EFFECTS OF UNSTEADY AERODYNAMICS ON VERTICAL-AXIS

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understanding, while having to share my time and attention with this work. For that, .... axis wind turbine (HAWT), drag-based Savonius design, and the lift- ... flow passes through the turbine, with the greatest drops occurring ..... the turbine, the blade pushes against the flow and slowing it down. ...... Air Vehicle Applications.
EFFECTS OF UNSTEADY AERODYNAMICS ON VERTICAL-AXIS WIND TURBINE PERFORMANCE

BY PETER KOZAK

Submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical and Aerospace Engineering in the Graduate College of the Illinois Institute of Technology

Approved Advisor

Chicago, Illinois May 2014

ACKNOWLEDGMENT I would like to thank my advisor, Professor Dietmar Rempfer, for his guidance and support. Many of the ideas and concepts contained in this thesis originate from our discussions, for which I am tremendously grateful. I would also like to express my gratitude to Professors David Williams and Kevin Meade for taking the time and effort to serve on my thesis committee. My colleagues in the Fluid Dynamics Research Center and IIT in general are owed a debt of gratitude for their willingness to serve as sounding boards for my ideas. Throughout this research, I’ve relied on a great deal of encouragement from my family and friends. Over the past year, they have shown incredible patience and understanding, while having to share my time and attention with this work. For that, I am truly thankful.

iii

TABLE OF CONTENTS Page ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . . .

iii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiv

CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . .

1

1.1. Background . . . . . . . . . . . . . . . . . . . . . . . 1.2. Previous Research . . . . . . . . . . . . . . . . . . . . 1.3. Scope . . . . . . . . . . . . . . . . . . . . . . . . . .

1 16 25

2. METHODOLOGY AND VALIDATION . . . . . . . . . . . .

28

2.1. Finite Volume Simulation . . . . . . . . . . . . . . . . . 2.2. Estimating the Effective Angle of Attack . . . . . . . . .

28 41

3. OBSERVATIONS OF UNSTEADY EFFECTS . . . . . . . . .

48

3.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Performance Limiting Phenomena . . . . . . . . . . . . .

48 58

4. TURBINE PERFORMANCE IMPROVEMENTS

. . . . . . .

67

4.1. Procedure for Turbine Optimization . . . . . . . . . . . . 4.2. Fixed Non-Zero Blade Pitch . . . . . . . . . . . . . . . . 4.3. Variable Pitch . . . . . . . . . . . . . . . . . . . . . .

67 69 73

5. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . .

85

5.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Future Research Topics . . . . . . . . . . . . . . . . . .

85 90

APPENDIX

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

A. SUGGESTED CRITERIA FOR VAWT SIMULATIONS . . . . A.1. Overset Stability Criteria . . . . . . . . . . . . . . . . . A.2. Accurate Wall Treatment Criteria . . . . . . . . . . . . .

92 93 93

iv

B. EFFECTIVE ANGLE OF ATTACK FOR SEVERAL AIRFOILS . . . . B.1. NACA 0012 . . . . . . . . . B.2. NACA 0015 . . . . . . . . . B.3. NACA 0021 . . . . . . . . .

CORRELATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CURVES . . . . . . . . . . . . . . . . . . . .

C. EQUATIONS FOR BLADE PITCH C.1. Iteration 1 . . . . . . . . . . C.2. Iteration 2 . . . . . . . . . . C.3. Iteration 3 . . . . . . . . . .

. . . .

. . . .

. . . .

101 102 103 103

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

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94 95 97 99

LIST OF TABLES Table

Page

1.1

Advantages / Disadvantages of VAWT Analysis Methods

. . . . .

18

2.1

VAWT Geometry . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.2

VAWT Simulation Parameters

. . . . . . . . . . . . . . . . . .

37

2.3

Oscillating Airfoil Simulation Parameters . . . . . . . . . . . . .

45

4.1

Pitch Offset Performance . . . . . . . . . . . . . . . . . . . . .

69

4.2

Harmonic Pitch Cases

. . . . . . . . . . . . . . . . . . . . . .

76

4.3

Harmonic Pitch Performance . . . . . . . . . . . . . . . . . . .

77

4.4

Variable Pitch Performance . . . . . . . . . . . . . . . . . . . .

80

5.1

VAWT Performance Evaluation . . . . . . . . . . . . . . . . . .

88

B.1 NACA 0012 CP Data . . . . . . . . . . . . . . . . . . . . . . .

95

B.2 NACA 0015 CP Data . . . . . . . . . . . . . . . . . . . . . . .

97

B.3 NACA 0021 CP Data . . . . . . . . . . . . . . . . . . . . . . .

99

C.1 Variable Pitch 1 (N=8) . . . . . . . . . . . . . . . . . . . . . .

102

C.2 Variable Pitch 2 (N=5) . . . . . . . . . . . . . . . . . . . . . .

103

C.3 Variable Pitch 3 (N=6) . . . . . . . . . . . . . . . . . . . . . .

103

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LIST OF FIGURES Figure 1.1

Page The contributions of all electricity sources in the United States during the 2012 year [82]. Due to recent growth, wind energy is now responsible for 3.5% of US electricity production, more than any other non-hydroelectric renewable source. . . . . . . . . . . . .

2

Histogram showing exponential growth of wind energy’s contribution to the US electricity production [81]. . . . . . . . . . . . .

3

The major wind turbine types including the propeller-type horizontalaxis wind turbine (HAWT), drag-based Savonius design, and the liftbased Darrieus and H-rotor vertical-axis wind turbines (VAWTs). VAWT diagrams originate from (Eriksson, 2008) [23]. . . . . . . .

4

The left subfigure shows the 2-D geometry and layout of a Darrieus turbine or H-rotor. The blades are typically mounted at the quarter or half-chord point equidistant from the others. The sub-figure on the right demonstrates the velocity components and aerodynamic forces on one of the blades located at an azimuthal angle of 90o . .

5

The maximum angle of attack present at an azimuthal angle of 90o for a typical range of tip-speed ratios for a vertical-axis machine. .

7

Example of net torque curves for a single blade (a) and the entire turbine (b) for a 3-blade VAWT at a tip-speed ratio of TSR = 2.5

9

2-D variation in the time averaged U-velocity for at turbine operating at TSR = 3.0. The figure shows that kinetic energy is lost as the flow passes through the turbine, with the greatest drops occurring as the flow passes through the blade path. There is also a variation in the speed in the vertical direction. At the top of the turbine, the blade pushes against the flow and slowing it down. At the bottom of the figure, the blade pulls the flow faster. . . . . . . . . . . .

10

Actuator disk representation of a wind turbine. State 0 and State 3 are the far field conditions upstream and downstream and the States 1 and 2 are the flow in the vicinity (but outside) of the turbine. . .

11

Vertax Wind Ltd. proposed multi-megawatt turbines. These seabased turbines would rely on fewer moving parts than horizontal-axis machines, allowing a longer lifespan and less maintenance [39]. . .

15

1.10 A VAWT blade oriented with a positive (tow out) pitch β. . . . .

22

1.2 1.3

1.4

1.5 1.6 1.7

1.8

1.9

vii

2.1

Layout of (a) the physical domain of the FVM simulation with boundary conditions and (b) the overset grids surrounding the turbine blades. . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.2

Boundary between the overset and underset grids. . . . . . . . .

31

2.3

Fully elliptic overset grid containing a VAWT blade surface.

. . .

32

2.4

Topology of the overset grid layout used to maximize orthogonality of the element edges. . . . . . . . . . . . . . . . . . . . . . .

33

Closeup view of the overset grid near the blade’s surface, including the boundary layer. . . . . . . . . . . . . . . . . . . . . . . .

34

2.6

View of the overset grid near the turbine blades trailing-edge. . . .

35

2.7

Plots comparing the average power coefficient with respect to tipspeed ratio for (a) all simulations and (b) only low Y+ wall treatment FVM simulations. . . . . . . . . . . . . . . . . . . . . .

39

Lift coefficient with respect to angle of attack (neglecting stall) for a NACA 0015 airfoil obtained using JavaFoil [34]. . . . . . . . .

42

Pressure coefficient with respect to location along the chord line at angles of attack between 0 and 12o for a NACA 0015 airfoil obtained using JavaFoil [34]. . . . . . . . . . . . . . . . . . . . . . . .

43

2.10 Correlation curves of the effective angle of attack with respect to the pressure coefficient ratio at 0.20 c. . . . . . . . . . . . . .

44

2.11 Physical domain of the rotating NACA 0015 simulation with boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . .

45

2.12 Validation of the oscillating airfoil simulation, comparing the (a) lift and (b) drag with Piziali’s experimental results. . . . . . . . . .

46

2.13 Effective angle of attack with respect to the geometric angle of attack for Piziali (corrected for Re = 300,000) using the lift and that of the FVM simulation using the pressure ratio. . . . . . . . . . . . .

47

2.5

2.8 2.9

3.1 3.2 3.3

Average power coefficient with respect to tip-speed ratio for the FVM simulation. . . . . . . . . . . . . . . . . . . . . . . . .

48

Blade torque coefficient with respect to azimuthal angle for various tip-speed ratios. . . . . . . . . . . . . . . . . . . . . . . . .

49

Turbine torque coefficient with respect to azimuthal angle for various tip-speed ratios. . . . . . . . . . . . . . . . . . . . . . . . . .

50

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3.4

Effective angle of attack with respect to the azimuthal angle for various tip-speed ratios. Sudden drops in magnitude correspond to full leading-edge separation and the jumps correspond to reattachment.

52

The sequence shows the vorticity magnitude distribution as the turbine passes through one third of a cycle at a tip-speed ratio of TSR = 3.0. The turbine maintains fully attached flow throughout the entire cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

The sequence shows the vorticity magnitude distribution as the turbine passes through one third of a cycle at a tip-speed ratio of TSR = 2.0. Around the 180o point, the blades undergo separation and reattachment. . . . . . . . . . . . . . . . . . . . . . . . . . .

54

Average power coefficient curves for the BEM model and the FVM simulation, with respect to tip-speed ratio. . . . . . . . . . . . .

56

Comparison of the effective angle of attack for the streamtube (BEM) model and the FVM simulation. . . . . . . . . . . . . . . . . .

57

Close up view of the vorticity magnitude distribution around a turbine blade at an azimuthal angle of 180o . (TSR = 2.0) Near the end of the half-cycle, the blade stalls and the separation bubble is swept away as the angle of attack changes from negative to positive. . .

59

3.10 Two views of the flow downstream of a VAWT blade at an azimuthal angle of 0o and for a tip-speed ratio of TSR = 2.0. . . . . . . . .

62

3.11 Close up view of the vorticity magnitude distribution for tip-speed ratios of TSR = 2.0 and 3.0, demonstrating the higher likelihood of blade / wake interaction at higher tip-speed ratios. . . . . . . . .

63

3.12 Close up view of the vorticity magnitude distribution around a turbine blade as it passes through wake structures. (TSR = 3.0) . . .

65

4.1

Effective angle of attack curve for the TSR = 2.0 case. . . . . . .

67

4.2

Torque curve for the TSR = 2.0 case.

67

4.3

The procedure used to develop an effective variable blade pitch regime.

68

4.4

Comparison of the effective angle of attack for several constant blade pitches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

Comparison of the blade torque coefficient for several constant blade pitches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

Target angle of attack with respect to the azimuthal angle. . . . .

73

3.5

3.6

3.7 3.8 3.9

4.5 4.6

ix

. . . . . . . . . . . . . .

4.7

Blade pitch vs azimuthal angle for the three iterations. . . . . . .

74

4.8

Harmonic variable blade pitch curve with a maximum pitch angle of 5o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

Blade torque coefficients with respect to azimuthal angle for the two harmonic variable blade pitch cases. . . . . . . . . . . . . . . .

78

4.10 Effective angle of attack with respect to azimuthal angle for the two harmonic variable blade pitch cases. . . . . . . . . . . . . . . .

79

4.11 Comparison of the effective angle of attack for the several variable blade pitch iterations. . . . . . . . . . . . . . . . . . . . . . .

81

4.12 Comparison of the blade torque coefficients for the several variable blade pitch iterations. . . . . . . . . . . . . . . . . . . . . . .

82

4.13 Comparison of the blade torque coefficients for the third iteration variable pitch curve and the zero pitch case. . . . . . . . . . . .

83

4.14 Comparison of the effective angle of attack curve for the third iteration variable pitch case and the desired angle of attack. . . . . .

84

4.9

5.1

Average power coefficients for several VAWT cases compared to a Mod-5B horizontal-axis machine [75]. . . . . . . . . . . . . . .

89

B.1 NACA 0012 airfoil shape. . . . . . . . . . . . . . . . . . . . .

95

B.2 Pressure coefficient ratio versus angle of attack for the NACA 0012 blade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

B.3 NACA 0015 airfoil shape. . . . . . . . . . . . . . . . . . . . .

97

B.4 Pressure coefficient ratio versus angle of attack for the NACA 0015 blade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

B.5 NACA 0021 airfoil shape. . . . . . . . . . . . . . . . . . . . .

