DEVELOPMENT OF A PROCEDURE FOR ...

DEVELOPMENT OF A PROCEDURE FOR PREDICTING POWER GENERATED FROM A TIDAL CURRENT TURBINE FARM by YE LI B.Eng. Shanghai Jiaotong University, 2000 M.A.Sc. The University of British Columbia, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering)

THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2008 © Ye Li, 2008

ABSTRACT A tidal current turbine is a device for harnessing energy from tidal currents and functions in a manner similar to a wind turbine. Similar to a wind turbine farm, a tidal current turbine farm consists of a group of tidal current turbines distributed in a site where high-speed tidal current is available. The accurate prediction of power generated from a tidal current turbine farm is important to the justification of planning and constructing such a farm. The existing approaches used to predict power output from tidal current turbine farms oversimplify or even neglect the hydrodynamic interactions between turbines, which significantly affects the power output. The major focus of this dissertation is to study the relationship between turbine distribution (the relative position of the turbines) and the hydrodynamic interactions between turbines, and its impact on power output and energy cost of a tidal current turbine farm. A new formulation of a discrete vortex method with free wake structure (DVM-UBC) is proposed to describe the behavior of turbines and flow mathematically, and a numerical model is developed to predict the performance (force, torque and power output) and the unsteady wake structure of a stand-alone turbine using the DVM-UBC. Good agreement is obtained between the results obtained with DVM-UBC and published numerical and experimental results. The results also suggest that DVM-UBC can predict turbine performance 50% more accurately than traditional DVM does with comparable computational efforts. Then, a numerical model is developed to estimate the acoustic emission from a tidal current turbine. Benefiting from the stand-alone turbine analysis and simulation, another numerical model is developed to predict the performance and wake structure of a two-turbine system using DVM-UBC. This model can also be used to quantify the effect of the relative positions (relative distance and incoming flow angle) and relative rotating direction of the two turbines on the performance of the system. The results show that the power output of a two-turbine system with optimal relative position is about 25% more than two times the power output of a stand-alone turbine under the same operating conditions. In addition, the torque fluctuation of a two-turbine system with optimal relative position is about 50% less than that of a stand-alone turbine. The results also suggest that the acoustic emission of a two-turbine system with optimal relative position and rotating direction is 35% ii

less than that of a corresponding stand-alone turbine. As an extension to the study of the two-turbine system, a numerical procedure is developed to estimate the efficiency (power output) of an N-turbine system (a tidal current turbine farm) by using perturbation theory and linear theory together with the numerical model developed for the two-turbine system. By integrating the hydrodynamic models for predicting power output and a newly-developed Operation and Maintenance (O&M) model to predict O&M cost, a tidal farm system model is framed to estimate the energy cost using a scenario-based cost-effectiveness analysis. This model can be used to estimate the energy cost more accurately than the previous models because it breaks down turbine’s components and O&M strategies in much greater details than previous models and it incorporates the hydrodynamic interactions between turbines. Overall, this dissertation provides a design tool for tidal current turbine farm planners and designers, and provides information for investors to make decisions on investing in ocean energy and for governments to make subsidy decisions and set environmental assessment guidelines, and shed light on other disciplines such as environmental sciences and oceanography in terms of studying flow fluctuation. Key words: Discrete Vortex Method, Energy Cost, Environmental Impact, Acoustic (Noise) Emission, Ocean Energy, Operation and Maintenance, Tidal Current, Tidal Current Turbine, Torque Fluctuation

iii

TABLE OF CONTENTS ABSTRACT………………………………………………………………………………………...……...….… ii TABLE OF CONTENTS……………………………………………………………………………………….. iv LIST OF TABLES……………………………………………………………………………………………….ix LIST OF FIGURES……………………………………………………………………………………………. xii NOMENCLATURE…………………………………………………………………………………………...xvii ACRONYMS AND ABBREVIATIONS……………………………………………………………………. xxiii ACKNOWLEDGEMENTS…………………………………………………………………………………. xxiv DEDICATION……………………………………………………………………………………………….. xxvi CHAPTER 1

INTRODUCTION...................................................................................................................1

1.1

RESEARCH BACKGROUND AND MOTIVATIONS ........................................................................................1

1.2

RESEARCH FOCUS AND EXPECTED CONTRIBUTIONS ...............................................................................5

1.3

RESEARCH METHODS .............................................................................................................................7

1.4

OUTLINE OF THE DISSERTATION ..............................................................................................................9

CHAPTER 2

A DESCRIPTION OF TIDAL CURRENT TURBINES ....................................................10

2.1

TIDAL POWER POTENTIALS ..................................................................................................................10

2.2

TIDAL CURRENT TURBINES ..................................................................................................................12

2.2.1

Drag-Driven Devices...................................................................................................................15

2.2.2

Lift-Driven Devices .....................................................................................................................16

2.2.3

The Vertical Axis Tidal Current Turbine in This Study.................................................................17

CHAPTER 3

HYDRODYNAMIC ANALYSIS OF A STAND-ALONE TIDAL CURRENT TURBINE.. .................................................................................................................................................21

3.1

A REVIEW OF PREVIOUS RESEARCH .....................................................................................................21

3.1.1

Research on a Stand-alone Wind Turbine....................................................................................22

3.1.2

Research on a Stand-alone Tidal Current Turbine ......................................................................24

3.2

A DISCRETE VORTEX METHOD WITH FREE WAKE STRUCTURE FOR A STAND-ALONE TIDAL CURRENT

TURBINE ...........................................................................................................................................................24 3.2.1

The History of the Discrete Vortex Method..................................................................................25

iv

3.2.2

Assumptions of the Model............................................................................................................26

3.2.3

Turbine Working Principle ..........................................................................................................27

3.2.4

Blade Bound Vortex .....................................................................................................................28

3.2.5

Summary of the Traditional Discrete Vortex Method...................................................................32

3.2.6

Vortex Decay................................................................................................................................32

3.2.7

Lamb Vortices ..............................................................................................................................34

3.2.8

Shedding Frequency ....................................................................................................................36

3.2.9

Nascent Vortex .............................................................................................................................37

3.2.10

Hydrodynamic Characteristics of a Tidal Current Turbine .........................................................38

3.2.11

Non-dimensionalization...............................................................................................................40

3.2.12

Computational Procedure............................................................................................................41

3.3

PARALLEL NUMERICAL AND EXPERIMENTAL RESEARCH IN THE NAVAL ARCHITECTURE LABORATORY AT

UBC

.............................................................................................................................................................44

3.3.1

Numerical Simulation..................................................................................................................44

3.3.2

Experimental Test ........................................................................................................................44

3.4

VALIDATION OF DVM-UBC AND CASE STUDIES ..................................................................................47

3.4.1

Validation ....................................................................................................................................47

3.4.2

Comparison with Other Potential Flow Methods........................................................................61

3.4.3

Three-Dimensional Wake Structure .............................................................................................65

3.4.4

An Example Vertical Axis Tidal Current Turbine.........................................................................67

3.5

ACOUSTIC MODEL ................................................................................................................................76

3.5.1

Sound in the Ocean......................................................................................................................77

3.5.2

Turbine Noise Prediction.............................................................................................................79

3.5.3

Power Spectrum Analysis ............................................................................................................80

3.5.4

Computational Procedure............................................................................................................81

3.5.5

Case Study ...................................................................................................................................82

3.6 3.6.1

DISCUSSION AND CONCLUSIONS ...........................................................................................................85 Discussion ...................................................................................................................................85

v

3.6.2

Conclusions .................................................................................................................................89

3.6.3

Future Work .................................................................................................................................90

CHAPTER 4 4.1

HYDRODYNAMIC ANALYSIS OF A TWO-TURBINE SYSTEM.................................91

REVIEW OF PREVIOUS RESEARCH .........................................................................................................91

4.1.1

Incompressible Aerodynamic Interactions between Two Wind Turbines .....................................91

4.1.2

Marine Hydrodynamic Interactions between Two Tidal Current Turbines..................................93

4.2

NUMERICAL MODEL FOR A TWO-TURBINE SYSTEM ..............................................................................95

4.2.1

Review of the Stand-alone Turbine Model ...................................................................................95

4.2.2

Assumptions for the Two-turbine Model......................................................................................96

4.2.3

Program Flowchart .....................................................................................................................96

4.2.4

Parameters of a Two-turbine System ...........................................................................................98

4.3

PERFORMANCE OF THE TWO-TURBINE SYSTEM IN DIFFERENT LAYOUTS ............................................105

4.3.1

The Performance of an Example Two-turbine System ...............................................................108

4.3.2

Generalized Results for the Performance of a Two-turbine System .......................................... 117

4.3.3

Summary....................................................................................................................................136

4.4

WAKE AND ACOUSTIC IMPACTS ..........................................................................................................138

4.4.1

Wake ..........................................................................................................................................138

4.4.2

Acoustic Emission......................................................................................................................139

4.5

A PROCEDURE FOR SIMULATING AN N-TURBINE SYSTEM ...................................................................142

4.5.1

N-Turbine System Formulation .................................................................................................142

4.5.2

System Simplification.................................................................................................................144

4.5.3

A Case Study – the Relative Efficiency of an N-turbine System.................................................145

4.6

DISCUSSION AND CONCLUSIONS .........................................................................................................150

4.6.1

Discussion .................................................................................................................................150

4.6.2

Conclusions ...............................................................................................................................156

4.6.3

Future Work ...............................................................................................................................157

CHAPTER 5 5.1

TIDAL CURRENT TURBINE FARM SYSTEM MODELING .....................................158

INTRODUCTION ...................................................................................................................................158 vi

5.2

ASSUMPTIONS.....................................................................................................................................160

5.3

DESCRIPTION OF THE INTEGRATED TIDAL CURRENT TURBINE FARM SYSTEM MODEL .......................161

5.3.1

Energy Output ...........................................................................................................................163

5.3.2

Total Cost...................................................................................................................................167

5.4

HYDRODYNAMIC MODULE .................................................................................................................169

5.5

OPERATION AND MAINTENANCE (O&M) MODULE .............................................................................170

5.5.1

Emergency Maintenance Cost ...................................................................................................171

5.5.2

Routine Maintenance Cost ........................................................................................................172

5.5.3

Service Sub-module and Farm Attribute Sub-module ...............................................................174

5.6

COMPUTATIONAL PROCEDURE OF THE FARM SYSTEM MODEL ............................................................174

5.7

OPTIMAL DESIGN OF A TIDAL CURRENT TURBINE FARM IN THE QUATSINO NARROW.........................177

5.8

DISCUSSION AND CONCLUSIONS .........................................................................................................184

5.8.1

Discussion .................................................................................................................................184

5.8.2

Environmental Impact Assessment ............................................................................................186

5.8.3

Conclusions ...............................................................................................................................190

5.8.4

Future Work ...............................................................................................................................191

CHAPTER 6

SUMMARY, CONCLUSIONS AND FUTURE WORK ..................................................192

6.1

RESTATEMENT OF THE RESEARCH PROBLEMS .....................................................................................192

6.2

SUMMARY OF THE METHOD AND ITS APPLICATIONS ...........................................................................193

6.2.1

Summary of the Work.................................................................................................................194

6.2.2

Sample Applications ..................................................................................................................197

6.3

CONTRIBUTIONS AND INSIGHTS ..........................................................................................................199

6.4

FUTURE RESEARCH DIRECTIONS ........................................................................................................200

REFERENCES .................................................................................................................................................202 APPENDICES ..................................................................................................................................................217 A. EXTENDED HYDRODYNAMIC ANALYSIS .....................................................................................................217 B. EXTENDED ANALYSIS ON OPERATION AND MAINTENANCE MODULE .........................................................225 C. ERROR ANALYSIS........................................................................................................................................229 vii

D. SOFTWARE DESCRIPTION ............................................................................................................................230

viii

LIST OF TABLES Table 1-1

Comparison of various energy sources (Adapted from Li 2005) ...................................2

Table 2-1

The main parameters of the turbine and its working environment...............................18

Table 3-1

Cases used to validate DVM-UBC...............................................................................48

Table 3-2

Basic specification of case 1 for kinematics validation................................................48

Table 3-3

The relative deviation of the results with DVM-UBC from the experimental results..49

Table 3-4

Basic information of Case 2 for kinematics validation ................................................50

Table 3-5

The relative deviation of the results generated with DVM-UBC from the results

generated by using the conformal mapping method ....................................................................50 Table 3-6

Basic information of Case 3 for kinematics validation ................................................52

Table 3-7

Comparison of wake geometry obtained with DVM-UBC and Fluent for Case 3.......52

Table 3-8

Basic specification of Case 1 for dynamics comparison ..............................................54

Table 3-9

Comparison of power coefficient obtained with different methods (DVM-UBC vs.

