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Development of a procedure for power generated from a tidal current turbine farm Li, Ye 2008

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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  Power spectrum of the noise intensity at location 2 of Case 1 ....................................84  Figure 3-26  Power spectrum of the noise intensity at location 3 of Case 1 ....................................85  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  The torque fluctuation coefficient of the two-turbine system in Case 2 ....................124  Figure 4-23  The relative efficiency of the two-turbine system in Case 3......................................126  xiv  Figure 4-24  The torque fluctuation coefficient of the two-turbine system in Case 3 ....................126  Figure 4-25  The relative efficiency of the two-turbine system in Case 4......................................128  Figure 4-26  The torque fluctuation coefficient of the two-turbine system in Case 4 ....................128  Figure 4-27  The relative efficiency of the two-turbine system in Case 5......................................133  Figure 4-28  The torque fluctuation coefficient of the two-turbine system in Case 5 ....................133  Figure 4-29  The relative efficiency of the two-turbine system in Case 6......................................135  Figure 4-30  The torque fluctuation coefficient of the two-turbine system in Case 6 ....................135  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  Power spectrum of the noise intensity of the two-turbine system at Reciever 2 .......141  Figure 4-35  Power spectrum of the noise intensity of the two-turbine system at Reciever 3 .......142  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 wake structure with the  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 wake structure with the  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  <4%  <10%  Case 2  <7%  <14%  Case 3  <6%  >20%  In validating DVM-UBC, we compared its results with that obtained with other methods (traditional DVM, RANS CFD, conformal mapping, and experimental test). Financially, experimental test and RANS CFD are both more costly than the potential flow method for unsteady flow. As to the performance, good agreements are obtained between DVM-UBC and experimental results. DVM-UBC can predict the performance 50% more accurately than the traditional DVM. To the best of our knowledge, no report shows that conformal mapping method is able to predict the performance of a three-blade tidal current turbine yet. Thus, one can say that DVM-UBC is a cost-effective method among the four methods.  3.4.2 Comparison with Other Potential Flow Methods In order to identify the cost-effectiveness of DVM-UBC, we compare the power coefficient predicted by DVM-UBC and by using other potential flow methods as mentioned in Section 3.1.1 which are single streamtube method, multiple streamtube method, and fixed-wake method. The results are shown in Figure 3-15. The basic information of the case is given in Table 3-15. We do not compare the wake structure because all the other potential flow methods do not have a vortical wake.  - 61 -  0.90 Experiment (Templin 1974) DVM-UBC Single streamtube method Multiple streamtube method Fixed wake method  0.80 Power coefficient (Cp)  0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 -0.10 1  3  5 Tip Speed Ratio (TSR)  7  9  Figure 3-15 Power coefficient predicted by using different potential flow methods and one experiment test Table 3-15 Basic information of the case for comparing power coefficient predicted by using different potential flow methods Parameters  Values  Number of blades  3  Blades type  NACA 0015  Solidity  0.25  Reynolds number  360,000  Table 3-16 presents the results of the power coefficients obtained with different methods. It can be seen that the power coefficient generated by using DVM-UBC shows the most accurate prediction (i.e., less than 10% relative deviation from the experimental results), the power coefficient generated by using fixed-wake method is the second to the most accurate, and the power coefficient generated by using momentum method (the streamtube method  - 62 -  here) is the least accurate one. A much more detailed comparison of the results by using the different methods is given in Table 3-17. In general, we would like to say that DVM-UBC is the most cost20-effective one for predicting the tidal