99

B.6 Pressure coefficient ratio versus angle of attack for the NACA 0021 blade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100

x

LIST OF SYMBOLS Symbol

Definition

Roman A AR BEM c

Cross sectional or ”swept” area of the turbine Aspect ratio Blade element method Blade chord length

CD

Drag coefficient

CL

Lift coefficient

Co/u

Overset slip condition number

Cp

Pressure coefficient

CP

Power coefficient

CT

Torque coefficient

CFD DDomain

Computational fluid dynamics Physical extent of the flow in the x-direction

DRotor

Turbine rotor diameter

FVM

Finite volume method

F+ f

Reduced frequency Frequency

HDomain

Physical extent of the flow in the y-direction

HAWT

Horizontal-axis wind turbine

i LEV

Iteration step Leading-edge vortex xi

lo

Characteristic length of a boundary element of an overset grid

lu

Characteristic length of a boundary element of an underset grid

n

Number of turbine blades

N

Number of Fourier terms

p

Pressure

P

Power

R

Turbine rotor radius

r

Residual

RANS Re SST T

Reynolds averaged Navier-Stokes turbulence model Reynolds number Shear stress transport turbulence model Torque

TEV

Trailing-edge vortex

TSR

tip-speed ratio

u

Velocity component in the x-direction



Friction velocity at the wall

Uinduced U∞ VAWT

Velocity component induced by blade rotation Free-stream velocity Vertical-axis wind turbine

x

Direction in-line with the free-stream velocity and corresponding to an azimuthal angle of 0o

y

Direction corresponding to an azimuthal angle of 90o

xii

+ Ywall

Turbulent flow parameter used to quantify how well the boundary layer is resolved

Greek α αef f ective

Angle of attack Effective angle of attack

αmax

Maximum angle of attack

β (θ)

”Tow out” blade pitch

βo

Constant ”tow out” blade pitch offset

Γ

Circulation



Turbulence production

∆t

Simulation time step

θ

Azimuthal angle corresponding to the location of a VAWT blade in the turbine cycle

κ

Turbulent kinetic energy

ν

Kinematic velocity

σ

Turbine solidity

φ

Any arbitrary quantity



Turbine angular velocity

ω

Vorticity

Miscellaneous 6 skew

Skew angle

xiii

ABSTRACT Vertical-axis wind turbines (VAWTs) offer an inherently simpler design than horizontal-axis machines, while their lower blade speed mitigates safety and noise concerns. As a result, VAWTs can be used to open up more populated areas for large-scale wind energy development. While vertical-axis turbines do offer significant operational advantages, development has been hampered by the difficulty of modeling the aerodynamics involved, along with their rotating geometry. This thesis presents results from a simulation of a baseline VAWT computed using Star-CCM+, a commercial finite volume (FVM) code. Overset grid techniques are used to model the VAWT’s complex and moving geometry. VAWT aerodynamics are shown to be dominated at low tip-speed ratios by dynamic stall phenomena and at high tip-speed ratios by wake-blade interactions, using flow visualization and blade angle of attack. An iterative procedure to optimize the VAWT’s geometry is developed using blade pitch to mitigate the adverse effects of dynamic stall for a tip-speed ratio of 2.0 case. Relying on both a constant blade pitch offset as well as a variable blade pitch as a function of azimuthal angle, power output was shown to be increased by 17% and 38%, respectively, compared to the baseline case. Emphasis is placed on the modeling techniques used in the FVM simulation and the optimization process.

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1 CHAPTER 1 INTRODUCTION 1.1 Background Global energy consumption has expanded since 1980 and remains closely related to quality of life issues such as clean water access, longevity, and personal income levels [58]. As a result, new energy sources must be developed in order to meet demand for a growing world population. Energy production in the United States has traditionally come from a variety of sources including petroleum, coal, natural gas, nuclear fission reactors, and hydroelectric dams. For most of the last decade, coal fired power plants have supplied over 70% of the US’s electricity, but recent increases in the cost of coal as well as concerns about carbon emissions have caused that contribution to drop to 57% in 2012 [19]. Likewise, uncertainty over the supply and cost of petroleum has discouraged consumption. New extraction methods, including horizontal drilling and hydraulic fracturing, have resulted in a significant decrease in the price of natural gas, bringing up its relative contribution to the US electricity supply. Nevertheless, concerns over carbon emissions and the exhaustion of easily exploitable gas reserves will eventually contribute to future rises in the price of natural gas [48]. Over the next few decades, the proportional contribution from hydroelectric sources is expected to decrease simply because most profitable sites in the US have already been exploited. Growth in nuclear power plants is uncertain due to concerns regarding waste disposal, safety, and high capital costs [19]. These recent developments in the last 20 years have made wind energy sources comparatively more attractive than in previous years. The profitability of wind energy is highly dependent on the cost and availability of fossil fuel derived sources. In the 1970s and 1980s there was a boom in wind turbine development due to energy shortages at the time [48]. However, as the price

2

Figure 1.1. The contributions of all electricity sources in the United States during the 2012 year [82]. Due to recent growth, wind energy is now responsible for 3.5% of US electricity production, more than any other non-hydroelectric renewable source. of fossil fuels dropped in the 1990s, commercial wind turbines fell out of use. This resulted in the failure of all but a handful of wind energy developers throughout the 1990s [75]. Rapidly growing global energy demand, political instability in energy rich nations, and environmental concerns have lowered the relative cost of wind energy, as evidenced by exponential increases in wind energy investments over the last decade. Over the last five years, wind power generation has risen steadily by about 30% every year [57]. This growth is not expected to slow in the near future with wind providing an estimated 14% [19] to 20% [48] of US electricity by the year 2030. Wind turbines may be categorized by their axes of rotation into vertical-axis wind turbines (VAWTs) and horizontal-axis wind turbines (HAWTs), as well as by the dominant aerodynamic forces used to provide a turbines power including lift-based and drag-based turbines. HAWTs are lift-based machines that operate similarly to an airplane propeller in reverse. The set of VAWTs currently undergoing commercial development is much more diverse. These include drag-based Savonius turbines which

3

Figure 1.2. Histogram showing exponential growth of wind energy’s contribution to the US electricity production [81]. use scoops to capture the wind and generate torque [57]. Other VAWTs represent lift-based machines such as the curved-blade Darrieus turbines which are optimized for the centrifugal loads on the blades and the straight bladed H-rotor [56]. Hrotors are unique in that they combine the benefits of simple manufacturing along with the ability to use individual pitch control of the blade geometry. Except for small applications, lift-based turbines are exclusively used due to their enhanced efficiency [75]. The first windmills invented in Persia were of the drag-based, vertical-axis variety which were used for simple purposes such as grinding and water pumping. Up until this century, the majority of wind powered machines made use of dragbased horizontal-axis designs [75]. Logically, this historical dominance of horizontalaxis machines may explain why there has been comparatively less development and

4

Figure 1.3. The major wind turbine types including the propeller-type horizontalaxis wind turbine (HAWT), drag-based Savonius design, and the lift-based Darrieus and H-rotor vertical-axis wind turbines (VAWTs). VAWT diagrams originate from (Eriksson, 2008) [23]. interest in VAWTs. During the energy crises of the 1970s and 1980s, development began on industrial scale VAWTs as the US and Canada began searching for viable alternatives to fossil fuels. This research culminated in the construction of over 600 large-scale VAWTs intended to demonstrate the performance characteristics of those machines [41] [75]. While interest in wind energy and VAWTs evaporated once energy prices dropped throughout the following decade, development of vertical-axis turbines has returned as energy become more expensive once again. Today, VAWTs such as Savonius turbines and H-rotors are produced for use close to the ground and on top of buildings, where winds tend to be low speed and highly turbulent. As a result, these tend to be much smaller machines operating in the 5 - 20 kW range, unlike large-scale HAWT designs that often exceed 100 kW. However, applications for larger, industrial-scale VAWTS merit greater investigation. 1.1.1 Basics of Vertical-Axis Wind Turbines. VAWTs are differentiated from HAWTs by the axis of rotation, which is perpendicular to the free-stream velocity. This has several operational advantages, but also results in more complicated aerody-

5 namics. Major features of VAWT aerodynamics include non-constant blade angle of attack, dynamic stall, and wake dynamics, all of which make modeling and optimization of these turbines challenging. There are two ways of looking at the dominant two-dimensional aerodynamics of VAWTs: (1) examination of the individual blade aerodynamics and (2) observation of the kinetic energy loss of the airflow as it passes through the turbine. Turbine performance is also affected by three-dimensional flows near the ends of the turbine that are analogous to the wingtip vortices produced by aircraft. These effects cause an induced drag on the turbine blades, which cannot be predicted in a two-dimensional analysis. This, however, is best considered only after the dominant aerodynamic features have been dealt with.

(a) Turbine Geometry

(b) Individual Blade Aerodynamics

Figure 1.4. The left subfigure shows the 2-D geometry and layout of a Darrieus turbine or H-rotor. The blades are typically mounted at the quarter or half-chord point equidistant from the others. The sub-figure on the right demonstrates the velocity components and aerodynamic forces on one of the blades located at an azimuthal angle of 90o .

As seen in Figure 1.4 (a), which shows the two dimensional cross section of a Darrieus turbine or H-rotor, an azimuthal angle is determined by its orientation away from the vertical. A blade’s position is designated by the turbine radius and its azimuthal angle. The solidity of a turbine is defined as the number of turbine blades

6 multiplied by their chord-lengths and divided by the turbine diameter,

σ=

nc . D

(1.1)

The angle of attack of each turbine blade is determined by two main velocity components. One velocity component is the free-stream wind whose average flow is constant in magnitude and direction. The other velocity component is induced by the blade’s rotation around the central pivot point, resulting in a vector which is always anti-parallel to the blade’s velocity vector and whose magnitude is equal to the turbine radius multiplied with the angular velocity. This velocity component is defined as the blade’s induced velocity,

Uinduced = ΩR .

(1.2)

If the blade pitch angle is zero, the induced velocity vector alone cannot provide the blade with an angle of attack. Therefore, a blade’s geometric angle of attack must rely on its position with respect to the free-stream wind vector and the ratio of that vector’s projection normal to the blade’s tangent with respect to the induced velocity (see Figure 1.4. (b) ). This ratio is defined as the tip-speed ratio,

T SR =

ΩR Uinduced = . U∞ U∞

(1.3)

It is easy to see that the changing direction of the turbine blades will result in a variable angle of attack oscillating between positive and negative maxima at the front end (the left side of the turbine in the figure) and back end, respectively. At the front and back end of the turbine, the induced velocity and free-stream vectors

7 are perpendicular, so that the maximum angle of attack is equal to the arctangent of the tip-speed ratio’s inverse,

αmax = tan−1 (

1 ). T SR

(1.4)

One natural consequence of this is that VAWTs typically have poor starting characteristics, since the induced velocity produced by turbine rotation is necessary to produce a finite angle of attack [40]. This problem has been solved in the past by using an electric motor to force start turbine rotation or by adding a smaller Savonius turbine, which has good starting characteristics.

Figure 1.5. The maximum angle of attack present at an azimuthal angle of 90o for a typical range of tip-speed ratios for a vertical-axis machine.

The tip-speed ratio and azimuthal angle also have an effect on the magnitude and direction of the lift vector. When the free-stream wind vector and the induced velocity combine to produce the relative velocity the blade sees with respect to the blade-centered frame of motion, this causes the lift vector to be pushed forward

8 producing an induced thrust, known as the Katzmayr effect [42]. It is therefore this lift, projected onto the rotor tangent which is responsible for the usable torque of the turbine. The instantaneous power, P, produced by the turbine may then be calculated by multiplying the net torque on the blades,

T =

n X

(F~i · tˆ) ∗ R ,

(1.5)

i=1

with the angular velocity, Ω, of the the turbine rotor,

P =T ∗Ω .

(1.6)

One result of the oscillating angle of attack is a phenomenon known as torque ripple, where the torque produced by a single blade oscillates with the same frequency. This oscillation is exacerbated by interaction between the wakes produced at the front end as the blades pass through the back half of the cycle, resulting in torque losses. As shown in Figure 1.6, the torque profile of a blade as it rotates can be divided into the motor half cycle (front end) and the resistant half-cycle. Combined, theses effects on each blade produce an oscillating power coefficient whose frequency throughout a cycle is equal to the number of turbine blades. As such, more blades have been shown to reduce the effects of torque ripple, while increasing the overall solidity of the turbine [57]. It should be noted that while many sources divide the turbine cycle into the motor and resistant half cycles, significant amounts of energy can be extracted during the resistant half-cycle. Therefore, the naming convention for the future will refer to the ”front” and ”back” half cycles, with the ”front” half cycle being that half of the turbine facing into the wind. Another important way of understanding basic wind turbine performance is to

9

(a) Blade torque produced with respect to azimuthal angle.

(b) Net torque produced for the entire turbine with respect to azimuthal angle.

Figure 1.6. Example of net torque curves for a single blade (a) and the entire turbine (b) for a 3-blade VAWT at a tip-speed ratio of TSR = 2.5

10

Figure 1.7. 2-D variation in the time averaged U-velocity for at turbine operating at TSR = 3.0. The figure shows that kinetic energy is lost as the flow passes through the turbine, with the greatest drops occurring as the flow passes through the blade path. There is also a variation in the speed in the vertical direction. At the top of the turbine, the blade pushes against the flow and slowing it down. At the bottom of the figure, the blade pulls the flow faster.

look at the kinetic energy of the airflow as it passes through the turbine, since power output is equal to the energy extracted from the free-stream. It is best to start by assuming that the dominant behavior of the air is one-dimensional, flowing from left to right. Since energy is conserved, power produced by the turbine must result in a slowing of the airflow. As shown in Figure 1.7, the velocity remains nearly that of the free-stream until the flow reaches the front end (motor half-cycle) of the blade path. Once the flow enters the turbine, the airflow slows and contains less energy

11 that may be extracted by the back end (resistant half-cycle). It should also be noted that kinetic energy of the flow can also be lost even if the turbine is not producing useful power. Flow energy can be lost due to viscous effects such as blade drag and wake dynamics as well as due to other phenomena such as dynamic stall. These along with blade orientation give a two dimensional variation in the flow speed, since the turbine blades may be moving in the same or opposite direction of the free-stream.

Figure 1.8. Actuator disk representation of a wind turbine. State 0 and State 3 are the far field conditions upstream and downstream and the States 1 and 2 are the flow in the vicinity (but outside) of the turbine.

Using this idea as a starting point, it becomes possible to determine the absolute theoretical limit for the power that may be extracted by any given wind turbine. Obviously, no wind turbine could possibly extract all of the kinetic energy from the wind since no kinetic energy would mean no velocity, resulting in total blockage of the turbine. Therefore, the ideal wind turbine would have to balance maximum energy extraction with the requirement that the wind still has enough velocity to carry it through the turbine. One natural consequence of this is that far downstream the flow must return to the ambient pressure of the surroundings. Since the initial and final states of the airflow are the subject of interest, the turbine itself may be compressed

12 into a single actuating disk as shown in Figure 1.8 [27] [85]. Since an ideal case is being considered, viscous effects may be neglected and a steady state may be assumed. The Bernoulli equation may then be used,

p0 +

ρu2 ρu20 = p1 + 1 2 2

(1.7)

p2 +

ρu22 ρu2 = p3 + 3 . 2 2

(1.8)

and

It may be assumed that the pressure at p0 and p3 are equivalent as are u1 and u2 . Thus, with the conditions and a definition for the velocity at State 3 can be defined in terms of the velocity at State 0 and an arbitrary interference factor, a,

u3 = u0 (1 − 2a) .

