance of the complex flow potential, w, can be written as

w(s)≡ wsv(s)+iγv

2πlog(s− sv) = wzv(z)+

iγv

2πlog(z− zv)≡ w(z)

where wzv(z) and wsv(s) denotes the complex flow potential induced from all other causes except the vortexwith center at zv and sv respectively. From this expression Clements [36] used a Taylor series expansion toshow that the velocity of the vortex can be written as

vzv =dwzv

dz

∣∣∣∣zv

=dwsv

ds

∣∣∣∣sv

dsdz

∣∣∣∣zv

+iγv

2π

12

d f−1

dz

∣∣∣zv

d f−1

dz

∣∣∣zv

(3.26)

Using the implicit function theorem (3.26) is rewritten as

vzv =dFzv

dz

∣∣∣∣zv

=

(dFsv

ds

∣∣∣∣sv

− iγv

2π

12

f ′′(sv)

f ′(sv)

)1

f ′(sv)(3.27)

However, the algorithm for satisfying the boundary conditions is using the circle plane. Therefore it

24

would be convenient if the velocity of the vortices in this plane also could be found. We write

vs = limh→0

f−1(zv + vzh)− f−1(zv)

h

= limh→0

f−1(zv)+ vzhd f−1

dz

∣∣∣zv+ 1

2 v2z h2 d2 f−1

dz2

∣∣∣zv+O(h3)− f−1(zv)

h

= vzd f−1

dz

∣∣∣∣zv

= vz1

f ′(sv)(3.28)

Finally, by inserting (3.27) in (3.28) the velocity of the vortex particles, expressed in the circle place isobtained

vs =

(dFsv

ds

∣∣∣∣sv

− iγv

2π

12

f ′′(sv)

f ′(sv)

)1

| f ′(sv)|2(3.29)

3.9 Convection in Circle PlaneEquation (3.28) gave the relationship between how a vortex move in the physical plane and how the corre-sponding vortex moves in the circle plane. It is obvious that the two expressions

zv,k(t) =∫ t

0vz(zv,k(t ′))dt ′ (3.30)

sv,k(t) =∫ t

0vs(sv,k(t ′))dt ′ (3.31)

are equivalent in the sense thatf (sv,k(t)) = zv,k(t) ∀t > 0

However, this is not necessarily true for a numerical integration scheme. By comparing the predicted motionof a single vortex around an airfoil it was discovered that (3.31) gave predictions of higher quality for thesame time-step. An illustration of this interesting fact can be viewed in figure (3.6)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Physical plane ∆t=0.01Circle plane ∆t=0.1Physical plane ∆t=0.1Panel method ∆t=0.01

Figure 3.6: Single vortex moving in counter-clockwise path around a Joukowski airfoil using the midpoint integrationmethod and three different ways of calculating the velocity

The experiments visualized in figure 3.6 indicate that (3.31) can be used to calculate the paths of thevortices instead of (3.30) without disadvantage. In this particular case it is interesting to note that (3.31)even performs better than (3.30) at the specified time-step. Why this is so, and whether or not this is ageneral property remains to investigate.

25

Chapter 4

Numerical Validation of the Model

In this chapter the validity of the flow model is examined by comparing computational results with valuesobtained by experiments and other models.

4.1 Steady flowThe steady flow around a NACA0018 airfoil has been computed, as a first means for validation of the model.For this simple case it is sufficient to compare the results with those of the well known XFoil code withviscous modeling enabled[37]. The most suitable parameter for comparison is the pressure coefficient thatis defined as

Cp =p− p∞

12 ρV 2

∞

where p∞ V∞ and ρ∞ are the undisturbed pressure, velocity and density respectively. The results are shownin figure 4.1. The agreement is excellent.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−Cp

chord

Present modelXfoil

Figure 4.1: Streamlines and pressure coefficient over a NACA0018 airfoil. Angle of attack 8 degrees. Re = 6 ·105

§11

4.2 Blade LoadsThe operational loads on the turbine blades has been computed as a means for the validation of the completenumerical model.