Templin’s (1974) experiment test) ...............................................................................................55 Table 3-10

Basic specification of Case 2 for dynamic validation ..................................................56

Table 3-11

Basic results of Case 2 for the dynamic validation ......................................................57

Table 3-12

Basic specification of Case 3 for dynamic validation ..................................................58

Table 3-13

Basic results of Case 3 for the dynamic validation ......................................................59

Table 3-14

Summary of the validations..........................................................................................61

Table 3-15

Basic information of the case for comparing power coefficient predicted by using

different potential flow methods ..................................................................................................62 Table 3-16

Results for comparing the power coefficient generated by using different potential

flow methods................................................................................................................................63 Table 3-17 Comparison of different numerical methods......................................................................64 Table 3-18

Basic specification of the case used to generate three-dimensional wake trajectory ...65

Table 3-19

Results of the case for three-dimensional wake trajectory ...........................................66

Table 3-20

Basic specification of an example tidal current turbine ...............................................67

Table 3-21

Locations of four points on a turbine for induced velocity investigation.....................72 ix

Table 3-22

Basic specification of the location of the three receivers of a stand-alone turbine ......82

Table 3-23

Differences in modeling conditions with different methods ........................................86

Table 3-24

Fish acoustic information .............................................................................................88

Table 4-1

The relationship between the incoming flow angle and the layout of the two turbines ... ....................................................................................................................................106

Table 4-2

Basic specification of the two-turbine system............................................................108

Table 4-3

The layout of the two-turbine system.........................................................................108

Table 4-4

Scenario description for a canard layout two-turbine system.....................................109

Table 4-5

Scenario description for a diagonal layout two-turbine system ................................. 112

Table 4-6

Basic specification of the two-turbine system for torque comparison ....................... 114

Table 4-7

The basic specification for Case 1.............................................................................. 118

Table 4-8

The relative deviation of the value in the negative plane from the corresponding value

in the positive plane in Case 1....................................................................................................121 Table 4-9

Basic specifications of a set of two-turbine systems ..................................................122

Table 4-10

The performance of the NACA 0015 blade two-turbine systems in different cases ..131

Table 4-11

Comparison of the performance of the NACA 63(4)-021two-turbine system.............136

Table 4-12

Comparison of the performance of two-turbine systems with two different blade types. ....................................................................................................................................137

Table 4-13

Basic specification of the two-turbine system for wake prediction............................139

Table 4-14

Location of three receivers.........................................................................................140

Table 4-15

Typical computation time of simulating an N-turbine system by using DVM-UBC .144

Table 4-16

Basic specification of the case for estimating the efficiency of the farm ...................146

Table 4-17

Summary of the results for predicting the performance of the tidal current turbine farm

as described in Table 4-16 and Figure 4-37................................................................................148 Table 4-18

Effective distance between the two turbines in Case 4, Section 4.3.2 at different

incoming flow angles .................................................................................................................151 Table 4-19

Results of the first peak in the two cases....................................................................154

Table 4-20

Results of the second peak in the two cases ...............................................................154

x

Table 5-1

Inputs for each of the sub-modules ............................................................................162

Table 5-2

Basic turbine specifications........................................................................................177

Table 5-3

Farm specifications ....................................................................................................178

Table 5-4

The relative deviation of the energy cost of Case 2 with respect to the energy cost of

Case 1

....................................................................................................................................179

Table 5-5

The ideal governmental subsidy in British Columbia, Canada ..................................182

Table 6-1

The basic specification included in the cases that we summarize here ......................198

Table 6-2

Results of the sample application............................................................................... 198

xi

LIST OF FIGURES Figure 1-1

Conceptual rendering of a tidal current turbine farm (Courtesy of Peter Frankel from

Marine Current Turbine Ltd.).........................................................................................................3 Figure 1-2

The integrated model for estimating energy cost ...........................................................7

Figure 2-1

World-wide potential tidal power sites (Adapted from Charlier (1993)) ..................... 11

Figure 2-2

Potential tidal power sites in Canada (Cornett, 2006)..................................................12

Figure 2-3

A vertical axis tidal current turbine ..............................................................................13

Figure 2-4

A horizontal axis tidal current turbine (Courtesy of Marine Current Turbine Ltd.) .....14

Figure 2-5

An example Savonius turbine (ENA, 2007).................................................................15

Figure 2-6

An example Darrieus turbine (Ecopower, 2007)..........................................................16

Figure 2-7

A sketch of a Davis turbine (BE, 2007)........................................................................17

Figure 2-8

The vertical axis tidal current turbine used in this study..............................................19

Figure 2-9

An illustration of azimuth angle ( φ ) ...........................................................................19

Figure 2-10

An illustration of blade arm angle ...............................................................................20

Figure 3-1

An illustration of turbine working principle with a bird-eye view on one of the blades.. ......................................................................................................................................28

Figure 3-2

An illustration of a two-dimensional time-dependent (unsteady) vortex wake structure. ......................................................................................................................................29

Figure 3-3

An illustration of a three-dimensional time-dependent (unsteady) vortex wake

structure ```..................................................................................................................................30 Figure 3-4

An illustration of the dimensionless tangential velocity of Lamb vortices ..................35

Figure 3-5

Flow chart of the DVM-UBC computational procedure for estimating the performance

(torque and power) and the wake structure of a stand-alone tidal current turbine .......................43 Figure 3-6

An illustration of the scheme of one of the turbines designed at UBC ........................45

Figure 3-7

(a ) A snap shot of the towing tank at UBC; (b) an scheme of the carriages used to test

the turbine (Rawlings 2008).........................................................................................................46 Figure 3-8

An illustration of the experimental setup: the turbine test frame, the turbine and the xii

measuring instruments (Rawlings 2008)......................................................................................47 Figure 3-9

A comparison of the two-dimensional wake trajectory by Strickland (1976) (While

Bubble) and DVM-UBC (line and circle) ....................................................................................49 Figure 3-10

A comparison of the two-dimensional wake trajectory by using conformal mapping

method (Deglaire 2007) (Red) and DVM-UBC (Blue)................................................................51 Figure 3-11

(a) Turbine wake velocity generated by using DVM-UBC; (b) Turbine wake velocity

generated by using Fluent ............................................................................................................53 Figure 3-12

A comparison of the power coefficient of a stand-alone tidal current turbine by using

different methods (DVM-UBC vs. Templin’s (1974) experiment test) under different scenarios55 Figure 3-13

A comparison of power coefficient of a stand-alone tidal current turbine by using

different methods (DVM-UBC, traditional DVM, Fluent and experiment).................................57 Figure 3-14

The relationship between torque and azimuth angle obtained by using different

methods under (a) scenario 1, and (b) scenario 2.........................................................................60 Figure 3-15

Power coefficient predicted by using different potential flow methods and one

experiment test .............................................................................................................................62 Figure 3-16

Three-dimensional wake structure for one blade of an example turbine .....................66

Figure 3-17

The relationship between power coefficient and TSR obtained with DVM-UBC.......68

Figure 3-18

Torque curve of a tidal current turbine on a polar diagram..........................................70

Figure 3-19

Torque of a stand-alone turbine in frequency domain .................................................71

Figure 3-20

Induced velocity at four points on a vertical axis tidal current turbine........................72

Figure 3-21

Dimensionless induced velocity at four points on a tidal current turbine: (a) at point 1;

(b) at point 2; (c) at point 3; and (d) at point 4.............................................................................75 Figure 3-22

Computation program’s flowchart of the noise emission estimation model................82

Figure 3-23

Locations of three receivers of a stand-alone turbine ..................................................83

Figure 3-24

Power spectrum of the noise intensity at location 1 of Case 1 ....................................84

Figure 3-25


Figure 3-26


Figure 4-1

The Cronalaght wind farm in Donegal, Ireland (EMD 2006) ......................................92

Figure 4-2

An offshore wind farm in Denmark (Sandia National Lab 2003) ................................92 xiii

Figure 4-3

A conceptual horizontal axis tidal current turbine farm (Courtesy of Marine Current

Turbine Ltd.) ................................................................................................................................94 Figure 4-4

A conceptual vertical axis tidal current turbine farm (Courtesy of Professor Coiro at

the University of Naples) .............................................................................................................94 Figure 4-5

Flowchart of the computation program for the two-turbine model ..............................98

Figure 4-6

A sketch of the conceptual layout of turbines in a tidal current turbine farm (not

representative of the real number of turbines in a farm) ............................................................100 Figure 4-7

A sketch of two turbines in canard layout ..................................................................101

Figure 4-8

A sketch of two turbines in tandem layout .................................................................102

Figure 4-9

A sketch of two turbines in diagonal layout ...............................................................103

Figure 4-10

An illustration of the phase shift between two turbines.............................................105

Figure 4-11

An illustration of the incoming flow angle (ψ ) and the relative distance ( Dr ) of a

two-turbine system.....................................................................................................................107 Figure 4-12

An illustration of the three scenarios of the canard layout two-turbine system.........109

Figure 4-13

The relative efficiency of a canard layout two-turbine system under different rotation

scenarios ................................................................................................................................... 110 Figure 4-14

The relative efficiency of a tandem layout two-turbine system................................. 111

Figure 4-15

The relative efficiency of a diagonal layout two-turbine system............................... 112

Figure 4-16

The relative efficiency of individual turbines in a tandem layout co-rotating

two-turbine system..................................................................................................................... 113 Figure 4-17

Torque of a two-turbine system in the frequency domain ......................................... 116

Figure 4-18

Torque of the upstream and the downstream turbines in a two-turbine system in the

frequency domain....................................................................................................................... 116 Figure 4-19

The relative efficiency of the two-turbine system in Case 1...................................... 119

Figure 4-20

The torque fluctuation coefficient of the two-turbine system in Case 1 ....................120

Figure 4-21

The relative efficiency of the two-turbine system in Case 2......................................124

Figure 4-22


Figure 4-23


xiv

Figure 4-24


Figure 4-25


Figure 4-26


Figure 4-27


Figure 4-28


Figure 4-29


Figure 4-30


Figure 4-31

The wake of a canard layout two-turbine system ......................................................139

Figure 4-32

Locations of three receivers.......................................................................................140

Figure 4-33

Power spectrum of the noise intensity of the two-turbine system at Reciever 1 .......141

Figure 4-34


Figure 4-35


Figure 4-36

An illustration of a linearlized turbine system...........................................................145

Figure 4-37

A hypothetical rectangular turbine farm site..............................................................147

Figure 4-38

A case study for an N-turbine system: (a) the change of the total efficiency with the

total number of turbines in a column; (b) the change of the total efficiency with the dimensionless relative tandem distance between two adjacent turbines ....................................149 Figure 4-39

An illustration of the upstream turbine wake (not representative of the real structure of

the wake) and the downstream turbine.......................................................................................152 Figure 5-1

Tidal turbine farm system model (the same as Figure 1-2) ........................................162

Figure 5-2

The structure of the hydrodynamic module ...............................................................170

Figure 5-3

The structure of the O&M module (Left) and an expansion of the emergency

maintenance sub-module (Right) ...............................................................................................171 Figure 5-4

The routine maintenance sub-module ........................................................................173

Figure 5-5

The flowchart of TE-UBC..........................................................................................176

Figure 5-6

Energy cost: (a) no hydrodynamic interaction; (b) with constructive hydrodynamic

interactions .................................................................................................................................180 Figure 5-7

2005-2006 Electricity price in several regions in North America (BCHydro 2007) ..182

xv

Figure 5-8

Energy cost in a certain year for a 20-year life time’s medium farm(blue bar: the

energy cost is the 5th cost; red bar: the increment with respect to )............................................183 Figure 5-9 Sensitivity analysis on the effects of five factors on energy cost (a) technicians salary, (b) technician workload, (c) farm offshore distance, (d) extreme wind and wave conditions, and (e) extreme fog condition ................................................................................................................185

xvi

NOMENCLATURE Greek Symbols

α

Angle of attack

–

αo

Vortex core constant

–

β

Blade arm angle

–

Δβ

Pitching angle

–

ε ρ

Critical convergent deviation value

–

Density

kg / m3

ρw

Sea water density

kg / m3

Γ

Vortex strength

m2 / s

ΓB

Bound vortex strength

m2 / s

ΓS

Spanwise vortex strength

m2 / s

ΓT

Trailing vortex strength

m2 / s

ω

Angular velocity

1/ s

τ

Time

s

φ

Azimuth angle

–

ψ

Incoming flow angle

–

θ

Induced velocity angle

–

π η

The ratio of a circle’s circumference to its diameter

–

Turbine efficiency

–

ηS

Standalone turbine efficiency

–

η

Turbine relative efficiency( η

λ

Tip speed ratio (

σm

Maximum radius of a vortex core

m

σc

Cut-off radius of a vortex core

m

ηS )