(1.9)

The pressure difference across the actuator disk may be found to be

p1 − p2 = 2ρu20 a(1 − a) .

(1.10)

The force exerted on the actuator disk is defined by the propeller impulse equation,

F = (∆p)A = m∆u ˙ = ρAu1 (u0 − u3 ) .

(1.11)

13 where A is the area of the disk. Since the power is simply the force multiplied with the velocity and the power is also equal to the change in kinetic energy from State 0 to State 3,

P =

ρAu1 2 (u0 − u23 ) . 2

(1.12)

Equation 1.7 and 1.8 can be substituted into Equation 1.10 to show that u1 is the average of the far field velocities and results in

u1 = u0 (1 − a) .

(1.13)

Substituting Equation 1.11 into 1.10, the total power is found to be

P =(

ρu30 A )4a(1 − a)2 2

(1.14)

and the power coefficient,

CP = 4a(1 − a)2 .

The power coefficient is maximized when a =

1 3

for a CP =

(1.15)

16 . 27

This quantity is known

as the Betz limit, the maximum power coefficient that can be achieved for a wind turbine under ideal conditions [10]. It should be noted that while this model is complete enough for an initial estimate there are particular problems with actuator disk theory when applied to VAWTs. Due to the VAWT’s orientation, two-dimensional characteristics play an important part of the turbine’s aerodynamics. A study by

14 Agren et al. [1], suggests that the Betz limit may be significantly underestimated when assuming only one-dimensional flow.

1.1.2 Advantages of VAWTs.

The primary advantages of vertical-axis wind

turbines over horizontal-axis machines stems from their lower tip-speed ratios and omni-directionality. A lower tip-speed ratio results in higher magnitude angles of attack, less influence of parasitic drag, and less noise production. However, the problem of dynamic stall also becomes apparent at low tip-speed ratios and can severely limit turbine power output. HAWTs usually operate at tip-speed ratios between six and ten whereas VAWTs operate between one half and four [57]. For an average wind speed of 15 mph that equates to VAWT blades speeds of only about 30 mph and HAWT tip-speeds of over 120 mph! These high tip-speeds can create safety hazards such as ice being launched from the blades or catastrophic failure in the event of a load imbalance [70]. HAWTs also have been shown to be hazardous to birds and bats by striking the animals directly or from barotrauma due to high pressure gradients in the vicinity of the blade tips [4] [72]. HAWT operation also produces significant amounts of noise which makes them unsuitable for use in populated areas [60]. While the majority of the noise is created near a HAWT’s blade tips, it is the lower frequency noise-component caused by wake/tower interactions that cause the greatest disruption. During each revolution, the wake created by the blade strikes the tower structure used to support the turbine. Low-frequency noise has been shown to produce significantly higher response from human subjects from simple annoyance to serious impairment [8]. Complaints about turbine noise are a contribution to public opposition to wind development near populated areas. Low-frequency noise produced by VAWTs is minimized because the distance between turbine blade and structure is typically much higher, giving the wakes time to dissipate [75]. It is in these cases that VAWTs, with their relatively low tip-speeds, could be utilized.

15

Figure 1.9. Vertax Wind Ltd. proposed multi-megawatt turbines. These sea-based turbines would rely on fewer moving parts than horizontal-axis machines, allowing a longer lifespan and less maintenance [39].

Vertical-axis machines benefit significantly from their orientation in terms of their aerodynamic performance as well as overall operation. VAWTs are relatively simple to construct and maintain, since their omni-directionality negates the need for a yaw mechanism and permits most of the heavy machinery to be built at ground level. Since a HAWTs transmission and gear system must be mounted at the top of the tower, the machinery must be both light and compact, which adds to the overall capital cost of the turbine [57]. Turbine gear systems, transmissions, and yaw control motors are complex devices with many moving parts that require regular maintenance. However, the location of critical mechanisms makes maintenance very difficult and requires technicians to work hundreds of feet above the ground. VAWTs’ omni-directionality makes them better suited for placement in built-up areas or loca-

16 tions where geographical features produce highly variable, turbulent wind conditions, where the wind may quickly change direction in time or altitude [75]. A vertical-axis of rotation also eliminates the oscillating gravity loads that are exerted on HAWT blades, which create bending moments in one direction on the up-swing and then the other direction on the downswing. This problem quickly grows with turbine size and is considered one of the main limits on the scale of large HAWTs [56] [59]. It is for this reason that VAWTs are being considered for very large scale applications where HAWTs would be impractical, including multi-megawatt turbines [49]. Other benefits of VAWTs include ease of blade manufacturing (for H-rotors) and the ability to directly couple the turbine shaft to the generator, eliminating the need for a gearbox [23]. Clearly, VAWTs are deserving of interest and further development from the wind engineering community. Once vertical-axis machines have benefited from development and optimization, they could play a large part in making wind energy more versatile and open up more populated areas to wind development. 1.2 Previous Research 1.2.1 Experiments and Modeling Methods.

There are several criteria for

determining the appropriateness of a model or experiment: material cost, investment in time, and accuracy. All methods require some degree of compromise and emphasize or neglect particular aspects of the flow physics. For example, flow visualization can be extremely time consuming and difficult when conducting wind tunnel experiments; whereas in a finite volume simulation it is a rather straightforward process that only requires already known information to be displayed. However, experiments allow an investigator to see the actual physics being studied whereas a simulation must rely on a model. Therefore, it is essential that simulations be validated by comparison with experimental data. Some of the investigational methods used to understand the aerodynamic behavior of VAWTs include particle image velocimetry [25], pitot

17 ports [71], and direct measurement of forces/moments or power output [13]. Experimental studies of VAWT aerodynamics have been done in the past by placing scaled-down models in wind tunnels or by building full-scale turbines in the field. In a recent study, Castelli et al. [13] conducted experiments in the Politecnico di Milano wind tunnel in order to validate computational models. While the wind tunnel data was qualitatively similar to the computational results, the wind tunnel data could only serve as a rough estimate since tunnel blockage effects for VAWT experiments are distinct and not well understood. These tunnel effects were studied further by Battisti et al. [6] who concluded that the effects of tunnel blockage could be mitigated by utilizing open-section wind tunnels, which better facilitate bypass flow around the model. Battisti concurs with Castelli that the standard tunnel blockage corrections are not applicable for VAWT wind tunnel studies and attributes this to unexpected entrainment of the flow along the axis of rotation. As mentioned earlier in the chapter, a large scale study was conducted in cooperation with Sandia National Laboratories by Kadlec [41] and Sheldahl et al. [71] in which over 600 Darrieus turbines were built. This massive undertaking involved building turbines as large as 200kW in order to collect basic performance data and to determine the feasibility of VAWTs for large scale energy production. Due to the high cost and difficulty of experimental investigations, various aerodynamic models have been developed as alternatives. These models can be separated into three main types: blade element methods, vortex methods, and NavierStokes/grid methods. Blade element methods (BEM) have been the most popular approach to studying VAWT aerodynamics because of their usefulness for giving rough but reasonably accurate predictions of VAWT performance, while remaining relatively easy to implement. Blade element methods model the momentum loss of the free-stream flow as it passes through the turbine [66] [79]. BEM models make

18

Table 1.1. Advantages / Disadvantages of VAWT Analysis Methods

use of actuator disk theory which was first applied to wind turbines by Glauert [27]. Lift and drag forces are estimated from the angle of attack and empirical data, which is then used to compute the momentum lost by the airflow. The interference factor, the velocity divided by the free-stream velocity, is then used to compute the angle of attack and the aerodynamic forces on the turbine blades downstream. These methods tend to be relatively cheap in terms of computational power and provide good accuracy for moderate tip-speed ratios. The downside is that BEM models are heavily reliant on empirical airfoil data which neglects unsteady aerodynamics, dynamic stall behavior, and wake dynamics - major features of VAWT aerodynamics [57]. The first dynamic stall models applied to BEM models for analysis of VAWTs were by Gormont [28] and later modified by Strickland [76], Paraschivoiu [57], and

19 Masse & Berg [7]. Claessens [18] made extensive use of BEM models to study the effects of airfoil shape on the dynamic stall behavior of VAWT blades. McCoy et al. [53] optimized a two dimensional turbine for power coefficient, solidity, and their corresponding interference coefficients. Combining BEM models with grid-centered methods, Castelli et al. [15] provided detailed performance characteristics of a low solidity VAWT at various tip-speed ratios. Periodic passage of turbo-machinery is sometime referred to as a cascade [35]. Similar to boundary element models, cascade models make use of the Bernoulli equation and empirical data to approximate the flow velocity in the near wake of the turbine blades. While this method is the simplest and cheapest method of accurately predicting VAWT performance, is heavily dependent on empirical data [40]. Vortex methods are more computationally expensive than BEM, but provide more information about the two and three dimensional behavior of the flow through a VAWT. These methods assume an ideal flow and use local circulation around the turbine blades by substituting the blade elements with bound vortex line. Due to Kelvin’s theorem which states that the total circulation within the flow must remain constant, there must be a blade tip vortex which is equal and opposite to the circulation that is produced by the bound vortex lines. Furthermore, additional span-wise vortex lines must be shed whenever the strength of the bound vortex changes and whose magnitude is equal to the change in circulation. These vortex lines may then be used to approximate the surrounding velocity potential field [40]. As a result of the inviscid assumption, the overall flow approximates high Reynolds number cases while wake dynamics and viscous drag are neglected [29]. The first researcher to use vortex methods to model VAWT aerodynamics was Larsen [46]. Others have continued to make improvements to vortex methods, including Strickland et al. [77], who added dynamic stall models and fully extended the model to include three-dimensional effects as well as Masse [50] who approximated wake effects in the turbine operation,

20 which compared favorably to experimental observations. Grid-centered methods include finite difference (FDM) [65] , finite element (FEM) [62] , and finite volume methods (FVM) [33]. These methods work by discretizing the Navier-Stokes equations as well as the physical domain and solving the resultant system of equations. They are more computationally expensive than BEM or vortex methods, but they can approximate the physics more faithfully with minimal modeling and empirical data. While grid-based methods can be relatively straightforward for simple problems, solution accuracy becomes highly dependent on proper discretization of the physical domain and choice of turbulence model, once complex phenomena such as turbulence and separation are introduced. As of 2013, the most popular method utilized by computational fluid dynamics software is the finite volume method. Usage of FVM software is also likely to grow as computational resources become cheaper and its computational expense declines compared to other models. [84] One of the first researchers to use a grid-centered approach to investigate VAWT aerodynamics was Rajagopalan et al. [65] , who used a finite difference model. Ponta et al. [62], combined a local circulation model with a finite element model in order to better approximate the instantaneous forces on the turbine blades as well as the near wakes. More recently, research has emphasized FVM as the primary tool for grid-based VAWT studies. Castelli et al. [13] developed a procedure for modeling a low-solidity VAWT at moderate tip-speed ratios. Hamada et al. [33], used FVM models to observe the effects of dynamic stall and wake dynamics for a set of turbines. Howell et al. [37] conducted case studies for various parameters such as tip-speed ratio, solidity, and blade surface finish. In his doctoral dissertation, Ferreira [24] investigated the two and three dimensional behavior of flows in the near wake of a VAWT, utilizing FVM models as well as other methods. At this point, the FVM approach seems to be the likely successor as the most predominant tool for VAWT aerodynamic research.

21 1.2.2 VAWT Performance Studies.

Due to the complexity of VAWT aero-

dynamics and the sparsity of research in previous years, the number of parameters to be optimized for VAWTs remains immense. These parameters include tip-speed ratio, solidity, blade number, blade shape and camber, as well as constant or variable blade pitch offsets. While there have been several studies to investigate the effects of these parameters on overall turbine performance, there is still a tremendous amount of work to be done to optimize these parameters with respect to each other. As a result, comparisons should not be made between modern horizontal-axis turbines which have benefited from years of interest and vertical-axis ones whose performance characteristics often rely on 40-year old data [23]. Of the various parameters that have been studied, very little investigation has been done on the influence of the number of turbine blades on turbine performance, holding all other parameters constant. In the reference, Wind Turbine Technology [75], it is noted that blade numbers of three or more minimize the effects of torque ripple. Castelli et al. [14] ran a case-study with three, four, and five blade turbines which also demonstrated the dampening effect of additional turbine blades on torque ripple. It was also shown that blade number has little effect on overall power output at low tip-speed ratios, but the three-blade turbine was shown to be superior at higher tip-speed ratios. The effects of turbine solidity on VAWT performance characteristics are better known. Paraschivoiu [57] has shown that solidity has the greatest influence at very high or very low tip-speed ratios, as well as near the optimal tip-speed ratio. At high tip-speeds, solidity has an unfavorable effect on turbine output due to blockage effects. However, higher solidity does seem to dampen the effects of dynamic stall making turbines with large chord-lengths relative to their diameters more attractive at very low tip-speed ratios. Since three-dimensional effects such as induced drag are inversely

22 proportional to the aspect ratio of the turbine blades, larger chord-lengths (and higher solidity) can be expected to exacerbate these phenomena. Howell et al. [37] show in their experimental observations that higher solidity results in unfavorable changes in lift to drag ratios of the turbine blades and overall turbine efficiency. Li et al. [86] present similar results that limit the use of high solidity VAWTs to low tip-speed ratios, but suggest that higher solidity may improve self-starting capability.

Figure 1.10. A VAWT blade oriented with a positive (tow out) pitch β.