The forces can be calculated numerically from integration of the unsteady Bernoulli equation.

p =−ρ

(|v|2

2+

∂φ

∂t

)+ p0 (4.1)

26

where v denotes the velocity relative to the airfoil and ∂φ

∂t is the time derivative of the velocity potential.When the chord-radius ratio is small it is believed that the ∂φ

∂t term can be neglected. For example, in thecase of a single blade at tip-speed-ratio five and c/R = 0.25 the error of this approximation is small (seefigure 4.2). None-the-less this can be improved in a future version of the code.

0 100 200 300 400−35

−30

−25

−20

−15

−10

−5

0

5

10

15

Azimutal angle (deg) − 1080

Non

−dim

ensi

onal

nor

mal

forc

e

Exact formulaUnsteady BernoulliBernoulli relative velocity

(a)

0 100 200 300 400−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Azimutal angle (deg) − 1080

Non

−dim

ensi

onal

tang

entia

l for

ce

Exact formulaUnsteady BernoulliBernoulli relative velocity

(b)

Figure 4.2: Comparison of blade loads calculated with an exact formula; unsteady Bernoulli principle and staticBernoulli principle

We thus calculate the pressure force on a wing by

Fn =−ρ

∫∂Tn

|v|2

2nds

By convention this is then divided in tangential and normal parts using a cylindrical coordinate system.

FN,n = Fn · r (4.2)

FT,n = Fn · θ (4.3)

the force FT is providing the torque that produces power. The normal force is important for the study ofthe structural loads on the turbine. For ease of comparison the forces are then nondimensionalized by the

27

dynamic pressure of the undisturbed wind speed, blade chord and blade length

CN =FN,n

12 ρv2

∞cL(4.4)

CT =FT,n

12 ρv2

∞cL(4.5)

The aerodynamic efficiency is defined as the power excerpted to the shaft divided by the power in thewind that flows through the projected frontal area.

Cp =P

12 ρDLV 3

∞

=T β

12 ρDLV 3

∞

=

where P denotes the total power delivered to the shaft whereas T denotes the average torque on the shaft.Although it has not been done in this diploma work. It is possible to estimate the losses due to the finite

span of the blades through the formulas for lift- and drag coefficients of finite span wings [38]

CL =Clo

1+ a0πAR

CD =Cdo +C2

LπAR

where AR is the aspect ratio of the blade (length divided by chord) and Clo and Cdo are the infinite span lift-and drag coefficients respectively. a0 is the slope of the CL versus α graph. For a rectangular blade one canwrite

a0 = 1.8π(1+0.5t/c)

where t is the thickness of the blade.

4.3 Strickland/Klimas TurbineSandia National Laboratory built several straight-bladed experimental giromill that was operated in a towtank. The performance of one of them is included by Klimas [39]. The device was equipped with straingages on the blades that were able to record the operating loads continuously. The details of the setup canbe viewed in the following table.

Wing-section NACA0012Chord 9.14 cmNumber of blades 2Turbine Radius 0.61 mTurbine Height 1.1 mTip-speed-ratio 5Blade Reynolds Number 40000Attachment point Presumably 1/4-chord

The results of the simulation can be found in figure 4.3. The agreement is good. It should be notedthat the agreement is somewhat better than could be expected considering that viscous effects have not beenaccounted for in the simulation. This could perhaps be due to measurement uncertainty.

4.4 Strickland/Oler TurbineAnother Sandia turbine was presented in [40]. As with the previous setup the device was equipped withstrain gages that were able to record data of the operating loads continuously. The turbine had one NACA0015blade mounted at the center chord.