Rω ) U∞

– –

xvii

English Symbols A

Turbine frontal area

m2

b

Length of foil segment

m

c

Hydrofoil chord length

m

c

Sound Velocity

m/s

C

Stiffness

kg / ms 2

Cenergy

Energy cost

$ / KWH

Ct

Tangential force coefficient

–

Cn

Normal force coefficient

–

CD

Drag coefficient(

CL

Lift coefficient(

CP

Turbine power coefficient(

CTF

Torque

D

1 ρ AU R2 2 L

1 ρ AU R2 2

fluctuation

–

)

–

)

P

1 ρ AU R3 2

coefficient,

–

) ΔTpeak Δf peak

,See dBS

Eq.(3.47) CW

Wake growth coefficient, , SeeEq.(3.46)

–

cap

Capital cost

$

D

Drag

kgm / s 2

Dr

Relative distance

–

Ed

Energy demand per house

KWh

Eideal

Ideal energy output from a farm

KWh

Edown

Downtime energy loss of a farm

KWh

EC

Emergency maintenance cost

$

EEC

Emergency equipment cost

$

xviii

ELC

Emergency labor cost

$

EMC

Emergency material cost

$

ETC

Emergency transmission cost

$

f

General coefficient

–

fB

Strike frequency

1/s

fe

Electrical coefficient

–

fm

Mechanical coefficient

–

ft

Tidal coefficient

–

ftotal

Total conversion coefficient

–

f peak

frequency of the peak of a torque power spectrum

1/s

Fn

Normal force

kgm / s 2

Ft

Tangential force

kgm / s 2

H

Turbine height

m

i

Square root of -1

–

I

Noise intensity

dB

J

Periodigram

–

k

Number of computational cycle

–

K

Signal section

–

Kc

Vortices strength decay coefficient

1/ s

Kd

Vortices strength decay coefficient

s

l

General length

m

lN

Critical number of the computational loops

–

lW

Length of the wake

m

L

Lift

kgm / s 2

L

Length of a signal sequence

–

levco

Levelized cost

$

M

Torque

kgm 2 / s 2

n

Signal sequence number

–

xix

nB

Number of blade

–

N

Total number of sequences

–

N

Total number of turbines in a farm

–

NC

Number of columns in a farm

–

Nh

Number of house holds

–

Nr

Number of rows in a farm

–

P

General power output

Watt

Pm

Power output of a mechanical system

Watt

Pe

Power output of an electrical system

Watt

Phydro

Power output of a hydrodynamic system

Watt

Pout

Final power output from a farm

Watt

r

General distance

m

rc

Radius of vortex core

m

R

Turbine radius

m

RC

Routine maintenance cost

$

REC

Routine equipment cost

$

RLC

Routine labor cost

$

RMC

Routine material cost

$

RTC

Routine transmission cost

$

SG

Governmental subsidy

$/KWh

St

Strouhal number

–

t

Time increment in integration

year or day

Tpeak

Magnitude of the peak of the torque power dB spectrum

u

Velocity in the x direction

m/s

U

General velocity U (u, v, w)

m/s

Ui

Induced velocity

m/s

U iB

Induced velocity at Blade

m/s

xx

UP

General induced velocity

m/s

UR

Relative velocity

m/s

Ut

Tip velocity

m/s

UV

Vortex convection velocity

m/s

U∞

Freestream incoming flow

m/s

ViPT

Induced velocity by trailing edge vortex wake

m/s

ViPS

Induced velocity by spanwise vortex wake

m/s

Winitial

Initial width of the wake

m

Wend

End width of the wake

m

Xd

The x axis relative distance of a two-turbine

m

system

( xv , yv )

Location of a wake vortex

(x

Location of the upstream turbine of a two-turbine ( m , m )

up

, yup )

(m ,m )

system

( xdown , ydown )

Location of the downstream turbine of a ( m , m ) two-turbine system

Yd

The y axis relative distance of a two-turbine

m

system Z

Acoustic impendence

kg / m 2 s

xxi

Σ

Special Symbols Summation

∂

Partial differential

D

Total differential

iff

If and only if

Δ

Difference

~

Fluctuation/relative value

^

Dimensionless

i

Product and vector dot

∇

Gradient operator

→

Vector

∀

For all

xxii

ACRONYMS AND ABBREVIATIONS 2D

Two Dimensional

3D

Three Dimensional

BC

British Columbia (Canada)

CFD

Computational Fluid Dynamics

CFE

Center for Energy

DVM

Discrete Vortex Method (traditional)

DVM-UBC

Discrete Vortex Method with free wake structure developed in this dissertation

FFT

Fast Fourier Transformation

MCT

Marine Current Turbine Ltd.

NACA

National Advisory Committee for Aeronautics

NRC

National Research Council

O&G

Oil and Gas

O&M

Operational and Maintenance

PSM

Power Spectrum Magnitude

TE-UBC

Tidal turbine farm system model for Energy cost estimation developed in this dissertation

TSR

Tip Speed Ratio

UBC

The University of British Columbia

UK

The United Kingdom

US

The United States

USACE

US Army Corp of Engineers

xxiii

ACKNOWLEDGEMENTS There are many people who deserve credits for being of help in this research. First of all, I would like to thank my advisor, Dr. Sander M. Calisal, for giving me such a great opportunity to learn from him again. I knew him in person since 2003 when I began to pursue my Master of Applied Science degree in the Department Mechanical Engineering at the University of British Columbia under his supervision. At that time, he brought me into the world of advanced research. During my doctoral study, he guided me to investigate the very fundamentals of the important problems in mechanical engineering and taught me how to think and solve general engineering problems from the basics of the physics and in a rigorous mathematical way. He is not only my advisor but also my mentor and friend. I sincerely hope I can have chances to work with him and learn more from him in the future. Besides Dr. Calisal, I have also benefited greatly from the advices and thoughts of my research committee members and other faculty. I would like to thank Dr. Barbara J. Lence for her guidance in system modeling and optimization, Dr. Samuel, S, Li for his instruction in simulating ocean circulation, Dr. Gouri Bhuyan for his comments and insights on ocean power technologies, Dr. Farrokh Sassanni for his advice on mechanical design and industrial engineering, Dr. H. Keith Florig for his guidance on developing electricity integration models and system operation and maintenance strategies, Dr. Murray Hodgson for his advice on estimating acoustic emission, and Mr. Jon Mikkelsen for his suggestions on preparing good presentations and help in dissertation development. I would also like to take this opportunity to thank my colleagues in the Naval Architecture and Offshore Engineering Laboratory at the University of British Columbia for the friendship and help. They are Mahmoud Alidadi, Kelvin Gould, Voytek Klaptocz, James McRoberts, Yasser Nabavi, and George Rawings from the University of British Columbia and Florent, Perrier and Thomas who are French visiting students. Also, I would like to thank my families who always support the decisions that I made about my life. I am forever grateful for their understanding. The University of British Columbia is really a great university with extremely free academic environment and many

xxiv

outstanding scientists. I spent five unforgettable years here for both my master and doctoral studies. Finally, acknowledgements are given to the following agencies for providing me financial support: the University of British Columbia; Natural Sciences and Engineering Research Council, Canada; Institute of Electrical and Electronics Engineers(IEEE)-Oceanic Engineering Society; IEEE-Industrial Electronics Society; International Society of Offshore and Polar Engineering; American Society of Mechanical Engineers, Ocean, Offshore and Arctic Engineering Division, USA; The Society of Naval Architects and Marine Engineers, USA; John Davies Foundation, Canada; Canadian Transportation Research Forum, Canada; Transportation Association of Canada, Canada; Dieter Family Foundation, USA as well as numerous agencies for travel support during my study.

xxv

To Professor Sander M. Calisal, who has taught me honesty in scientific research and guided me to understand the nature in a philosophical way

xxvi

Chapter 1 Introduction In this dissertation, we present the results of research on harnessing energy from tidal currents by using tidal current turbines. In this chapter, we introduce the research background and motivations, define the problems, briefly describe the methods adopted to address these problems and present the outline of the dissertation.

1.1 Research Background and Motivations The depletion of traditional energy sources (fossil fuels) and the degradation of the environment as a result of fossil fuels consumption urge the global community to seek alternative energy sources, especially renewable sources. A variety of alternative renewable energy resources, such as wind, sun (solar energy), ocean wave and tidal current, are being explored. A comparison of the different energy resources across several dimensions (renewable, predictable, visual impact, environmental impact, capital cost and maintenance cost) is shown in Table 1-1. Tidal current as an energy source has the comparative advantages of being renewable and predictable, having low visual impact and low environmental impact, and incurring low maintenance cost. Tidal current turbines1, which are analogous to wind turbines, are promising devices for extracting energy from tidal currents (see Lang 2003). A set of tidal current turbines, distributed schematically at an offshore site, is called a tidal current turbine farm (Figure 1-1) which is analogous to a wind (turbine) farm. Unlike the wind power industry, the tidal current power industry is still in its infancy, and no commercial tidal current turbine farm has been constructed yet. However, there have been commercial tidal power facilities in France and Canada that harness energy from tidal ranges (the elevation difference between high tides and low tides).

1

Tidal current turbines can be driven by tidal current flow, gulf stream, or other current. Thus, sometimes, tidal current turbines are called marine current turbine or ocean current turbine while the basic technologies are the same. In this dissertation, we use the term “tidal current turbine” to refer the turbines driven by any ocean flow while we focus on tidal flow. 1

Table 1-1

Sources

Renewable

Comparison of various energy sources (Adapted from Li 2005)

Predictable

Visual

Capital

Maintenance

Environmental

impact

cost

cost

impact

Fossil

No

Yes

High

Low

High

High

Nuclear

No

Yes

High

Medium

High

High

Wind

Yes

No

High

High

Low

Medium

Solar

Yes

No

High

High

Low

Low

Hydro

Yes

Yes

High

High

Low

Medium

Wave

Yes

No

Medium

High

Low

Low

Tidal range

Yes

Yes

High

High

Medium

Medium

Tidal current

Yes

Yes

Low

High

Low

Low

2

Figure 1-1

Conceptual rendering of a tidal current turbine farm (Courtesy of Peter Frankel from Marine Current Turbine Ltd.)

Significant research has taken place in recent years to study the various aspects of tidal current turbine farms across a variety of fields. In the electricity generation (energy) field, research has been conducted to predict the energy output from a tidal current turbine farm (Triton 2002; Meyers and Bahaj 2005; Fraenkel 2002; Bedard 2006). In the industrial engineering field, research has been conducted to estimate the energy cost of a tidal current turbine farm (Frankel 2002; Bedard, 2006). The former papers, which estimate the energy output from a tidal current turbine farm, all use the efficiency of a stand-alone turbine to represent the efficiency of any tidal current turbine in the farm and neglect the hydrodynamic interactions between turbines. The latter papers, which estimate energy cost of tidal current turbine farms, all assume that operation and maintenance (O&M) cost is equal to a fixed percentage (e.g., 3 - 5%) of the capital cost of the tidal current turbine farm, which makes the total cost (sum of capital cost and O&M cost) proportional to the capital cost. The results based on these simplifications and assumptions are not convincing to investors (Campell 2006), and this situation is considered as one of the largest barriers to the industrialization of tidal current turbine farms (Bregman et al. 1995; Eaton and Harmony 2003). In the -3-

oceanography field, research has been conducted to study the impact of a tidal current turbine farm on current flow (Garret and Cummings 2004). This research treats the tidal current turbine farm as a black box, as if it were one big turbine, which means that the impact of the hydrodynamic interactions between turbines on current flow is neglected. In the environmental and policy fields, research has been conducted to assess the environmental impacts of a tidal current turbine farm (Van Walsum 2003). This research treats the environmental impacts of any individual turbine in a farm as the same as those of a stand-alone turbine. The research on tidal current turbines in all of the above-mentioned fields is important in facilitating the industrialization of tidal current turbine farms, and each study requires a good understanding of the hydrodynamic interactions between turbines and between tidal current turbines and tidal current flow. In fact, in the ocean engineering field, research has focused mainly on studying a stand-alone turbine rather than turbine farms (Davis et al. 1984; Coiro et al. 2005). It is clear that all of the above-mentioned investigations either neglect or simplify the hydrodynamic interactions involved in the process of harnessing tidal energy with tidal current turbines. Therefore, the core of this study is an investigation of the hydrodynamic interactions involved in the process of generating power from a tidal current farm. We focus on predicting the power output from a tidal current turbine farm, since an accurate predication of power output from a tidal current turbine farm is important for accurately estimating the energy cost of such a farm, which is in turn important for the economic justification of constructing such a farm. This systematically analytical process and its results are expected to shed light on related research in other fields, such as oceanography, industrial engineering, and environmental impact assessment and policy making.

