IMPROVEMENT OF VERTICAL-AXIS WIND TURBINE PERFORMANCE VIA ..... Wind turbines are categorized based on their axes of rotation, into Horizontal-.
IMPROVEMENT OF VERTICAL-AXIS WIND TURBINE PERFORMANCE VIA TURBINE COUPLING
BY PAYAM MEHRPOOYA
DEPARTMENT OF MECHANICAL AND AEROSPACE ENGINEERING
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 _________________________ Adviser
Chicago, Illinois July 2014
ACKNOWLEDGEMENT I would like to express my greatest appreciation to my academic advisor, Professor Dietmar Rempfer, for his patient guidance, useful critiques and brilliant ideas. I would also like to thank Dr. Kevin Cassel both for seving on my thesis committee, and his valuable assistance throughout my entire study. I wish to acknowlegde the help provided by Dr. Suddhakar Nair and my colleagues in the Fluid Dynamic Research Center, and thank them all. I am truly grateful to my family for being tremendously patient and encouraging during the time I had to devote my attention to this work. Without them, I would never be able to tackle all the hardships associated with graduate school and studying abroad. Special thanks to Kara Easlick for taking time to review this thesis, sentence by sentence, and giving insightful feedbacks. My friends who contributed to this work by their encouragement are owed my deepest gratitude.
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TABLE OF CONTENTS Page ACKNOWLEDGEMENT
.......................................................................................
iii
LIST OF TABLES
...................................................................................................
vi
LIST OF FIGURES
.................................................................................................
vii
LIST OF SYMBOLS
...............................................................................................
x
ABSTRACT
............................................................................................................. xiii
CHAPTER 1. INTRODUCTION ................................................................................... 1.1 1.1 1.3 1.4
1
Background ............................................................................ Fundamentals of Vertical-Axis Wind Turbines .......................... Literature Review ....................................................................... Scope ...........................................................................................
1 6 10 16
2. METHODOLOGY AND VALIDATION ...............................................
19
2.1 2.2 2.3 2.4
Model Geometry ......................................................................... Domain Discretization ................................................................ Numerical Simulation ................................................................. Validation of Results ..................................................................
19 21 31 33
3. EFFECTS OF UNSTEADY AERODYNAMICS ...................................
37
3.1 Turbine Performance .................................................................. 3.2 Performance Limiting Factors ....................................................
37 43
4. TURBINE PERFORMANCE IMPROVEMENT ...................................
53
4.1 Power Enhancement ................................................................... 4.2 Wake Recovery ...........................................................................
56 69
5. CONCLUSION ........................................................................................
76
5.1 Summary ..................................................................................... 5.2 Future Work ................................................................................
76 82
iv
APPENDIX A. TIME-AVERAGED VELOCITY CALCULATIONS ............................
84
A.1 Time-averaged Velocity Distribution at the Vicinity of the Turbine ...................................................................................... A.2 Time-averaged Velocity along the Trajectory of the Blades ....
85 86
BIBLIOGRAPHY
....................................................................................................
v
88
LIST OF TABLES Table
Page
2.1. Model Geometry ..............................................................................................
19
2.2. Grid Size Growth Ratio ...................................................................................
28
2.3. Main Simulation Parameters ............................................................................
32
4.1. Spacing between the Pair of Turbines .............................................................
55
vi
LIST OF FIGURES Figure
Page
1.1 Schematic of horizontal-axis, Savonius drag-based, Darrieus curved-blade, and Giromill H-Rotor wind turbines. [27, 44] .............................................................
2
1.2 The left figure shows the VAWT layout (view from the top) and the schematic of the model geometry. The right figure demonstrates the velocity vectors and aerodynamic forces on a blade located at the front half-cycle. .............................
7
1.3 Geometric angle of attack with respect to azimuthal angle, for multiple tip speed ratios ranging from 1.5 to 3.0. ....................................................................
8
2.1 layout of the physical domain, boundary conditions and the interface between the two zones. (a) Wind tunnel zone (b) Rotor zone. ............................................
20
2.2 Non-conformal grid interface boundary between the sliding and stationary subgrids. ................................................................................................................
24
2.3 Hybrid grid, with structured prism layer mesh around the airfoil, and unstructured triangular mesh for the rest of the domain. ......................................
27
2.4 Prism Layer mesh around the airfoil and the concentration of grid nodes at the trailing edge. ..........................................................................................................
28
2.5 Grid within the control ellipse for balde section. ..................................................
29
2.6 Turbine average power coefficient with respect to tip speed ratio for (a) ! ! simulations with high 𝑌!"## treatment and (b) simulation with low 𝑌!"## treatment and DSTM model. .................................................................................
35
3.1 Turbine average power Coefficient with respect to tip speed ratio. ......................
37
3.2 Blade torque coefficient with respect to Azimuthal Angle for four different tip speed ratios. ...........................................................................................................
40
3.3 Turbine net torque coefficient with respect to azimuthal angle corresponding to one third of a cycle, for tip speed ratios ranging between 1.5 and 3.0. .................
41
3.4 Vorticity magnitude distribution for turbine running at TSR = 2.0. .....................
44
3.5 Vorticity magnitude distribution for turbine running at TSR = 3.0. .....................
45
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3.6 Sequence of formation of dynamic stall in the case with TSR = 2.0. Vorticity distribution is shown in grey scale, and the streamlines with velocity magnitude in color. ................................................................................................
48
3.7 Vorticity magnitude distribution, showing the blade-wake Interactions within the rear half-cycle of the turbine running at TSR = 3.0. .......................................
50
3.8 Distributions of time-averaged (a) streamwise and (b) cross-stream velocities, normalizez by the freestream velocity, in the vicinity of the turbine. ...................
51
4.1 Layout of the geometry for the pair of turbines. ...................................................
54
4.2 Average power coefficient with respect to tip speed ratio for a single turbine, and pairs of turbines with 1.65D, 1.50D and 1.35D spacing. ...............................
57
4.3 Blade torque coefficient with respect to azimuthal angle, for a single turbine, and pairs of turbines with 1.65D, 1.50D and 1.35D spacing. TSR = 3.0 for all cases.......................................................................................................................
59
4.4 Turbine torque coefficient with respect to azimuthal angle, for a single turbine, and pairs of turbines with 1.65D, 1.50D and 1.35D spacing. TSR = 3.0 for all cases.......................................................................................................................
60
4.5 Vorticity magnitude Distribution for the case with 1.65D spacing and TSR = 3.0. The sequence starts from the top left, and ends at the top right .........
62
4.6 Vorticity magnitude Distribution for the case with 1.50D spacing and TSR =3.0. The sequence starts from the top left, and ends at the top right ..........
63
4.7 Vorticity magnitude Distribution for the case with 1.35D spacing and TSR =3.0. The sequence starts from the top left, and ends at the top right ..........
64
4.8 Blade-Wake intersection for cases with (a) 1.65D and (b) 1.50D spacing. ..........
65
4.9 Distribution of time-averaged streamwise velocity, normalized by freestream velocity, for the pair of turbines with 1.50D spacing and TSR = 3.0....................
67
4.10 Time-averaged streamwise velocity along the blades’ path, normalized by the freestream velocity, for the single turbine and the pair of turbines with 1.50D spacing, at TSR = 3.0. ...........................................................................................
68
4.11 Distribution of streamwise velocity for (a) a single turbine, and pair of turbines with (b) 1.65D (c) 1.50D and (d) 1.35D spacing, at TSR = 3.0. ...........................
72
4.12 Distribution of cross-stream velocity for (a) a single turbine, and pair of turbines with (b) 1.65D, (c) 1.50D and (d) 1.35D spacing, at TSR = 3.0. .......................... 73 viii
4.13 Streamlines for cases with (a) a single turbine, and pair of turbines with (b) 1.65D spacing, (c) 1.50D spacing and (c) 1.35D spacing, at TSR = 3.0...............
75
A.1 The probe points located at the inner and outer side of the blade, with respect to the center of the rotor ............................................................................................
87
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LIST OF SYMBOLS
Symbol
Definition
Roman APP
Average power produced by the wind farm
ASwept
Turbine rotor swept area
APlanform
Planform area above the wind turbine array
AFrontal
Wind farm frontal area
BEM
Blade element method
c
Chord Length
CCOE
Capital cost of electricity
CFD
Computational Fluid Dynamics
CCW
Counter clockwise
CW
Clockwise
CP
Turbine power coefficient
CAP
Array power coefficient
DES
Detached eddy simulation
DSTM
Double streamtube model
HDomain
Extent of the model geometry in y-direction
FVM
Finite Volume Method
LES
Large eddy simulation
LEV
Leading edge vortex
n
Number of turbine blades
P
Power
p
Pressure
PD
Turbine array power density
PArray
Array of turbine power output
PSingle
Single turbine power output
R
Turbine rotor radius
Re
Reynolds Number
x
r
Residual
S
Spacing between pair of turbines
SST
Shear Stress Transport
T
Torque
t
Time
TEV
Trailing edge vortex
TSR
Tip speed ratio
URANS
Unsteady Reynolds-averaged Navier-Stokes
Uinduced
Velocity component induced by blade rotation
URelative
Relative velocity of the flow after contact with rotor
U∞ Uτ
Free-stream velocity Friction velocity at the wall
VAWT
Vertical-axis wind turbine
vʹ′
Turbulent velocity in y-direction
wʹ′
Turbulent velocity in z-direction
x
Direction along the free-stream velocity and corresponding to azimuthal angle of 0°
y
Direction along the free-stream velocity and corresponding to azimuthal angle of 0°
! 𝑌!"##
Non-dimensional distance from the wall quantifying how well the boundary layer is resolved
Greek α
Angle of attack
αmax
Maximum angle of attack
ε
Rate of dissipation of turbulent kinetic energy
θ
Azimuthal angle corresponding to the location of a VAWT blade in the turbine cycle
κ
Turbulent kinetic energy
𝜇!
Turbulent viscosity
𝜇
Molecular dynamic viscosity
ν
Kinematic viscosity
ρ
Fluid density xi
σ
Turbine solidity
τ
Shear stress
Ω
Turbine angular velocity
ω
Specific rate of dissipation of turbulent kinetic energy
xii
ABSTRACT While vertical-axis wind turbines (VAWTs) have a simpler design than the horizontal-axis wind turbines, their development has been hindered due to their unsteady aerodynamics and complex flow field. In this thesis, a parameterized study is conducted to simulate a baseline VAWT using STAR-CCM+, a commercial finite volume code. A hybrid grid scheme, with structured prism layer mesh at the surface of the blades, is used to properly resolve the turbulent boundary layers on the blades. The flow was highly unsteady due to the rotating geometries. Thus, a sliding mesh technique is implemented at the interface of rotating and stationary zones. The dominant factors limiting the performance of the VAWTs are investigated for a range of moderate tip speed ratios, by visualizing the flow field and modeling the individual blade aerodynamics. The VAWT aerodynamics is shown to be dominated by the dynamic stall, at low tip speed ratios, and by the blade-wake interactions and the wake blockage effects, at high tip speed ratios. The concept of turbine coupling is used to improve the performance of the VAWTs by their internal aerodynamic interactions. Two counter-rotating turbines are placed in close proximity, and simulated over the same range of tip speed ratios as before, and for a set of different spacing between them. The effects of spacing and the tip speed ratio on their overall power output and their wake recovery characteristics are then investigated. A cluster of turbines with spacing equal to 1.50 turbine diameters and tip speed ratio of three is shown to have the quickest wake recovery and highest power enhancement, increasing the turbine average power coefficient by 22%.
xiii
1 CHAPTER 1 INTRODUCTION 1.1
Background Wind turbines are categorized based on their axes of rotation, into Horizontal-
Axis Wind Turbines (HAWTs) and Vertical-Axis Wind Turbines (VAWTs). HAWTs have their axis of rotation parallel to the flow direction, while this axis is perpendicular to the flow direction for VAWTs. In addition to these categories, wind turbines are categorized based on their dominant aerodynamic forces used to generate power, into Drag-Based and Lift-Based turbines. HAWTs are lift-based, and similar to the wings of airplanes, they use the lift component of the aerodynamic force to generate power. VAWTs, however, have more diversity, and are commercially available in three main types [60]: •
Drag-based Savonius turbine, that uses scoops, vertically mounted on the main shaft, to capture the wind energy.
•
Lift-based Darrieus turbine with curved blades, that are optimized for centrifugal forces on the blades, as the curvature of the blades allows them to be stressed only in tension at high rotating speeds.
•
Lift-based Giromill H-rotor turbines with straight blades, which are the most recent and common VAWTs in use. A schematic of all the aforementioned turbines can be seen in Figure 1.1. Lift-
based turbines are usually preferred over drag-based turbines for most of the applications, due to their higher efficiency. Drag-based turbines, such as Savonuis scoops, have better self-starting characteristics, and are usually attached to a lift-based turbine as a starter.