current turbine’s performance among all potential methods discussed in this study. Table 3-16 Results for comparing the power coefficient generated by using different potential flow methods Maximum  Relative deviation from  TSR corresponding to  CP  the experimental result  the Maximum CP  DVM-UBC  0.53  9.8%  5  Fixed-wake  0.65  10.1%  4  Single Streamtube  0.81  37.6%  3.75  Multiple Streamtube  0.76  28.8%  3.77  Experiment (Templin 1974)  0.59  -  4.95  Methods  20  Cost here refers to the computational cost. - 63 -  Table 3-17 Comparison of different numerical methods Methods  Single Streamtube  Multiple(double) Fixed-wake streamtube vortex method  Conformal mapping method  Free-wake DVM developed in this study  CFD (Fluent)  Computational time per case by using a standard PC  Within a few seconds  Within a few seconds  About ten to twenty seconds  About half minute to one minute  One to two minutes  Two to three days depending on the size of the grid  Agreement with experimental results  Unacceptable Acceptable21  Good  Acceptable  Very good  Very good with near-field flow results  Whether can be used to predict wake?  No  No  Yes (Independent Yes (Time of time) dependent)  Yes (Time dependent)  Yes (Time dependent)  Code structural complexity  Low  Low  Medium  High  High  N/A22  Program flexibility  Low  Low  Medium  High  High  High  Attentions received in tidal current turbine studies  Low  High  N/A  Low  Medium  High  Properties  21  In some case, it is reported that double and multiple streamtube can achieve results with similar accuracy as the discrete vortex free wake method (Paraschivoio 2002). 22 The CFD package is not an open source software although it has an add-on options called User Defined Function (UDF) which allows user to write their own open source code. - 64 -  3.4.3 Three-Dimensional Wake Structure In validating DVM-UBC, we compared the two-dimensional wake structure generated by using DVM-UBC with that generated by experiment and the conformal mapping method. There is no investigation on the three-dimensional wake structure by using the latter two methods. The DVM-UBC can also describe the three-dimensional wake structure, which is shown by using an example three-blade turbine, the basic specification of which is given in Table 3-18. Table 3-18 Basic specification of the case used to generate three-dimensional wake trajectory Parameters  Values  Number of blades  3  Blades type  NACA 0015  Solidity  0.25  TSR  4  Reynolds number  360,000  The turbine height to radius ratio  2.5  Rotating direction  clockwise  The height to radius ratio of blade element’s 1.5 location for wake trajectory tracking Figure 3-16 shows a snapshot of the three-dimensional wake trajectory of one turbine blade element23. At this time, the blade element’s position is [-1, 0, 1.5] and it rotates in clockwise direction. If we look from the top of the wake structure, this wake structure will be a two-dimensional wake structure, which is similar to that in Figure 3-10. The numerical results are summarized in Table 3-19. One can notice that the range of z direction wake elevation within the first two cycles is less than four percent of the turbine height, and this range of the eighth cycle is less than fourteen percent of the turbine height. One can say that 23  For illustration purpose, we only show the first two cycles of the wake structure for this blade element: the blue curve represents the first cycle while and the red curve represents the second cycle. - 65 -  the three-dimensional effect is not significant for the case studied above24.  Figure 3-16 Three-dimensional wake structure for one blade of an example turbine  Table 3-19 Results of the case for three-dimensional wake trajectory Maximum elevation  Maximum elevation  Maximum elevation  in the 1st cycle  in the 2nd cycle  in the 8th cycle  Elevation  0.03  0.08  0.35  Elevation to  1.2%  3.2%  14%  turbine height ratio  24  The three-dimensional effect of a curvature blade turbine is more significant than that of a straight blade turbine. More discussions on Three-Dimensional effect are given in Appendix A. - 66 -  3.4.4 An Example Vertical Axis Tidal Current Turbine After validating DVM-UBC, we use this method to predict the performance (power output and torque) of an example tidal current turbine. The basic specification of this turbine is given in Table 3-20. These parameters are selected according to previous study and UBC experimental test. For example, 160,000 is the average Reynolds number during UBC experimental test, and NACA0015 blade has been used for turbine tests by Coiro (2007). Table 3-20 Basic specification of an example tidal current turbine Parameters  Values  Number of blades  3  Blades type  NACA 0015  Solidity  0.375  Reynolds number  160,000  TSR for torque and induced velocity investigation  4.75  Power Output By using DVM-UBC, the maximum power efficiency is estimated to be around 48% when the TSR is around 3.8 as shown in Figure 3-17. The slope of the efficiency curve is steep when the TSR varies from 3 to 4 and from 5.5 to 7.  - 67 -  Figure 3-17 The relationship between power coefficient and TSR obtained with DVM-UBC  Dimensionless Torque The torque is a concern of mechanical and electrical engineers who design the generator, for it is easy to design a generator for a constant turbine torque. Figure 3-18 shows the polar plot of the dimensionless torque of the turbine as described in Table 3-20 when TSR is equal to 4.75 and the dimensionless torque of an ideal turbine which generates the same power as the turbine described in Table 3-20. The torque of the ideal turbine is a constant as expected by the mechanical system designers. As shown in Figure 3-18, the torque curve of an ideal turbine is a circle. For the example turbine, it is clear that the torque fluctuates significantly so that the metal fatigue of this turbine’s structural components and the fatigue of the generator should be investigated in details for reliability purposes. In order to investigate the variation of torque systematically, we evaluate the torque output in a frequency domain by using Power Spectrum Analysis25, which is a method for estimating spectral density of a random signal in a frequency domain. In our context, the signal is the torque generated by the turbine and the analysis method that we use is the Welch method. By 25  In order to keep the integrity of the discussion on marine hydrodynamic, the theory of Power Spectrum Analysis is shown in Section 3.5.3 with the discussion on underwater acoustics as it is a signal processing theory. - 68 -  using the Welch method, we transform Figure 3-18 (torque vs. azimuth angle) into Figure 3-19 (power spectrum magnitude the torque vs. dimensionless frequency). For the ideal turbine, it is clear that only one signal (the DC component) exits as expected by the generator designers. For the example turbine, the first peak represents the main torque while the rest represent the fluctuations. That is to say, if the magnitude of first peak is much greater than the magnitude of second’s and the third’s peaks, the torque output is more stable. However, the second peak is very close to the first peak here. Furthermore, even the third, fourth and fifth peaks are all almost as high as the second peak. In order to quantify the fluctuation of torque during one resolution, we define a new dimensionless coefficient which is the partial differential of the power spectrum magnitude (PSM) with respect to the frequency as follows,  ⎧⎪ ΔT ⎫⎪ CTF = ⎨ peak : first few peaks ⎬ ⎩⎪ Δf peak ⎭⎪  (3.47)  where Tpeak and f peak denotes the torque PSM and the frequency respectively, and ΔTpeak denotes the change in Tpeak over the change in frequency. Thus, it is clear that the larger the CTF is, the more stable the torque is. By using Eq.(3.47), we get CTF = 38dBS for the stand-alone turbine case given in Table 3-20. This result and definition will be used in Section 4.4 again for further study regarding torque fluctuation of the two-turbine system.  - 69 -  Figure 3-18 Torque curve of a tidal current turbine on a polar diagram26  26  The angle here in the polar diagram is the azimuth angle. - 70 -  (Hz)  Figure 3-19 Torque of a stand-alone turbine in frequency domain  Investigation on Induced Velocity A good understanding of the induced velocity is important for predicting power output and for researches related to flow change in other disciplines (such as oceanography). For example, oceanographers need to know the flow velocity to analyze the changes in ocean flow which is important to study the chemical and biological processes in the ocean. Induced velocity can be calculated by using the numerical model developed in this chapter. In this section, an example is given for investigating the induced velocity at four special points on a three-blade turbine as given in Table 3-20, and the locations of the four points are given in Table 3-21. Figure 3-20 shows the induced velocities at four special points on such