-4-

1.2 Research Focus and Expected Contributions Predicting power output from a tidal current turbine farm The major purpose of this study is to provide a systematic procedure for predicting power output from a tidal current turbine farm. Specifically, we would like to identify the relationship between turbine distribution and the power output from a tidal current turbine farm by incorporating the hydrodynamic interactions involved in the process. Mathematically, we plan to quantify the relationship between the turbine power output, Pi , and farm parameter (e.g., turbine distribution and individual turbine configuration) and incoming flow condition Eq. (1.1). F ( Pi , incoming flow condition, farm parameter ) = 0

(1.1)

where Pi denotes the power output of turbine i in a tidal current turbine farm. In detail, we intend to fulfill the following tasks: Task A. Develop a numerical model to accurately predict the power output from a stand-alone tidal current turbine in an infinite fluid domain as well as the vortical wake flow behind the turbine, and estimate the acoustic emission from such a turbine Task B. Develop a numerical model to predict the power output, turbine hydrodynamic interactions and the vertical wake flow from a two-turbine system and extend it to an N-turbine system (a tidal current turbine farm), with an emphasis on the hydrodynamic analysis of a two-turbine system. By conducting this research, we expect to make contributions in providing a tool to z

Predict the power output from both a stand-alone turbine and an N-turbine system by incorporating the hydrodynamic interactions involved in the process

z

Predict the torque fluctuation and acoustic emission of a stand-alone turbine and a two-turbine system

z

Quantify the effects of the important factors (e.g., turbine design parameters and turbine -5-

distribution parameters) on turbine performance z

Improve the design of turbines and the design of the experimental test

z

Help researchers in other disciplines to study issues related to tidal current turbines by providing necessary information (e.g., wake velocity and torque fluctuation) regarding the estimation of turbine noise and current flow around the farm.

Estimating energy cost from a tidal current turbine farm In addition to our major research focus, which is to predict power output from a tidal current farm by incorporating the hydrodynamic interactions between turbines, we also plan to develop an approach for estimating energy cost from a tidal current farm. Energy cost information is useful for investors and policy makers in making investment and subsidy decisions. Mathematically, energy cost is defined as the ratio of the total cost to the total energy output over the lifetime of a tidal current turbine farm, given as follows,

cenergy

=

Total cost Energy

(1.2)

More specifically, we plan to perform the following task: Task C. Formulate an integrated model for estimating energy cost by integrating a hydrodynamic module for estimating energy output from a tidal current turbine farm (which is developed from Task A and Task B) with an O&M module for estimating O&M cost (Figure 1-2).

-6-

Figure 1-2

The integrated model for estimating energy cost

By conducting such research, we expect to contribute in providing a tool to z

Predict the energy cost of a tidal current turbine farm by incorporating the hydrodynamic interactions

z

Estimate the O&M cost of a tidal current turbine

z

Quantify the effects of the important factors (e.g., turbine distribution, turbine lifetime and farm size) on the energy cost of a tidal current turbine farm.

1.3 Research Methods In Section 1.2, we have identified three major tasks related to harnessing energy from tidal currents using tidal current turbines. To fulfill each of these tasks, we employ the methods listed below. The details of these methods are discussed in the main body of the dissertation.

Methods for addressing Task A We assume that the viscous effects are limited to the near field where no slip boundary conditions exist. We then decide to represent the wake with a potential flow, including -7-

vorticity and uniform flow. Based on the governing equations of potential flow, we develop a numerical model to predict the performance (torque and power) and the wake structure of a stand-alone tidal current turbine by using a discrete vortex method with free wake structure. In order to validate the developed numerical model, we carry out experimental tests in a towing tank and use a commercial Reynolds Average Navier-Stokes equations (RANS) Computational Fluid Dynamics (CFD) package to obtain the corresponding information. Additionally, power spectrum analysis is used to evaluate the turbine’s torque fluctuation and acoustic emission.

Methods for addressing Task B We develop a numerical model to predict the performance and wake structure of a two-turbine system by using a discrete vortex method with free wake structure based on the stand-alone turbine model developed in Task A. Then, we investigate the performance of various configurations of two-turbine systems under different operating conditions. Using a perturbation theory and a linearity assumption, we simplify an N-turbine system as a linear hydrodynamic system. The results obtained for the two-turbine system are then extrapolated to an N-turbine system.

Methods for addressing Task C By using cost-effectiveness as a metric, we define the objective function to minimize the energy cost (i.e., the ratio of total cost to total energy output) subject to local information (e.g., labor behavior and weather) and the hydrodynamic relationships between turbines. The total energy output is calculated by integrating the power output with respect to time. By using statistical analysis and life cycle analysis we obtain the formulation to estimate the total cost. Then we develop an integrated model to estimate the energy cost by integrating both the energy output calculation and cost estimation. By using a scenario-based analysis, we minimize the energy cost.

-8-

1.4 Outline of the Dissertation The body of this dissertation is divided into six chapters. Following this brief statement of problems in Chapter 1, Chapter 2 briefly discusses the world-wide tidal power potential and the state-of-the-art of vertical axis tidal current turbines. Chapter 3 develops and validates a numerical model to predict the performance of a stand-alone turbine by using a discrete vortex method with free wake structure, and develops another model to predict the acoustic emission from a stand-alone turbine. Chapter 4 illustrates the development of a numerical model to predict the performance of a two-turbine system as well as the acoustic emission of the system. Additionally, a procedure for predicting the performance of an N-turbine system is extrapolated from the two-turbine system analysis. Chapter 5 frames an integrated model to predict and optimize the energy cost of a tidal current turbine farm. This model includes the hydrodynamic models developed in Chapter 3 and Chapter 4, and a new turbine farm O&M model developed in this chapter. Finally, Chapter 6 concludes the dissertation and suggests future works.

-9-

Chapter 2 A Description of Tidal Current Turbines In this chapter, we briefly introduce tidal power potentials in the world and the history and state-of-the-art of tidal power technologies with a focus on tidal current turbines.

2.1 Tidal Power Potentials There are two types of tidal power technologies, with one harnessing potential energy from tide by using semi-permeable barrages and the other harnessing kinetic energy from tidal current flow by using tidal current turbines. The first attempt to harness tidal energy can be dated back to thousands years ago, when the Persians built their first water wheel to extract kinetic energy from water flow. In the 20th century, many oceanic countries, such as Canada, China, France, Indian, Russia, United Kingdom, and United States, started to develop their modern devices for extracting tidal energy. The earliest modern tidal power technology is the barrage technology. Barrages are expensive in small scale and have negative impacts on fish populations and the environment (Rulifson and Dadswell 1987). As a consequence, the evolution of tidal power technology slowed down and was on hold finally in the 1980’s. In the 1990’s, tidal current turbine technology regained people’s attention in exploring tidal current as an alternative energy source (Lang 2003). In general, harnessing energy from tidal current with tidal current turbines has a few advantages which make itself attractive, which include 1) tidal power is green and it neither produces toxic chemicals nor emits greenhouse gases, 2) tidal power is renewable since tidal current is naturally self-replenishable, 3) tidal power is highly predictable so that it is with low fluctuation when electricity integration is concerned, and 4) tidal current turbine farms have little visual impact since tidal current turbines and the auxiliary facilities have their main parts submerged underwater or within other offshore structures such as floating platforms and floating bridges. World-widely, there are quite a few tidal sites with high power potentials, as shown in Figure 2-1. In this figure, the red circles position the potential sites in North America while the blue - 10 -

circles position the potential sites in the rest of the world. Canada, as an oceanic country, has four most abundant tidal power sites (Figure 2-2). With such an abundant resource, several tidal power companies chose Canadian coast to test their tidal current turbines (Pearson 2005; Nova Scotia Power, 2007). As we focus on harnessing energy from tidal current with tidal current turbines, hereafter, when “tidal power” is mentioned, we refer to the power generated from tidal current by using tidal current turbines unless otherwise stated.

Figure 2-1

World-wide potential tidal power sites (Adapted from Charlier (1993))

- 11 -

Figure 2-2

Potential tidal power sites in Canada (Cornett, 2006)

2.2 Tidal Current Turbines The working principle of a tidal current turbine is similar to that of a wind turbine except that a tidal current turbine is driven by ocean flow instead of wind. Typically, a tidal current turbine consists of a few (normally three to five) blades, a shaft and some other add-on components (e.g., generator, brake, and damp) with blades being connected to the shaft. Tidal flow forces the blades to rotate around the central spinning shaft2. This rotating motion generates hydrodynamic power from ocean flows. The generator converts hydrodynamic power into mechanical power and then into electrical power which will be transmitted to local electricity loading centers through underwater and on-land cables. According to the axis direction, tidal current turbines can be classified into vertical axis tidal current turbines (Figure 2-3) and horizontal axis tidal current turbines (Figure 2-4). A 2

The working principles of vertical axis tidal current turbines are detailed in Chapter 3. - 12 -

horizontal axis tidal current turbine has its shaft located in a horizontal plane and a vertical axis tidal current turbine has its shaft located in a vertical plane. In this study, we focus on vertical axis tidal current turbines, which are also the research focus of the Naval Architecture and Offshore Engineering Laboratory (Naval Architecture Laboratory) at the University of British Columbia (UBC).

Figure 2-3

A vertical axis tidal current turbine

- 13 -

Figure 2-4

A horizontal axis tidal current turbine (Courtesy of Marine Current Turbine Ltd.)

The evolution of vertical axis tidal current turbines always follows the footprints of vertical axis wind turbines because vertical axis tidal current turbines share the same working principles with vertical axis wind turbines. Therefore, we review both vertical axis wind turbines and vertical axis tidal current turbines. The earliest vertical axis device is called Panemones, which is designed to harness energy from wind by Persians in A.D. 1300 (Paraschivoiu, 2002). It is not a cost-effective device from our points of view nowadays, but is the seminal device which leads to the development and evolution of similar technologies. Technically, according to the driven principle, modern vertical axis devices can be sorted as drag-driven devices and lift-driven devices. In the following sections, we describe each type of devices with a typical example.

- 14 -

2.2.1 Drag-Driven Devices As indicated by its name, the rotation of a drag-driven device is driven by drag. The well-known drag-driven device is the Savonius turbine. It is invented by the Finnish engineer S. J. Savonius in 1922 to harness energy from wind (Figure 2-5). It is regarded as one of the simplest vertical axis turbines. It consists of two or three scoops and a shaft. The scoops work as the blades of the turbine, driven by the incoming flow, and rotate around the shaft. Attributed to their special curvature, the scoops bear less drag when moving against the flow. The electrical power generator is usually located at either the top or the bottom of the scoop structures. The mechanical configuration of a drag-driven device is simpler than that of a lift-driven device, and generally the efficiency of a drag-driven device is lower than that of a lift-driven device.

Figure 2-5

An example Savonius turbine (ENA, 2007)

- 15 -

2.2.2 Lift-Driven Devices Again, as indicated by its name, the rotation of a lift-driven device is driven by lift. The well-know lift-driven device is the Darrieus turbine. It is invented by Georges Jean Marie Darrieus, a French aeronautical engineer, in 1931 to harness energy from wind (Figure 2-6). A Darrieus turbine typically has two symmetrical curvature blades. The two blades rotate around the shaft, which is located at the center of the turbine. Additionally, two support arms are installed at a certain height between two blades to reinforce the structural strength according to the distribution of the blade loads. This configuration has several notable advantages: 1) A Darrieus turbine can be placed on the ground for easy-servicing, and 2) The support tower is much lighter than that of a Savonius turbine because much of the force on the tower goes to the bottom.

Figure 2-6

An example Darrieus turbine (Ecopower, 2007)

Barry Davis, a Canadian veteran aerospace engineer, revised the Darrieus turbine and applied it to harness energy from tidal current in the early 1980’s (Davis 1981). This revised turbine is named as Davis turbine (Figure 2-7). It has four straight blades rotating around the shaft at the center of the turbine. This shaft connects with the power generator at the top of the turbine. Unlike the Darrieus turbine, a Davis turbine has three support arms.