2
Figure 1.1. Schematic of Horizontal-Axis, Savonius drag-based, Darrieus curved-blade, and Giromill H-Rotor wind turbines. [27, 44] The first windmills were invented in Persia. They were drag-based and verticalaxis turbines used for simple purposes, such as grinding and water pumping. Over the time, however, the HAWTs became the dominant type of wind turbines in use, due to their enhanced efficiency. The development of VAWTs was then hindered by the public tendency towards the HAWTs. In the recent century, there has been a renewed interest in the VAWTs, as the cost of power generation became of high importance. One of the most important factors that determines the appropriate technology to generate electricity is the Capital Cost of Electricity [23]. This factor is the ratio of the total amount of money invested into the technology, to the average power it produces, and can be calculated as, 𝐶𝐶𝑂𝐸 =
$ , (1.1) 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑃𝑜𝑤𝑒𝑟 𝑃𝑟𝑜𝑑𝑢𝑐𝑒𝑑
3 and the ultimate goal is to minimize this factor by maximizing the denominator. Average Power Produced (APP) by the wind technology can be estimated as,
APP =
Wind power into wind farm
×
Total turbine swept area
×
Power conversion efficiency
(1.2)
where the first parameter is the amount of wind energy entering the wind farm. The second parameter is the total number of turbines that can be placed in the wind farm multiplied by their corresponding area that interacts with the wind. The third parameter is the amount of interacting energy that is converted to electricity. After the last energy crisis, most of the efforts for maximizing the APP have been focused on enhancing the last parameter. A comparison between the power conversion efficiency of a HAWT with that of a VAWT, clearly demonstrates the advantage of the former over the latter, with a remarkably higher efficiency. Therefore, the development of the VAWTs was hindered, as the HAWTs became the dominant wind turbine technology used in the wind farms. However, Over the past past 30 years, as previously mentioned, there has been a renewed interest in the VAWTs, as researchers realized the presence of multiple factors that had yet to be considered, when choosing the appropriate wind technology. VAWTs present several advantages over HAWTs, making them more suitable for some specific applications. These advantages and their corresponding applications will be discussed below. HAWTs emit a low-frequency noise generated by the interaction between the blades and the supporting tower. The blades of the turbine generate pressure waves,
4 which strike the tower and produce a disruptive noise, when the blades pass by the tower [63]. This, however, is not an issue in the VAWTs, since the distance between the blades and the tower is relatively large and always constant. VAWTs usually work at tip speed ratios between one half and four, thus have relatively lower rotational speed than HAWTs, which have tip speed ratios ranging from six to ten [60]. Tip-speed ratio is an indicator of how fast the blades rotate relative to the wind speed, and will be further explained in the following section. Lower rotational speed will reduce the frequency of the noise emitted from the turbine. Moreover, it enhances their safety by decreasing the risks of mechanical threats, such as fatigue and catastrophic failure due to load imbalances. This advantage makes the VAWTs more suitable for highly populated areas, where noise level and safety is of high importance. VAWTs are Omni-directional [49]. This means they are insensitive to the wind direction. However, HAWTs require a yaw system that frequently tracks the wind direction, and places the rotor accordingly. Omni-directionality removes the yaw mechanism from the turbine and makes the design much simpler. VAWTs have a higher potential for scalability [36]. Due to their constant gravitational and inertial forces on the blades, they are more appropriate than HAWTs at large scales (10 MW+), where these forces become excessively high and much more difficult to deal with. In HAWT design, these forces are alternating with time in terms of quantity and direction, making the design much more complex and expensive. VAWTs usually last longer than HAWTs. First of all, their blades do not experience alternating gravitational forces and thus are less vulnerable to fatigue damages. Secondly, they are mechanically better able to withstand gust conditions through changing stall behavior.
5 In addition, VAWTs are less sensitive to cross-wind and turbulence, and can operate in tumultuous wind conditions [49]. Therefore, a small-scale VAWT can be mounted close to the ground, or on a rooftop, which makes it a better choice for residential applications or in urban areas, where the flow is more turbulent than in an open wind farm. VAWTs’ simpler design reduces their manufacturing and maintenance costs. This is first of all, due to their straight constant-section blades, compared to HAWT blades with a three-dimensional design, skewed profile and variable pitch. In addition, VAWTs have a lower center of mass by having their electric generator at the ground level. This reduces the maintenance costs, since the entire system is more accessible. Moreover, simpler design allows mass production, which will additionally lower the cost of production. VAWTs have better environmental impact. Their three-dimensionality along with their capability to be mounted closer to the ground makes them more bird-friendly. Birds are more vulnerable to strike the blades of HAWTs, since they rotate faster and in higher altitudes. Finally, previous studies have hypothesized faster wake recovery and reduced turbine spacing, which will result in optimized array configurations and higher power densities [80, 21, 42]. VAWT wind farms, as a whole, can potentially be more efficient than HAWT wind farms, since the aerodynamic interactions between the turbines may actually improve their individual and net array power coefficients. This hypothesis, initiated the basic idea of this study, and will be further discussed in chapter 4. In conclusion, considering all the advantages that the VAWTs present over HAWTs, it may be fruitful to better study the aerodynamics of these turbines, and
6 investigate whether they can potentially change the future of wind technology. 1.2
Fundamentals of Vertical-Axis Wind Turbines The most common vertical-axis wind turbine currenly in use is the H-rotor type
[60]. As could be seen in Figure 1.1, this turbine consists of a number of straight blades vertically mounted on some struts, which are attached to the main shaft. The number of the blades depends on the solidity of the turbine. Solidity is the ratio of the turbine blade area to the turbine swept area, and can be calculated as,
𝜎 =
𝑛 𝑐 , (1.3) 𝐷
where n is the number of the blades, c is the chord length of the blade, and D is the diameter of the turbine. The blades rotate around the main shaft relative to an axis, which is perpendicular to the wind direction. An azimuthal angle θ is defined by its orientation from the vertical position [44], and it varies from 0 to 180 degrees on the front-half cycle, and from 180 to 360 on the rear half-cycle. As illustrated in Figure 1.2 (a), the front half-cycle corresponds to the part of the cycle in which the blade is traveling along the upwind portion of the turbine circumference, and the rear half-cycle corresponds to its downwind portion. The individual blade aerodynamics have been shown Figure 1.2 (b). 𝑈! is the velocity vector of the free-stream wind and is constant both in magnitude and direction. 𝑈!"#$%&# is the velocity vector induced by the blade’s rotation around the main axis and is always anti-parallel to the blade’s velocity vector. Its magnitude can be determined as,
7
Figure 1.2. The left figure shows the VAWT layout (view from the top) and the schematic of the model geometry. The right figure demonstrates the velocity vectors and aerodynamic forces on a blade located at the front half-cycle.
𝑈!"#$%&# = 𝑅 Ω, (1.4)
where R is the turbine’s radius, and Ω is the turbine’s angular velocity. The ratio of turbine induced velocity to the free-stream velocity is called the turbine Tip Speed Ratio (TSR) and can be calculated as,
𝑇𝑆𝑅 =
𝑈!"#$%&# 𝑅 Ω = . (1.5) 𝑈! 𝑈!
As will be shown later, tip speed ratio is one of the most important factors is wind turbine design.
8 If the blade’s pitch angle is zero, the induced velocity cannot provide the blade with an angle of attack. When the VAWT is spinning, the blades are moving forward through the air in a circular path. The resultant of the oncoming free-stream velocity vector and the blade’s induced velocity vector is called the relative velocity, which creates a varying angle of attack to the blade. The blade's geometric angle of attack, 𝛼, is then defined as the angle between the relative velocity vector and the induced velocity vector. Clearly, the changing direction of the blades will result in a variable angle of attack, oscillating between its positive and negative maxima, at the front and rear halfcycles, respectively. The variation in the geometric angle of attack within an entire cycle has been demonstrated in Figure1.3, for several tip speed ratios [1].
Figure 1.3. Geometric Angle of Attack with respect to Azimuthal Angle, for multiple tip speed ratios ranging from 1.5 to 3.0.
9
It can be easily seen in this figure that as the tip speed ratio increases, the extrema of the curves decrease in value and take place at lower azimuthal angles for its maximum and higher azimuthal angles for its minimum. The maximum value of the angle of attack is an important parameter in studying the blades in pitching motion and stall conditions, and can be calculated as [60],
𝛼!"# = 𝑡𝑎𝑛!!
! !"#
, (1.6)
and shows how the higher tip speed ratio results in a lower maximum angle of attack. The component of the aerodynamic force exerted on the blade, parallel to the relative velocity vector, is called the drag force, and it’s perpendicular component is the lift force. As shown in Figure 1.2 (b), when the absolute value of the angle of attack is greater than zero, the lift force is pushed forward, generating an induced thrust, known as Katzmayr effect [41]. It is the lift force, projected to the rotor tangent that produces the desired torque to run the turbine and generate electricity. Clearly, as the angle of attack increases, the lift force is further pushed forward, and its component along the rotor tangent increases in magnitude, until it reaches its maximum at the maximum angle of attack. The instantaneous turbine power output, P, can be calculated by multiplying the resultant torque of all the blades with the turbine’s angluar velocity,
𝑃 = 𝑇 . Ω , (1.7)
10
where, the net torque can be calculated as [60], !
𝑇 =
𝑓! . 𝑡 ∗ 𝑅 . (1.7) !!!
It is clear that turbines with different sizes have different power outputs. A largescale turbine, used in a large wind farm, has a power output several orders of magnitude greater than a small-scale turbine used in urban areas. In order to measure how efficiently each of these turbines converts the wind energy into electricity, and to be able to have a comparison between them, their power outputs have to be normalized by the amount of wind energy entering the turbine. Turbine Power Coefficient, also known as Turbine Efficiency, is the parameter that facilitates this comparison, and can be calculated as,
𝐶! =
𝑃 1 ! 2 𝜌 𝐴!"#$% 𝑈!
, (1.8)
where, 𝜌 is the air density, A is the swept area of the wind turbine, and 𝑈! is the freestream velocity [60]. The denominator is the amount of wind energy that enters the swept area of, and interacts with the turbine. Turbine power coefficient is the main parameter determining whether the use of the turbine will provide the best advantage. 1.3
Literature Review There are several approaches to study the aerodynamics of the VAWTs and their
inherently unsteady flow field. Several criteria, such as total cost, investment in time, and accuracy have to be considered to determine whether an approach is appropriate for this
11 application. Experimental studies have always been conducted for this purpose, as they are necessary to investigate the actual flow physics. The majority of the experimental works in this area are focused on the performance of the H-Rotor Darrieus turbine, rather than the conventional Darrieus rotor with curved blades [56]. Ferreira et al. [28] used a two-dimensional Particle Image Velocimetry (PIV) method to visualize the development of dynamic stall on a blade in a pitching motion, at different tip speed ratios. Armstrong et al. [7] conducted an experimental study on a full size turbine to investigate the aerodynamics of a high-solidity VAWT. To observe the flow characteristics on a large scale, they utilized light-weight tufts, attached to the inner surface of the blades, which they claimed to be less challenging and expensive than the PIV and laser Doppler velocimetry. Tescione et al. [77] recently conducted a near wake flow analysis of a VAWT, using stereoscopic particle image velocimetry in an open-jet wind tunnel. They studied the wake dynamics by obtaining the phase-locked measurments at the turbine mid span. In addition, they used several vertical planes, aligned with the free-stream, to visualize the evolution of the blade tip vortices and the 3D wake geometry. Experimental studies are inevitable in designing and developing any technology. However, they are extremely time-consuming and expensive, in particular for flow visualization purposes. There are numerous factors and conditions that have to be considered and satisfied, in order to have a reliable result, especially if the model under study has complex flow or geometry. Since vertical-axis wind turbines are characterized by their complex, unsteady and turbulent flow field, researchers have always been trying to introduce numerical or analytical models that can predict the performance of these turbines at lower costs. These models are in three main categories: Blade Element Methods, Vortex methods, and Navier-Stokes (grid) methods.