Studies have been conducted on the shape of turbine blades in order to minimize drag, correct for virtual camber effects resulting from the curved blade trajectory, and to improve self-starting characteristics of VAWTs. In their paper, Howell et al. [37] state that blade surface roughness can help reduce drag at lower Reynolds numbers by encouraging transition from laminar to turbulent flow within the boundary layer. Another parameter that was demonstrated was the blade thickness, which was demonstrated to delay the onset of stall at high blade angles of attack but decreases the lift-to-drag ratio overall. Claessens [18] was able to increase the performance of a VAWT by changing the blades from a NACA 0018 shape to a DU 06-W-200 airfoil, resulting in a larger lift to drag ratio and discouraging the formation of laminar separation bubbles at high blade angles of attack. The effect of camber on the overall

23 performance of a VAWT was investigated by Danao et al. [20] who found that the optimal camber line should follow the path line of the blade, since this eliminates the virtual camber produced by the curved trajectory of the turbine blades. They also saw interesting effects of blade camber on how energy is extracted from the airflow throughout the turbine cycle. In particular, it was observed that a negative camber resulted in almost all of the energy being extracted at the front half of the turbine cycle. Beri et al. [9] found that VAWTs with cambered blades showed greater potential to self-start, though this also results in reductions in peak efficiency. For most cases, VAWT blades are mounted at 25 or 50% of their chordlengths along the tangent line of the swept circle. By offsetting the blade’s attitude away from the tangent line, either in the ”tow out” or ”tow in” direction, the geometric angle of attack may be adjusted down or up, respectively (see Figure 1.10). Altering the blade angle of attack affects dynamic stall behavior, the balance of energy extraction between the front and back halves of the turbine, and self starting potential. Using wind tunnel observations, South et al. [73] reported increases up to 11 % in power coefficients for a low solidity VAWT with a pitch offset of 4o (tow out), compared to the zero pitch case. Klimas et al. [44] demonstrated that with a pitch offset of 2o (tow out), a VAWT produced a 3% higher power coefficient with a lower optimal tip-speed ratio, when compared to the zero pitch case. Fiedler et al. [26] similarly shows significant improvements in power coefficient for a VAWT as the pitch angle increases, but with diminishing returns for angles greater than about 4o . These studies consistently show improved performance for modest tow out blade pitches, though the optimal pitch offset and the flow behavior are highly dependent on other parameters such as solidity, blade number, and airfoil shape. They also show that a tow-in blade pitch is detrimental to turbine performance. This would suggest that turbine performance is improved by lowering the maximum angle of attack (magnitude) on the front turbine half-cycle in order to prevent separation and increasing the maximum

24 angle of attack on the back half cycle, which serves to smooth torque ripple. Dynamic stall plays a large part in VAWT aerodynamics, particularly at low tip-speed ratios. Features of dynamic stall include vortex shedding, periodicity, and hysteresis delay in response to changing angles of attack. [54] All of these behaviors can be unpredictable since they are highly dependent on parameters such as angle of attack, pitching rate, and acceleration of the free-stream. Baez Guada [5] used FVM models to simulate the behavior of flow separation for a flat plate at high angles of attack. Changes in lift were correlated to the formation of leading or trailing-edge vortices as well as the location of the stagnation point. Huellmantel et al. [38] conducted a series of wind tunnel tests looking at the process of flow reattachment for several airfoils at different pitch rates. Preceding reattachment, observations showed a constant velocity pressure wave that traveled from the leading-edge to the trailing-edge, independent of airfoil shape and reduced pitch rate. Ahmed et al. [2] investigated the flow reattachment process of an oscillating airfoil using experimental techniques. It was concluded that the reattachment process began near the static stall angle but was not complete until the angle of attack was equal to about 6o , due to hysteresis delays. It was also found that the most dramatic rise in lift occurred just as the angle of attack passed below the static stall angle. Using advanced experimental methods and accurate grid-centered simulations, efforts have been made in the last decade to observe the dynamic stall behavior present in VAWT aerodynamics. Ferreira et al. [25] used particle image velocimetry (PIV) to visualize flow over the section side of the VAWT blades and quantify circulation changes due to leading-edge separation. Hamada et al. [33] demonstrated that at low tip-speed ratios the greatest torque exerted on VAWT blades occurred at angles of attack much larger than the static stall angle. Individual blade pitch control is already used on some HAWTs and becomes

25 more feasible as turbines increase in scale. VAWTs equipped with blade pitch control can utilize the self-starting capabilities of turbines with blade pitch offsets, optimize blade angle of attack in order to maintain attached flow, maximize average power output at low tip-speed ratios, and mitigate torque ripple effects. Therefore, blade pitch regimes must be developed based on an understanding of VAWT performance characteristics for fixed pitch cases as well as dynamic stall effects and reattachment phenomenon, including hysteresis delay. Two such studies were published in the early 1990’s by Kirke et al. [43] and Lazauskas [47], who investigated using preset blade pitch regimes and self-stabilizing blades to maintain attached flow throughout the turbine cycle. Other active flow control devices such as trailing-edge flaps [69] and co-flow jets [87] could potentially be used to maintain attached flow for VAWTs. Acoustic devices such as plasma actuators [31], zero mass-flux jets [30] have also been studied. Greenblatt et al. [32] and Amitay et al. [3] have found that the introducing periodic perturbations (with a reduced frequency, F + =

fc , U∞

of about 1.0) are effective

at increasing average lift of airfoils at high angles of attack. 1.3 Scope As mentioned earlier in the introduction, HAWTs have benefited from decades of case studies and optimization. In fact, so much so that the majority of HAWT development today is now focused on turbine placement and its interaction with the terrain and surrounding turbines, rather than on the machines themselves. [75] VAWT aerodynamics, on the other hand, is not nearly as well understood and presents dozens of different parameters yet to be optimized. Because of this, an exhaustive investigation of different turbine solidities, blade types, etc will require many more years to complete. Therefore, the research contained in the work will be limited to the modeling and first iteration optimization of a low solidity, three-bladed, H-rotor turbine in 2-D for a set of constant tip-speed ratios. In the interest of brevity, this

26 study was also limited to blade pitch offsets and individual blade pitch controls. 1.3.1 Modeling Aerodynamics. Since identifying unsteady phenomena involved in VAWT aerodynamics and the ability to conduct parameterized studies were necessary for this investigation, a grid-centered simulation was the natural choice. For grid-centered methods, the two major challenges remain: (1) adequate discretization of the physical domain and (2) the problem of turbulence. The first challenge is made difficult by the complex and moving geometry that will be involved in the modeling of any VAWT. The second relies on determining which turbulence model could properly approximate the flow physics with reasonable accuracy. The goal for the first half of this study was to develop a procedure that could be easily implemented while also remaining flexible enough to work for a diverse set of turbine geometries. The greatest indicator of success was the ability of a simulation to correctly predict the time and location of turbulent separation of the flow around the turbine blades since this is the most important factor on overall turbine performance at low tip-speed ratios. This also required identifying a set of criteria that could be used to ensure a reasonable degree of stability and accuracy for an initial assessment of unsteady VAWT aerodynamics. The turbine geometry used for this study was borrowed from Castelli et al. [15], who has well documented the turbine performance parameters which were used for validation purposes. A simple BEM model was also used to compare with the grid-centered simulation along with other sources. 1.3.2

Performance Studies.

Many of the advantages of VAWTs over more

traditional designs is the ability to run at lower tip-speed ratios, albeit with a lower power coefficient. Therefore, VAWTs could be made more competitive by mitigating the flow phenomena such as dynamic stall and wake effects that limit performance at lower tip-speed ratios. One of the most easily implemented ways to do this is to offset the turbine blades or to implement a variable pitch regime that will maintain

27 attached flow and decrease blade drag. Other benefits of implementing this solution are balancing the extraction of energy between the front and back half-cycles and improve the self starting characteristics. The first step in this process was to investigate the overall flow behavior for the baseline cases in order to understand which phenomena were the most disadvantageous to the overall turbine performance. A pitch offset case study was completed first and, in combination with previous observations and insights provided by the BEM model, variable pitch regimes were then considered. The objective of the second half of this study was to develop an iterative procedure for finding the optimal variable pitch trajectory for a given turbine case.

28 CHAPTER 2 METHODOLOGY AND VALIDATION

2.1 Finite Volume Simulation 2.1.1 Summary.

The basic geometry of the turbine under study was borrowed

from Castelli et al. [15]; the main features can be found in Table 2.1. This VAWT was chosen because it is similar to other proposed designs that have been shown to operate for a wide variety of conditions. Furthermore, low solidity turbines are ideal for variable pitch studies since the blades are spaced far enough apart to prevent interaction between them. This investigation was limited to two-dimensional aerodynamics in order to simplify the model and to minimize computational expense.

Table 2.1. VAWT Geometry Blade shape n [-]

NACA 0021 3

DRotor [mm]

1030

c [mm]

85.8

Blade mounting point

0.25 c

HDomain /DRotor [-]

30

LDomain /DRotor [-]

50

Because the purpose of this research was to look at unsteady aerodynamic phenomena such as dynamic stall and wake interactions, as well as to conduct a parametric study with respect to blade pitch, a finite volume method approach was most appropriate. As mentioned in the previous chapter, grid-centered methods require more effort to set up the analysis and often require the researcher to go back and repeat steps based on the results.

29

(a) Physical domain

(b) Overset grids

Figure 2.1. Layout of (a) the physical domain of the FVM simulation with boundary conditions and (b) the overset grids surrounding the turbine blades.

30 Analysis of a fluid problem using a grid-centered approach typically involves four major steps:

1. Modeling of the physical domain and formulating assumptions 2. Discretization of the physical domain 3. Solving the finite volume problem computationally 4. Post-processing of the data obtained

The two-dimensional VAWT geometry was modeled by mirroring a NACA 0021 airfoil three times around the axis of rotation. This geometry is then placed in a larger physical domain and conditions at the boundaries are defined. The turbine geometry is presented in Figure 2.1. Deciding on appropriate assumptions used to simplify the physics of the problem required more consideration. Since the flow speed was well below Mach = 0.3 with minimal blockage effects, compressibility effects were neglected. Other assumptions that were made regarding turbulence are discussed later in this chapter. In order to solve the finite volume problem, the physical domain must be split into many control volumes (i.e. grid cells) with the overall discretization referred to as a grid or mesh. In order to ensure accuracy of the simulation, size and shape of the cells must be dictated by the magnitude and direction of gradients in the flow. This often requires corrections to the grid once a preliminary solution is determined. The simulation must then be run until the flow has reached a periodic, quasi-steady state. At that point, data may then be extracted from the simulations and observations regarding the aerodynamics may be made. 2.1.2 Overset Grid Methods. The two greatest challenges of grid-centered computational fluid dynamics methods are: (1) adequate discretization of complex and

31

Figure 2.2. Boundary between the overset and underset grids.

moving geometries while minimizing computational expense and (2) accurately describing the effects of turbulence in moderate to high Reynolds number flows. While the ability to deal with the turbulence problem is seriously hampered by the intractability of the governing equations and limitations with respect to computational resources [63] , there have been recent advances in the ability to deal with complex and moving geometries. The most recent advance has been in the form of overset or chimera grid methods, beginning to be implemented in commercial codes only in the last few years. Overset grid methods have already been used to study the aerodynamics of helicopter rotors [12] [21], which share many geometric features with VAWTs such as their axis of rotation, relative to the bulk flow. Hoke et al. [36] found that overset grid methods compared favorably to other methods such as conformal meshes and grid deformation techniques.

32

Figure 2.3. Fully elliptic overset grid containing a VAWT blade surface.

The procedure for implementing the overset grid method is as follows [68]:

1. Grid generation 2. Overlaying the overset grid onto the underset grid with hole-cutting 3. Grid coupling with chimera interpolation

A set of suggested criteria for the overset grid and boundary layer discretization is contained in Appendix A. Fully structured grids were created using Gridpro [22],

33

Figure 2.4. Topology of the overset grid layout used to maximize orthogonality of the element edges. an elliptic grid generation software. Gridpro allows the user to ensure orthogonality of the grid near the boundaries and provides control over cell shape by adjusting the grid topology. One of the strict requirements of overset grid methods is that the cells on the overset boundary be nearly uniform in size and shape. It is for that reason that a circular shape was chosen for the outer boundary of the overset grids. The portion of the underset grid that was overlaid by the overset grid was uniform with the cells growing outside of that region. The cells on boundary of the overset grid should be smaller than those of the underset grid and the ratio of the two must fall within

34

1 lo 2 ≤ ≤ . 3 lu 3

(2.1)

Figure 2.5. Closeup view of the overset grid near the blade’s surface, including the boundary layer.

Accuracy of the simulation is highly dependent on the discretization of the boundary layer and near stagnation points, where large velocity gradients are present. Near the blade surface and within the boundary layer, the non-dimensional length scale,

+ Ywall =

y uτ , ν

(2.2)

is critical to determining if the boundary layer is adequately resolved. For FVM simulations, the velocity is solved at the center of each grid cell and + a no-slip boundary condition is defined at the blade surface. Ywall is a measure of the

shear on the wall and can be used to determine the smallest sublayer of the turbulent

35 boundary layer that is resolved. This quantity will play an important role later in the chapter.

Figure 2.6. View of the overset grid near the turbine blades trailing-edge.

For some commercial CFD solvers with overset grid capability, the hole-cutting and grid coupling steps are performed automatically without the need for user input. Hole cutting is done once the overset grid has been placed on top of the underset grid and most of the underset cells covered by the overset grid are removed. This can be seen in Figure 2.2 where the circular overset grid is placed on a uniform underset grid. The computational code then uses chimera interpolation to couple the grids together and set the boundary conditions for both grids at the overset boundary to be equal. This permits the computational code to solve for the flow in all regions simultaneously.

36 2.1.3 Computation and Turbulence. The computational solution was obtained using Star-CCM+, a commercial finite volume code with overset capability [16]. The most important simulation parameters are listed in Table 2.2. The simulations made use of Star-CCM+’s implicit solver which permitted much larger time steps at a relatively low computational expense. For validation purposes, as well as to get a complete picture of the turbine performance at low to moderate tip-speed ratios, the turbine was run at tip-speed ratios of TSR = 1.5, 2.0, 2.5, and 3.0. Since the blade Reynolds number was defined to be 300,000 and the simulation geometry was limited to two-dimensional flow, a turbulence model was required. This, however, was not an obvious choice. Some investigators had chosen the two equation k- model [45] due to its insensitivity with respect to grid dimension within the boundary layer [13] and relatively good approximation of pure shear flows, which is useful for modeling wake dynamics. [63] In contrast, Paraschivoiu [57] demonstrated that the one-equation Spalart-Allmaras turbulence model [74] performed better at reproducing Piziali’s [61] experimental data for rotating airfoils. Since there is little difference between the k- model and Spalart-Allmaras in predicting the behavior of pure shear flow, the results of Paraschivoiu’s study indicate Spalart-Allmaras to be better at predicting dynamic stall behavior. Because of this and the marginally lower computational expense of one fewer equation, the Spalart-Allmaras model was chosen for the simulation. After some initial runs and adjustments to the mesh, two sets of simulations were produced. One + with Ywall > 5, that relied on wall modeling to approximate the flow in the boundary + layer. The simulation utilizing a low Ywall wall treatment had a more refined grid that + resolved the laminar sub-region (Ywall ≈ 1) and required only the governing equations

to solve the boundary layer flow.