a turbine when the azimuth angle is equal to zero (i.e., at the end of each revolution). It is obvious that the induced velocity at point 1 is the largest because it is at the center of downstream area of the - 71 -  turbine where the torque of the turbine is around its peak value and the induced velocity at point 3 is the smallest at the center of upstream where the torque is around its trough value (where azimuth angle is equal to 270o, see Figure 3-18). Crossing flows are observed at both point 2 and point 4 which shows agreement with the results generated by using Fluent (Nabavi, 2008). Table 3-21 Locations of four points on a turbine for induced velocity investigation Point 1  Point 2  Point 3  Point 4  Coordinates  (1,0)  (0,1)  (-1,0)  (0,-1)  Azimuth angle  90o  180o  270o  0o  Figure 3-20 Induced velocity at four points on a vertical axis tidal current turbine In order to investigate these induced velocities systematically, we analyze their magnitudes and directions in time sequence during one revolution (when azimuth angle varies from 0o to 360o) as shown in Figure 3-21. In this figure, the direction of the induced velocity is defined - 72 -  as the induced velocity angle, θ , which can be expressed as  θ = arctan  Vp, x  (3.48)  Vp, y  where V p , x and V p , y denote the induced velocities at a given point in x direction and y direction, respectively. It is noted that y  At point 1, both induced velocity and induced velocity angle are almost constant, except that there is a dramatic change when azimuth angle is between 250o and 300o  y  At point 2, the maximum values of both induced velocity and induced velocity angle are obtained when the azimuth angle is around 0o and the minimum values are obtained when azimuth angle is around 200o  y  At point 3, the magnitude of the induced velocity keeps to be 0 except when the azimuth angle is 100 o where induced velocity is about 0.5, and the induced velocity angle keeps to be -2 except when azimuth angle is 75o where the induced velocity angle is around 2  y  At point 4, the induced velocity fluctuates as the azimuth angle changes while the average induced velocity is around 0.2, and the induced velocity angle keeps to be -3 when azimuth angle is between 0o and 160o and 340o and 360o while it keeps to e 3 when azimuth angle is between 160o and 340o.  y  Additionally, the dimensionless induced velocities at point 2 and point 4 are both around 0.2 while their temporal variation patterns are different, which also shows the asymmetry of the wake.  - 73 -  (a)  (b) - 74 -  (c)  (d) 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 -  3.5 Acoustic Model In physical oceanography, one important application of utilizing induced velocity is to predict the acoustic emission from a tidal current turbine. Turbines generate noise while rotating. The noise from turbines may pose risk to human beings and animals if they are exposed to the noise (Shepherd 1983). In the past two decades, research has been conducted on the mechanism of turbine noise generation from wind turbines and propellers, the impact of the noise on animal hearing and communication, and the mitigation of the noise. Hanson (1985) proposed a wake model to predict noise generated by counter-rotating propellers (turbines). Madsen et al. (2007) presented a method to predict downstream turbine noise in a wind farm. Lowson (1992) delineated different noise sources (different components of a wind turbine), compared the factors affecting the noise, and provided a theoretical analysis of the noise control method. Elliot et al. (1989) evaluated the acoustic impact of a specific wind turbine in the coastal zone of south Wales. Particularly, Betke et al. (2004) provided a comprehensive analysis of underwater noise emission from an offshore wind turbine farm. However, no research has been reported on the acoustic impact from tidal current turbines. Technically, the noise from a tidal current turbine comes from two sources: 1) the hydrodynamic noise, which is generated when the fluctuating hydrodynamic lift acts on the turbine blades and when the unsteady load acts on the shaft, and 2) the mechanical noise, which is produced by the gearbox and the bearings. The hydrodynamic noise mainly propagates under the water surface while the mechanical noise propagates above water surface. It is easier to control mechanical noise than to control hydrodynamic noise. The mechanical noise can be reduced by modifying the gearbox (e.g., by inserting damping material