- 16 -

Figure 2-7

A sketch of a Davis turbine (BE, 2007)

2.2.3 The Vertical Axis Tidal Current Turbine in This Study A newly-designed vertical axis tidal turbine has been built in the Naval Architecture Laboratory at UBC based on Davis turbine. The new turbine consists of three straight blades, a shaft and two support arms (Figure 2-8). The arm is connected with the blade at the location of 1 chord length from the nose of the blade. All electrical components are 4 connected to the top of the turbine. The main parameters of the turbine and its working environment are given in Table 2-1. Figure 2-9 shows the turbine’s position when the azimuth angle is equal to φ . It also depicts where the 0o azimuth angle is, when the arm is perpendicular to the free stream incoming flow direction. The blade arm angle is depicted in Figure 2-10, and mathematically it can be expressed as follows,

β=

π 2

+ Δβ

(2.1)

Where β and Δβ denote the blade arm angle and the blade arm pitching angle. In this study, the pitching angle is fixed during the turbine rotation, and this type of turbine is - 17 -

called fixed pitch turbine3. Table 2-1

The main parameters of the turbine and its working environment

Parameters

Symbol

Angular velocity

ω

Blade arm angle

β

Blade chord length

c

Blade number

nB

Current velocity

U∞

Frontal area

A = 2 RH

Pitching angle

Δβ

Azimuth angle

φ

Solidity

Nc R

Turbine height

H

Turbine radius

R

Tip speed ratio (TSR)

λ=

Power coefficient (Cp)

Cp =

3

Rω U∞ P 1 AU 3 ∞ 2

Those turbines with adjustable pitching angle during their rotations are called variable pitch turbines. The efficiency of variable pitch turbine can be higher than that of fixed pitch turbine if the pitching angle is optimally controlled by the pitching angle controller, although the total cost of the turbine will increase. Interest reader can refer to Pawsey (2002). - 18 -

Figure 2-8

The vertical axis tidal current turbine used in this study

Figure 2-9

An illustration of azimuth angle ( φ ) - 19 -

Figure 2-10 An illustration of blade arm angle

- 20 -

Chapter 3 Hydrodynamic Analysis of a Stand-alone Tidal Current Turbine In studying the hydrodynamics of a stand-alone tidal current turbine, we focus on understanding the physics of the turbine. We model the rotation of a stand-alone turbine and predict its performance (e.g., power, forces and torque) as well as its wake structure. In order to understand these characteristics thoroughly, we analyze the hydrodynamic principles of a stand-alone tidal current turbine theoretically. In this chapter, as a first step, previous research on incompressible aerodynamics of a stand-alone wind turbine and hydrodynamics of a stand-alone tidal current turbine are reviewed. In Section 3.2, we develop a hydrodynamic model for studying the behavior of a stand-alone turbine by using a discrete vortex method with free wake structure. Section 3.3 then shows the recent relevant research on tidal current turbines in the Naval Architecture Laboratory at UBC. The hydrodynamic model developed in this chapter is validated by comparing the kinematic and dynamic results obtained with this model and with experimental tests and other numerical methods. We then apply the hydrodynamic model to predict the performance of an example turbine. Then, we extend this hydrodynamic model to predict the acoustic emission from the turbine.

3.1 A Review of Previous Research Numerous numerical investigations on wind turbines were conducted by using incompressible aerodynamic theory which shares the same principles with marine hydrodynamic theory for studying tidal current turbines. Thus, in this section, the numerical methods for modeling both a stand-alone wind turbine and a stand-alone tidal current turbine are summarized. As described in Chapter 2, the focus of this study is on vertical axis turbines.

- 21 -

3.1.1 Research on a Stand-alone Wind Turbine Although patented quite a few decades ago (Darrieus 1931), vertical axis wind turbines did not see extensive research on power prediction until the 1970’s when Canadian National Research Council (South and Rangi 1973; Templin 1974) and United States Sandia National Laboratory (Blackwell 1974; Strickland 1976) conducted their seminal experimental tests and numerical modeling. Numerically, two sets of methods are commonly used to study the design and behavior of vertical axis turbines: potential flow method and Reynolds Averaged Navier-Stokes equation (RANS) method.

Potential Flow Method Based on the way how flow structure and a turbine are described, the potential flow method can be divided into the momentum method (actuator disc theory) and the vortex method. 1.

Momentum Method

The first momentum method is the single streamtube method (single actuator disc theory), which was proposed by Templin (1974). The single streamtube method is the simplest momentum method, enclosing a turbine within a streamtube. With this method，the incoming flow velocity is assumed to be constant upstream in the computational domain, and the flow velocity around the turbine is then related to the undisturbed incoming flow velocity by equating the drag force on the turbine to the change in fluid momentum through the turbine. Strickland (1975) advanced the single streamtube method into the multiple-streamtube method (multiple-actuator disc theory) in which a series of streamtubes are assumed to pass through the turbine. Later, attentions were shifted from seeking new methods towards modifying the multiple-streamtube method so as to improve the accuracy in terms of power prediction (Paraschivoiu, 1981; Berg, 1983). The multiple-streamtube method is still inadequate in describing the flow field. However, it can predict turbine power output more accurately and yield a more realistic distribution of blade forces, compared with the single streamtube method.

- 22 -

2.

Vortex Method

Vortex method suggests using vortices to describe turbine blades and flow. According to the way in which the wake structure is described, the vortex method can be divided into fixed-wake vortex method and free-wake vortex method. The fixed-wake vortex method uses a time-independent wake structure (Wilson 1976). Force, torque and power are determined by using the Kutta-Joukowski theory and local circulations. This method has not been investigated as extensively as the streamtube method, but its formulation has been improved by a number of researchers (e.g., Wilson and Walker, 1983; Jiang et al., 1991). On the contrary, the free-wake method suggests the use of discrete, force-free, and time-dependent wake structure (Strickland et al., 1979). The force and power are calculated in the same way as in the fixed-wake method. In the 1980’s, the fixed-wake method and the free-wake method were extended with some modification for specific purposes. For example, either the free-wake method (Oler, 1981) or the fixed-wake method (Masse, 1986) was combined with the hybrid-local-circulation-method to simulate a turbine and the flow around it, with a focus on the detailed curve of the curvature blade. Compared with the momentum method, the vortex method can predict turbine performance more accurately, but requires longer computational time.

RANS RANS methods have been widely employed to simulate wind turbine’s rotation since the 1990’s and with the advent of computational technologies that use commercial Computational Fluid Dynamics (CFD) packages (e.g., Fluent and Star-CD). Younsi et al. (2001) and Takeuchi et al. (2003) used Fluent, and Nakhla et al. (2006) used Star-CD to predict a wind turbine’s performance. It is easier to set up and initialize a case with complicated turbine geometry by using commercial CFD packages, compared with the potential flow method. Furthermore, one can get a more specific description on near field flow by using RANS methods. Notwithstanding, RANS methods are much more computationally costly than the potential flow method.

- 23 -

3.1.2 Research on a Stand-alone Tidal Current Turbine Preliminary experimental test on a stand-alone tidal current turbine was carried out in the early 1980’s and sponsored by the National Research Council of Canada (Davis et al., 1981, 1982 and 1984). This experimental test was conducted in a restricted area and the results obtained can not be extrapolated to represent turbine’s behavior in open areas. Recently, Coiro et al. (2005) built a prototype turbine and carried a sea test. Only a handful of papers reported numerical methods for simulating a stand-alone tidal current turbine in previous research. Commercial RANS CFD packages have been used to predict the performance of tidal current turbines although they are still limited by the associated high computational cost (Batten et al., 2006). RANS CFD packages tend to over-predict the forces in the marine hydrodynamic applications (Nabavi, 2008). In this study, commercial RANS CFD packages are mainly used for evaluating the performance of turbine components with new shapes at the design stage, to investigate very near-field flow, and validate models developed by using potential flow method4. Regarding the potential flow method, the momentum method has been used to predict power output from a stand-alone tidal current turbine by Camporeale and Magi (2000), and the vortex method has been used to predict power output from a stand-alone tidal current turbine (Li and Calisal 2007a). An extended version of the work of Li and Calisal (2007a) is detailed in Section 3.2.

3.2 A Discrete Vortex Method with Free Wake Structure for a Stand-alone Tidal Current Turbine Vortex method has been used to predict power output from a stand-alone wind turbine. However, this procedure can not directly be transferred to predict power output from a stand-alone tidal current turbine, because there are a number of fundamental differences between a wind turbine and a tidal current turbine in the design, operation and working environment (air vs. ocean water). One major difference is the Reynolds number (with respect to turbine blade chord length and incoming flow velocity) which can induce 4

A detailed description on research related to tidal current turbine in the Naval Architecture Laboratory at UBC is shown in Section 3.4. - 24 -

differences in load on blades, boundary layer separation, vortex decay, vortex growth, and vortex shedding frequency. Another major difference is cavitation. To understand these physical differences, the behavior of a tidal current turbine is described in a rigorous way according to the physics of the flow in this study. The vortex method is chosen to describe tidal current turbines and the unsteady flow. Among all numerical methods, which have been applied to simulate vertical axis tidal current turbines, the discrete vortex method with free wake structure is considered as the most suitable one for two reasons: z

It can describe the unsteady nature of wake structure (compared with momentum method and fixed-wake vortex method)5

z

It can be relatively easily coded and of course it is relatively inexpensive (compared with commercial RANS CFD package).

3.2.1 The History of the Discrete Vortex Method If the evolution of vortex sheet is simulated, the velocity field and other quantities of interest, such as time-dependent forces acting on the body that sheds the vortices, can be determined accordingly. Modeling the evolution of vortex sheet mathematically, however, is relatively difficult, given the complicated vortex structure. Rosenhead (1931) proposed a method, called Discrete Vortex Method (DVM), to solve this problem by discretizing shedding vortices. Specifically, this paper showed an example on using discrete vortices to represent the large features of separated shear layers shed from bluff bodies. This method (DVM) approached a sinusoidally-perturbed vortex sheet by using twelve two-dimensional line vortices and obtained a smooth roll-up of the shear layer into discrete vortex clusters spaced one wavelength apart. Westwater (1935) followed this attempt by applying this method to study the shedding of the vortex sheet from an elliptically loaded wing. At that time, this work was quite laborious given that the necessary computational requirements were not available. Even after the 1970’s, such a calculation still requires significant computational effort. Clements and Maull (1975) proposed a modified DVM, which helps limit the induced 5

The flow is unsteady and the wake changes all the way as the turbine rotates. RANS CFD is able to simulate the unsteady flow more accurately, but its computational cost is too high. - 25 -

velocities by amalgamating any pair of vortices that are very close to each other. However this modified method tends to increase the separation of the vortices. Fink and Soh (1974) developed a rediscretization scheme to distribute the vortex sheet in equidistant position after each time step in the numerical procedure. This scheme increases the computational cost significantly, but can obtain the result of vortex sheet more orderly and stable as a pioneering method. Additionally, Graham (1977) applied the DVM to calculate the vortex shedding from a sharp edge in oscillatory flow. In wind turbine studies, Strickland et al. (1979) used the DVM to predict the performance of a vertical axis wind turbine. In the Naval Architecture Laboratory at UBC, Wong (1990; 1995) used the DVM to calculate vortex shedding from a sharp edge.

3.2.2 Assumptions of the Model To formulate a rigorous mathematical model to describe the hydrodynamics of a tidal current turbine and the unsteady flow, we made the following assumptions: z

The tidal current turbine works as a stand-alone turbine. There are no auxiliary structures (such as ducts and anchors) and other turbines around the studied turbine

z

The incoming flow is uniform

z

The lift and drag on a blade element are calculated by using steady state lift and drag coefficients obtained by using experimental methods; at different azimuth angles, they are calculated according to the angle of attack during the revolution of the turbine

z

Each turbine blade is divided into several finite segments (elements) along the span of a blade with a given geometry

z

Each blade element is represented as a bound vortex

z

In the wake of the blade, the production, convection, and interaction of the vortex system shed from individual blade elements are modeled by using the induced velocity concept. The generation of vortex shedding obeys Kelvin’s theorem which can be expressed as follows

z

DΓ (3.1) =0 Dt In the wake, the velocity at a single point can be simplified by superimposing all the

- 26 -

induced velocities upon the undisturbed incoming flow velocity z

The effects from the turbine shaft, supporting arm and blade controller on the turbine performance are obtained from experimental test data. There is no shaft and shaft induced wake structure being considered in this formulation.

3.2.3 Turbine Working Principle When the DVM is used for simulating the flow around the turbine, velocity induced by a vortex filament should be presented first. According to the Biot-Savart law (e.g., Anderson 2006), given a vortex filament of an arbitrary shape with a strength of Γ and a length of l (e.g., a turbine blade), the induced velocity at point p (but not on the filament) can be

calculated as follows6, Γ r × dl (3.2) 4π ∫l r 3 where r denotes the position vector from a point on the filament to a point p . U iP ,l =

To model a turbine, we need to understand the turbine’s working principle. Figure 3-1 depicts a working turbine with one zoomed-in blade element (cross section). The turbine rotates at a certain angular velocity ( ω ) driven by the force acting on the blades by the incoming flow. At this angular velocity, the tip velocity of the turbine blade can be written as follows, U t = Rω

(3.3)

In detail, the incoming flow and the turbine rotation introduce an angle of attack ( α ), the angle between the blade local relative velocity and the blade chord line, which can be obtained by resolving the blade local relative velocity ( U R ) and the chord line of the blade according to their vectorial relationship as shown in Figure 3-1. One can calculate the angle of attack with respect to the nose of the foil or the location of the

1 chord length from the 4

nose of the blade. In this dissertation, we calculate it with respect to the location of the 1 chord length from the nose of the blade. The blade local relative velocity is the flow 4 6

Readers who are interested in the fundamental mathematical formulation of induced velocity are referred to Appendix A. - 27 -

velocity seen by the blade element, which is a function of free stream incoming velocity ( U ∞ ), induced velocity at the blade ( U iB ), and tip velocity of the blade. Mathematically, it can be written as: U R = U ∞ + U iB + U t

(3.4)

The lift and drag generated by the incoming flow are directly related to this angle of attack. The resultant force on a blade element can be calculated by summing the lift and the drag. The projection of this resultant force on the blade chord line is called the tangential force.