12 Due to the unsteady and turbulent nature of the flow, analytical studies are restricted to fundamental assumptions that tredemendously simplifies the problem. Brahimi et al. [13] compared the validity and accuracy of several aerodynamic analysis models, such as streamtube models and numerical simulations, with experimental data. They demonstrated a fairly good agreement in prediction of the aerodynamic loads and rotor performance as well as dynamic stall. One of the basic models to calculate the local forces on a propeller or a wind turbine, is the Blade Element-Momentum (BEM) theory. The use of Blade ElementMomentum (BEM) models dates back as far as the nineteenth century, when Glauert [31] introduced BEM Theory for the first time, in order to predict the structural dynamics and performance of airplane propellers. Templin [76] then adjusted this theory for VAWT aerodynamics by introducing the most basic BEM model, called the Single Stream-tube model. This theory simplifies the problem by assuming that the turbine is a permeable ideal disc, called the actuator disc. Ideal means that the disc is frictionless, and there is no rotational velocity component in the wake [52]. This model breaks the turbine down into smaller parts, called the streamtubes, through which the flow passes and reaches the turbine. The actuator disc creates a discontinuity of pressure in the streamtube of the flow. This model uses linear momentum theory and assumes a control volume, in which the boundaries are the surface of a streamtube and the two cross-sections of the streamtube [26, 32]. Since there is only one streamtube in this model, the entire freestream has a uniform velocity, before and after the disc. This is not consistent with the actual flow, in which the velocity varies in cross-stream direction. Therefore, this method was further developed by introducing double multiple-streamtube models [60], in which
13 the turbine is replaced by an upwind and a downwind actuator disc, and the flow is split by multiple streamtubes rather than one. The main disadvantage of the BEM and vortex models is their dependency on empirical airfoil data, which neglects unsteady aerodynamics, dynamic stall, and wake dynamics [32, 75]. Habtamu et al. [33] and Paraschiviou [59] modified this model by incorporating these unsteady effects in their works and enhanced the accuracy of the model. However, both of these models neglect the viscous effects on the aerodynamic forces applied on the turbine blades. VAWTs have highly turbulent flows with high Reynolds numbers. Due to the presence of the blades, these flows are wall-bounded, and there exists a turbulent boundary layer on each blade, that significantly impacts the blade’s aerodynamics. Within the turbulent boundary layer, viscous effects dominate the flow. Hence, neglecting these effects remarkably reduces the accuracy of the model, which still is the main drawback from BEM and vortex models. Grid methods, better known as Numerical Simulations, discretize the domain into smaller control volumes, called cells, and resolve the Navier-Stokes equations at each cell, at a specific time. These simulations require a tremendous amount of time and memory. However, when the unsteady and viscous effects are not negligible, they are the most appropriate model to use. Grid methods include Finite difference (FDM) [68], Finite Element (FEM) [65] and Finite-Volume methods (FVM) [34]. Within the past few years, the finite volume method has become more popular than the other two, as the recent improvements in computer technology significantly reduced their cost [79]. Ferreira et al. [29] used FVM to simulate dynamic stall in a VAWT and compared the results with their experimental data. They compared the accuracy of the commonly used turbulence models, URANS Spalart-Allmaras and k-ε, Large Eddy Simulation (LES) and Detached
14 Eddy Simulation (DES). They demonstrated that DES was better able to predict the generation, shedding and convection of vorticity. Castelli et al. [67, 16] simulated a lowsolidity vertical-axis micro wind turbine at moderate tip speed ratios, using URANS k-ε turbulence model. They statistically illustrated that the accuracy of the simulation can be enhanced by refining the grid within the turbulent boundary layer. They also compared their results with their wind tunnel study conducted in Politecnico de Milano. While the wind tunnel data was qualitatively similar to the numerical results, it could only be considered as a rough estimate, as the tunnel blockage effects were neglected in their experiments. Howell et al. [36] conducted a wind tunnel study along with a two- and a threedimensional numerical study of a small VAWT, and investigated the effects of wind speed, tip speed ratio, solidity and surface finish on the accuracy of the results. They proposed that the turbine’s performance is highly dependent on the number of blades and their surface roughness. Moreover, they showed some discrepancies between the twodimensional results and the experimental data, and related them to the incapability of the 2D methods of simulating the large tip vortices. Amet et al. [1] performed 2D numerical simulation of blade-vortex interaction in a VAWT. They demonstrated that turbines at lower tip speed ratios produce stronger dynamic stall, while the flow remains attached to the blade at higher tip speed ratios. They also illustrated that neither the Momentum nor the Vortex model can predict the physics beyond stall. Li et al. [49] investigated the feasibility and accuracy of three FVM models, namely 2D URANS, 2.5D URANS, and 2.5D LES, in prediciting the characteristics of high-angle-of-attack flows. They defined their 2.5D geometry as a 3D VAWT with periodic boundary conditions at either ends of the blades. With that done, they demonstrated that the dominant factor affecting the
15 accuracy of the 2D FVM simulations is neither the effect of tip vortices nor the vertical flow divergence, but their inherent limitations in vortex modeling. They proposed the 2.5D LES simulation as the most accuarate model to predict the performance of the VAWTs. Apart from accurately predicting the VAWT performance, there have been numerous studies concentrating on their improvement. Several approaches have been adopted to enhance the average power coefficient of the VAWTs within the past few years. Among these, fixed and variable pitch blades have shown a promising future for this goal. These methods try to control the variation of the effective angle of attack and keep it slightly below the stall limit of the airfoil, thus mitigating the negative effects of dynamic stall. While these studies focus on enhancing the individual turbine performance, recent studies are leaning towards improving the performance of VAWT arrays using the aerodynamic interactions between the turbines. Whittlesey et al. [80], Dabiri [21] and Kinzel et al. [42] suggested that in contrast to HAWT wind farms, closely spaced VAWTs may have higher individual turbine power coefficients, and yield higher power outputs per unit land area. In their bio-inspired study, Whittlesey used experimental data to develop a potential flow model of inter-VAWT interactions to investigate the effect of VAWT spatial arrangement on the array power output. Placing the turbines in a geometric arrangement, based on the configuration of shed vortices in the wake of schooling fish, was shown to remarkably enhance the array power coefficient. The fish tend to align themselves to optimize their forward propulsion, while spatial configuration of VAWT arrays aim at maximizing energy extraction. In addition to these array configurations, the aforementioned authors suggest that placing two counter-rotating turbines in close proximity may enhance their individual
16 performance. They associated this power enhancement to the flow acceleration and stream-tube contraction in the region between the two turbines. Dabiri also showed a faster wake recovery behind the coupled turbines than that of an isolated turbine. The concept of Turbine Coupling was based on previous research on the suppression of vortex shedding in the wake of counter-rotating cylinders [18, 25, 55]. This initiated from the fact that on large scales, the wake behind a VAWT is similar in structure to the one behind a rotating cylinder. Rajagopalan et al. [69] also conducted a study in cooperation with Sandia National Laboratory to invesitgate the effect of Turbine Coupling on the VAWT performance improvement. The concept was finally patented by Thomas [78]. 1.4
Scope As mentioned earlier in the chapter, HAWTs have been benefiting from years of
development and research studies, such that the modern HAWT’s average power coefficient is reaching the Betz limit [23]. On the other hand, VAWT’s development has been extremely gradual, and their aerodynamics is not quite well understood. In particular, the complexity of their unsteady aerodynamics has been a driving factor in hindering their progress. VAWT aerodynamics present a number of factors yet to be investigated, and several different parameters, such as solidity, blade shape and tip speed ratio, yet to be optimized. This requires a great deal of research that takes several years to complete. Consequently, this case study is limited to modeling the aerodynamics of a VAWT in 2-D, and then investigating the effect of turbine coupling on its performance. In order to identify the unsteady phenomena associated with the VAWT aerodynamics, and to be able to conduct a parametrized study, a grid-based approach is clearly necessary. Thus, FVM method will be implemented to simulate a low solidity, three-bladed, H-Rotor Darrieus wind turbine. This is a random geometry chosen as a base
17 for developing a general procedure that can be easily implemented on a diverse set of turbine geometries. It has been borrowed from Castelli et al. [67, 16] who has welldocumented all the parameters incorporated in their study, and will be used for validation purposes. The main goal in modeling the aerodynamics of the VAWTs is to identify the dominant factors limiting their performance. The flow physics that result in the formation of these factors as well as the effect of tip speed ratio on their nature, will be further explained. This will be done by visualizing the VAWT flow field for each case, which facilitates the observation of the wake structures, flow seperation, and blade-vortex interactions. Accurately predicting each of these phenomena is the best indicator of success for these simulations. After identifying the performance-limiting factors, the concept of Turbine Coupling will be used, in order to mitigate their negative effects, and to improve the performance of the VAWT. Turbine Coupling, as mentioned previously, is to place two turbines in close proximity, and set them to rotate in opposite directions. These counterrotating turbines will have aerodynamic interactions with each other that will result in the improvement of their individual performance. The effect of different spacing between these turbines as well as the effect of the tip speed ratio on the VAWT performance will also be investigated. Power enhancement is the first goal of this approach, which will be explored by comparing the turbine average power coefficient of the single turbine, with that of the pairs of turbines. To better explain this phenomenon, turbine and blade torque coefficients will be plotted and compared between all cases.
18 Wake recovery is the second goal of this approach, which will be examined by visualizing and comparing the wake behind the single turbine and that of the pairs of turbines. Wake recovery results in more efficient wind turbine arrays, and as will be shown in chapter four, is a key factor in lowering the cost of electricity generation.
19 CHAPTER 2 METHODOLOGY AND VALIDATION 2.1
Model Geometry The basic model geometry has been borrowed from Castelli et al. [67, 16] and its
main features have been shown in Table 2.1. This two-dimensional geometry consists of a rectangular outer zone, shown in light blue in Figure 2.1 (a), and a circular inner zone, shown in dark blue and magnified in Figure 2.1 (b). The outer zone is stationary and represents the wind tunnel. The inner zone represents the rotor and rotates counterclockwise with an angular velocity of Ω. The rotation is relative to the Z-axis of a coordinate system whose origin is located at the center of the rotor. The extents of the Wind Tunnel Zone are 30 and 45 turbine diameters in its width and length, respectively. These large extents are necessary to minimize the wind tunnel blockage effect and to prevent reverse flow from the outlet.
Table 2.1. Model Geometry Blade Profile Number of Blades [-]
NACA 0021 3
Drotor [mm]
1030
Chord Length [mm]
85.8
Blade Mounting Point
0.25 c
DDomain / DRotor [-]
45,000
HDomain / HRotor [-]
30,000
20
(a)
(b) Figure 2.1. layout of the physical domain, boundary conditions and the interface between the two zones. (a) Wind tunnel zone (b) Rotor zone. Two symmetry boundary conditions were chosen for the top and bottom boundaries of the model geometry to represent the wind tunnel walls. Symmetry boundary conditions are generally used to model the zero-shear slip walls in viscous
21 flows [4]. An inlet boundary condition on the left lets air flow into the wind tunnel with a speed of 9 m/s with a constant turbulent viscosity ratio of 0.1. Turbulent viscosity ratio is the ratio between the turbulent viscosity, 𝜇! , and the molecular dynamic viscosity, 𝜇, and is a convenient way to control the turbulence level in the unperturbed flow (freestream) from the inlet. It is especially useful for cases in which estimating any characteristic turbulent length scale is difficult, such as external aerodynamics. Flow-split outlet boundary condition was chosen for the outlet. This boundary condition gives you the option to specify a mass flow rate at the outlet, and is usually appropriate when the flow is assumed to be incompressible and the segregated flow solver is used [74]. By using this boundary condition at the outlet, we ensure that all the flow coming from the inlet would exit through the outlet. The wind tunnel and rotor zones are separated by a grid interface boundary condition that ensures continuity in the flow field. As will be explained later in this chapter, this interface is where the sliding mesh technique is implemented. All turbine blades are NACA 0021 symmetrical airfoils with 85.8 mm chord length. They all have been mounted at the quarter point from the leading edge with a zero offset pitch. No-slip wall boundary condition was chosen for the blades. Each blade is bounded by an oval interior boundary, called the control ellipse. These oval boundaries have no physical importance and they just help to have a better control over the quality of the grid around the airfoil [4]. 2.2
Domain Discretization The majority of the time spent on the computational studies is allocated to grid
generation and finding a grid independent solution. There is always a compromise between accuracy and computational expenses. Most often, grid quality should be
22 evaluated by trial and error and comparing the solution with experimental data or analytical solutions, before proceeding to the next step. In order to save time, usually a coarse grid should be generated first. This helps to have a rough estimate of the order of the grid size at each region of the computational domain. Then the grid should be refined at the critical regions that need higher resolutions. These regions are where the higher solution variable gradients take place. This way we can save computational expenses by ensuring higher grid concentration only at regions where it is needed. This is an iterative procedure, and every time the quality of the grid will improve and the solution will be more valid, until the solution does not vary by further refining the grid. This is when the grid independent solution is found and no more iteration is required. Once the appropriate grid setup is found, the process becomes straightforward. Discretization methods are highly problem-dependent. One method may not be applicable to all sorts of problems, and one problem may be solved using several methods. Discretization method should be chosen cleverly and carefully, as it can be remarkably influential on the final result. Each method has its own restrictions, advantages and disadvantages. As this work aims at observing the effects of unsteady aerodynamics on the flow field of the VAWTs, a sliding mesh method has been used for our simulations. Fundamentals of this method, its main characteristics and its application in our problem are described below. 2.2.1
Sliding Mesh Concept. Sliding mesh technique is a domain decomposition
method that allows two adjacent grid zones to move relative to each other along a grid interface [11, 2, 3, 10]. Thus, grid interface conformity is not necessary. Conformity is the condition that two meshes match in a way that every pair of adjacent elements can only intersect at a node or an edge, and no intersection faces are allowed [53]. It is
23 necessary to keep the size of the elements on either side of the grid interface about the same size, in order to keep the cell aspect ratio close to one [4]. This enhances the solution consistency and prevents possible instabilities. Due to its adaptivity, sliding mesh technique is especially valid when the two subdomains on either side of the grid interface have been constructed using different discretization methods, i.e. the hybrid grids [11]. Nonconforming Sliding Mesh method for solving the Unsteady Navier-Stokes equations in nonstationary geometries was initiated by Bazilevs et al. [9], and further developed by Anagnostou et al. [2, 3]. Being implemented in many researches, Sliding mesh method has shown to be suitable for moving geometries. McNaughton et al. [53] used sliding mesh technique for their CFD simulations of Tidal-Stream turbines. They implemented this method by treating the grid interface as an internal Dirichlet boundary condition, and calculating the interfacial values by interpolation from halo nodes. Their simulations are consistent with experimental results and alternative FVM methods. Sliding mesh method has widely been used to model wind turbine aerodynamics as well. Hsu et al. [37] used this method in their numerical simulations of Horizontal Axis Wind Turbines (HAWTs), using finite element arbitrary Lagrangian–Eulerian– variational multiscale formulation and validated the results with some experimental data. In another study, Hsu and Bazilevs [38] have used sliding grid in their 3D simulation of full-scale HAWT with nacelle and tower. Korobenko et al. [43] have also used this approach to simulate Vertical Axis Wind Turbines’ (VAWTs) unsteady aerodynamics, as well as their interactions on one another when they are placed in close proximity. Castelli et al. [67, 16, 17] have employed this technique in their numerous works on different factors affecting the performance of VAWTs. Howell et al. [36], Ferreira et al. [29] and
24 Nobile et al. [58] have all used this method in their visualization of the flow in the near wake of VAWTs. Before the simulation starts, both of the subdomains on either side of the grid interface have an interface zone. Once the solution is initialized, the two interface zones will be merged to generate the grid interface. During the simulation, the relative movement of the cell zones changes the number of faces on both interface zones. At each time step, the intersection between the two interface zones will be found and a new non-conformal grid interface will be generated. The flux across this grid interface will then be calculated. As can be seen in Figure 2.2, at the new time step, the original faces 𝑎𝑐, 𝑐𝑒, 𝑏𝑑 and 𝑑𝑓 will be removed and the new interface will be composed of smaller faces 𝑎𝑏, 𝑏𝑐, 𝑐𝑑, 𝑑𝑒 and 𝑒𝑓. To calculate the flux into the element III, for instance, data will be transferred across faces 𝑏𝑐 and 𝑐𝑑, from elements I and II, respectively [4].