between the gear box casing and the top of the power house module, by modifying the power house wall structure, by lowering the pressure angles of the gears, and by replacing the spur gears with helical ones). These methods have been well studied and widely adapted in some marine systems to reduce noise. For example, a helical gear system is used in marine propulsion (Hoppe and Vollmer 2007). The study on how to mitigate hydrodynamic noise is rarely reported as it has military implications. Theoretically, hydrodynamic noise may be reduced by an appropriate design of the turbine. Therefore, a - 76 -  clear understanding of the acoustic emission of the tidal current turbine is necessary. In this section, we briefly introduce the fundamentals of underwater acoustics and then formulate a model to predict the hydrodynamic noise generated from the tidal current turbine by using DVM-UBC.  3.5.1 Sound in the Ocean We explain the physical meanings of the acoustic intensity for those readers who never exposed to underwater acoustic in the past so as to evaluate the acoustic emission from a tidal current turbine. Sound in ocean was first studied in 1490, when Leonardo Da Vinci wrote "If you cause your ship to stop and place the head of a long tube in the water and place the outer extremity to your ear, you will hear ships at a great distance from you". The first mathematical treatment of underwater acoustics can be traced back to 1687 when Sir Isaac Newton published his Mathematical Principles of Natural Philosophy. Later, Lord Rayleigh published his Theory of Sound in two volumes and established modern acoustic theory in 1877-1878. In order to understand the physics of underwater acoustic emission, we give a brief introduction to the fundamentals of underwater acoustics, which is a synthesis of what you might find in many textbooks and reviews, such as Beranek (1983), Urick (1983), Kalmijn, (1988 and 1989), Rogers and Cox (1988), Burdic (2003), Medwin and Clay (1998), and Etter (2003). Sound propagates in water in the form of acoustic waves. An acoustic wave is generated by the longitudinal particle displacement of the carrier medium (e.g., rock, air and water). Such a displacement creates local pressure fluctuations, which create an acoustic wave that travels at sound velocity ( c ). This sound velocity is determined by the stiffness ( C ) and density of the carrier medium, given as follows,  c=  C  (3.49)  ρ  The intensity of the acoustic field is defined as the time-average vector product of local pressure fluctuation ( p ) and the particle velocity ( u ). Mathematically, acoustic intensity ( I )  - 77 -  is expressed as follows,  I = pu  (3.50)  The pressure fluctuation and particle velocity are related as follows,  p = Zu  (3.51)  where Z is the acoustic impedance of the medium. Eq. (3.50) is called Ohm’s law of acoustics because of its similarity to the Ohm’s law in electricity, which defines the relationships among voltage, current and resistance. In a free acoustic field (without reflecting boundaries and with a homogenous sound velocity), at a distance of several wavelengths from the sound source, acoustic impedance ( Z ) can be obtained by multiplying the density of the medium and sound velocity as follows,  Z = ρc  (3.52)  where ρ is the density of the medium. The theory assumes planar sound wave, and the ocean is assumed to be a free acoustic field. Thus, by plugging Eq.(3.52) and Eq.(3.51) into Eq.(3.50), the acoustic intensity can be expressed as a function of the particle velocity Eq.(3.53) or as a function of pressure fluctuation, Eq.(3.54).  I = ρ cu 2  I=  (3.53)  p2 ρc  (3.54)  The pressure fluctuation ( p ) can be measured by using a pressure transducer.  - 78 -  3.5.2 Turbine Noise Prediction There are quite a few studies on predicting turbine noise. However, most of them focus on predicting near-field noise intensity from some particular components of the turbine and their computational cost is relatively high (DeGagne et al. 2002; Bastasch et al. 2006; Zhou and Joseph 2005). Thus, these methods cannot be directly used for the purpose of this study on predicting turbine acoustic emission, and we develop another method. In this study, we use DVM-UBC to calculate the particle velocity and thus to predict acoustic intensity. The wake velocity can be calculated by summing the incoming flow velocity and induced velocity (i.e., by substituting Eq. (3.11) into Eq.