37

Table 2.2. VAWT Simulation Parameters Reynolds number [-]

300,000

Number of Elements [-]

2,000,000

lo [mm]

4.0

lu lo

≈3

[-]

10−4

∆t [s] Inlet velocity [m/s]

9

TSR [-]

1.5 - 3.0

Turbulence model

Spalart-Allmaras

At each grid-point and for each of the governing equations, a residual,

r = f − Lφ ,

(2.3)

is calculated, where f is the solution for the given equation at that point. Once the residuals have been calculated for each grid-point, the values are then normalized,

sP

rnormalized =

n cells i=1

r2 , n cells

(2.4)

providing a measure of how well the computational solution holds to the governing equations. During each time step, the code was set to iterate until the residuals of the continuity and momentum equations fell below 10−4 and the residual of the + turbulent viscosity transport equation fell below 10−3 . The simulations in each Ywall

range were also run with twice the number of grid elements in order to demonstrate grid convergence; differences in the average turbine power coefficient were found to be less than 5% for all cases. The flow was determined to have reached a quasi-steady

38 solution when the average turbine power coefficient of a full cycle was within 5% of the previous cycle. This typically occurred after about five full turbine cycles or about 2500 hours of CPU time. One other issue should be noted with respect to computational stability. Simulations that utilize overset grids typically benefit from increased flexibility and accuracy than other methods. However, one disadvantage became apparent very quickly, which was that the computational stability was very sensitive to the grid and simulation parameters. It became quite clear that the simulation would diverge if the time step exceeded a value that was inversely proportional to the velocity of the overset grid with respect to the underset grid, Vo/u . The maximum stable time step also decreased as

lu lo

increased. This suggested that when the boundary cells of the overset

grid moved too quickly with respect to the boundary cells of the underset grid, a large amount of error resulted that could cause the simulation to diverge. Therefore, it became necessary for this author to create a new criterion that could be used to ensure that a simulation using overset grids behaves in a predictable way. This quantity was named the Overset Slip Condition Number,

Co/u =

lu Vo/u ∆t ≤ 2 . lo2

(2.5)

Largely through trial and error, Co/u ≈ 1 was found to provide the best stability characteristics for the VAWT simulations. However, this is a good demonstration that overset grid methods are still maturing as a technique and require more robust criteria for proper use. 2.1.4 Validation of the Finite Volume Simulation.

The finite volume simu-

lation was validated by comparing the average power coefficient curves with respect to tip-speed ratio to data extracted from other simulations by Castelli et al. [15],

39

(a) All simulations

(b) Low Y+ treatment simulations

Figure 2.7. Plots comparing the average power coefficient with respect to tip-speed ratio for (a) all simulations and (b) only low Y+ wall treatment FVM simulations.

40 Mehrpooya [55], and Subashki et al. [78]. There are three different approaches taken in these studies. Castelli and Mehrpooya simulated the VAWT aerodynamics using + finite volume simulations with high Ywall treatment at the wall, which rely on wall

functions rather than the governing equations to resolve the flow within the boundary + layer. Mehrpooya also ran a FVM simulation with low Ywall treatment, which utilizes

the Spalart-Allmaras turbulence model all the way to the blade surface. Subashki modeled the VAWT performance with a BEM code without any dynamic stall model. As shown in Figure 2.7 (a), all simulations with the exception of the BEM model share the same basic characteristics. For all of the finite volume simulations, the average power coefficient peaks at a tip-speed ratio of TSR = 2.5, decreasing sharply as the tip-speed ratio decreases and falling more gradually as the tip-speed + ratio is increased. The simulations with high Ywall treatment at the wall generally + predict a higher average power coefficient at lower tip-speed ratios. The high Ywall

treatment FVM simulations also vary significantly from one another, while the two + low Ywall treatment simulations are more consistent, with less than 10% difference

at lower tip-speed ratios. The BEM model provides the most qualitatively different average power coefficient curve, predicting the lowest power between tip-speed ratios of TSR = 1.5 and 2.5, then predicting a much higher power output than the others at tip-speed ratios above TSR = 2.5. When different models predict qualitatively different performance and there is a dearth of experimental data to settle the issue, it becomes necessary to consider the critical assumptions of each computational model and the consistency between the various models. As mentioned previously, FVM simulations that utilize a high + Ywall treatment at the wall, must rely on wall functions that are based on empirical

observations, since the grid near the wall is not refined enough to resolve the boundary layer using the governing equations. The problem is that the wall functions are

41 developed from cases where the flow is attached and fully developed, which makes wall functions unsuitable for predicting flow separation at high angles of attack [17]. Since the wall functions will replicate the fully attached boundary layer flow from which they were developed, it is reasonable to conclude that implementing these wall functions at high angle of attacks will have the effect of delaying stall and increasing the average power coefficient of the turbine, as is seen in Figure 2.7 (b). Combined with the fact + that there is greater consistency between the two low Ywall treatment simulations,

which rely on different methods of discretization (Mehrpooya’s simulation is run using + an unstructured, deformable grid), this supports the low Ywall wall treatment as the

more accurate method of resolving the turbulent boundary layer for VAWTs operating at low tip-speed ratios. 2.2 Estimating the Effective Angle of Attack

2.2.1 Geometric vs Effective Angle of Attack. Measureing a blade’s angle of attack as it passes through the VAWT cycle is a valuable way to assess the turbine performance. However, in a flow that is constantly changing with respect to location and time with wake interactions, the concept of a geometric angle of attack becomes muddled. A better way to consider the problem is to look at the effect of the flow on the pressure distribution and forces on blade, since it is these factors that determine the turbine performance. A useful analogy can be used between the actual pressure distribution or lift force on the blade and the angle of attack at which those characteristics would be similar for the steady case. Therefore, the effective angle of attack may be determined for a blade by taking the lift coefficient for an unsteady blade and finding the corresponding steady angle of attack in Figure 2.8. Clearly, the effective angle of attack will have a different meaning for the blade aerodynamics since at very high geometric angles of attack where the flow has separated, the effective angle of attack will actually be very low or possibly even negative. This is observed for VAWT

42

Figure 2.8. Lift coefficient with respect to angle of attack (neglecting stall) for a NACA 0015 airfoil obtained using JavaFoil [34]. blades undergoing dynamic stall and is discussed in the next chapter.

Similarly, the pressure distribution on the top and bottom surfaces of an airfoil is unique for a given angle of attack (see Figure 2.9). The variation in the pressure is greatest near the leading edge, where the majority of the lift is produced. By taking a ratio of the pressures on the top and bottom surfaces of the airfoil at some location along the chord length,

αef f ective = f (

CPT OP − CPBOT T OM ). CPT OP

(2.6)

Since the majority of lift on a blade is generated near the leading-edge and

43

Figure 2.9. Pressure coefficient with respect to location along the chord line at angles of attack between 0 and 12o for a NACA 0015 airfoil obtained using JavaFoil [34]. separation typically begins near the trailing-edge, it is desirable that the pressure ratio be taken near the leading-edge. For the purpose of this study the 0.20c point was chosen. The pressure distributions along the blade were generated for many angles of attack using JavaFoil [34] and a curve fit was obtained (see Figure 2.10). Using the correlation curve, the effective angle of attack may then be found once the pressure ratio is determined using only two pressure measurements on the blade surface. While less information is required to find the effective angle of attack this way, there are more limitations on the validity of this method. This method is no longer valid if leading-edge separation occurs or flow separation begins to approach the 0.20c point from the trailing-edge. This method may also erroneously give extremely high effective angles of attack for the blade if the blade passes through shed vortices or wake structures that contain significant pressure changes. 2.2.2 Validation of Angle of Attack Estimation. All novel methods for data collection must be evaluated for accuracy and consistency, just as new simulations

44

Figure 2.10. Correlation curves of the effective angle of attack with respect to the pressure coefficient ratio at 0.20 c. must be validated. While the lift-based method of estimating angle of attack is rather straightforward, the effectiveness of the pressure based method must be confirmed. In order to do this, a simplified simulation was used to reproduce an experiment by Piziali [61], where lift and drag were measured for an oscillating NACA 0015 in a free-stream (see Figure 2.11) with a reduced frequency,

F+ =

fc , U∞

(2.7)

of 0.1. In order to spare some computational expense, the simulation was run at Re = 300, 000 instead of Re = 2, 000, 000 for Piziali. The slope of the average lift with respect to angle of attack was then shifted for Piziali to match that of the lower Reynolds number case, bringing the difference between the data sets from 10% to less than 3% (see Figure 2.12).

45

Table 2.3. Oscillating Airfoil Simulation Parameters Blade shape

NACA 0015

Reynolds Number [-]

300,000

Angle of Attack [deg]

4.0 ± 4.2

F

+

[-]

Blade pivot point

0.1 0.5 c

HDomain /c [-]

30

LDomain /c [-]

50

Piziali’s lift data was then be used to calculate the effective angle of attack and the same was calculated for the simulation using the pressure ratio at the 0.20c point. The error between the two calculations of the effective angle of attack were found be less than 3% and can be attributed primarily to differences between the

Figure 2.11. Physical domain of the rotating NACA 0015 simulation with boundary conditions.

46

(a) Lift comparison.

(b) Drag comparison

Figure 2.12. Validation of the oscillating airfoil simulation, comparing the (a) lift and (b) drag with Piziali’s experimental results.

47

Figure 2.13. Effective angle of attack with respect to the geometric angle of attack for Piziali (corrected for Re = 300,000) using the lift and that of the FVM simulation using the pressure ratio. simulation and experiment, rather than error due to the method. As seen in Figure 2.13, the effective angle of attack accurately shows the effects of hysteresis delay as the effective angle of attack, and thus the aerodynamic forces, lag behind as the airfoil undergoes changes in geometric angle of attack.

48 CHAPTER 3 OBSERVATIONS OF UNSTEADY EFFECTS 3.1 Summary 3.1.1 Performance.

The overall performance of the turbine is highly dependent

on the operating tip-speed ratio, with the highest power output occurring at about TSR = 2.5 (see Figure 3.1). The power output drops dramatically below this tipspeed ratio, due to the effects of dynamic stall and drops more slowly at higher tip-speed ratios where blockage and wake effects dominate the aerodynamics. The torque exerted on the turbine blades is highest during the front half of the turbine cycle, though this becomes more apparent as the tip-speed ratio increases. As shown in Figure 3.2, the torque invariably drops below zero near the azimuthal angles of 0o and 180o since the lift vector is in line with the moment arm at these points. Some effects of dynamic stall and wake effects can be identified in plots showing the blade torque, but looking at the effective angle of attack is usually more effective.

Figure 3.1. Average power coefficient with respect to tip-speed ratio for the FVM simulation.

49

(a) TSR = 1.5

(b) TSR = 2.0

(c) TSR = 2.5

(d) TSR = 3.0

Figure 3.2. Blade torque coefficient with respect to azimuthal angle for various tipspeed ratios.

50

(a) TSR = 1.5

(b) TSR = 2.0

(c) TSR = 2.5

(d) TSR = 3.0

Figure 3.3. Turbine torque coefficient with respect to azimuthal angle for various tip-speed ratios.

51 The torque on the full turbine is simply the sum of the torques on each blade. Because of this, the frequency of the torque ripple for this turbine is exactly three times that of the blade. Figure 3.3 demonstrates that the variation in the net torque decreases as the tip-speed ratio increases and becomes more symmetric and sinusoidal in shape. This is desirable since the variation in torque is a driving factor in the fatigue characteristics and design of VAWT structures. Whether the minimum torque on the turbine is above zero is also an important factor. When a turbine’s net torque drops below zero, the blades must then have enough momentum to keep the turbine moving until the net torque rises again. Otherwise, the turbine will stall and cannot continue to operate. It is for tip-speed ratios of TSR = 2.5 and higher, that the VAWT’s net torque remains positive throughout the entire cycle. While the torque and power outputs are useful for gauging the overall performance of the turbine and identifying if there are problems, the effective angle of attack and flow visualization are much better tools for diagnosing what those problems are. Figure 3.4 shows the effective angle of attack curves for various tip-speed ratios. Sudden drops in effective angle of attack correspond to the beginning of leading-edge separation and sudden jumps correspond to rises in lift due to reattachment. Comparing the torque and effective angle of attack figures, points of stall can be identified using the effective angle of attack that cannot be easily identified using the torque, near the 0o and 180o points. The tip-speed ratio of TSR = 2.5 case is one example. The effective angle of attack switches from negative to positive at the azimuthal angle of 150o , well before the 180o point where this would be expected to occur. This happens because the flow has separated and reattached at this point, decreasing lift at the end of the front half-cycle and significantly increasing drag at the 180o point, resulting in marginal power losses. Therefore, it can be concluded that stall effects are present for tip-speed ratios below TSR = 3.0, rather than TSR = 2.5 as the blade torque curves would suggest.

52

Figure 3.4. Effective angle of attack with respect to the azimuthal angle for various tip-speed ratios. Sudden drops in magnitude correspond to full leading-edge separation and the jumps correspond to reattachment. Flow visualization is a laborious process, but it is also an effective tool for understanding the aerodynamics of concern. Since the effective angle of attack is more useful for looking at dynamic stall rather than wake dynamics, the vorticity magnitude distribution of a flow can be helpful for tracking the path of wake structures and identifying where they interfere with turbine blades. Figures 3.5 and 3.6 show a sequence of vorticity magnitude distributions for tip-speed ratios of TSR = 3.0 and 2.0, respectively. The sequences can be viewed from state 1 through state 10, and show the flow characteristics which repeat every 1/3 of a turbine cycle. The first sequence clearly demonstrates that attached flow is maintained on the blades for the full cycle. However, due to the faster rotation rate of the turbine with respect to the free-stream velocity, the blade will intersect with almost twice as many wake structures than for the tip-speed ratio of TSR = 2.0 case. The wakes that are produced on the front end of the turbine for the faster rotation rate also contain more vorticity. The effect of the wakes on the individual blade aerodynamics is discussed later in the chapter.