Figure 2.2. Non-conformal grid interface boundary between the sliding and stationary subgrids.
25 There is a limit to how much one sub-domain can move with respect to another, without excessive information loss between the grids. Therefore, a conservative time-step is required, ensuring that the interface passes by no more than half a cell in a single interval. Sliding mesh technique decomposes the model geometry to at least two cell zones, based on zones’ relative velocity. For simplicity, we assume there are only two cell zones in our problem, one of which is stationary and the other moves with the velocity of v with respect to the former. In the stationary zone, the standard conservation equations for mass and momentum are solved. In the moving zone, however, the modified set of balance equations,
@ (uj @Xj
vj ) = 0
@ @ ⇢ui + ⇢ (uj @t @Xj
v j ) ui =
@p @⌧ij + @Xi @Xj
(2.1)
are solved. The first equation is the modified continuity and the second is the modified momentum equation. Here uj is the flow velocity in a stationary reference frame and vj is the velocity component originated from the mesh motion. Like before, p is the pressure and τij is the stress tensor. At the grid interface a conservative interpolation is used for both mass and momentum equations [8]. 2.2.2
Hybrid Grid. Based on the structure of the grid, discretization methods are
categorized in three different groups; namely, Structured, Unstructured and Hybrid.
26 Structured grid method uses efficient numerical algorithms and requires less computer memory than the others. Due to its accuracy, this method is suitable for resolving viscous sub-layers [82]. However, this method may not be applicable to many problems, as they require a fairly simple geometry to generate the grid. On the other hand, Unstructured grid method is suitable to generate grids around complex geometries due to their flexiblity [12, 39, 66]. In particular, the solution adaptivity is their greatest advantage [51, 64]. However, the unstructured grid method has been shown to be tremendously memory consuming and computationally expensive [30]. Moreover, there are limitations to the choice of flow solver, which affects the computational efficiency of the method. Clearly, these methods complement each other on weaknesses. Therefore, a Hybrid grid method that properly combines these two methods may facilitate having the same level of accuracy in problems with complex geometries. Considering the complexity of our model geometry, a hybrid grid scheme has been used to discretize the domain. As can be seen in Figure 2.3, it consists of a structured prism layer mesh around the airfoil and unstructured triangular meshes for the rest of the domain. The prism layer mesh helps to better resolve the flow and capture the solution parameter gradients inside the turbulent boundary layer. This will be further explained later in this section. The thickness of this layer was estimated to be equal to the thickness of the attached turbulent boundary layer. Using three size functions, we increased the number of grid nodes at the leading and trailing edges of the airfoil. This can also be seen in Figure 2.3. A finer mesh with higher concentration of nodes is necessary at these regions, as they are critical in observing the flow seperation and require higher resolution.
27
Figure 2.3. Hybrid Grid, with structured prism layer mesh around the airfoil, and unstructured triangular mesh for the rest of the domain. As shown in Figure 2.4, the prism layer mesh has 20 O-shaped block layers around the airfoil, with the first layer being 1% of the prism layer thickness. The thickness of the first row of cells on the surface of the airfoil determines the local ! 𝑌!"## values. These values have to be kept within a desired range, which will be
explained shortly. To prevent instabilities induced to the solution by singular points located at the sharp corners of the trailing edge, these corners have been rounded. Figure 2.4 also shows the round trailing edge and the smooth changes in its surface profile. Outside the prism layer mesh, unstructured triangular mesh begins with a cell aspect ratio of approximately one, with respect to the structured mesh. Going away from the airfoil
28
Figure 2.4. Prism Layer mesh around the airfoil and the concentration of grid nodes at the trailing edge. towards the oval interior boundaries, cell sizes grow gradually with a growth ratio equal to 1.05. This has been demonstrated in Figure 2.5. The same mesh generation process has been repeated for the rest of the domain, and their corresponding size functions have been presented in Table 2.2. [4] In order to obtain a grid independent solution, three different grids have been generated for our model geometry. Since we used the same size functions for all three
Table 2.2. Grid size functions growth ratios Region
Growth Ratio
Blade surface to control ellipse
1.05
Control Ellipse to circular grid interface
1.12
Grid interface to the farfield
1.20
29
Figure 2.5. Grid within the Control ellipse for balde section cases, the only difference between these grids was the number of grid nodes on the ! surface of the airfoils. The first grid, which ensured a value of 𝑌!"## within the range of 5
to 30, had 500 elements on the surface of the airfoil and 750,000 elements in total. This fairly coarse grid was the first attempt, and demonstrated the general behavior of the flow. It required wall functions to model turbulence in the region close to the airfoil where the flow is affected by the boundary layer. The second grid, with 900 elements on each airfoil and approximately 2.5 × 10! ! elements in total, kept the value of 𝑌!"## close to one, at the wall adjacent elements. This
remarkably finer grid makes the simulation computationally more expensive. However, its application is necessary to properly resolve the viscous sub-layer, where the velocity gradients are of high significance in predicting the occurrence of flow separation and dynamic stall. The third grid, with 1,500 nodes on the airfoil, was made to ensure a grid-
30 independent solution and did not show any significant improvement over the 900-node grid, and therefore was disregarded. ! Computationally, 𝑌!"## is an indicator of the grid size at the wall-adjacent cells.
Physically, this parameter is the non-dimensional distance from the wall. Similar to a local Reynolds number, it determines the sub-layer of the turbulent boundary layer, in which the wall-adjacent cells are located, and thus specifies the part of boundary layer they resolve. Y+ at regions near the wall is defined as [6],
Y+ ⌘
U⇤ y ⌫
(2.2)
where y is the distance to the nearest wall, and ν is the local kinematic viscosity of the fluid. U* is a characteristic velocity at the nearest wall, called the friction velocity, and is defined as,
U⇤ =
r
⌧! ⇢
(2.3)
where, 𝜏! is the wall shear stress, and ρ is the fluid density. According to the law of the wall [6] there are 3 sub-divisions of the near-wall region in a turbulent boundary layer: ! (a) 𝑌!"## < 5, Viscous (laminar) sub-layer. Velocity profile is assumed to be laminar, and
viscous stress dominates the wall shear. ! (b) 5 < 𝑌!"## < 30, Buffer layer. Both viscous and turbulent shear dominate, and flow
regime is in a transition. ! (c) 30 < 𝑌!"## < 300, Fully turbulent (log-law) region inner layer. Viscous stress
effects fade away and the turbulent shear predominates.
31 Presence of a wall has a significant effect on the turbulent flows. Since the viscosity-affected regions near the walls have large gradients in the solution variables, accurately presenting these regions is a key factor in successfully predicting wall! bounded flows. In our simulations with low 𝑌!"## , since the value of this number was
kept close to one, the first layer of cells on the surface of the airfoils, as described in the law of the wall, lies within the viscous sub-layer. Therefore, the governing equations were directly solved for the flow within this sub-layer. As we go away from the wall ! (airfoil), the value of the 𝑌!"## increases until we reach the inner layer of the fully
turbulent region, where the log law equations govern. This procedure significantly improves the accuracy of the simulations by properly predicting the dynamic stall. Dynamic Stall, as will be demonstrated in section 3.2.2, is the dominant factor affecting the performance of the wind turbines at low tip speed ratios. If the occurrence of this phenomenon is predicted with a lag/lead, the average power coefficient values will be higher/lower than the real value, respectively. 2.3
Numerical Simulation Numerical Simulations were carried out using Star-CCM+, a commercial finite
volume code. The main physical parameters are presented in Table 2.2. The fluid was assumed to be incompressible with constant density, which led us to use the segregated flow model. To solve the governing equations in the transient flow, implicit unsteady solver with second-order upwind temporal discretization was used. This method reduces the restrictions on the temporal discretization, i.e. Convective Courant Number. However, as mentioned previously, sliding mesh technique requires a conservative timestep, which in our simulations was kept as low as 5 × 10!! 𝑠 for all simulations. This will allow for more stable solution and better convergence.
32 At each grid point, a residual,
𝑟 = 𝑓 − ℒ𝜙 (2.4)
is calculated for each of the governing equations, 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 normalized [74], providing a measure of how well the computational solution holds to the governing equations. During each time-step the simulation is set to iterate, Table 2.3. Main Simulation Parameters Reynolds Number [-] Inlet Velocity [m/s]
300,000 9
TSR
1.5 – 3.5
Number of Elements [-]
2,500,000
Δt [s] Turbulence Model
5 × 10-5 Spalart-Allmaras
until the residuals fall below the predefined convergence criteria. In our simulations, these criteria were set to be 10!! for the Continuity and Momentum equations, and 10!! for the Turbulent Viscosity Transport equation. Considering the moderately high Reynolds Number value for our simulation with a two-dimensional geometry, a URANS turbulence model was required. In order to choose a suitable URANS turbulence model, results from simulations with Standard Spalart-Allmaras (S-A) [73], Standard K-epsilon (k-ε) [47] and Shear Stress Transport k-
33 omega (SST k-ω) [54] turbulence models were compared. S-A and SST k-ω demonstrated more consistent solutions than k-ε, as the latter is more suitable for internal flows. The main difference between these models was the tendency of the k-ε to delay the occurrence of dynamic stall. A delayed dynamic stall results in higher and unrealistic power coefficient values, which will be discussed further in section 2.4 in validation of the results. In addition to our approach of choosing the turbulence model, previous work done by Parachiviou [60] suggested that S-A turbulence model is more compatible with aerodynamic applications, especially in boundary layer flows. k-ω SST is a threeequation model and is computationally more expensive than the single-equation S-A. Therefore, Standard Spalart-Allmaras model was adopted as the turbulence model for our simulations. The flow was initially set to be at atmospheric pressure and to have zero velocity in the entire domain. As mentioned previously, the turbulence viscosity ratio of the inflow was set to be equal to 0.1. Simulations were set to run until they reached a quasisteady state with a periodic solution. The corresponding physical time required to reach this state was often the time required for 10 turbine revolutions and would vary in each case, according to their tip speed ratios. 2.4
Validation of the Results The finite volume simulations were validated by comparing their average power
coefficients within the mentioned range of tip speed ratios, with that of the FVM case studies of Castelli et al. [67] and Kozak et al. [44], and a double multiple-streamtube model (DSTM) developed by Kozak et al. [45]. Three different approaches have been adopted in these studies. Castelli, Kozak and Mehrpooya simulated the VAWT flow field ! using FVM simulations with high 𝑌!"## treatment at the wall. Mehrpooya and Kozak also
34 ! performed FVM simulations with low 𝑌!"## treatment at the wall. Mehrpooya’s
simulation utilizes a hybrid grid scheme with a sliding mesh technique at the interface of the stationary and rotary zones. Kozak’s FVM simulation, however, uses a structured overset grid floating on top of an underset grid [44]. Kozak has also used a double multiple-streamtube model code that accounts for the effects of dynamic stall on the performance of the turbine. As shown in Figure 2.6, all simulations share the same performance characteristics. The curves have similar shapes; as they all peak at a tip speed ratio of approximately 2.5, fall sharply as the tip speed ratio decreases, and decrease more ! gradually as the tip speed ratio increases. The FVM simulations, with high 𝑌!"##
treatment at the wall, show a considerably higher average power coefficient than the others, particularly at lower tip speed ratios. These curves also vary significantly from ! one another. However, the FVM simulations with low 𝑌!"## treatment at the wall are
much more consistent, and they also match better with the DSTM data, having less than 10% difference throughout the entire range of tip speed ratios. As could be seen, different models have qualitatively different predictions of the performance of the same turbine. In the absence of reliable experimental data, it may be necessary to further evaluate each model and all its crucial assumptions as well as the consistency between their results, in order to find the most accurate approach to study the aerodynamics of the VAWTs.