(3.13)) as follows, U = U ∞ + ∑∑ ViPT ,i , j + ∑∑ ViPS ,i , j i  j  i  (3.55)  j  where U ∞ denotes the freestream incoming velocity, VPT ,i , j denotes the velocity induced by the trailing edge wake vortices shed from blade element i at time step j , and VPS ,i , j denotes the velocity induced by the spanwise wake vortices shed from the same element at the same time step. We assume that the velocity fluctuation is fully induced by the vortex shedding effect (i.e., U ∞ is the mean particle velocity), given that the incoming flow is along the x direction.  Since vortex shedding is a periodic phenomenon, the particle velocity can be approximated as follows,  u ≈ u = (U ) x  (3.56)  Thus, by substituting Eq.(3.56) into Eq.(3.53), the acoustic intensity can be rewritten as follows,  I ≈ ρ c (U ) x  2  (3.57)  - 79 -  Similar to the discussion of marine hydrodynamics in Section 3.2.11, the acoustic intensity is nondimensionalized as follows,  Iˆ =  I ρ wcU ∞2  (3.58)  3.5.3 Power Spectrum Analysis We use power spectrum analysis to evaluate the noise intensity (noise intensity and acoustic intensity are used interchangeably here). Power spectrum analysis is a technique which can be applied to analyze signals and to determine how the power of a signal is distributed with respect to frequency. Power spectrum analysis has been widely used in electrical engineering and physical oceanography as well as mechanical engineering. The signal is usually a wave, such as an acoustic wave, a water wave, an electromagnetic wave or a random vibration. Interested readers are referred to Bartlett (1953), Blackman and Tukey (1958) and Oppenheim and Schafer (1975), etc., for a detailed description of the theory and application of various power spectrum analysis techniques. In this study, the power spectrum analysis employed is Welch’s method proposed by Welch (1967), which is a modification of the Bartlett procedure that is particularly well suited to estimate power spectrum by using Fast Fourier Transformation (FFT). If we have a sequence x ( n ) 0 ≤ n ≤ N − 1 , the data record is divided into K sections and the length of each section  is L . That is  KL = N After  applying  (3.59) the  window  ( w ( n ) , n = 1, 2 ⋅⋅⋅ L − 1 ),  we  form  a  sequence:  x(1) ( n ) w ( n ) , ⋅⋅⋅, x( K ) ( n ) w ( n ) . Then, the K modified periodograms can be written as 1 J (ω ) = LU j L  L −1  ∑ x (n) w( n) e ( j)  2 − iω n  , j = 1, 2, ⋅⋅⋅, K  n=0  - 80 -  (3.60)  i = −1  U=  (3.61)  1 L −1 2 ∑ w (n) L n=0  (3.62)  The spectrum estimate is the average of these periodograms. 1 L Pˆ (ω ) = ∑ J ( j ) (ω ) L i =1  (3.63)  3.5.4 Computational Procedure Eqs.(3.49) to (3.63) describe the mathematical formulations to estimate the noise intensity from a stand-alone tidal current turbine, which uses DVM-UBC for turbine wake prediction and Welch’s method for power spectrum analysis. Figure 3-22 shows the flowchart of the computational procedure of the numerical method for estimating the noise intensity. As the turbine hydrodynamic model, the noise estimation model also use turbine geometry and environmental conditions as inputs. Then, the main program will call the stand-alone turbine model to calculate the wake velocity by using Eq.(3.55). With the wake velocity, the particle velocity can be obtained by using Eq. (3.56). Thus, the noise intensity can be obtained by using Eq.(3.58) in time sequence. The final output is the power spectrum of the noise intensity which is obtained by using Welch’s Method as described from Eqs.(3.59) to (3.63).  - 81 -  Figure 3-22 Computation program’s flowchart of the noise emission estimation model  3.5.5 Case Study In this section, we show how the turbine acoustic model is employed to estimate turbine noise by conducting a case study. The turbine studied in this case is the typical turbine discussed in Section 3.4.4 and its specifications are given in Table 3-20. In this case, we evaluate the noise intensity at three receivers as shown in Figure 3-23, which are located at 1) the middle plane; 2) the positive plane; and 3) the negative plane, respectively. The locations of the turbine and receivers are also given in Table 3-22. Table 3-22 Basic specification of the location of the three receivers of a stand-alone turbine  Parameters  Values  Turbine shaft location  (0,0)  Location of receiver 1  (4,0)  Location of receiver 2  (3,1)  Location of receiver 3  (3,-1)  - 82 -  Figure 3-23 Locations of three receivers of a stand-alone turbine By using Welch’s method, we estimate the power spectrums of the turbine noise intensity of receivers 