53

Figure 3.5. The sequence shows the vorticity magnitude distribution as the turbine passes through one third of a cycle at a tip-speed ratio of TSR = 3.0. The turbine maintains fully attached flow throughout the entire cycle.

54

Figure 3.6. The sequence shows the vorticity magnitude distribution as the turbine passes through one third of a cycle at a tip-speed ratio of TSR = 2.0. Around the 180o point, the blades undergo separation and reattachment.

55 3.1.2 Comparison of Streamtube and Finite Volume Models.

With the

overall turbine performance evaluated using an FVM simulation, the results may then be compared with those of the streamtube model produced by Subashki et al. [78]. The BEM model was of the double-multiple streamtube variety without a dynamic stall model used to correct for unsteady effects. The two major limitations of the streamtube model are that the model fails to take into account the effects of dynamic stall and viscous effects. This has the effect of pushing the average power coefficient curve to the right and overestimates the power output at higher tip-speed ratios (see Figure 3.7). While these problems result in significant differences in turbine performance, they are not necessarily insurmountable. As shown in the previous chapter, the average power coefficient is in fact more consistent with that of the low + Ywall FVM simulation than the simulation that relies on wall functions to approximate

the boundary layer. Since the BEM model relies on steady lift and drag data, the model predicts that a turbine blade immediately loses lift once the effective angle of attack rises above the static stall angle. The FVM model shows that this is not necessarily so. When the blade surpasses the static stall angle, the flow needs time to form the separation bubble and trailing-edge vortex that reduce circulation around the blade, causing lift to drop. This results in the BEM model over-predicting stall on the front half of the turbine cycle and underestimating the amount of energy extracted on the front end. The problem of the streamtube model’s inability to model dynamic stall is mitigated by the fact that since the blade is extracting less energy on the front end, there is more kinetic energy contained in the flow that reaches the back half of the turbine, which is then recovered. This accounts for the relative accuracy of the streamtube model compared to some of the other simulations. With the major problems of the BEM simulation identified, improvements can be made to better capture aspects of the flow physics that had previously been

56

Figure 3.7. Average power coefficient curves for the BEM model and the FVM simulation, with respect to tip-speed ratio. neglected. In the introduction, dynamic stall models for BEM simulations were discussed. Adding a stall model would be very effective for improving the accuracy of the streamtube model at low tip-speed ratios, since wake effects are not as dominant. In order to better predict VAWT performance at high tip-speed ratios, a better drag model is required. While viscous effects do not change the overall performance of the turbine as dramatically as dynamic stall, neglecting unsteady viscous features of the flow does result in some important, qualitative changes. The streamtube model does take into account parasitic drag by utilizing static drag curve data. However, the streamtube model is not able to take into account the wake dynamics that affect turbine performance at higher tip-speed ratios. Figure 3.8 compares the effective angle of attack for the turbine blades with respect to azimuthal angle for tip-speed ratios of TSR = 2.0, 2.5, and 3.0. It should be noted that there is relatively good

57

(a) TSR = 2.0

(b) TSR = 2.5

(c) TSR = 3.0

Figure 3.8. Comparison of the effective angle of attack for the streamtube (BEM) model and the FVM simulation.

58 agreement between the two simulations near the top of the turbine cycle, around the 0 or 360o azimuthal angle. However, there is a large disparity near the 180o point, where blade/wake interactions are most likely to occur. Even for the highest tip-speed ratio case, where the flow remains attached for the entire cycle, the effective angle of attack is much lower on the back half cycle than the streamtube model predicts. Also, at higher tip-speed ratios the aerodynamics become more two-dimensional outside of the vicinity of the blade. In any case, it is clear that more energy is being extracted from the flow on the front end than is predicted in the BEM simulation and not necessarily in the form of additional lift. One of the observations that has been made is that steady airfoil data is not sufficient for predicting drag on a blade undergoing changes in angle of attack. 3.2 Performance Limiting Phenomena 3.2.1 Flow Separation and Circulation. Since efficient operation of VAWTs at low tip-speed ratios is desirable for a number of reasons, dynamic stall becomes the driving aerodynamic characteristic that determines performance. For all tip-speed ratios below TSR = 4.0, the blade angle of attack rises above the static stall angle of 12o , the angle of attack where stall effects begin to appear. However, full stall does not necessarily occur due to hysteresis delay, allowing the turbine to operate efficiently at much lower tip-speed ratios. Leading-edge separation does take place for this turbine at all tip-speed ratios below TSR = 3.0. The azimuthal angle at which separation begins is difficult to predict, since the phenomenon is highly dependent on many different variables. For a tip-speed ratio of 1.5, separation and loss of lift occurs at azimuthal angles of approximately 90o , 210o , and again at 300o . For 2.0, separation takes place at azimuthal angles of 140 and 180o . Separation only occurs once for the tip-speed ratio of TSR = 2.5 case, occurring at an azimuthal angle of 150o , with the angle of attack going from negative to positive as the flow reattaches

59

Figure 3.9. Close up view of the vorticity magnitude distribution around a turbine blade at an azimuthal angle of 180o . (TSR = 2.0) Near the end of the half-cycle, the blade stalls and the separation bubble is swept away as the angle of attack changes from negative to positive.

to the blade. While the specific dynamic stall characteristics are difficult to predict from only the tip-speed ratio, such as the azimuthal angle at which separation occurs, there are a few patterns that should be noted. One interesting behavior is that full leading-edge stall is limited to the front half-cycle of the turbines operation for most of the operating tip-speed ratios, with the exception of the 1.5 case at the low end of that range. It is then reasonable to conclude that stall mitigation is most important for azimuthal angles between 0o and 180o . The angle of attack can also be raised significantly on the back half-cycle without necessarily causing the blade to stall. Another obvious pattern demonstrates that the problem of dynamic stall becomes dramatically worse as the tip-speed ratio is reduced from TSR = 3.0, with the number of occurrences of separation increasing as well as the performance damaging effects. Aerodynamic forces on the blade change greatly as the flow becomes separated; the lift drops and the drag increases suddenly. Consequently, the performance of the

60 VAWT is altered not only by how often the blade stalls during a cycle, but also by the point in the turbine cycle where the blade loses lift. This is demonstrated by the TSR = 2.5 case, where the blade stalls near the end of the front half-cycle. Even though the flow over the blade becomes completely separated, this cannot be seen in the blade torque profile since the blade wouldn’t be positioned to produce any torque anyway. The turbine can then operate at a higher angle of attack at this tip-speed ratio without dynamic stall effects reducing the lift at critical points. Another important aspect of dynamic stall is the process of flow separation from the suction side of the blade and reattachment. In order for this to occur, the blade must be subjected to a high angle of attack for a significant amount of time. This can be seen in Figure 3.9, where the process of full leading-edge separation and reattachment can be seen for a turbine blade. As the blade angle of attack passes the static stall angle, separation begins at the trailing-edge and progresses toward the leading-edge. Lift drops significantly when the point of separation nears the quarter chord point since the majority of lift is generated near the leading-edge of the blade. The shear layer that separates the recirculation zone near the blade surface and the free-stream forms what is called the leading-edge vortex (LEV). As the lift and the circulation decreases for the blade, a trailing-edge vortex (TEV) is formed that is equal in magnitude and opposite in direction to the LEV. This is a consequence of Kelvin’s law, which states that for flows that can be approximated as inviscid with conservative body forces, the total circulation present in the flow,

DΓ =0, Dt

(3.1)

remains constant. If the blade remains at a high angle of attack or the angle of attack is suddenly decreased to below zero, the recirculation zone will eventually

61 separate from the blade, taking the TEV with it. This results in a sudden increase in circulation around the blade, resulting in a rapid rise in lift. This phenomenon helps to increase the turbine performance for the TSR = 2.5 case, where the angle of attack magnitude rises sharply following reattachment. From these observations some conclusions can be drawn regarding the dynamic stall effects on turbine operation. The blades will continue to produce lift at angles of attack above the static stall angle, but stall mitigation is needed to prevent separation at tip-speed ratios below TSR = 3.0. Separation is limited to the front half-cycle for tip-speed ratios above TSR = 1.5. The exact point at which separation occurs is difficult to predict, but separation at azimuthal angles less than 150o is harmful to the overall turbine performance. Lastly, the reattachment process may be used to boost the torque produced by a turbine blade but this will require a way to induce separation at a given azimuthal angle. 3.2.2 Wake Interactions and Parasitic Drag. As a turbine blade passes through a flow, a wake is produced downstream that is characterized by a low average velocity relative to the blade’s frame of reference as well as high concentrations of vorticity. Wakes are produced by the reaction forces exerted on the fluid by the body, which are equal and opposite to the sum of the lift and drag vectors. As a result, the drag on a body and wake production are directly linked. Therefore, it is necessary to examine the causes of drag on the turbine blades. For the two dimensional case, there are three main components of drag: (1) friction drag generated by shear stresses in the boundary layer, (2) pressure drag resulting from the presence of flow separation and regions of recirculating fluid, and (3) viscous induced drag produced by added mass effects due to the blades’ accelerating frames. The structure of the wake may be characterized in terms of the relative velocity and the vorticity distribution. These two methods of observation are compared in

62

(a) Relative velocity (m/s) distribution

(b) Vorticity distribution

Figure 3.10. Two views of the flow downstream of a VAWT blade at an azimuthal angle of 0o and for a tip-speed ratio of TSR = 2.0.

63 Figure 3.10 for a VAWT blade. Near the trailing-edge surface, the wake is made up by a region of nearly uniform low relative velocity flow. Further downstream, the wake begins to diffuse producing a relative velocity distribution that is parabolic. Vorticity of opposite directions is generated along the top and bottom surfaces of the blade which then combine downstream. Other features may be introduced into the wake, such as shed vortices caused by separated flow over a blade. This stream of slowly diffusing wake structures trails behind the blade and is pulled along by the free-stream.

Figure 3.11. Close up view of the vorticity magnitude distribution for tip-speed ratios of TSR = 2.0 and 3.0, demonstrating the higher likelihood of blade / wake interaction at higher tip-speed ratios.

64 Wake production is indirectly related to the overall turbine performance since it is proportional to the drag forces on the turbine blades, resulting in lower torque. However, there is also a direct effect on the turbine performance because the wake is swept along by the free-stream from the front half-cycle into the path of the blades on the back half-cycle. This affects the individual blade aerodynamics in several ways, as shown in Figure 3.11. The lower average velocity relative to the blades results in a loss of lift, hampering energy extraction from the flow on the back half-cycle. Likewise, when a blade intersects with a wake at an oblique angle, flow may be cut off to one side of the blade and cause the blade to partially stall. Evidence of this may be seen in the form of ripples in the blade torque curve for azimuthal angles greater than 180o . These effects grow as the tip-speed ratio increases and the blade paths intersect with more numerous and stronger wakes. The most straightforward way to limit wake interactions is to decrease wake production in the first place, which means limiting the drag forces on the blades during the front half-cycle. Because wake production is a necessary consequence of lift generation, it is necessary to identify and mitigate the other major variables that increase drag. Clearly, stall is undesirable because it both reduces lift and adds greatly to blade drag. Second only to separation, the driving factor responsible for wake production and drag is the induced velocity and, therefore, the tip-speed ratio. Total drag is proportional to the net drag coefficient, but is also proportional to the induced velocity squared,

2 Cd ∼ Uinduced .

(3.2)

This explains the sudden drop in performance for tip-speed ratios above TSR = 2.5, where the tip-speed ratio is high enough to prevent the harmful effects of stall while

65

Figure 3.12. Close up view of the vorticity magnitude distribution around a turbine blade as it passes through wake structures. (TSR = 3.0)

66 also minimizing the tip-speed ratio, and thus, the drag as well as wake interactions. Also, sudden changes in lift are also to be avoided since, due to Kelvin’s Law, vorticity must be shed from the blade equivalent to the change in circulation around the blade. The shed vorticity robs the surrounding flow of kinetic energy and produces drag. Lastly, the curvature of the blades’ trajectory has an effect on the individual blade aerodynamics, since it does impart a small angle to the flow near the leading and trailing-edge. This phenomenon is referred to as the ”induced camber” of the blade and is dependent on the turbine geometry. The effect of induced camber increases with the chord length relative to the turbine diameter, resulting in lift and drag being produced at azimuthal angles of 0o and 180o . While the losses from this are small, they are not negligible. The effects of dynamic stall allow VAWTs to operate at lower tip-speed ratios that raise the maximum blade angle of attack above the static stall angle, the effects of parasitic drag and wake interactions serve to reduce performance at higher tip-speed ratios. Therefore, the best way to mitigate both dynamic stall and wake interactions is to lower the tip-speed ratio below TSR = 2.5 and then attempt to prevent flow separation caused by sustained, high angles of attack. In order to do so, the effective angle of attack for the blade must be adjusted. The approach that was used to do so is described in the next chapter.

67 CHAPTER 4 TURBINE PERFORMANCE IMPROVEMENTS 4.1 Procedure for Turbine Optimization

Figure 4.1. Effective angle of attack curve for the TSR = 2.0 case.

Figure 4.2. Torque curve for the TSR = 2.0 case.

In order to minimize drag and wake effects, a relatively low tip-speed ratio of TSR = 2.0 was chosen for optimization using blade pitch. Since the flow over the suction side of the blades becomes fully separated during the turbine cycle, the blade pitch must be adjusted in order to reduce the magnitude of the maximum angle of attack on the front side of the VAWT, below the static stall angle. If it is assumed

68 that there are no limitations on the pitch of the blades during turbine operation, there are then an infinite number of ways to do this. Therefore, a procedure must then be developed to optimize the blade pitch and, consequently, the effective angle of attack. At a tip-speed ratio TSR = 2.0, the effective angle of attack reaches −15o on the front end of the turbine cycle and an average of 5o on the back end. Stall takes place at azimuthal angles of 150o and 180o , which represents the greatest loss in energy extraction. The lower angle of attack on the back end also results in significantly less torque produced during the back half-cycle. Since the goal was to lower the magnitude of the effective angle of attack on the front end, adding positive pitch was the first logical step in increasing the average torque for the turbine cycle.