35
(a)
(b) Figure 2.6. Turbine Average Power Coefficient with respect to tip speed ratio for (a) ! ! simulations with high 𝑌!"## treatment and (b) simulation with low 𝑌!"## treatment and DSTM model.
36 ! As mentioned previously, FVM simulations with high 𝑌!"## treatment at the wall
rely on the wall functions, rather than the governing equations, to resolve the flow within the turbulent boundary layer. Wall functions are based on empirical observations and have been developed from the cases with attached and fully-developed flow. Therefore, these functions may not be suitable for prediction of flow seperation at high angles of attack, as they tend to replicate the attached and fully-developed boundary layer charcteristics, while the boundary layers are never fully developed [19]. This may result in delaying the stall and increasing the average power coefficient of the turbine. On the ! other hand, the simulations with low 𝑌!"## treatment at the wall have a more refined grid
within the boundary layer, and are able to directly resolve the governing equations at that region. In addition, the two FVM simulations, which used two different algorithms to solve the governing equations, show better consistency between themselves and with the DSTM data. Therefore, it can be concluded that the predictions based on this method are more reliable and accurate than the others. Consequently, this method has been used in the simulations performed later in this study.
37 CHAPTER 3 EFFECTS OF UNSTEADY AERODYNAMICS 3.1
Turbine Performance Turbine overall performance is dependent on various factors. As demonstrated in
the previous chapter, one of the most influential factors is the turbine tip speed ratio (TSR). As can be seen in Figure 3.1, turbine average power coefficient has considerably lower values at low TSRs; it increases rapidly going towards the higher TSRs, until it reaches its maximum at appoximately TSR=2.5. Then it decreases gradually as the TSR increases. Since the rate of change of the turbine’s average power coefficient varies with TSR, the dominant factors that affect this parameter may be different at each TSR. In order to recognize these dominant factors, power extraction will be investigated with more details in the next section.
Figure 3.1. Turbine average power coefficient with respect to tip speed ratio
38 3.1.1
Blade Torque Coefficient. Figure 3.2 shows the curves of blade torque
coefficient with respect to azimuthal angle, for four different tip speed ratios. This blade starts from the vertical position, which corresponds to zero degrees of azimuthal angle. These curves show a phenomenon known as torque ripples. As can be seen in all cases, the majority of the power is generated within the front half-cycle, and the turbine has very low power outputs within the rear half-cycle. However, there are some fundamental differences between these curves: First of all, at higher tip speed ratios, the curves become smoother and show less fluctuation. In addition, the maximum torque coefficient value decreases as the tip speed ratio increases. At lower tip speed ratios, since the rotational speed of the turbine is relatively low, the maximum angle of attack of the turbine increases as described in the first chapter, and it may go beyond the stall limit of the airfoil. In this case the blade will suffer from dynamic stall and generate negative torque. Whether the minimum value of the turbine’s net torque coefficient remains positive is highly important. When the net torque is negative, the blades must have enough momentum to counteract the resisting torque and maintain their rotational movement. In this case, the net torque must eventually rise to positive values, otherwise, the turbine will fail to rotate and cannot operate anymore. At higher tip speed ratios, the maximum angle of attack has lower values and cannot go beyond the stall limit. As a result, the effect of dynamic stall will fade away and the curve becomes smoother. This is desirable, as the fluctuating torque values will result in mechanical threats, such as fatigue. On the other hand, as the maximum angle of attack decreases, the maximum lift coefficient of the blade decreases, which will result in lower maximum torque coefficient. Within the rear half-cycle, the curves are smooth with
39 low values. The reason why the torque coefficient remains low is the blade-wake interaction, which will be discussed in section 3.2.3. With the dynamic stall effects at the lower tip speed ratios, and blade-wake interactions at hgher tip speed ratios, it could be expected that the maximum average power coefficient would be observed at neither the lower or higher ends of the tip speed ratio spectrum. For the case with tip speed ratio of 1.5, the torque coefficient starts to drop rapidly at the azimuthal angle of 90°, and it falls below zero at about 100°. A second drop takes place at about 250°. Since the angle of attack is not high enough for occurrence of dynamic stall, this drop may be due to a previously shed vortex, generated by an upstream blade, that is now hitting this blade at the rear half-cycle. For the case with tip speed ratio of 2.0, the curve remains positive until higher azimuthal angles, and it falls below zero at about 130°. Thus, the total power generated by this blade will be higher. This is due to the fact that the maximum angle of attack is lower and has been reached at higher azimuthal angles. This maximum angle of attack, however, is still large enough to cause dynamic stall and bring the torque coefficient down to negative values. Therefore, dynamic stall is still the dominant factor that affects the turbine’s performance. Like the previous case, a second drop, this time less intense, can be seen at about 230°. Starting from the tip speed ratio of 2.5, the curve tends to remain positive almost throughout the entire cycle. This is where the average power coefficient reaches its highest value. The maximum angle of attack decreases compared to the previous cases, thus the effect of dynamic stall is more mitigated but still apparent. It would be desired for this maximum angle to remain slightly less than the stall limit of the airfoil. This way,
40
Figure 3.2. Blade torque coefficient with respect to azimuthal angle for four different tip speed ratios.
41 it is small enough to prevent the negative effects of dynamic stall, yet sufficiently high to let the blade generate high amounts of lift. At tip speed ratio of 3.0, the maximum torque coefficient is noticeably lower. Due to lower maximum angle of attack, no dynamic stall is expected to take place. Consistently, no sudden drop can be seen, and the changes are smooth. Since the blades rotate faster, they complete the cycle quicker, and will not allow the wake sheets to be convected downstream. Hence, they are likely to intersect with more wake sheets at the rear half-cycle. These sheets are the wakes of either the same or previous blades, which contain high turbulent fluctuations and have low mean velocity. Intersection of the blade and these wakes will adversely affect power generation. It is proposed that the low values of torque coefficient within the rear half-cycle are due to this phenomenon. Blade-wake interaction will be further discussed in the sub-section 3.2.3. 3.1.2
Turbine Net Torque Coefficient. Figure 3.3 demonstrates the turbine net
torque coefficient with respect to the azimuthal angle for 4 different tip speed ratios. Since the solution has reached a quasi-steady state, it shows periodic characteristics. This means the solution will repeat after each period, which in this case, is one complete cycle. In addition, since the turbine has three identical blades, the frequency of the net torque is three times that of a single blade. Respectively, the results shown below are only for one third of the cycle, corresponding to 120 degrees of azimuthal angle. These curves vary significantly with the tip speed ratio. The values indicated by the curve for tip speed ratio of 1.5 range from -0.15 to slightly more than 0.3. This range narrows as the tip speed ratio increases. Consequently, for tip speed ratios higher than 2.5, the curves remain positive throughout the entire cycle, but their maximum values decrease significantly. As mentioned before, the former is highly favorable, as the blades
42
Figure 3.3. Turbine net torque coefficient with respect to azimuthal angle corresponding to one third of a cycle, for tip speed ratios ranging between 1.5 and 3.0.
43 will not experience a resisting torque, but the latter decreases the turbine’s efficiency. In addition to their differences in their values, the maximum value of net torque coefficient tends to move toward the lower azimuthal angles as the tip speed ratio increases. 3.2
Performance Limiting Factors
3.2.1
Summary. While flow visualization is an expensive and memory-consuming
way of observing the flow field, it is highly informative, and can help us investigate the factors that limit the performance of VAWTs. Observing the phenomena that happen during one turbine revolution could be complementary to the information given by the previously shown torque curves. Figures 3.4 and 3.5 demonstrate the vorticity magnitude distribution for tip speed ratios of 2.0 and 3.0, respectively. Observing the vorticity magnitude distribution is an effective way of determining the structure of the flow field. It is also suitable to show the wake structures, their path, and their interactions with the blades. These sequences show the flow characteristics for one third of an entire cycle. Following one blade from sequence 1 to 10 would be the best way of observing these figures. As can be seen, the flow field is more complex at tip speed ratio of 2.0. The reason behind this fact is the occurrence of dynamic stall, in which a leading edge vortex will form, detach, and convect downstream of the turbine. On their way, these vortices interact with the blades (sequence 1 of Figure 3.4), and hamper the power generation. Therefore, at this tip speed ratio, dynamic stall and blade-vortex interaction are dominating the flow. These factors will be explained in detail in section 3.2.2.
44
1
10
2
9
3
8
4
7
5
6
Figure 3.4. Vorticity magnitude distribution for turbine running at TSR = 2.0. The sequence shows one third of a cycle, with 12 degrees of rotation at each step.
45
1
10
2
9
3
8
4
7
5
6
Figure 3.5. Vorticity magnitude distribution for turbine running at TSR = 3.0. The sequence shows one third of a cycle, with 12 degrees of rotation at each step.
46 By increasing the tip speed ratio, the flow tends to remain attached to the surface of the airfoil throughout the entire cycle. As a result, the flow becomes more orderly. On the other hand, since the turbine rotates faster than before, the blades would reach the wake sheets before they are convected downstream. Therefore, the blades are likely to intersect with more numerous wake sheets. Blade-vortex interaction will be replaced by blade-wake interaction, which will be the new dominating factor that affects the performance of the VAWTs. This factor will be discussed more in section 3.2.3. Apart from their direct interactions with the blades, the wakes impact the efficiency of the VAWTs by inducing drag to the flow. This is called wake blockage effect, and will be described in section 3.2.4. 3.2.2
Dynamic Stall. In spite of their simple design, VAWT’s aerodynamics and flow
field are quite complicated. During their operation, blades of the turbine work under both static and dynamic stall conditions. Thus, they are subjected to cyclic forces due to variation of instantaneous angle of attack. As a turbine blade rotates, its instantaneous angle of attack fluctuates rapidly. At lower tip speed ratios, the amplitude of this fluctuation is large enough to go beyond the stall limit of the airfoil [71], which manifests as dynamic stall. As demonstrated by Larsen et al. [48] and Nobile et al. [58], dynamic stall is characterized by the flow separation at the suction side of the airfoil. Figure 3.6 sequentially shows how dynamic stall takes place. First leading edge separation starts and gradually rolls up and turns into a leading edge vortex (LEV). This vortex then becomes larger and covers the entire surface of the airfoil. At this moment the blade generates the highest amount of lift. LEV, however, is highly unstable, and it quickly detaches from the surface of the airfoil, while a trailing edge vortex (TEV) starts to form. TEV has the same
47 strength as the LEV but with an opposite sign [40]. Once the LEV detaches, there will be a drastic drop in the value of the blade’s lift. Subsequently, the LEV breaks down, and the TEV rolls up and detaches from the surface of the blade. Both of these vortices will then be convected downstream of the turbine, and flow will reattach to the surface of the blade. Flow reattachment is highly favorable, as it rapidly boosts the lift. Researchers are currently using flow reattachment by active flow control to enhance the lift on a stalled blade [81]. It should be noted that the LEV usually becomes visible in cases with significantly high Reynolds Number [35]. Since Figure 3.6 has been extracted from our case with a fairly moderate Reynolds Number, the separation bubble tends to start from the regions closer to the trailing edge, then it expands towards the leading edge [50]. The rest of the process would be the same for both cases. The lift and power coefficients reach their maximum values when the LEV is on the surface of the blade. However, since LEV is extremely unstable, their values will drop sharply by the LEV detachment, which will generate oscillating forces. These unsteady forces can generate vibration, noise and fatigue characteristics, which reduces the effective life of the blades and the turbine. Hence, a great deal of research has been conducted in order to prevent dynamic stall at lower tip speed ratios [61, 57, 62]. 3.2.3
Blade-Wake Interactions. As the blades of the turbine pass through the flow,
they inevitably generate wakes [46]. Wake sheets are regions of circulating flow immediately after a moving or stationary rigid body. In other words, they are concentrations of high-
48
1
10
2
9
3
8
4
7
5
6
Figure 3.6. Sequence of formation of dynamic stall in the case with TSR = 2.0. Vorticity distribution is shown in grey scale, and the streamlines with velocity magnitude in color.