Figure 3-1

An illustration of turbine working principle with a bird-eye view on one of the blades7

3.2.4 Blade Bound Vortex The relationship between lift ( L ) and bound vortex strength ( Γ B ) on a blade segment is employed to derive the relationship between blade bound vortex strength and shedding vortex strength. The former relationship can be derived by using the Kutta-Joukowski law (e.g., Anderson, 2006) as follows,

L = ρU R Γ B

(3.5)

7

This illustration does not represent the configuration and the scale of a real turbine. Also, induced velocity is not depicted. - 28 -

According to the definition of the lift coefficient, the lift can be written as follows, 1 ρ CL cU R2 (3.6) 2 where CL and c denotes the lift coefficient and the chord length, respectively.

L=

By combining Eq.(3.5) and Eq.(3.6), we can express the bound vortex strength as follows, 1 Γ B = CL cU R 2

(3.7)

In a two-dimensional model, the structure of wake, expressed with discrete vortices, is depicted in Figure 3-2. Mathematically, the strength of the wake vortices can be written as

ΓW ,i = Γ B ,i − Γ B ,i −1

(3.8)

where ΓW ,i and Γ B ,i denote the strengths of wake vortices and blade bound vortex at time step i , respectively.

Figure 3-2

An illustration of a two-dimensional time-dependent (unsteady) vortex wake structure

- 29 -

In a three-dimensional model, the structure of the wake vortex system is shown in Figure 3-3. This vortex system is assumed to be of horseshoe shape initially. The spanwise vortex filament strength, Γ S , can be written as

Γ S ,i −1, j = Γ B ,i −1, j − Γ B ,i , j

(3.9)

where i is the index of the time step and j is the index of the blade element.

Figure 3-3

An illustration of a three-dimensional time-dependent (unsteady) vortex wake structure

Similarly, the trailing edge vortex shedding filament strength, ΓT , can be expressed as follows,

- 30 -

ΓT ,i −1, j = Γ B ,i , j − Γ B ,i , j −1

(3.10)

So, numerically, by summing all velocities induced by all vortex filaments using Eq.(3.2), the total induced velocity at any given point p can be written as follows, U iP = ∑∑ ViPT ,i , j + ∑∑ ViPS ,i , j i

j

i

(3.11)

j

where ViPT ,i , j denotes the velocity induced by the trailing edge wake vortices shed from blade element i at time step j , and ViPS ,i , j denotes the velocity induced by the spanwise wake vortices shed from the same element; they are both calculated by using Eq.(3.2). Then, by substituting the formulation of total induced velocity (i.e., Eq.(3.11)) and the formulation of tip speed velocity of the blade (Eq.(3.3)) into Eq.(3.4), we can have the relative local velocity as follows, U R = U ∞ + ∑∑ViPT ,i , j + ∑∑ViPS ,i , j + Rω i

j

i

(3.12)

j

The Eqs. (3.7) to (3.10) show how to calculate the strength of vortices in the unsteady wake of a stand-alone turbine. As to the motion of the vortices in the wake, regardless of the blade element, every portion of the vortex filament is convected with the local fluid in the flow. The velocity of the local fluid (at point p ) is the vectorial sum of the incoming flow velocity and the total induced velocity, given as follows, UV = U ∞ + U iP

(3.13)

From now on, we focus on three-dimensional simulation of the turbine rotation although the two-dimensional simulation is computationally less costly. Actually, the two-dimensional structure can be considered as a simplified version of the three-dimensional structure by considering a hydro foil as only one blade element in the three-dimensional structure.

- 31 -

3.2.5 Summary of the Traditional Discrete Vortex Method The Eqs. (3.1) to (3.13) show the basic formulation of the traditional DVM. Knowing the relative local velocity, force and torque on the blade can be calculated with lift and drag coefficient according to their definition (See Eq.(3.6))8. In traditional DVM, it can be seen that the key factor to be figured out is the velocity induced by vortices. For a given blade element, if the induced velocity can be obtained, all the variables in from Eqs. (3.2) to (3.13) can be calculated accordingly. In this study, the chord length c is a constant, and the blade’s lift coefficient is a function of angle of attack of the blade. This angle of attack is a function of blade local relative velocity and incoming flow velocity. The bound vortex strength is strongly affected by local relative velocity. As discussed earlier, this local relative velocity is a function of incoming flow velocity, angular velocity and vortex wake induced velocity. For a typical working tidal current turbine, the angular velocity can be treated as a constant in a certain period (a few revolutions as described in the assumptions) because the period of a dominant tidal cycle is approximately 12.42 hours (Pond, 1983). That is

ΔU ∞ ⎛ 2π ⎞ = 0 iff Δt ∼ O ⎜ ⎟ Δt ⎝ω ⎠

(3.14)

In Sections 3.2.6 to 3.2.9, we will present the unique formulations of the discrete vortex method with free wake structure developed in this study, which is called DVM-UBC and thus to distinguish it from the traditional discrete vortex method (traditional DVM) as summarized above.

3.2.6 Vortex Decay The mathematical description of the traditional discrete vortex method as mentioned above does not allow for viscous diffusion which can induce decay of vortices. In this study, to get a better approximation to the behavior of the flow and achieve more stable computational 8

In order to keep the continuity of the discussion, we present the force and torque calculation in Section 3.2.10, which is shared by both traditional DVM and the new DVM (i.e., DVM-UBC) developed in this dissertation. - 32 -

results, vortices are allowed to decay with time. The first quantitative research on vortex decay by using discrete vortex method is given by Graham (1980) that shows that the decay of vortices calculated by using the traditional discrete vortex method is overestimated by approximately thirty percent, and this error is related to the shape of the body, especially its trailing edge. Graham (1980) then proposed an equation, Eq. (3.15), to describe the decay of vortices. Kudo (1981) cut the vortex shedding at certain distance. Hansen (1993) gave a comprehensive review on studying viscous diffusion by using the discrete vortex method, and proposed an expression, Eq. (3.16) for describing vortex decay which is similar to Eq. (3.15):

Γ j = Γ0

(1 − e )

Γ j = Γ 0 e(

⎛ − Kd ⎞ ⎜ ⎟ ⎝ τ ⎠

(3.15)

− KC t )

(3.16)

where both K d and K c denote vortex strength decay coefficient, which can be obtained from experiments or predicted by using numerical methods for a body of a specific shape. Both Eq. (3.15) and Eq. (3.16) work well for simulating vortices shedding from a blade. In this study, we adopt the equation proposed by Graham (1980),Eq. (3.15), as it has been successfully used to address marine hydrodynamics problems by Wong (1990). In the calculations, the number of vortices increases as the turbine rotates. A large number of vortices require a large space of memory in the computer which significantly increases the computational cost. The vortex strength decays exponentially with distance, and it almost vanishes at a certain distance from the blades. Thus, we can set a critical distance to enforce the vortex die out criterion so as to reduce the computational cost. In this study, we use a critical number of turbine revolutions to substitute the critical distance, which means that the vortex vanishes after a number of turbine revolutions after it is shed. In this study, the critical number of turbine revolutions is set to ten.

- 33 -

3.2.7 Lamb Vortices In most research on turbines, researchers use potential vortices,Eq. (3.17), to express vortices. Vθ ( r ) =

Γ 2π r

(3.17)

One of the exceptions is the research by Vandenberghe and Dick (1987), who modified the potential vortices in the traditional DVM to predict the performance of a turbine (Eqs. (3.18) to (3.20)). Vθ ( r ) =

Γr

(3.18)

2πσ m2

σ m = max ( r , σ c )

(3.19)

σ c = 4ν t

(3.20)

where σ m and σ c denote the maximum radius and the cut-off radius of the vortex core, respectively. However, the expression of potential vortices can not precisely describe induced velocity. Besides, the numerical results by using potential vortices to describe induced velocity are not stable (Fink and Soh, 1974). When two vortices are closely located, they induce large velocities on each other. This also happens when a vortex approaches a rigid body too closely due to the influence of its image. Large mutually-induced velocity is one of the reasons that cause the instability of discrete vortex computation. In order to increase the computational stability, and make the results represent the behavior of the flow, we employ Lamb vortices. A Lamb vortex has a viscous core. As the distance from a point to the viscous core approaches zero, the tangential velocity of that point vanishes exponentially. Mathematically, Lamb vortices are defined by using Eq. (3.21) and Eq. (3.22). An example of the relationship between dimensionless tangential velocity of a Lamb vortex and the dimensionless distance at certain time stage is given in Figure 3-4. - 34 -

Vθ ( r ) =

⎛r⎞ −⎜ ⎟ ⎝ rc ⎠

2

Γ 1− e π r

(3.21)

rc = ν t

(3.22)

where rc denotes the vortex core radius.

Figure 3-4

An illustration of the dimensionless tangential velocity of Lamb vortices

Dalton and Wang (1990) showed that Lamb vortices have been successfully used to prolong the stability of their computation. In the Naval Architecture Laboratory at UBC, Lamb vortices have been used to address marine hydrodynamics problems (Wong 1990; Wong 1995). Besides these, other expressions for vortices have been proposed in the literature. For example, with the advent of high-speed computers and advanced measuring techniques, a more elaborate definition of the vortices was proposed by Devenport et al. (1996). In this - 35 -

definition, peak tangential velocity of the vortex instead of its total circulation is used and the mathematical expression is 2 ⎛ ⎛ 0.5 ⎞ r ⎡ ⎛ ⎞ ⎞ ⎜ −α o ⎜ r ⎟ ⎟ ⎤ Vθ = Vθ max ⎜ 1 + ⎜ r ⎟ ⎢ ⎝ c ⎠ ⎟⎠ ⎥ ⎝ ⎦ ⎝ α o ⎠ rc ⎣1 − e

(3.23)

where α o is a constant ( α o = 1.26 ), and the vortex core radius ( rc ) can be measured in experiments. The vortex expression proposed by Devenport et al. (1996) has its advantages in describing the near-field flow, but requires experimentally obtained vortex core value. If one uses this vortex expression to study tidal current turbines, the details (experimentally obtained vortex core value) of each of the blades have to be obtained, and therefore the computational cost is expected to be high. The vortex expression as suggested by Vandenberghe and Dick (1987) cannot match the Lamb vortices expression in describing the velocity field of vortex in marine hydrodynamics applications. Thus, in this study, the classical expression of Lamb vortices is employed.

3.2.8 Shedding Frequency In this model, in using the discrete vortex method with free wake structure, a few parameters need to be tuned to approximate the physical behavior of the viscous flow when viscosity is introduced by using vortex decay and Lamb vortices. Besides the decay factor and radius of the vortex core, vortex shedding frequency is also very important. The DVM-UBC allows the vortices to shed in any frequency. We find that numerically, a high frequency (over 10Hz) vortex shedding would not have significant effect on the performance of a stand-alone turbine and a low vortex shedding frequency can reduce the computational cost in our model. Vortex shedding frequency can either be obtained by experimental test or be estimated by using the Strouhal number ( St ) and the Reynolds number in the investigation of ocean energy conversion devices (Bernitsas et al. 2006a and 2006b). In this study, we use the Strouhal number to predict the shedding frequency, Eq. (3.24).

- 36 -

f =

StU R c

(3.24)

where St denotes the Strouhal number.