Figure 4.3. The procedure used to develop an effective variable blade pitch regime.

69 For simplicity, a series of constant positive pitch angles were chosen as the first pitch scheme. The effective angle of attack curves generated from these cases were then used to find the optimal angle of attack to be produced using variable blade pitch. The effective angle of attack is dependent not only on a blade’s azimuthal angle, but also on the free-stream velocity distribution, which is itself dependent on the kinetic energy extracted from the flow by the turbine. Therefore, an iterative approach is required in order to fine tune the variable pitch with respect to the azimuthal angle. The full procedure that was used to optimize the variable pitch curve is shown in Figure 4.3. The ultimate goal was to maintain attached flow, minimize wake interaction, and maximize energy extraction. 4.2 Fixed Non-Zero Blade Pitch 4.2.1 Performance. VAWT simulations were run with a set of positive constant blade pitches in order to find the optimal case. Stall was the greatest performance reducing factor with blade drag also contributing. Blade pitch less than 5o was found not to significantly mitigate the effects of separation and actually resulted in a net reduction in performance compared to the zero pitch case due to an overall increase in drag. At an optimal pitch of 5o , the average power coefficient of the turbine sharply

Table 4.1. Pitch Offset Performance β [deg]

CP [−]

0

0.2642

4

0.2073

5

0.3087

6

0.2965

8

0.1933

12

0.0375

70

(a) β = 4◦

(b) β = 5◦

(c) β = 8◦

Figure 4.4. Comparison of the effective angle of attack for several constant blade pitches.

71

(a) β = 4◦

(b) β = 5◦

(c) β = 8◦

Figure 4.5. Comparison of the blade torque coefficient for several constant blade pitches.

72 rises from 0.2642 to 0.3087, or an approximately 17% improvement in performance over the baseline TSR = 2.0 case. For blade pitches above 5o , the power coefficient gradually drops as the average effective angle of attack decreases and the drag rises. 4.2.2 Optimal Angle of Attack.

There are several interesting aspects to the

aerodynamics for the 5o blade pitch case that suggest desirable features that the variable blade pitch case should emulate (see Figure 4.4). There are also some clear improvements that can be made. While the optimal blade pitch does have the effect of mitigating stall, full leading-edge separation does occur at an azimuthal angle of 160o . There are also irregular spikes in the effective angle of attack as the blade passes through the rear half-cycle. Compared to the other cases, there is less variation in the effective angle of attack. This suggests that maintaining a constant angle of attack through each half-cycle is desirable and that the angle of attack on each half-cycle should be equal and opposite. This notion is supported by results of a simple BEM model created by Rempfer [67], who optimized the blade angle of attack for given tip speed ratios. The exact determination of the optimal blade angle of attack is unlikely for simple BEM models due to their reliance on static airfoil data. However, the qualitative aspects of the results are useful and agree with the conclusions made from the FVM simulations. Maintaining a steady angle of attack produces two other positive effects, including flattening of the torque profile and more even extraction of kinetic energy from the flow. The fact that separation still occurs on the front end where the maximum magnitude of the angle of attack is slightly above the static stall angle implies then that the optimal effective angle of attack should be large, but below the static stall angle. Therefore, a constant angle of attack magnitude of 10o was chosen as the target for the variable blade pitch case, with 10o being the angle of attack with the highest lift over drag ratio.

73

Figure 4.6. Target angle of attack with respect to the azimuthal angle. 4.3 Variable Pitch 4.3.1

Development of Variable Pitch Scheme.

Since the target angle of

attack must switch signs and little torque is produced near the point where the front and back half-cycles meet, a smooth transition is desirable. Figure 4.6 contains the optimal effective angle of attack curve that should be produced by the variable pitch scheme. The variable pitch curve was formulated by first subtracting the effective angle of attack from the desired angle of attack,

β (θ) = αoptimal (θ) − αef f ective (θ) .

(4.1)

The effective angle of attack data was then smoothed to remove any sudden drops due to stall effects. One rule of thumb that was followed was that at no point should the pitch drop much below 0o , since previous research has shown that this produces the opposite effect than what was intended. Following the creation of a discrete data set for the pitch with respect to the azimuthal angle, a continuous function was needed

74

(a) Iteration 1

(b) Iteration 2

(c) Iteration 3

Figure 4.7. Blade pitch vs azimuthal angle for the three iterations.

75 for the individual blade pitch. Due to the periodic nature of the turbine, the data was fitted with a Fourier series,

β (θ) =

N X

[ ai cos(iθ) + bi sin(iθ) ] .

(4.2)

i=0

The formulation of the data series for the blade pitch as well as the aerodynamics naturally favored a smooth curve, so the Fourier series created to fit the data generally required less than eight terms. Because adding blade pitch affects the free-stream velocity distribution and, therefore, the effective angle of attack, the optimization process required an iterative approach. Once a variable pitch curve was formulated, the turbine was run and the effective angle of attack was measured. The new variable pitch curve then is generated by subtracting the new effective angle of attack from the desired angle of attack and then adding to the previous pitch curve. This was repeated for a total of three iterations. The variable blade pitch curves for each iteration is displayed in Figure 4.7, showing the progression from first to last. The first iteration included a higher overall pitch magnitude than the second and third. For the second iteration, the pitch curve was deliberately kept above zero and smoothed out. The last iteration fine-tuned the pitch curve on the front end and near the azimuthal angles of 0o and 180o . Generally, the variable blade pitch curves for each of the three iteration all shared a number of features. Each maintained a higher pitch for the front half-cycle than for the rear and went to zero as the blade reached the boundary between the two half-cycles. During the rear half-cycle, the blade pitch curves seemed to correct for wake interaction as evidenced by the oscillations in the curve within that region. An alternative method of developing a variable pitch curve was also pursued

76

Figure 4.8. Harmonic variable blade pitch curve with a maximum pitch angle of 5o .

with the intention of simplifying the process. An optimal blade pitch had been found to be 5o , but this pitch offset was kept constant for the entire cycle, including the transition points between the front and back half-cycles. A sinusoid could be used to bring the blade pitch down close to zero where little torque is produced and then back up to the optimal pitch angle near the azimuthal angles of 90o and 270o , where the highest torque was produced. Therefore, a harmonic function for the blade pitch was developed as a function of the azimuthal angle, maximum blade pitch, and a pitch offset used to correct for the curvature of the blade trajectory,

β (θ) =

βmax − βo [1 − cos(θ)] + βo . 2

(4.3)

Table 4.2. Harmonic Pitch Cases Case

βmax [deg]

βo [deg]

1

5

-0.7

2

7

-0.7

The reasoning for this was that by doing so, the drag could be reduced at azimuthal

77 angles of 0 and 180o , while at the same time maintaining the benefits of the blade pitch, such as reducing stall and bringing up the power produced by the rear halfcycle. This was done for two cases of a maximum blade pitch of 5o and 7o .

Table 4.3. Harmonic Pitch Performance Method

Cp [−]

No blade pitch

0.2642

Constant pitch (5 o )

0.3087

Harmonic pitch (5 o max)

0.3013

o

Harmonic pitch (7 max)

0.2611

4.3.2 Results and Discussion for Harmonic Blade Pitch. While an intriguing idea, implementation of the harmonic blade pitch proved to be ineffective at both reducing drag and mitigating stall. As shown in Table 4.3, while the harmonic pitch case with a maximum blade pitch angle of 5o did compare favorably to the zero pitch case, it actually underperformed compared to the optimal constant blade pitch offset. There are a number of reasons why this would be so and the blade torque coefficient sheds some light on them. The blade torque profiles for the harmonic pitch cases can be seen in Figure 4.9. The plot for the maximum pitch of 5o shows that like the constant pitch offset case, stall occurs during the front half-cycle. However, full stall occurs near an azimuthal angle of 120o , which is earlier than for the constant pitch case. The torque plot also clearly shows a large drop in power during the rear half-cycle, similar to that for the 5o offset. It was suggested that perhaps the fact that the pitch only reaches its maximum at the highest angle of attack, makes the maximum blade pitch of 5o too low for the harmonic pitch case. In order to improve upon the stall characteristics of the turbine on the front half-cycle, the maximum angle of attack for the harmonic blade pitch was increased to 7o . This did work as intended, Figure 4.10 shows that this also had the effect of

78

(a) Case 1

(b) Case 2

Figure 4.9. Blade torque coefficients with respect to azimuthal angle for the two harmonic variable blade pitch cases. raising the maximum blade angle of attack well above the static stall angle, resulting in full stall and a significant power loss. Because of these results, it was concluded that a purely harmonic pitch curve was inappropriate for the purposes of turbine optimization. This shows that even though the harmonic pitch was rejected, it does show what will and what won’t work for optimization of the blade pitch. In order to maintain a constant effective blade angle of attack throughout the turbine cycle, the blade pitch must be considerably lower for the rear half-cycle in order to prevent stall. The behavior of the flow for the harmonic pitch case also suggests that the

79

(a) Case 1

(b) Case 2

Figure 4.10. Effective angle of attack with respect to azimuthal angle for the two harmonic variable blade pitch cases.

maximum blade pitch must rise well above the optimal pitch offset angle to counter hysteresis delay caused by the changing pitch. These are useful observations and serve to provide good rules of thumb as the variable pitch curve changes through the optimization process.

80 4.3.3 Results and Discussion for Iterative Blade Pitch. As mentioned earlier in the chapter, variable blade pitch curves were developed and adjusted for three iterations, using the procedure previously described. Table 4.4 displays the power coefficient values for each iteration, starting with the zero pitch case. Each new variable pitch curve resulted in about a 9% improvement over the previous iteration, monotonically decreasing slightly each time. Since the purpose of this work was to develop a procedure for optimizing a given VAWT rather than optimizing this particular turbine, and in the interest of brevity, the optimization process was limited to three iterations. The final iteration resulted in a 38% increase in power coefficient compared to the zero pitch case as well as a 18% rise over the optimal pitch offset case. This demonstrates a dramatic improvement in performance even with a truncated optimization process. Several conclusions may then be made regarding VAWT aerodynamics and the use of variable blade pitch for low tip-speed ratios.

Table 4.4. Variable Pitch Performance Iteration

CP [−]

0

0.2642

1

0.2985

2

0.3343

3

0.3647

While dynamic stall effects limit VAWT performance at relatively low tipspeed ratios and wake-blade interactions reduce power output at higher tip-speed ratios, separation due to high effective angle of attack is more easily mitigated. Using variable blade pitch as a tool to prevent separation, turbines can be run at significantly lower tip-speed ratios while providing comparable performance to that of the same turbine with zero blade pitch operating at its optimal tip-speed ratio. Assuming at this point that the lower limit of the range of operational tip-speed ratios is the blade

81

(a) Iteration 1

(b) Iteration 2

(c) Iteration 3

Figure 4.11. Comparison of the effective angle of attack for the several variable blade pitch iterations.

82

(a) Iteration 1

(b) Iteration 2

(c) Iteration 3

Figure 4.12. Comparison of the blade torque coefficients for the several variable blade pitch iterations.

83 pitch (βmax ≈ 45o ), there is convincing justification for optimization at tip-speed ratios of less than one, relying on blade pitch to keep the flow attached to the blade surface.

Figure 4.13. Comparison of the blade torque coefficients for the third iteration variable pitch curve and the zero pitch case.

Figures 4.11 and 4.12 show the progression of the effective angle of attack and blade torque coefficient curves, respectively. These show that by the second iteration, full leading-edge separation no longer occurs. However, even the third iteration does show some stall taking place near the azimuthal angle of 150o , similar to the turbine operating at a tip-speed ratio of TSR = 2.5 with zero blade pitch. The effective angle of attack curves are shown to become more smooth with each iteration, though wake interactions remain apparent on the back half-cycle. The comparison of the blade torque coefficient between the final variable pitch iteration and the zero pitch case shows that even marginal improvements in maintaining lift on the turbine blades can result in significant increases in energy extraction (see Figure 4.13). Furthermore, Figure 4.14, which compares the effective angle of attack for the final variable pitch iteration and the desired blade angle of attack demonstrates that there are still large

84 variations that can be resolved. These features combined with the continued existence of dynamic stall effects on the front half-cycle suggests that significant improvements in performance can be made with additional iterations.

Figure 4.14. Comparison of the effective angle of attack curve for the third iteration variable pitch case and the desired angle of attack.

85 CHAPTER 5 CONCLUSION 5.1 Summary Vertical-axis wind turbines benefit from several operational advantages. The omni-directionality of the turbines along with the ability to keep much of the equipment at ground level allows VAWT designs to remain simple and cost effective. Gravity loads on the blades remain constant along with uniform blade geometry would permit vertical-axis machines to be built at larger scales than HAWTs. Large VAWTs could be installed much closer to populated areas than similarly sized HAWTs due to lower blade speeds and their positive effect on safety and noise. These features make VAWTs attractive for further research and development, which will be needed to optimize performance. This means that VAWTs must be evaluated using a method that takes into account the unsteady and viscous aerodynamics that dominate turbine performance. Once the main issues that limit performance are identified, a reasonable procedure for turbine optimization must be devised and implemented. 5.1.1 VAWT Modeling and Simulation. A low solidity, three-blade VAWT was chosen because of its well known performance characteristics. The turbine aerodynamics were simulated using FVM software for a variety of tip-speed ratios. The physical domain was discretized using overset grids, which were optimal for the complex, moving geometry. A one-equation Spalart-Allmaras turbulence model was chosen due to its relative accuracy for a wide variety of turbulent flows and its adoption for + similar applications. A low Ywall wall treatment was determined to be most suitable

for modeling the turbulence within the blade boundary layer, due to the non-uniform nature of the flow and high angle of attack of the blades for lower tip-speed ratios. While there is little experimental data that can be relied upon for validation, the power coefficient with respect to tip-speed ratio was compared for several different

86 + simulations, with good agreement between the cases that utilized a low Ywall wall

treatment. A set of suggested criteria was established for accurate VAWT simulations and to ensure stability of simulations that make use of overset grids. These criteria are contained in Appendix A. One valuable way to evaluate the aerodynamics, including the flow speed and direction near the blade, is to measure the blade angle of attack. However, due to the non-uniform and unsteady nature of the flow through the VAWT, the traditional notion of the angle of attack, that is the orientation of the blade with respect to a free-stream, becomes difficult to measure directly. Therefore, rather than observe the flow around the blade, an effective angle of attack was measured by considering the effect of the flow on the blade. This was done by taking the lift coefficient or the ratio of pressure coefficients at points on the top and bottom surfaces of the blade and correlating it to the angle of attack for the steady airfoil case. This method was validated for a pitching airfoil case and was found to be accurate within 3%. 5.1.2 VAWT Aerodynamics and Unsteady Effects.