49
magnitude vorticity and they contain low mean velocity and high turbulent fluctuations [24]. As the time goes on, the wake sheets generated within the front half-cycle are swept along by the freestream towards the rear half-cycle of the turbine. That is where they intersect with the rotating blades. Figure 3.7 shows the sequence of blade-wake interaction within the rear half-cycle of the turbine for the case with tip speed ratio of 3.0. Blade-Wake interactions immensely hinder the power generation, due to the reasons to be discussed below. The turbine’s average power coefficient is directly proportional to the kinetic energy provided by mean stream-wise velocity of the flow. Since these wake sheets contain considerably lower mean velocity than the freestream, they have less extractable kinetic energy. Therefore, blade-wake interaction within the rear half-cycle will result in loss of lift, which could be clearly observed in the torque ripples previously shown. Moreover, when the intersection of the blade and the wake sheet creates an oblique angle, flow separation, however less intense, is likely to happen. This is called partial flow separation, which can be observed at sequence 4 of Figure 3.7, and may negatively impact the blades’ lift coefficient. 3.2.4. Wake Blockage Effect. Wake sheets are concentrations of high vorticity and they deplete their surrounding flow of kinetic energy and result in drag [46]. Drag force is generated in three main ways: •
Surface Friction Drag: caused by shear stresses within the boundary layer.
•
Pressure (form) Drag: induced by flow separation and recirculation regions.
•
Viscous Induced Drag: produced by added mass effects [44].
When the unperturbed freestream approaches the turbine, it experiences some resistance
50
1
4
2
5
3
6
Figure 3.7. Vorticity magnitude distribution, showing the blade-wake interactions within the rear half-cycle of the turbine running at TSR = 3.0. due to the presence of drag, and tends to bypass the turbine towards the regions with less resistance. This can be clearly seen in Figure 3.8, which shows the mean stream-wise and cross-stream velocities, normalized by the freestream velocity, for a VAWT running at tip speed ratio of TSR = 3.0. Appendix A explains the method used to obtain these timeaveraged velocity distributions in detail. Looking at Figure 3.8 (a), it can be easily noticed that the mean stream-wise velocity drops rapidly after the freestream hits the front-half-cycle blades. The wake has significantly lower mean velocity and extends far
51
(a)
(b) Figure 3.8. Distributions of time-averaged (a) streamwise and (b) cross-stream velocities, normalized by the freestream velocity, in the vicinity of the turbine. behind the turbine. Flow with lower mean velocity contains lower kinetic energy and limits the power generation, when interacting with the turbine blades [20].
52 Figure 3.8 (b) better illustrates the bypassed flow. The flow in front of the turbine has higher cross-stream velocity, and it tangentially goes around the turbine. This phenomenon is called the wake blockage effect and is similar in nature to the solid blockage effect [70]. Wake blockage reduces the turbine’s mass flow rate, which negatively impacts the input power to the turbine. Both the blade-wake interactions and the wake blockage effect are more serious at higher tip ratios. This is due to the presence of more numerous and stronger wakes. At tip speed ratios higher than 3.0, the flow remains attached to the surface of the blade throughout the entire cycle. Therefore, it is these two effects, rather than the dynamic stall, that limit the overall performance of the turbine and dominate the flow. Each of the aforementioned performance limiting factors require their own way of treatment in order to mitigate their negative effects. Some researchers prefer to stay at lower tip speed ratios and subdue the effects of dynamic stall. While their works aim at improving the efficiency of an isolated turbine, the main goal of the current work is to find a way to improve the efficiency of both the isolated turbine and wind turbine arrays. This has been accomplished via turbine coupling, which mitigates the negative effetcs of wake blockage. This concept will be discussed in detail in the next chapter.
53 CHAPTER 4 TURBINE PERFORMANCE IMPROVEMENT In order to convert the wind power to electricty with lower cost, more efficient wind farms are required. Clearly, improving the efficieny of indiviual turbines is necessary to improve the efficiency of the entire wind farm. As mentioned previously, this work and many other studies have been conducted towards improving the performance of wind turbines and enhancing their power output. While producing more efficient wind turbines is an effective way to achieve this goal, it may be fruitful to look at the efficiency of the wind farm as a whole. The array power coefficient CAP is the ratio of the average power coefficient of an array of VAWTs to that of a single VAWT [80]. This ratio can be calculated using
!
𝐶!" = !!""!# , !"#$%&
(4.1)
where Psingle can be calculated using equation (1.7). CAP > 1 indicates that the array of VAWTs enhances the total power output , and CAP < 1 shows that the array reduces the total power. Therefore, this study concentrates on raising this ratio above one by applying the concept of turbine coupling. To have a better sense of the problem, Figure 4.1 shows a schematic of the model geometry. Two identical counter-rotating VAWTs have been placed in close proximity, with the same rotational rate (tip speed ratio). In the current work, the turbines are synchronized with no phase difference. This means both of the turbines have the same frequency and start their cycle from the same azimuthal angle. Therefore, the geometry is completely symmetrical with respect to the line that
54 perpendicularly halves the centerline. The centerline is the line that connects the centers of the turbines, and its length (S) is equal to the spacing between the two turbines. The optimal spacing, at which the total average power coefficient of the couple is maximum, is then to be sought.
Figure 4.1. layout of the pair of turbines. Considering this symmetry, a legitimate question here would be, “Why should both turbines be included in the simulation?” The reason to keep both turbines is to be capable of manipulating more parameters, such as wind direction, turbine frequencies and phase differences in our future works, that would revoke the problem’s symmetry. Another parameter incorporated in this study is the turbine speed ratio. For each turbine spacing, the pair of turbines has been simulated for three different tip speed ratios. These are the same tip speed ratios as for the simulations with a single turbine, with the
55 exception of TSR = 1.5. The reason for eliminating this case will be explained later in this chapter. To be able to compare the results from these cases to the ones from a single turbine, all simulation parameters, as well as the mesh generation process, were kept the same throughout the entire process. All the simulations have been run until they reached a quasi-steady state and demonstrated a periodic solution, as explained previously. Three different spacings between the turbines have been studied in this work, which their values have been presented in Table 4.1. These spacings were chosen after communicating with John Dabiri [22]. In their work [21, 42], the turbines have been placed 1.65 turbine diameters away from each other. This is the closest they could be placed due to the constraints that the measuring tools, such as anemometers that were installed in between the turbines, imposed on the problem. Since there is no such a constraint in our numerical simulations, the effect of closer spacing on the performance of turbines may be of high interest, and will be further evaluated in the next two sections.
Table 4.1. Spacing between the pair of turbines Case Number
Spacing between the turbines
1
1.65 D*
2
1.50 D
3
1.35 D * D is the diameter of the turbines
56
In section 4.1, after showing the results of the simulations and visualizing the flow field, the effect of turbine coupling on the VAWT power output will be investigated. Wake recovery is the second goal of this method, which will be discussed afterwards, in section 4.2. 4.1
Power Enhancement Figure 4.2 demonstrates the turbine’s average power coefficient with respect to
the tip speed ratio. There are four curves representing four different cases: one for a single turbine and the other three for pairs of turbines with three different spacings. The curves have similar shapes at first glance, but they have notable differences. As it can be seen and was explained previously, the average power coefficient of a single turbine reaches its maximum at about TSR = 2.5. The power coefficient rapidly drops by decreasing the tip speed ratio, but gradually decreases by increasing the tip speed ratio. However, the pair of turbines does not repeat the exact same pattern. The rapid drop can be observed again for all of the pairs, but this time it is considerably steeper. Therefore, as the pair’s average power coefficient at TSR = 1.5 would be drastically lower than the one for a single turbine, thus counterproductive, this particular tip speed ratio was not considered in this approach in order to save time. At the other end of the spectrum, each curve behaves differently. For the case with 1.65D spacing, the curve has slightly lower slope after its maximum, than a single turbine, and it maintains its value until higher tip speed ratios. Thus, the maximum
57
Figure 4.2. Average power coefficient with respect to tip speed ratio for a single turbine, and pairs of turbines with 1.65D, 1.50D and 1.35D spacing. takes place somewhere in between TSR = 2.5 and TSR = 3.0. Average power coefficient has remarkably increased at higher tip speed ratios, and the method has effectively improved the performance of the turbine. For the case with 1.50D spacing, the method progressively enhances the power generation at higher tip speed ratios. The curve’s slope after its maximum continues to decrease, and as a result the average power coefficient has a slightly higher value at TSR = 3.0 than TSR = 2.5. When the turbines are placed 1.35D apart, their corresponding curve does not reach its maximum within this spectrum of tip speed ratios, and it continues to increase. This means the maximum will be reached at a tip speed ratio higher than TSR = 3.0. Therefore, the curve has relatively lower values than the previous spacing within this range.
58 Clearly, the interaction between the turbines is more effective at higher tip speed ratios, and the method is actually more suitable for faster turbines. With regards to power enhancement, the case with 1.50D spacing and TSR = 3.0 turns out to be the most effective spacing among the three. Having CAP = 1.22, this case improves the array average power coefficient by 22%. Following the same pattern and considering the variation in the slope of the curve, it can be presumed that the pair of turbines’ average power coefficient may continue to improve by decreasing the spacing and running the turbines at higher tip speed ratios. However, this may not be favorable, since as explained in the first chapter, one of the most significant advantages of VAWTs over HAWTs is their relatively lower angular velocity. It is preferred not to jeopardize this advantage for the sole purpose of power enhancement, as it may cause design, acoustic and environmental issues. 4.1.1
Torque Coefficients. As previously demonstrated, observing blade torque
coefficient can provide us with some insightful ideas about what the blades experience while going through a turbine cycle. Figure 4.2 shows the Blade Torque Coefficient with respect to the Azimuthal angle for one complete cycle. Turbine tip speed ratio is equal to three for all the cases, as explained previously. The first torque curve represents the case with a single turbine, and has been reshown here as a reference to observe the amount of improvement gained through pairing the turbines. For 1.65D spacing, within the front half-cycle, the maximum torque coefficient has remarkably increased compared to the single turbine. Within the rear half-cycle, the curve moves slightly above zero and gains positive values. The change, however, is quite insignificant and the turbine generates very little amount of power. Overall, the method enhances power generation and mitigates the negative torque.
59
Figure 4.3. Blade torque coefficient with respect to azimuthal angle, for a single turbine, and pairs of turbines with 1.65D, 1.50D and 1.35D spacing. TSR = 3.0 for all cases.
60 For 1.50D spacing, within the front half cycle, power enhancement does not show a significant improvement from the previous case, but the method is still effective. Within the rear half cycle the method shows more improvement, which makes the net torque coefficient higher than the previous case. As previously shown, this spacing provides the highest turbine net average power coefficient among the three. The maximum value of the torque curve reduces at the next step, with 1.35D spacing. Although the method continues to enhance the power generation within the rear half-cycle, the overall torque coefficient will reduce due to lower values within the front half-cycle. Similar to a single turbine, no effect of dynamic stall is apparent in the torque curves. Therefore, the Blade-Wake interactions remains the dominating factor affecting the performance of the turbines. Differences between the torque curves in Figure 4.3 are subtle and nearly undiscernable. To make these differences more visible, and to give a better sense of the amount of improvement gained after every step of decreasing the spacing, pair of turbines’ net torque coefficients with respect to azimuthal angle have been compared in Figure 4.4. 4.1.2
Flow Visualization. As previously demonstrated, torque curves can sufficiently
reflect the presence of dynamic stall. However, observing the pair of turbines’ flow field may be necessary to better understand the dominating factors and also the effects of reducing the spacing between the turbines. Therefore, Figures 4.5 to 4.7 are presented here to show the distribution of vorticity magnitude for cases with 1.65D, 1.50D and 1.35D spacing, respectively. The sequences show one third of an entire turbine cycle.