3.2.9 Nascent Vortex The traditional DVM focuses on describing the vortex far away from the blade in the wake. However, the initial motion of the vortex (near the blade) shed from the blade, called nascent vortex, is an important concern considering the viscosity of the flow. Several studies on nascent vortex were reported in the past few decades. Wong (1990) argues that the nascent vortex is located at the trailing edge and there is a pushing velocity with a magnitude of one to two times the normal velocity in the ζ plane. Streitlien (1995) stated that the nascent vortex starts at a certain point determined by interpolating the trailing edge and the last vortices. In this study, we try to simulate the physics of the flow which is closer to the blade than that the traditional DVM can simulate. Wong (1990)’s argument is not suitable for this study since we have to avoid the singularity issues near the rigid boundary. Streitlien (1995)’s suggestion is more suitable for small motion applications. To approximate the physics of the turbine motion and the unsteady flow, we suggest that the nascent vortex velocity be half of the last vortex velocity. That is, at time to , the velocity of nascent vortex (vortex m ) is 1 UV , m t =t = UV , m −1 o 2 t = to

(3.25)

where UV , m t =t denotes the velocity of vortex m at time to . o

At the same time, the velocity of the last vortex (vortex m − 1 ) can be obtained by substituting the nascent vortex definition into Eq.(3.13) as follows, UV , m −1 t =t = U ∞ + U iP ,m −1

(3.26)

o

- 37 -

At the next time step, vortex m is not a nascent vortex any more, and its velocity is written according to Eq.(3.13), and is given as follows, UV , m t =t

o +Δt

= U ∞ + U iP , m

(3.27)

Also, we suggest that the location of the nascent vortex be set in the middle point between the trailing edge and the last vortex. That is, if the location of the trailing edge of a blade at time to is ( xto , yto ), the location of the nascent vortex (vortex m ) can be given as follows,

( (

1 ⎧ x + xV , m −1 t =t ⎪⎪ xV ,m t =to = 2 to o ⎨ 1 ⎪ yV ,m = yto + yV ,m −1 t =to t t = o ⎪⎩ 2

) )

(3.28)

Then, since the next time step, vortex m travels at local flow velocity as given in Eq.(3.27) and its position can be predicted accordingly.

3.2.10 Hydrodynamic Characteristics of a Tidal Current Turbine By using the methods for describing and calculating vortices as shown above, we can calculate the performance (force, torque and power) of a stand-alone tidal current turbine. For example, after vortex shedding and induced velocity are analyzed, lift can be obtained by using Eq. (3.6). After obtaining the angle of attack, we can obtain the normal force coefficient and tangential force coefficient with lift and drag coefficient as follows, ⎛ Cn ⎞ ⎛ CD ⎞ ⎜ ⎟ = D⎜ ⎟ ⎝ CL ⎠ ⎝ Ct ⎠

(3.29)

⎛ -sinα -cosα ⎞ D=⎜ ⎟ ⎝ -cosα sinα ⎠

(3.30)

where Cn , Ct and CD denote the normal force coefficient, tangential force coefficient and drag coefficient, respectively. - 38 -

Then, we can use these normal force coefficient and tangential force coefficient to calculate tangential force and normal force according to their definition, given in Eqs.(3.31) to (3.34) and thus to calculate the power output ( P ) and torque ( M ) by using Eq.(3.35) and Eq.(3.36), respectively. Ft = ∑ Ft ,i

(3.31)

Fn = ∑ Fn ,i

(3.32)

1 Ft ,i = Ct ρ bi cU R2 2

(3.33)

1 Fn ,i = Cn ρ bi cU R2 2

(3.34)

P = M ⋅ω

(3.35)

M = F × r = Ft R

(3.36)

i

i

where Ft and Fn denote the tangential force and the normal force, respectively; Ft ,i , Fn ,i and bi denote the tangential force, normal force, and blade element length of segment i, respectively. It is important to note that the lift and drag coefficients are obtained by assuming that the system is in quasi steady state. Specifically, the formulation is based on the assumption that the effect of wake vortex shedding on the lift coefficient is negligible. This assumption has been widely accepted in using vortex methods to address problems in wind turbines and marine propellers (Wilson, 1976; Strickland, 1975). In this study, in order to approximate the physics of the flow, although DVM-UBC is a potential flow method, the drag can be calculated accordingly because of following reasons: 1) a few new formulations of viscosity (e.g., vortex decay, lamb vortices, nascent vortex) are introduced to simulate the unsteady wake, and 2) the turbine blades are divided in several segments. Thus, UBC-DVM can predict the turbine performance more accurately9.

9

Although one can calculate the lift by using integral in the pressure field, it is more complicated and it would not significantly increase the accuracy since there is no major separation in this study. - 39 -

Additionally, there are several studies on investigating the behavior of a turbine in dynamic state by using dynamic stall method (Cardona,1984;Brochier,1985;Leishman and Beddoes, 1989; Major 1992). Without considering the dynamic stall, a turbine model can not accurately predict the turbine performance when the turbine is operating at very low TSR or very high TSR due to the viscous effects. The dynamic stall method, however, is more appropriate for studying wind turbines which always face unstable incoming flow and quite often wind turbines work at very low TSR. Thus, it is not introduced into the DVM-UBC in our study.

3.2.11 Non-dimensionalization To express the equations mentioned above in a dimensionless format to simplify their relationships, we select U ∞ (incoming flow velocity), Rmax (maximum radius of the turbine) 10 and ρ w (water density) as independent variables. We, then, use these independent variables to non-dimensionalize the physical parameters involved in this study, as shown in the following equations: U ∀U (velocity), we have Uˆ = U∞

(3.37)

l ∀l (length), we have lˆ = Rmax

(3.38)

U ∀t (time), we have tˆ = t ∞ Rmax

(3.39)

∀ρ (density), we have ρˆ =

ρ ρw

(3.40)

This will not change the definition of those dimensionless coefficients such as lift coefficient and drag coefficient as shown in the following equations: CL =

10

L

(3.41)

1 ρ AU R2 2

If the blades are not curved, the reference for length would be just R . - 40 -

CD =

D

(3.42)

1 ρ AU R2 2

where A denotes the turbine frontal area. With the basic dimensionless relationships,Eqs.(3.37) to (3.40), those more complicated equations can also be non-dimensionalized. For example, tangential force,Eq.(3.33) can be rewritten as

Fˆt =

Ft 1 ρ wbcU R2 2

= CtUˆ R2

(3.43)

So far, we have analyzed the physical parameters of a stand-alone tidal current turbine and showed how to use mathematical models to describe the stand-alone tidal current turbine and its behavior. In the next section, we present the numerical procedure for simulating a stand-lone turbine and its behavior.

3.2.12 Computational Procedure Figure 3-5 shows the flow chart of the computational procedure of DVM-UBC for estimating the performance (torque, force and power) and the wake of a stand-alone tidal current turbine given the design parameters of the turbine and the environmental conditions. This calculation procedure starts with the inputs and initial conditions (how a turbine starts to rotate), which include turbine components, component configuration and control parameters as well as environmental conditions, as detailed below: z

Incoming free stream velocity

z

Water and boundary conditions (e.g., water density and depth)

z

Turbine blade geometry (e.g., blade curvature and blade element shape)

z

Turbine configuration and dimension (e.g., blade numbers, turbine solidity and radius)

z

Operation parameters (e.g., critical number of turbine revolutions and initial azimuth angle). - 41 -

In the very beginning, the strength of the bound vortex is set to be zero (as a pseudo value) considering the computational stability issue. By using the relationship between the lift and the blade bound vortex, Eq. (3.7), the bound vortex strength can be obtained. With this calculated vortex strength, the induced velocities can be calculated by using Eq. (3.11). A new blade local relative velocity can be obtained by using Eq. (3.12). Then, with this induced velocity, the new bound vortex strength can be predicted by using Eq. (3.7) again. The induced velocity can be continuously revised by using the last predicted vortex strength and the vortex strength can be continuously predicted by using last induced velocity, which forms a calculation loop. This loop procedure can be repeated until a convergence criterion is satisfied. The convergence can be set in two ways: z

The deviation of the current value of strength of blade bound vortex from the value in the last loop is less than a certain value, described as follows, Γ kB − Γ kB−1 < ε , k = 2,3, 4 ⋅⋅⋅

z

(3.44)

The number of calculation cycles of the loop achieves the critical number of loops ( lN ), as follows, k = lN

(3.45)

where k denotes the number of computational cycles of the loop, Γ kB denotes the blade bound vortex strength of the k th cycle of the loop, and ε denotes the critical converge deviation value, respectively. These convergence criteria are determined according to specific simulation and computational capability requirements. In this study, we set the number of calculation cycles as four as an example ( lN = 4 ). Then, the blade forces and turbine performance are calculated by using the basic governing equations Eqs.(3.5) to (3.7) and the hydrodynamic characteristic equations Eq.(3.29) and Eq.(3.36). The wake structure can be calculated by using wake vortices system relationships, Eq.(3.13). Power coefficient is calculated accordingly, and the

- 42 -

program, then goes to the next time step. The program will end when the shaft revolution number reaches a pre-set value. In this study, we set a revolution value as fifteen.

Figure 3-5

Flow chart of the DVM-UBC computational procedure for estimating the

performance (torque and power) and the wake structure of a stand-alone tidal current turbine - 43 -

3.3 Parallel Numerical and Experimental Research in the Naval Architecture Laboratory at UBC A few years ago, strong interests and collaboration with local industry fostered the marine hydrodynamic research on vertical axis tidal current turbines, the Naval Architecture Laboratory at UBC. The research focus in the lab is on designing high-efficiency and high-reliability turbines and turbine systems by conducting both numerical simulations and experiments.

3.3.1 Numerical Simulation In the Naval Architecture Laboratory, almost all numerical methods (momentum method, vortex method, and RANS method) for studying turbines (as mentioned in section 3.1) have been used at the early stage of the research. Eventually, vortex method and RANS CFD package (Fluent 6.2)11 are selected for further studies. Three research topics are identified, which are z

studying turbine hydrodynamic interactions by using DVM-UBC (this study)

z

optimizing duct shapes for ducted turbine by using Fluent (Nabavi 2008)

z

studying the effects of towing tank walls on the performance of a turbine during the experiment by using traditional 2-Dimensional DVM (Alidadi, in progress).

3.3.2 Experimental Test In parallel with the numerical investigation, a series of vertical axis tidal current turbines have been designed, built and tested in the towing tank at UBC. Figure 3-6 shows an example turbine designed in the Naval Architecture Laboratory and Figure 3-7(a) shows a snapshot of the towing tank. The towing tank is 67 m long, 3.7 m wide and 2.4 m deep and there is a carriage running on the rails along the side of the tank, which was designed and built for testing ship models. The turbine is installed on a mounting frame (Figure 3-6), and both the turbine and the mounting frame are made of aluminum. However, the ship testing carriage is 11

A brief introduction to Fluent is given in Appendix D. - 44 -

not able to support the turbine mounting frame. Thus, we designed a secondary carriage which is attached to the original one with a diagonal brace for providing additional support (Figure 3-7 (b)). This secondary carriage is made of welded aluminum c-channel in two halves which is then bolted together. Two v-grooved wheels run along the inner rails which are closer to the tank, and two rubber wheels rest on the outer rails which are further from the tank. Specific instruments (e.g., torque meters) are connected to the turbine to measure the angular velocity and torque of the turbine, and a drive-train is used to control the angular velocity of the turbine (Figure 3-8). Moreover, one of the blades is also instrumented with pressure transducer to measure pressure on that blade. In this dissertation, the results from the experimental test at UBC are used for validation purpose. Detailed information on the experimental setup and data acquisition techniques can be found in Rawlings (2008).

Figure 3-6

An illustration of the scheme of one of the turbines designed at UBC - 45 -

(a)

(b) Figure 3-7

(a ) A snap shot of the towing tank at UBC; (b) an scheme of the carriages used to test the turbine (Rawlings 2008) - 46 -

Figure 3-8

An illustration of the experimental setup: the turbine test frame, the turbine and the measuring instruments (Rawlings 2008)

3.4 Validation of DVM-UBC and Case Studies The program of DVM-UBC is developed with Matlab for predicting the performance of a stand-alone vertical axis tidal current turbine. The results obtained are then compared with the results obtained with experimental tests and other numerical methods.