Once the VAWT simu-

lations had been completed, the overall performance was evaluated, the results were compared with a simple BEM simulation, and the sources of power losses for the VAWT were identified. For the baseline turbine geometry, the optimal power output was found to be at a tip-speed ratio of TSR = 2.5 with the power coefficient dropping suddenly as the tip-speed ratio is decreased and falling more gradually as the tip-speed ratio was increased. The steep drop in power output for lower tip-speed ratios was attributed to the effects of dynamic stall, which was the dominant aspect of VAWT aerodynamics. Hysteresis delay allowed the turbine to operate well at tipspeed ratios that brought the blade angle of attack above the static stall angle, with full leading-edge separation occurring at tip-speed ratios below TSR = 3.0 . Flow separation from the suction side of the blade was found to result in a drop in lift and

87 an increase in drag. Dynamic stall also contributed to the production of wakes, which often interacted with the blades during the rear half-cycle. Like dynamic stall effects at low tip-speed ratios, wake interactions were also found to play a significant role in decreasing power output at moderate to high tipspeed ratios. The primary sources of wake production were identified as blade drag, which increases with tip-speed ratio, and flow separation at low tip-speed ratios. There is little that can be done to decrease the drag on the blades and, furthermore, the number of intersections between the blade-path and the wakes increases at high tip-speed ratios. Therefore, it was concluded that power output could be raised at lower tip-speed ratios by mitigating the effects of dynamic stall using blade pitch. This important result illustrates one of the main limitations of BEM simulations. Despite the fact that dynamic stall models can be introduced to BEM models in order to increase their accuracy at low tip-speed ratios, blade/wake interactions are neglected, resulting in much higher power output at high tip-speed ratios. 5.1.3 VAWT Performance Improvements.

Wake effects are best mitigated

by lowering the tip-speed ratio, but dynamic stall greatly decreases the power output at the high angles of attack that accompany low tip-speed ratios. Dynamic stall was mitigated by adjusting the blade pitch and lowering the magnitude of the maximum angle of attack. Since there were many ways to do this, it was necessary to develop a procedure for generating blade pitch curves with respect the azimuthal angle. A tip-speed ratio of TSR = 2.0 was chosen to be a test case for improving VAWT performance. First, the optimal constant pitch offset was determined and the effective angle of attack for that case was examined. It was found that for the optimal case, the magnitude of the effective angle of attack remained nearly constant, just below the static stall angle. An optimal angle of attack curve was generated (αef f ective =

+

10o )

and subtracting the angle of attack for the previous run from the desired angle of

88 attack, the blade pitch with respect to the azimuthal angle was produced using Fourier series expansion. Since pitch alterations have the effect of changing the VAWTs flow field overall, an iterative approach was required to achieve the desired angle of attack. Repeating the subtraction of the effective angle of attack from the desired angle of attack, a new pitch curve was generated for three iterations. Table 5.1. VAWT Performance Evaluation Method

CP [−]

No blade pitch

0.2642 o

Constant pitch (5 )

0.3087

Variable pitch

0.3647

The power coefficients for each case were measured and compared to the baseline performance for the TSR = 2.0 case. The performance for each case is contained in Table 5.1. For the optimal pitch offset, there was a 17% improvement in power output compared to the baseline case. The optimal pitch offset was in the ”tow out” direction, which raised the angle of attack. Since the highest magnitude of the occurs during the front half-cycle, when the angle of attack is negative, this has the effect of lowering the magnitude of the angle of attack on the front end and raising the angle of attack during the rear half-cycle. This mitigates the effect of dynamic stall caused by large magnitude angles of attack and evens out the energy extraction from the flow between the front and rear half-cycles. There are problems with this configuration, since the blade pitch remains in place during points in the cycle when the blade cannot produce any positive torque, at azimuthal angles of 0o and 180o . This results in unnecessary drag production which results in power losses. Additionally, the blade pitch that was optimal for eliminating separation on the front half-cycle was not necessarily the optimal pitch for maximizing the power extraction during the rear half-cycle.

89 These problems were resolved once the variable blade pitch was implemented. Each iteration produced an approximately 9% improvement in power output, compared to the previous case. For the third and final iteration, the power coefficient had made a 38% improvement over the zero pitch case and 18% improvement over the optimal pitch offset case. A comparison of the different cases is made in Table 5.1. The effective angle of attack curve for the final iteration showed that more adjustment is needed for the blade pitch, which suggests that significant gains can be made with additional iterations.

Figure 5.1. Average power coefficients for several VAWT cases compared to a Mod-5B horizontal-axis machine [75].

Figure 5.1 shows the average power coefficient with respect to tip-speed ratio for several VAWT cases compared to a common horizontal-axis turbine. As can be seen in the plot, the zero pitch case performs favorably with respect to the HAWT. Some conclusions may then be made regarding the feasibility of these proposed improvements and the competitiveness of VAWTs compared to HAWTs. Without any blade pitch, the VAWT performs comparably to the HAWT, but the effective range of tip-speed ratios is much narrower. Introducing a blade pitch offset has been shown to

90 significantly improve overall performance for cases where dynamic stall is a dominant feature. However, blade pitch offsets alone are not enough to expand the effective range of tip-speed ratios. This makes pitch offsets a practical improvement for applications where implementing variable blade pitch controls would not be feasible, such as smaller scale or roof mounted VAWTs that must operate in highly variable wind conditions. On the other hand, variable blade pitch control can be used to enhance the overall performance and expand the effective operating range of tip-speed ratios. Following the blade pitch optimization procedure, it is apparent that VAWTs can be made to perform competitively when compared to HAWTs, at much lower tipspeed ratios. While additional research and development will be required to complete optimization, the ability to operate VAWTs with similar efficiency at much lower tip-speed ratios will open up many more opportunities for wind energy development. 5.2 Future Research Topics 5.2.1 Turbine Optimization. This thesis has covered the proper simulation, evaluation, and a procedure for optimization of VAWT aerodynamics. However, questions still persist regarding full optimization of the turbine as well as expanding to include more generalized cases. As a result, future work must include the completion of the optimization process for the turbine chosen for this research. This includes following through with the iterative process for the TSR = 2.0 case and finding the optimal pitch offset for various tip-speed ratios. These results will be needed in order to determine the answers to several questions:

1. Is the optimal blade pitch offset a function of tip-speed ratio? 2. Must the variable blade pitch change with tip-speed ratio as well? 3. What effect on performance does a variable blade pitch regime have on tip-speed

91 ratios other than the one for which it was optimized?

The answers to these questions will help to determine whether a the blade pitch can be held only as a function of the azimuthal angle or if a more generalized way of controlling the blade pitch is necessary. If the VAWT can perform competitively with respect to HAWTs used for similar applications, the blade pitch could be controlled using a simple cam system. If, however, the variable blade pitch must take into account not only the azimuthal angle but the tip-speed ratio, a more sophisticated control system must then be developed. This would be an ambitious project that would require a combination of aerodynamics and control system design. 5.2.2 Three Dimensional VAWT Aerodynamics. One of the least understood aspects of VAWT aerodynamics is the three dimensional aspects of the flow. The main reason for this is that simulating these effects are both computationally expensive and difficult. In order to gain a better understanding of the flow physics, novel computational techniques suitable for highly complex and moving 3-D geometries will likely be needed, such as the Lattice-Boltzmann method. Once a baseline case has successfully been simulated, it will be important to visualize the flow and understand the three dimensional aerodynamic effects on VAWT performance. While a reasonable assumption would be to assume that the three dimensional flow would be similar to that near the wingtips of an aircraft, the influence of oscillatory loads on the blades and their curved trajectories suggest that there may be much more interesting flow physics involved. Once the effects of the three dimensional aerodynamics on VAWT performance, measures to mitigate losses may then be evaluated. Fascinating and complex aerodynamics govern the operation of vertical-axis wind turbines, clearly there remains a great deal of work to be done.

92

APPENDIX A SUGGESTED CRITERIA FOR VAWT SIMULATIONS

93 The following lists of criteria are a set of conditions that the author has found to be necessary, but not sufficient, to ensure that a VAWT simulation will converge at low tip-speed ratios and that the solution will be relatively accurate. While not a guarantee of effectiveness for a simulation, these are easy to implement standards that will at least eliminate some of the largest contributors to problems with accuracy and stability.

A.1 Overset Stability Criteria

1. AR of all cells near the overset boundary ≈ 1 2.

6 skew

3.

1 3



of all cells near the overset boundary ≈ 0

lo lu



1 3

4. Co/u ≈ 1

A.2 Accurate Wall Treatment Criteria

1. ”Low Re” Spalart-Allmaras turbulence model 2. Low Y + wall treatment specified 3.

6 skew

of all cells within the boundary layer ≈ 0

4. Y + ≤ 1.5

94

APPENDIX B EFFECTIVE ANGLE OF ATTACK CORRELATION CURVES FOR SEVERAL AIRFOILS

95 As described in Chapter 2, a simple way to find the effective angle of attack for a VAWT blade was to find it as a function of the pressure coefficients at the top and bottom surface of the blade at the 20% chord point. Below are tables containing pressure coefficient data for several airfoil shapes.

RCP =

CPT OP − CPBOT T OM CPT OP

(B.1)

B.1 NACA 0012

Figure B.1. NACA 0012 airfoil shape.

Table B.1. NACA 0012 CP Data α [deg]

CPT OP [−]

CPBOT T OM [−]

RCP [−]

0

-0.38720

-0.38720

0.000000

1

-0.47790

-0.29856

0.375267

2

-0.57054

-0.21208

0.628282

3

-0.66500

-0.12787

0.807714

4

-0.76119

-0.04603

0.939529

5

-0.85897

0.03334

1.038814

6

-0.95823

0.11015

1.114952

7

-1.05885

0.18430

1.174057

8

-1.16071

0.25569

1.220288

9

-1.26368

0.32425

1.256592

10

-1.36763

0.38989

1.285084

11

-1.47244

0.45253

1.307333

12

-1.57799

0.51209

1.324520

96

Figure B.2. Pressure coefficient ratio versus angle of attack for the NACA 0012 blade.

97 B.2 NACA 0015

Figure B.3. NACA 0015 airfoil shape.

Table B.2. NACA 0015 CP Data α [deg]

CPT OP [−]

CPBOT T OM [−]

RCP [−]

0

-0.49265

-0.49265

0.000000

1

-0.58904

-0.39840

0.323645

2

-0.68745

-0.30639

0.554309

3

-0.78776

-0.21675

0.724853

4

-0.88984

-0.12958

0.854378

5

-0.99358

-0.04498

0.954729

6

-1.09884

0.03693

1.033608

7

-1.20550

0.11606

1.096275

8

-1.31342

0.19232

1.146427

9

-1.42248

0.26560

1.186716

10

-1.53254

0.33583

1.219133

11

-1.64347

0.40291

1.245158

12

-1.75513

0.46677

1.265946

98

Figure B.4. Pressure coefficient ratio versus angle of attack for the NACA 0015 blade.

99 B.3 NACA 0021

Figure B.5. NACA 0021 airfoil shape.

Table B.3. NACA 0021 CP Data α [deg]

CPT OP [−]

CPBOT T OM [−]

RCP [−]

0

-0.71325

-0.71325

0.000000

1

-0.82143

-0.60738

0.260582

2

-0.93177

-0.50393

0.459169

3

-1.04415

-0.40304

0.614002

4

-1.15843

-0.30483

0.736859

5

-1.27446

-0.20941

0.835687

6

-1.39211

-0.11691

0.916020

7

-1.51123

-0.02744

0.981843

8

-1.63168

0.05890

1.036098

9

-1.75332

0.14200

1.080989

10

-1.87598

0.22175

1.118205

11

-1.57799

0.29807

1.149070

12

-0.38720

0.37085

1.174615

100

Figure B.6. Pressure coefficient ratio versus angle of attack for the NACA 0021 blade.

101

APPENDIX C EQUATIONS FOR BLADE PITCH

102 The following tables contain the coefficients for the three Fourier series used to describe the blade pitch with respect to angle of attack. Fourier series take the following form.

β (θ) = a0 +a1 cos(θ)+b1 sin(θ)+a2 cos(2θ)+b2 sin(2θ)+...+aN cos(N θ)+bN sin(N θ) (C.1)

The Fourier series were generated to fit the discrete data sets produced using the procedure outlined in Chapter 4. Fourier coefficients were calculated using MATLAB’s [52] curve fitting tool.

C.1 Iteration 1

Table C.1. Variable Pitch 1 (N=8) i

ai [rad]

bi [rad]

0

0.03346

-

1

-0.00017

0.00836

2

-0.05146

-0.05166

3

0.00451

0.00115

4

-0.01127

-0.03633

5

0.00808

0.01404

6

0.00055

-0.01341

7

-0.00213

0.01244

8

0.00399

-0.00886

103 C.2 Iteration 2

Table C.2. Variable Pitch 2 (N=5) i

ai [rad]

bi [rad]

0

0.04735

-

1

-0.00346

0.01309

2

-0.02701

-0.02763

3

0.00085

-0.00396

4

-0.00737

-0.00794

5

0.00361

0.00232

C.3 Iteration 3

Table C.3. Variable Pitch 3 (N=6) i

ai [rad]

bi [rad]

0

0.04428

-

1

0.00930

0.01873

2

-0.04840

-0.00879

3

-0.00111

-0.01259

4

-0.00237

-0.00381

5

-0.00416

-0.00362

6

0.00247

0.00012

104 BIBLIOGRAPHY

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