61
Figure 4.4. Turbine net torque coefficient with respect to azimuthal angle, for a single turbine, and pairs of turbines with 1.65D, 1.50D and 1.35D spacing. TSR = 3.0 for all cases. Looking at these sequences, it can be noticed that the flow is more complex for the case with 1.65D spacing, and it becomes more ordered, as the turbines are placed closer to each other. In all cases, by tracking one blade throughout the entire cycle, we can clearly notice the absence of any major flow separation. However, some partial flow separation can be observed in the case with 1.65D spacing, which illustrates that the wakes have different structures. As can be seen in Figure 4.8, blade-wake intersections become more orthogonal as the spacing is decreased. Partial flow separation is more likely to happen at less orthogonal intersections, and will induce more chaos into the flow.
62
Figure 4.5. Vorticity magniture distribution for the case with 1.65D spacing and TSR = 3.0. The sequence shows one third of a cycle, with 12 degrees of rotation at each step. It starts from the top left, and ends at the top right.
63
Figure 4.6. Vorticity magniture distribution for the case with 1.50D spacing and TSR = 3.0. The sequence shows one third of a cycle, with 12 degrees of rotation at each step. It starts from the top left, and ends at the top right.
64
Figure 4.7. Vorticity magniture distribution for the case with 1.35D spacing and TSR = 3.0. The sequence shows one third of a cycle, with 12 degrees of rotation at each step. It starts from the top left, and ends at the top right.
65 Moreover, Figure 4.8 shows that the wake structures are different between these cases such that by decreasing the spacing, the wakes are less deformed and tend to retain their circular shapes as they are convected downstream. As previously mentioned, wake sheets are concentrations of high vorticity and they contain high levels of turbulence. In a
(a)
(b) Figure 4.8. Blade-wake intersection for cases with (a) 1.65D and (b) 1.50D spacing. VAWT flow field, the wake sheets are carried by the mean flow and their movement is directly related to the local velocity vectors [72]. Considering that in all cases the wakes have been generated by the same identical blades, from the same turbines, and with the
66 same physical conditions, they must have the same structures. The only difference between these cases is the spacing between the turbines. Therefore, it can be inferred that this spacing has a direct influence on the local flow inside the turbine, by changing the local velocity vectors both in magnitude and direction that eventually affects the wake structures. So far, we only know when two counter-rotating turbines are placed in close proximity, their local velocity vectors vary both in magnitude and direction. Streamlines may be the best tools to show the variation in the vectors’ directions. They are the curves that are instantaneously tangent to the velocity vectors of the flow. The streamlines for all the cases will be shown and further described in the following section. Thus, now we only focus on the variation in the magnitude of the local velocity vectors. Figure 4.9 demonstrates the time-averaged velocity distribution for a pair of turbines with 1.50D spacing. We obtained the time-averaged velocity distribution to remove the turbulent fluctuations and only focus on the mean velocity. Therefore, the velocity vectors in the vicinity of the turbines have been averaged over one turbine cycle. The details for this operation have been explained in Appendix A. As explained previously, the effects of solid and wake blockage have a negative impact on the performance of the VAWTs. The blades of the turbines, as well as the wake sheets they generate, block the area through which the flow is about to pass. This produces a significant amount of drag. When the flow approaches the turbines, it changes its path to bypass the resisting drag force. Therefore, the flow tends to go around rather than through the turbine. This was made apparent in Figure 3.8 as the cross-stream velocity component increased in front of the turbine.
67
Figure 4.9. Distribution of time-averaged streamwise velocity, normalized by freestream velocity, for the pair of turbines with 1.50D spacing and TSR = 3.0. One potential way to mitigate this effect is to confine the bypassed flow and force it to go through the turbine [5]. If we compare Figure 3.8 with Figure 4.9, we can realize how the presence of the second turbine confines and accelerates the flow in between the turbines. The pressure gradient generated by this accerated flow pushes the wakes towards the turbines and results in wake contraction. Therefore, the wakes block less area, and generate less drag. This phenomenon will be further explained in section 4.2. Figure 4.9 may not clearly show the change in the magnitude of local velocities within the turbines themselves. To better observe this, Figure 4.10 compares the mean stream-wise velocity along the trajectory of the blades for the single turbine to that of the
68 case with 1.50D spacing. The methodology to determine these curves has been included in Appendix A. Within the front half of the turbine, the curves do not show significant differences. Beginning at 𝜃 = 150° , the mean stream-wise velocity becomes greater for the pair of turbines than for the single turbine. This is due to the accelerated flow in between the turbines, which could be seen at Figure 4.9. The turbines are provided with higher kinetic energy flux at this region. However, since the blades are almost parallel to the flow direction, they may not generate any significant lift. Thus, the improvements in the mean streamwise velocity may not influence the power generation until higher azimuthal angles, at which the flow has higher angles of attack. Therefore, turbine
Figure 4.10. Time-averaged streamwise velocity along the blades’ path, normalized by the freestream velocity, for the single turbine and the pair of turbines with 1.50D spacing, at TSR = 3.0. coupling is responsible for the variation in the magnitude of the local velocity vectors of the flow inside the turbine, which is more apparent in this figure.
69 4.2
Wake Recovery As Kinzel et al. [42] presented in their work, wind farms are supplied with kinetic
energy flux from two different sources. Frontal (Horizontal) Kinetic energy is provided by the energy flux entering the turbine from upwind. This source is better understood, as it has been used in wind turbine analysis and calculating the upper bound of wind turbine average power coefficient, better known as Betz Limit. The power obtained from this source can be found by replacing the rotor swept area Aswept in equation (1.8) by the horizontal projected area of the wind farm Afrontal. This is the dominant source of power for individual turbines and small wind farms. However, since the turbines placed at the first few rows upstream of the wind farms deplete the horizontal kinetic energy of the flow, this energy flux does not reach the majority of the turbines placed farther downstream. In a large scale wind farm, it is the planform flux of turbulent kinetic energy that supplies most of the power [42]. For this to be the case, the length of the wind farm should exceed the height of the atmospheric boundary layer. This energy vertically enters through the top of the wind turbine array, and its power can be estimated as [14, 15],
P!"#$%&'( = − ρ A!"#$%&'( < u′w′ > (4.2)
where Aplanform is the planform area above the wind turbine array and u′ and w′ are the turbulent velocities. The inner product of these velocities forms the Reynolds Stress. Since the estimation of this energy is out of the scope of this work, the reader is referred to [42] for more details. In their work, Kinzel demonstrated that the VAWTs may have higher levels of planform turbulent kinetic energy flux than the HAWTs. This is due to the fact that they
70 are mounted closer to the ground, where the turbulent fluctuations have higher magnitudes. These fluctuations help the flow recover faster in the wake region behind the turbine pairs. Faster wake recovery will reduce the footprint of the VAWTs and allow closer turbine spacing along the flow direction. Closer spacing increases the power density of the wind turbine array. Power density is the power generated by the array of turbines per unit land area, and can be calculated as,
𝑃𝐷 =
𝑃!""!# . (4.3) 𝐴!""!#
Having larger power density in large wind farms would reduce the cost of electricty generation. Kinzel has also shown that the wake behind the VAWT pairs requires 4 to 6 turbine diameters to be recovered. Generally, the wake region behind the turbine is recovered once the streamwise velocity of the flow recovers to 95% of the unperturbed velocity. As its second goal, this work aims at observing the effect of turbine coupling on the VAWT wake recovery. First, the wake region behind the turbines will be visualized and compared, in order to analyze the effect of the turbine spacing on the wake recovery and to find the optimal spacing. Then, the physics behind this phenomena will be investigated, in order to better understand the contributing factors. Figure 4.11 illustrates the streamwise velocity of the flow in the near wake of the pairs of turbines as well as the single turbine. Comparing their wakes, it can be easily realized that the pair of counter-rotating turbines has significantly faster wake recovery than the single turbine, regardless of the spacing between them. Among the last three
71 cases, the wake of the case with 1.65D spacing is larger than the ones with 1.50D and 1.35D spacing. This indicates that the aerodynamic interactions between the turbines are stronger when they are placed closer to each other. The wake behind the pair of turbines with 1.50D spacing extends approximately 10 turbine diameters behind the turbine and turns out to be the shortest wake among all. Having the highest average power coeffcient, as demonstrated previously, and fastest wake recovery, this case once again proves to be the optimal spacing between the turbines at this tip speed ratio. 1.35D spacing has a slightly longer wake than 1.50D. As mentioned previously, this spacing may be the optimal spacing for turbines running at higher tip speed ratios. More work is required to find the optimal spacing at each tip speed ratio. Figure 4.12 demonstrates the cross-stream velocity of the flow in the vicinity of the turbines. While Figure 4.11 is more suitable to observe the wake recovery itself, Figure 4.12 better illustrates the phenomena that result in the faster wake recovery of the pair of counter-rotating turbines. It can be clearly observed in this figure how the recirculation regions diffuse as they are convected downstream. In the wake of a single turbine, the adjacent vortices have opposite signs as they are generated by different blades. According to Helmholtz’s and Kelvin’s Circulation Theorems, vortices with opposite signs tend to absorb and cancel each other out [46]. However, since their generating blades have a phase difference equal to one third of a turbine cycle, these vortices are not synchronous. Therefore, they are consecutively swept along by the mean velocity without interfering with one another, and will eventually dissipate by the effect of the viscous forces.
72
(a)
(b)
(c)
(d)
Figure 4.11. Distribution of streamwise velocity for (a) a single turbine, and pair of turbines with (b) 1.65D (c) 1.50D and (d) 1.35D spacing, at TSR = 3.0.
73
(a)
(b)
(c)
(d)
Figure 4.12. Distribution of cross-stream velocity for (a) a single turbine, and pair of turbines with (b) 1.65D, (c) 1.50D and (d) 1.35D spacing, at TSR = 3.0.
74 By coupling the turbines, the phase difference will be removed due to the symmetry of the problem. Vortices with the same strength but opposite signs are generated at the same time. When these vortices meet each other farther downstream, they counteract and cancel each other out. Therefore, faster wake recovery is accomplished by the supression of vortex shedding behind the VAWTs. The wake recovery length is the distance from the center of the turbines to where the vortices are terminated due to their interactions. This is the same as the previously mentioned distance required for the mean velocity to recover to 95% of the unperturbed velocity. This parameter depends both on the tip speed ratio and the spacing between the turbines. Another way of observing the wake recovery is to plot the streamlines. As mentioned previously, streamlines are the curves that are instantaneously tangent to the velocity vectors of the flow. Figure 4.13 demonstrates the streamlines for the single turbine and the three pairs of turbines. It can be seen that when the flow approaches the single turbine, the streamlines start to curve and lead the flow to bypass the turbine. As explained previously, this bypassed flow is caused by the turbine blockage effect. This curvature, however, cannot be observed in the streamlines passing in between the pair of turbines. The existence of a counter-rotating turbine imposes a symmetry that straightens the streamlines within this region. The streamlines are also consistent with the velocity distributions shown previously, and clearly demonstrate faster wake recovery for cases with counter-rotating turbines, in particular the last two cases with smaller spacing. The streamlines for the pair of turbines with 1.50D spacing return to its original straight shape faster than that of other cases.
75
(a)
(b)
(c)
(d)
Figure 4.13. Streamlines for cases with (a) a single turbine, and pair of turbines with (b) 1.65D spacing, (c) 1.50D spacing and (c) 1.35D spacing, at TSR = 3.0.
76 CHAPTER 5 CONCLUSION 5.1
Summary Within the past few years, there has been a resurgence of interest in vertical-axis
wind turbines, due to various operational advantages they present over horizontal-axis wind turbines. VAWTs are omni-directional and insensitive to wind direction. This removes the yaw mechanism from the system, and simplifies the design. VAWTs usually operate at tip speed ratios significantly lower than HAWTs, which reduces their noise level as well as their vulnerability to mechanical threats, such as fatigue and catastrophic failure. VAWTs are more scalable. They are easier to deal with when designing largescaled wind turbines, as they are subjected to constant gravitational and inertial forces. At the same time, these turbines are suitable for small-scaled applications, such as residential areas, as they are less sensitive to flow turbulence, and can be mounted closer to the ground. HAWTs should be placed far from each other, in order to reduce their mutual aerodynamic interferences. On the other hand, VAWTs can take advantage of the aerodynamic interactions between themselves, to enhance their individual power coefficients as well as their wake recovery characteristics. The former increases the efficiency of the wind turbines themselves, while the latter increases the efficiency of the entire wind farm as a whole, by increasing the power density of the wind farm. Considering all these advantages, VAWTs prove to be worthy of further research and development for improving their performance. First, all the dominant unsteady and viscous aerodynamics that limit the performance of these turbines should be identified.