28

50 100 150 200 250 300 350 400 450−40

−35

−30

−25

−20

−15

−10

−5

0

5

Azimutal Angle (deg)

Non

−Dim

ensi

onal

Nor

mal

For

ce

SimulationExperimental

50 100 150 200 250 300 350 400 450

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Azimutal Angle (deg)

Non

−Dim

ensi

onal

Tan

gent

ial F

orce

SimulationExperimental

Figure 4.3: Forces on one of the blades of the Strickland/Klimas turbine after the third revolution

29

Wing-section NACA0012Chord 15.24 cmNumber of blades 1Turbine Radius 0.61 mTurbine Height 1.1 mTip-speed-ratio 5.1Blade Reynolds Number 67000Attachment point 1/2-chord

The results of the simulation can be found in figure 4.4. The agreement is fair for the normal force.There is a notable phase shift. This is thought to be an effect of neglecting ∂φ

∂t in the unsteady Bernoulli(4.1). The disagreement in the tangential force is greater than in the normal force. This is natural because thetangential force is affected more severely by three error sources: 1) The tangential force is one magnitudesmaller than the normal force. Therefore it is more sensitive to errors in measurement. 2) Drag has beenneglected in the simulation. And 3) the three dimensional effects have been neglected in the simulation.

0 50 100 150 200 250 300 350

−35

−30

−25

−20

−15

−10

−5

0

5

10

15

Azimutal Angle (deg)

Non

−Dim

ensi

onal

Nor

mal

For

ce

SimulationExperimental

50 100 150 200 250 300 350−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Azimutal Angle (deg)

Non

−Dim

ensi

onal

Tan

gent

ial F

orce

SimulationExperimental

Figure 4.4: Forces on one of the blades of the Strickland/Oler turbine after the third revolution

30

Chapter 5

Application to the Marsta 12kWExperimental Wind Turbine

The Center for Renewable Electric Energy Conversion at Uppsala University has constructed an experi-mental H-rotor type VAWT, which is located in Marsta, a few kilometers north of Uppsala (see figure 5.1).Performance predictions were made for the Marsta turbine, as an application of the numerical model.

Figure 5.1: Uppsala University 12 kW experimental Wind turbine

During the numerical experiments it was found that it takes longer than expected for the blade forces tonormalize. If the Cp is calculated from the forced during the 6th revolution the calculation yields a Cp of0.684. If, on the other hand the forces are calculated during the 25th revolution the Cp is the more realistic(because inviscid flow has been assumed) 0.537, which is well under the Betz limit. A plot of the evolutionof average torque can be viewed in figure 5.4. An interesting consequence of the high initial torque is thatwe can expect a slight increase in generated power when the wind is changing direction. This effect shouldprobably be investigated further.

From figure 5.2(b) that depicts the velocity field of the developed wake of the Marsta turbine. Thewake becomes unstable after approximately 40 meter, which accounts to about seven rotor diameters. Thisis interesting because the simulation provides an upper limit for this distance. In reality the wake is alsoweakened by shear induced turbulence at all sides of the wake (including from underneath and from above).

Figure 5.3 shows varying nature of the blade forces during fourteen revolutions. In addition, figure 5.5

31

shows the varying force acting on the shaft. Note the net crosswind force.

Wing-section NACA0021Chord 25 cmNumber of blades 3Turbine Radius 3 mTurbine Height 5 mTip-speed-ratio 4Wind speed 12 m/sAttachment point Quarter chord

0 10 20 30 40 50 60 70 80 90 100

−10

−5

0

5

10

(a) Vortex particles

0 20 40 60 80 100 120 140

−10

−5

0

5

10

2

4

6

8

10

12

14

16

(b) Velocity field

Figure 5.2: Vortex distribution and velocity field in the wake of Marsta VAWT, TSR=4

32

14500 15000 15500 16000 16500 17000 17500 18000 18500 19000

0

0.5

1

1.5

2

2.5

3

3.5

Azimutal Angle (deg)

Non

−Dim

ensi

onal

Tan

gent

ial F

orce

14500 15000 15500 16000 16500 17000 17500 18000 18500 19000

−25

−20

−15

−10

−5

0

5

10

Azimutal Angle (deg)