3.4.1 Validation To conduct a systematic validation, the results obtained with DVM-UBC is compared with experimental results and other numerical results in both dynamic and kinematical ways with six cases as described in Table 3-1. In kinematics validation, we compare the geometrical characteristics of the wakes. In dynamics validation, we compare the power coefficient (as a function of TSR) and the dimensionless torque (as a function of azimuth angle). - 47 -

Table 3-1

Case 1

Case 2

Cases used to validate DVM-UBC

Kinematics validation

Dynamics validation

Compare the wake structure with the

Compare the power coefficient with

experimental results from Strickland

the experimental results from Templin

(1976)

(1974)


Compare the power coefficient with

results obtained with conformal

the results obtained with UBC

mapping method in Deglare (2007)

experimental test and numerical results with Fluent and traditional DVM

Case 3


Compare the torque with the results

results obtained by using Fluent

obtained with UBC experimental test, Fluent and traditional DVM

Kinematics Validation Case 1: we compare the two-dimensional wake trajectory generated by using DVM-UBC

with experimental measurement (Strickland, 1976). The basis specification of this case is shown in Table 3-2. The wake of the experimental work is represented by white bubbles while the wake of DVM-UBC is represented by line and circle. Table 3-2

Basic specification of case 1 for kinematics validation

Parameters

Values

Number of blades

2

Rotating direction

Counter-clockwise

Blades type

NACA 0015

Solidity

0.3

TSR Reynolds number

5 12,13

160,000

12

The Reynolds number in the numerical simulation is calculated with respect to the average blade relative local velocity at the design TSR. Design TSR refers to the TSR range that the turbine is designed for, under - 48 -

We compare the values of the first five cross- x -axis points, which are x1 , x2 , x3 , x4 and x5 , and the values of the four extreme y points in the first two revolutions, which are y1 , y2 , y3 and y4 in Figure 3-9. The comparison results are shown in Table 3-3. In general, the results generated with DVM-UBC are comparable with the results obtained in Strickland (1976). The difference between the experimental results and the results generated with DVM-UBC is within 4% of the experimental value. One can say that good agreement is obtained between the results with DVM-UBC and those in Strickland (1976). Table 3-3

The relative deviation of the results with DVM-UBC from the experimental results

Position

x1

Relative deviation14 4%

Figure 3-9

x2

x3

x4

x5

y1

y2

y3

y4

3%

1%

4%

1%

1%

1%

2%

3%

A comparison of the two-dimensional wake trajectory by Strickland (1976) (While Bubble) and DVM-UBC (line and circle)

which, the power coefficient of the turbine is around its maximum value. 13 In the calculation, we use Reynolds number to estimate the corresponding lift and drag coefficients. 14 The relative deviation here is defined as the ratio of the difference between the numerical value and experimental value to the experimental value. For example if the experimental value is 0.5 and the numerical value is 0.51, the relative deviation is 2%. - 49 -

Case 2: we compare the two-dimensional wake trajectory generated by using DVM-UBC

with that generated by using a conformal mapping method of Deglaire (2007). The basic information for this case is shown in Table 3-4. Table 3-4

Basic information of Case 2 for kinematics validation

Parameters

Values

Number of blades

1

Rotating direction

Clockwise

Blades type

NACA 0018

Solidity

0.2

TSR

5

Reynolds number

2,000,00015

Again, we compare the values of the first five cross- x -axis points and the value of the four extreme y points in first two revolutions (Figure 3-10). The comparison results are shown in Table 3-5. In general, the results generated with DVM-UBC are comparable with the results obtained with a conformal mapping method of Deglaire (2007). The differences between results by using these two different numerical methods are within 7% of the value generated by using the conformal mapping method. Also, the stability of the core vortices seems to be higher in Delglare (2007)’s representation. However, Delglare (2007) can not predict the turbine performance as accurately as DVM-UBC16. Table 3-5

The relative deviation of the results generated with DVM-UBC from the results generated by using the conformal mapping method

Position

x1

Relative deviation 1%

15 16

x2

x3

x4

x5

y1

y2

y3

y4

4%

5%

7%

2%

1%

3%

2%

2%

In Deglaire (2007), Reynolds number is infinite. The performance predicted by Delglare (2007) is about 30% higher than the experimental result. - 50 -

Figure 3-10 A comparison of the two-dimensional wake trajectory by using conformal mapping method (Deglaire 2007) (Red) and DVM-UBC (Blue) Case 3: we compare the growth of the wake structure generated by using DVM-UBC with

that generated by using Fluent (Figure 3-11). The basic specification of this case is shown in Table 3-6. Particularly, there are 300,000 cells used in Fluent, structured and unstructured. The domain is five time turbine diameter from upstream, ten time turbine diameter downstream, and seven time turbine diameter downstream on sides. Spalart-Allmaras model is used for simulating the turbulence effect17. More detailed information of Fluent setup can be found in Nabavi (2008). In this case, we are not able to compare the wake velocity distribution in details because we did not track each point’s velocity in Fluent. In order to numerically compare the wake geometry, we define the wake growth coefficient as CW =

winitial − wend lw

(3.46)

where winitial and wend denote the initial width of the wake (in y axis direction) and the end width of the wake (where wake velocity is almost the same as the free stream velocity) respectively, and lw denotes the length (in x axis direction) between the measure points of winitial and wend . 17

Both

k −ε

and k

−ω

models are also tried and the results are similar to the results presented in Figure 3-11. - 51 -

Table 3-6

Basic information of Case 3 for kinematics validation

Parameters

Values

Number of blades

3

Rotating direction

Clockwise

Blades type

NACA 63(4)-021

Solidity

0.435

TSR

2.75

Reynolds number

160,000

Special Note: There is a shaft in Fluent18. The comparison results are summarized in Table 3-7. It is noted that the relative deviation of the results obtained with DVM-UBC from the Fluent result is less than 6%. Table 3-7

18

Comparison of wake geometry obtained with DVM-UBC and Fluent for Case 3 DVM-UBC

Fluent

Relative deviation from Fluent result

winitial

2

2

-

wend

4.2

4.3

2.3%

lw

6.8

6.7

1.5%

CW

0.325

0.343

5.2%

The shaft diameter is 3% of the turbine diameter. - 52 -

(a)

(b) Figure 3-11 (a) Turbine wake velocity generated by using DVM-UBC; (b) Turbine wake velocity generated by using Fluent

- 53 -

Dynamics Validation Case 1: We compare the power coefficient ( CP ) obtained by using DVM-UBC with the

experimental result from one of the classical vertical axis turbine tests as reported in Templin (1974). The basic specification of this case is shown in Table 3-8. Table 3-8

Basic specification of Case 1 for dynamics comparison

Parameters

Values Scenario 1

Scenario 2

Number of blades

3

1

Blades type

NACA 0015

NACA 0015

Solidity

0.25

0.0833

Reynolds number

360,000

360,000

Figure 3-12 shows the relationship between power coefficient and tip speed ratio for case 1 by using different methods (DVM-UBC vs. experiment). Table 3-9 shows the maximum power coefficient and the TSR at which the power coefficient reaches its maximum for different scenarios with different methods. It can be seen that in scenario 1, the relative deviation of the power coefficient obtained with DVM-UBC from the experimental result is about 10%, and in scenario 2, the relative deviation of the power coefficient obtained with DVM-UBC from the experimental results is about 2.5%. Also, the power coefficient obtained with DVM-UBC is lower than the experimental result when the TSR is lower than the design TSR, and is higher than the experimental result when the TSR is higher than the design TSR.

- 54 -

0.80

Scenario 1- Experiment Scenario 1- DVM-UBC Scenario 2- Experiment Scenario 2- DVM-UBC

Power Coefficient (Cp)

0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 1

3

5 7 Tip Speed Ratio (TSR)

9

11

Figure 3-12 A comparison of the power coefficient of a stand-alone tidal current turbine by using different methods (DVM-UBC vs. Templin’s (1974) experiment test) under different scenarios Table 3-9

Comparison of power coefficient obtained with different methods (DVM-UBC vs. Templin’s (1974) experiment test) Methods

Scenario 1

Maximum CP

TSR corresponding to the Maximum CP

Experiment

0.59

4.95

DVM-UBC

0.53

5

9.8%

1%

Experiment

0.41

5.35

DVM-UBC

0.40

5.3

2.5%

1%

Relative deviation from the experimental result Scenario 2

Relative deviation from the experimental result

- 55 -

Case 2: We compare the power coefficient obtained by using our DVM-UBC with

experimental result from our in-house towing tank test and with numerical results generated by using other numerical methods (traditional 2-D DVM and Fluent). The basic specification of this case is shown in Table 3-10. Table 3-10 Basic specification of Case 2 for dynamic validation Parameters

Values

Number of blades

3

Blades type

NACA 63(4)-021

Solidity

0.435

Reynolds number

160,000

Special Note: There are two arms and a shaft in the experimentally-tested turbine and there is a shaft in Fluent. Figure 3-13 shows the relationship between power coefficient and TSR for case 2 by using different methods (DVM-UBC, traditional DVM, Fluent and experiment). Table 3-11 shows the maximum power coefficient with these methods. It can be seen that the results obtained with the four methods have good agreement when the TSR is around 2.75 at which the turbine achieves its maximum power coefficient. The relative deviation of the maximum power coefficient obtained with DVM-UBC from the maximum power coefficient obtained with experiment is about 14%. Additionally, if the Fluent result is regarded as the exact result, comparing the relative deviation of the DVM-UBC’s result from Fluent result (6.7%) and that of the traditional DVM’s result from the Fluent result (13.5%), one can note that the results generated by DVM-UBC are much closer to the exact results.

- 56 -

0.50 0.45 Power Coefficent (Cp)

0.40 0.35 0.30 0.25

DVM Traditional DVM

0.20

Fluent Experiment

0.15 0.10 1.75

2

2.25 2.5 2.75 Tip Speed Ratio (TSR)

3

3.25

Figure 3-13 A comparison of power coefficient of a stand-alone tidal current turbine by using different methods (DVM-UBC, traditional DVM, Fluent and experiment) Table 3-11 Basic results of Case 2 for the dynamic validation Methods

Maximum

Relative deviation

Relative deviation

TSR

CP

from experimental

from Fluent result

corresponding to

result

the maximum CP

DVM-UBC

0.395

14%

6.7%

2.76

Traditional

0.419

22%

13.5%

2.77

Fluent

0.369

7.6%

-

2.79

Experiment

0.343

-

-7%

2.75

DVM

Case 3: We compare the torque obtained by using DVM-UBC with the experimental result

from our in-house towing tank test and with numerical results generated by using other - 57 -

numerical methods (traditional DVM and Fluent). The basic specification of this case is shown in Table 3-12. Table 3-12 Basic specification of Case 3 for dynamic validation Parameters

Values Scenario 1

Scenario 2

Number of Blades

1

3

Blades type

NACA 63(4)-021

NACA 63(4)-021

Solidity

0.145

0.435

Reynolds Number

160,000

160,000

Special note: There are two arms and a shaft in the experimental turbine and there is a shaft in the turbine in the Fluent case. Figure 3-14 shows the relationship between the torque and the azimuth angle for case 3 obtained with different methods (DVM-UBC, traditional DVM, Fluent and experiment). Table 3-13 shows the maximum torque and the corresponding azimuth angle. It can be seen that in scenario 1, the torque generated by using DVM-UBC is closer to the experimental results than that generated with other numerical methods when the azimuth angle is high (over 200o). One may see two differences: z

There is a 15° or so phase shift between the numerical and experimental results in scenario 1 and a 20° or so phase shift in scenario 2 (Figure 3-12 (b)).

z

The maximum torque of the experiment is higher than that generated with the DVM and the minimum torque of the experiment is lower than that generated with the DVM while the average torque of the experiment is almost equal to that generated with the DVM.

These two differences are probably caused by following reasons: z

The error in experimental set-up: there was no dynamic calibration on the system so that sometimes signal amplification may cause such a phase shift; this issue is quite often observed during ship motion study - 58 -

z

The mounted frame effect: the mounted frame is not modeled in any of the numerical methods

z

The DC motor: the turbine angular velocity controller was unable to maintain a constant angular velocity for a stand-alone turbine during the turbine’s rotation, which may be responsible this phase shift. On the other hand, the DC motor was able to maintain a constant angular velocity for a ducted turbine. In the ducted turbine case, the phase shift was much less than that of a stand-alone turbine19

z

The towing tank wall effect (we have found that the turbine wake is asymmetric, and the width of the towing tank is only 3.7 meter. Thus, infinite vortex images are created, which could shift the torque). Table 3-13 Basic results of Case 3 for the dynamic validation Methods Scenario 1

Scenario 2

19

Maximum

Azimuth angle corresponding

torque(Nm)

to the maximum torque(Nm)

Experiment

78

90

DVM-UBC

46

75

Traditional DVM

38

76

Fluent

72

75

Experiment

122

110

DVM-UBC

66

88

Traditional DVM

58

89

Fluent

85

90

Refer to Appendix A for the details. - 59 -

100.00 Traditional DVM 80.00

Fluent Experiment DVM-UBC

Torque (Nm)

60.00 40.00 20.00 0.00 -20.00 -40.00 0

100

200

300

400

Azimuth angle (Degree)

(a) 160 Traditional DVM Experiment

140

Fluent DVM-UBC

Torque (Nm)

120 100 80 60 40 20 0 -20 0

50

100

150

200

250

300

350

400

Azimuth angle (Degree)

(b) Figure 3-14 The relationship between torque and azimuth angle obtained by using different methods under (a) scenario 1, and (b) scenario 2 - 60 -

Summary of the validations Table 3-14 summarizes the validation results as presented above. It is noted that the differences in kinematical validation is much less than the difference in dynamic validation. Besides Case 3 for the dynamic validation, DVM-UBC is able to provide acceptable results. Detailed analysis and discussion of these deviations are given in Section 3.5.1. Table 3-14 Summary of the validations Relative deviation in the

Relative deviation in the

kinematics validation

dynamics validation

Case 1

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