77 Once these factors are well understood, a reasonable procedure to mitigate these factors should be devised and implemented, in order to improve the performance of the VAWTs. 5.1.1
Modeling VAWT Unsteady Aerodynamics. A low-solidity three-bladed
VAWT with zero pitch angle was simulated for a range of moderate tip speed ratios, using a finite volume code. This turbine was chosen for our parametrized study due to its well known performance characteristics. The physical domain was discretized using a hybrid grid scheme, with structured prism layer mesh on the surface of the blades, and unstructured triangular mesh for the rest of the domain. A sliding mesh technique was implemented at the interface of the stationary and rotary zones, due to its versatility in modeling the unsteady flows. One-equation Spalart-Allmaras turbulence model seemed to be suitable for this application due to its prior adoption for similar applications. After a comparison between the results of different turbulence models, S-A was chosen as the turbulence model used in the FVM simulations, since it presented the same level of accuracy as the three-equation Shear Stress Transport k-ω. An approach with low 𝒀! 𝒘𝒂𝒍𝒍 wall treatment was found to be the most appropriate method to capture all the solution variable gradients within the turbulent boundary layer. Directly resoving the governing equations for the flow within this region is necessary to accurately predict the flow seperation and dynamic stall. This conclusion was reached after validating the results by comparing the turbine average power coefficient with respect to tip speed ratio for simulations with different solution algorithms. All the models showed similar performance characteristics, by having their maximum at about TSR = 2.5, decreasing gradually by increasing the tip speed ratio, and falling rapidly by decreasing the tip speed ratio. This illustrated that the dominant factors affecting the
78 performance of turbine may be different at each tip speed ratio, and it requires more investigation to be better understood. Individual blade and turbine aerodynamics were investigated in detail by plotting the blade and turbine torque coefficients. These curves demonstrated that the majority of the power is extracted from the wind within the front half-cycle. The turbines running at lower tip speed ratios showed two sudden drops in their curves, which brought the curve down to negative values. The first drop, which was at the front half-cycle, was expected to be due to the occurrence of dynamic stall. Within this half-cycle, the instantaneous angle of attack may go beyond the stall limit of the blade, and result in dynamic stall. The second drop, which was at the rear half-cycle, was expected to be due to a blade-vortex interaction. The turbines with higher tip speed ratios, however, showed a smoother curve, with positie values almost throughout the entire cycle. Torque coefficient value were approximately zero within the rear half-cycle for these turbines, which required more details to identify the reason. Flow visualization facilitated the recognition of the performance limiting factors at each tip speed ratio. Dynamic stall, as expected, was the dominant factor lowering the blade lift coefficient for turbines at lower tip speed ratios. This occurrence of this phenomenon was shown in a sequence. It started by formation of a leading edge vortex, which quickly rolled up and detached from the surface of the blade. Once this vortex was shed, the blade’s lift value droped rapidly, which could be seen in the torque ripples. Blade-wake interaction was found to be the factor affecting the performance of the turbines at higher tip spees ratios. Since the blades rotate faster at higher tip speed ratios, they are subject to intersect with more number of wake sheets within the rear halfcycle. Wake sheets are concentrations of high vorticity and contain low mean velocity,
79 thus have extremely low kinetic energy values. Therefore, blade-wake interactions significantly reduces the blade lift coefficient. Another factor affecting the performance of VAWTs was shown to be the wake blockage effect. VAWTs regardless of their tip speed ratio, generate wake. This wake induces a great amount of drag to the flow, which acts as a resisting force against the upcoming flow. When the flow reaches the wake, it tends to bypass the turbine to pass through the less-resistant regions, which results in lower mass flow rate across the turbine. Therefore, the turbne is provided with less kinetic energy, and generates less power. This phenomenon was made apparent by showing the time-averaged stream-wise and cross-stream velocity distributions in the vicinity of the turbine. The cross-stream component of the velocity suddenly increased as the flow reached the turbine. 5.1.2
VAWT Performance Improvement. After identifying all the performance
limiting factors, the concept of Turbine Coupling was used to improve the efficiency of the VAWTs. Turbine coupling is to place two counter-rotating turbines in close proximity, to take advantage of the aerodynamic interactions between them. The distance between the centeral axes of the turbines was called the Turbine Spacing. The effect of turbine spacing along with the effect of tip speed ratio on the performance of these turbines was then investigated. Three cases with three different spacings, namely 1.65D, 1.50D and 1.35D, were simulated over the same range of tip speed ratios used priorly for modeling the single turbine. A comparison between turbine average power coefficient with respect to tip speed ratio of a single turbine, and that of the three cases with pair of turbines clearly illustrated the advantage of the turbine coupling. All the pair of turbines, regardless of the spacing between them, showed higher average power coefficients than the single turbine, at tip speed ratio of TSR = 3.0. The pair with 1.50D spacing provided
80 the highest power coefficient among all, increasing by 22% with respect to the single turbine. While the method successfully enhanced the turbine average power coefficient at TSR = 3.0, it failed to improve this coefficient at lower tip speed ratios. This means that the method is actually more effective on faster-rotating turbines. In addition, the slope of the power coefficient curve for cases with pair of turbines kept on increasing its value by decreasing the turbine spacig, thus having their maximum values at higher tip speed ratios. Therefore, it can be inferred that the average power coefficient may be further enhanced by further decreasing the spacing, and setting the turbines to run at higher tip speed ratios. This, however, requires more investigation. Flow visualization was the next step, providing insightful information about the structure of the wakes, blade-wake iteractions and flow seperation. In consistence with the single turbine, flow remained attached to the surface of the blade throughout the entire cycle, and no major seperation was apparent. The flow field was more complex for the case with 1.65D spacing, and became more orderly by decreasing the spacing. This means that the wakes kept their circular shape in the cases with lower spacings, resulting in more orthogonal blade-wake intersections. When the intersection of the blade with the wake creates an oblique angle, partial flow seperation and more power loss are likely to take place. To better understand the reason behind the power enhancement via turbine coupling, the time-averaged stream-wise velocity distribution for the case with 1.50D spacing was plotted and compared with that of the single turbine. This figure demonstrated a flow acceleration in the region between the turbines. The flow is confined to pass through the turbines, due to the symmetry imposed to the flow by the presence of
81 the other turbine. To better illustrate this confinement, the time-averaged steam-wise velocity along the trajectory of the blades was determined for the same cases. This velocity showed higher values within the rear half-cycle for the pair of turbines, providing the turbines with higher mass flow rate and kinetic energy. In other words, turbine coupling improves the efficiency of the VAWTs by alleviating the turbine blockage effect. Apart from the improvement in the average power coefficient of the VAWTs, turbine coupling presented another advantage by improving the wake recovery behind the pair of turbines. Previous works on rotating cylinders demonstrated a suppression of vortex shedding behind the pair of counter-rotating cylinders. Since the two turbines placed next to eachother are identical, counter-rotating, and with no phase difference, their wake structures on large scale resemble the vortex street behind the pair of rotating cylinders. The recirculation regions, generated by the blades of the both turbines, diffuse gradually as they are swept along by the mean velocity, and create large vortical regions. These vortices have the same magnitude but opposite direction, and will eventually cancel each other out, and result in faster wake recovery. This was made apparent by plotting the stream-wise and cross-stream velocity components as well as the flow streamlines for the single turbine and for all pairs of turbines. All figures clearly illustrated faster wake recovery behind the pair of turbines. Among all, the pair with 1.50D spacing showed the shortest distance required for recovering the wake, and once again proved to be the optimal spacing within the range of tip speed ratio under study. Streamline were also able to show the flow confinement in the region between the turbines, as the induced symmetry produced more straight streamlines and suppressed their curvature.
82 5.2
Future Work The concept of turbine coupling is a recent-introduced approach, aiming at
improving the performance of the vertical-axis wind turbines. Therefore, this concept presents a number of parameters yet to be optimized, and a complex flow physics yet to be analyzed. Clearly, this demands a great deal of research and study, and cannot be included all in one work. While this study attempted to investigate the effects of turbine spacing and tip speed ratio on the performance of the VAWTs to a good extent, it did not incorporate a number of contributing factors, in the interest of brevity. In the simulations performed in this study, wind direction was set to have a constant value, and its direction was always perpendicular to the inlet, and the center line connecting the centers of the pair of turbines. This, however, may not be the case all the time. Thus, investigating the effect of wind direction on the performance of the coupled turbines, in particular their wake recovery characteristics, is certainly an interesting topic for future studies. In addition, as mentioned previously, the turbines were identical, with the same rotational rate and with no blade phase difference. While this cluster of turbines makes the fluid dynamics easier to understand, it is not essentially necessary for this application. The turbines can be different in size, frequency of rotation and blade phase difference. Any of these changes will certainly change the entire aerodynamics of the turbines, and require more research for their effects to be understood. All the simulations in this study were two-dimensional, and as mentioned in section 1.2, were not able to model some important phenomena, such as evolution of tip vortices and vertical flow divergence. Generally, simulating three-dimensional aspects of the flow is tremendously expensive and difficult. However, performing these simulations
83 are highly recommended for future works, as it can significantly enhance the accuracy of the results and provide us with better understanding of the flow physics. Threedimensional simulations that model the turbulence using Dettached Eddy Simulation, can better model the flow seperation, vortex generation and wake structures. Considering the fact that evaluating VAWTs’ performance is highly dependant on accurately predicting these phenomena, the necessity of these simulations can be better justified. In addition to modeling these factors, 3-D simulations may provide us with insightful ideas about the intercations between the pair of turbines placed in wind turbine arrays. The vertical flux of turbulent kinetic energy, which is the main source of power to the turbines placed farther downstream of large wind farms, can not be determined in 2-D simulations. Therefore, incorporating all the parameters influencing the aerodynamics of the VAWTs and their complex flow physics requires a great deal of work to be done.
84
APPENDIX A TIME-AVERAGED VELOCITY CALCULATIONS
85 Since the vertical-axis wind turbine’s flow field has an unsteady nature, the mean (time-averaged) quantities have often been found necessary to minimize the effect of unsteadiness and have a rough estimate of the flow pattern in general. In order to visualize the mean quantities obtained from our simulations, a probe field was created for each case. Each probe field was a grid of probe points spread over the region under study. Probe points would provide us with any desired flow quantity at the space they are located, and at a specific time instant. We can calculate the mean value of any quantity at a point in the space using
1 𝑈! 𝑥! = lim !→! 𝑛
𝑛
𝑢𝑖 𝑘 (𝑥𝑗 , 𝑡! ) , (𝐴. 1) 𝑘=1
where, n is the number of the times the quantity has been determined at point 𝑥! . At each time, the velocity 𝑢! (𝑥! , 𝑡! ) is determined at the same place and at instant tk. In our calculations, we let the turbine rotate for one revolution and obtained the results after every two degrees of rotation (n=180). At each instant, a matrix was generated giving the coordinates of all of the probe points and their corresponding values of the desired quantity. A.1
Time-averaged Velocity Distribution at the Vicinity of the Turbine For this case, a Cartesian grid of probe points in the vicinity of the turbine was
created, which spread over all of the rotating zones and some parts of the neighboring stationary zones. These were the regions of the domain, where the flow was affected the most, by the presence of the turbine rotor.
86 As the blades of the turbine moved with time, they coincided with a few probe points, which would eliminate that point from the resulting matrix, and leave us with matrices with different dimensions. This issue was addressed by a MATLAB code, which would compare the coordinates of all of the points given in a matrix with that of a matrix without any eliminated probe points. This code then would find the missing points, add them to the matrix and assign a NaN (Not a Number) value to their corresponding values. This way, all the matrices would have the same dimensions and be ready for averaging and visualizing their contours. However, an inevitable issue was associated with this method, which were the artifacts appearing in the final contours, along the blade’s path, where the NaN values were added to the matrices. These artifacts did not seem to have a significant effect on the flow pattern and the final results. After all these steps, the timeaveraged values of velocity in stream-wise and cross-stream directions for the turbine running at TSR = 3.0 were determined, which was shown in figure 3.8. The stream-wise velocity distribution was also determined for the pair of turbines with 1.50D spacing and TSR = 3.0, which was shown in figure 4.9. A.2
Time-averaged Velocity Along the Trajectory of the Blades For this case, the probe field was created using polar coordinate system. A probe
line was placed at every 5 degrees of azimuthal angle, resulting in total of 72 probe lines. As can be seen in Figure A.1, each of them contained two probe points, one slightly above the blade, and the other at the same distance below the blade. This was done in order to avoid balde-probe intersection, as mentioned in the previous section. Using a MATLAB code, the values of the stream-wise velocity were interpolated between the two probe points of each probe line, in order to calculate this quantity at the tarjectory of
87 the blades (blades’ path). This could be seen in Figure 4.10 for the case with a single turbine and the case with a pair of turbines with 1.50D spacing, both at TSR = 3.0.
Figure A.1. The probe points located at the inner and outer side of the blade, with respect to the center of the rotor.
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