Non

−Dim

ensi

onal

Nor

mal

For

ce

17200 17400 17600 17800 18000 18200 18400 18600

0.5

0.75

1.0

1.25

1.5

Azimutal Angle (deg)

Sha

ft To

rque

(kN

m)

Figure 5.3: Blade force on a single blade of the Marsta VAWT, TSR=4, after 40 revolutions

33

0 5 10 15 20 250

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Revolution

Sha

ft To

rque

(kN

m)

Instantanious torqueTorque averaged over current revolution

Figure 5.4: Simulation of the evolution of average shaft torque on the Marsta Turbine starting impulsively with TSR=4.For performance estimations it is necessary to simulate at least 20 revolutions unless the asymptote can be estimated ina reliable way.

7600 7700 7800 7900 8000 8100

−0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Azimutal angle (deg)

Forc

e (k

N)

Windwise forceCrosswind force

Figure 5.5: Simulation of forces acting on the the shaft on the Marsta Turbine.

34

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

0.48

0.5

0.52

0.54

0.56

0.58

0.6

Tip−Speed−Ratio

Aer

odyn

amic

effi

cien

cy (C

p)

1−bladed Nc/R=0.152−bladed Nc/R=0.23−bladed Nc/R=0.254−bladed Nc/R=0.2

Figure 5.6: Calculated aerodynamic efficiency for a number of straight bladed VAWTs. All turbines have symmetricblade sections. The 3-bladed turbine is the Marsta Turbine. The calculations were done neglecting drag.

35

Chapter 6

Conclusions

A tool for aerodynamic simulation of straight bladed vertical axis wind turbines has been developed. Thevalidity of the model relies on the assumption that there is no boundary layer separation at the blades. Suchan assumption can be justified when the tip-speed-ratio is larger than 4. At lower tip-speed-ratios the largerangle of attack experienced by the blades, leads to stall phenomena. Fortunateley for low tip-speed-ratiosthe streamtube models are quite reliable. Thus the two types of models complement each other.

As shown, for example, in figure 4.3 the model predictions compares well with experimental data. Tofurther increase the accuracy of the model there are a number of effects that can be easily included. Theeffect of the skin friction drag from the struts and blades could be included. This effect plays a larger rolefor small experimental turbines than for full-sized turbines that operates at much larger Reynolns numbers.

For future versions of the code it is also good to implement force calculations with the full UnsteadyBernoulli Equation, instead of the steady Bernoulli approximation used in 5.

There are several specific aspects of the aerodynamic behavior that deserves special attention. Many ofthese aspects can be addressed with a free vortex code. For example, it is possible to study the performanceof VAWTs with different configurations as in figure 5.6. As can be seen from figure 5.3 that is displayingthe shaft torque acting on the Marsta turbine. The maximum torque is as much as three times the minimumtorque over one revolution. This calls for an overdimensioned shaft in order to cope with fatigue relatedproblems. The same argument is of course valid also for the blades. It is believed that the variance can bedecreased by varying the pitch of the blades.

Because the developed tool is capable of studying this effect. It is believed that the outcome of thisdiploma work might be of practical use in the future.

It is beneficent to move the vortex particles in the circle plane, as was found in chapter 3.6. Whether ornot this is a general property has not been investigated in this project.

Even though very little effort has been put into optimizing the code for speed. The resulting code stillperforms very well in comparison to other methods capable of modeling the wake structure. This is due tothe inherent mesh free features of vortex methods and the use of the efficient Fast Multipole Method forsolving the resulting N-body problem. In addition, the speed is due to the access to some very fast routineswritten in C99. Also, convergence properties of the algorithm is perceived to be good. Even for complicatedgeometries it most often sufficient with less than 10 iterations to reach convergence, as illustrated in figure3.4.

Nonetheless, in future versions of the code some effort ought to be put in optimizing the code for speed.This is mainly due to the findings in chapter 5 where it was found that it takes more than 20 revolutionsbefore the blade forces normalize. There are several ways to deal with the performance issue. One idea isto let the code build up the large structures first using a larger time-step. Only then could the time-step beprogressively refined. Another obvious way is to optimize the code.

36

Acknowledgements

I would like to thank my supervisor Paul Deglaire. He concieved the idea of this model and he has beenconstantly willing to take time to explain the theory and discuss new ideas. I would also like to thank Dr.Hans Bernhoff, who is leading the wind power programme at Uppsala University and was the initiator ofthis MSc-project. Without his help the project would not have been realized.

Professor Mats Leijon is also acknowledged for welcoming me at the department.Also, I would like to express my gratitude to the department’s MSc students who has given me insight

into their projects and helped me with mine.

37

Appendix A

Method of Images

A.1 The Circle TheoremMilne-Thompson [20] shows how to, given a potential F(s) create a new potential F ′(s) in which the circless = b is a streamline but with the asymptotic behavior of F(s). He gives the formula:

F ′(s) = F(s)+F(b2

s)

For our purposes it is necessary to extend the theorem slightly. We want to be able to introduce a circleanywhere in the s-plane. I here present a modified theorem. The proof closely follows the proof of theoriginal theorem in Milne-Thomson.

Theorem A.1.0.1 (Circle Theorem). Let there be irrotational two-dimensional flow of incompressible in-viscid fluid in the s-plane. Let there be no rigid boundaries and let the complex potential of the flow be F(s),where the singularities of F(s) are all at a distance greater then b from the point sc. If a circular cylinderC, |s− sc|= b is introduced into the field of flow the complex potential becomes

F ′(s) = F(s)+F(b2

s− sc+ sc)

Proof. We proceed exactly as in Milne-Thomson. On the circle we have that (s− sc)(s− sc) = b2. Then,on the circle we thus have:

F ′(s) = F(s)+F(b2

s− sc+ sc)

= F(b2

s− sc+ sc)+F(

b2

b2

s−sc+ sc− sc

+ sc)

= F(b2

s− sc+ sc)+F(s− sc + sc)

= F(b2

s− sc+ sc)+F(s)

= F(b2

s− sc+ sc)+F(s)

= F ′(s)

We conclude that Im(F ′(s)) = 0 = const. Hence the circle is a streamline.Since all the singularities of F(s) are by hypothesis exterior to C, all the singularities of F( b2

s−sc+ sc)

are interior to C; in particular F( b2

s−sc+ sc) has no singularity at infinity since F(s) has none at s = 0. Thus

F ′(s) has exactly the same singularities as F(s) and so all the conditions are satisfied.

38

A.2 Image of a VortexA point vortex at s = sv induce the complex potential

F(s) =iγ2π

log(s− sv)

If we introduce the circle C, |s− sc|= b the circle theorem yields

F ′(s) =iγ2π

log(s− sv)−iγ2π

log(b2

s− sc+ sc− sv) (A.1)

We can rewrite (A.1) as

F ′(s) =iγ2π

log(s− sv)−iγ2π

log(b2 +(s− sc)(sc− sv)

s− sc)

=iγ2π

log(s− sv)−iγ2π

log(b2 +(s− sc)(sc− sv))+iγ2π

log(s− sc)

=iγ2π

log(s− sv)−iγ2π

log(b2

sc− sv+ s− sc)+

iγ2π

log(s− sc)+ const

The potential is only defined up to a constant we can therefore safely disregard the constant part andredefine F ′(s) to be:

F ′(s) =iγ2π

log(s− sv)−iγ2π

log(b2

sc− sv+ s− sc)+

iγ2π

log(s− sc)

=iγ2π

log(s− sv)−iγ2π

log(s− (b2

sv− sc+ sc))+

iγ2π

log(s− sc)

The inverse point s∗ is defined such that (s∗− sc)(s− sc) = b2. We see that the image system of a vortexis a vortex at the center with strength γ and and a vortex at the inverse point with strength −γ.

39

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42