1 to 3 as shown in Figures 3-24 to 3-2627, respectively. The peaks in each figure represent the noise induced by the velocity fluctuation. It is noted that the frequencies corresponding to the first peak (main noise frequency) at the three locations are all around 4 Hz. Particularly, the noise intensity at location 1 only has one peak, which corresponds to a frequency of 4 Hz, while there are multiple peaks at the remaining two locations. For example, the secondary peak at location 2 corresponds to a noise frequency of 18 Hz and the secondary peak at location 3 corresponds to a noise frequency of 31 Hz.  27  It is noticed that the power spectrum is negative here, because the acoustic intensity is dimensionless, i.e., a value smaller than 1. - 83 -  Figure 3-24 Power spectrum of the noise intensity at location 1 of Case 1  Figure 3-25 Power spectrum of the noise intensity at location 2 of Case 1 - 84 -  Figure 3-26 Power spectrum of the noise intensity at location 3 of Case 1  3.6 Discussion and Conclusions This chapter mainly describes the discrete vortex method with free wake structure (DVM-UBC) for estimating the performance of a standalone tidal current turbine.  3.6.1 Discussion The results and analysis in this Chapter show that DVM-UBC is a numerical method that can represent the effects of an unsteady wake on the turbine with an acceptable accuracy. The percentage differences in dynamics validation are about two times those in kinematics validation, and this may be because the dynamics result is one order higher than the kinematics result 28 . Some differences may be caused by the noise and error in the computations and experiments (Refer to Appendix C), but some may be attributed to the differences in the modeling conditions as summarized in Table 3-23. Also, it is difficult to 28  The dynamic result is proportional to the velocity square while the kinematics result is proportional to the velocity. - 85 -  estimate the sources of errors in an unsteady flow. For the turbine described in Table 3-20, it is noted that the slope of the power coefficient (See Figure 3-17) around the design TSR (around 3.8) is very steep. Thus, the reliability of the angular velocity controller should be very high if the designer tries to keep the peak efficiency. In Figure 3-21, it is noted that the induced velocity when the azimuth angle is equal to 0o is almost equal to the induced velocity when azimuth angle is equal to 350° at point 1, point 2 and point 3. At point 4, there is a difference between those two values (the velocity at 350o is about 30% higher than the velocity at 0o) which may be caused by the unsteady nature of the flow and the turbine rotation; if we consider the average value of the induced velocity, this difference is significantly reduced. More investigation on the unsteady nature of the flow around point 4 is expected. Table 3-23 Differences in modeling conditions with different methods  DVM-UBC State  Quasi  Strickland  Templin  (1976)  (1974)  Unsteady  Unsteady  Fluent Unsteady  Deglaire  UBC  (2007)  experiment  Steady  Unsteady  steady Testing  Unrestricted Restricted  Unrestricted Restricted  Unrestricted Restricted  No  Yes  No  No  No  Yes  Turbine  Fully  Half  Fully  Fully  Fully  Fully  position  submerged  submerged submerged  submerged submerged  submerged  Number  0  1  0  0  0  2  No  No  No  Yes  No  Yes  Variable  Variable  Infinite  Variable  area Free surface  of arms Shaft  Reynolds Fixed number  at Variable  design TSR - 86 -  In the acoustics study, it is noted that the main turbine noise frequency is 4 Hz. This frequency just represents the main noise frequency of the turbines with such a design as described in Table 3-22. The main noise frequency may change as the turbine design or the geometry of the two-turbine system changes. In this acoustic model, we neglected turbulence, shaft and arm effects and (refer to the assumptions in Section 3.2.2). Therefore, we might not accurately predict the noise intensity. Particularly, during a revolution, the blade poses a strike on the shaft when blade and shaft are in a line paralleling to the incoming flow direction, which generates a noise. Its frequency, f B , can be estimated by using Eq. (3.64).  fB =  nBω 2π  (3.64)  Where nB denote the number of the blades. The intensity of this noise can be measured by experimental test and estimated by using empirical data. Despite of the simplification of this model, it is helpful for indentifying the noise which might interfere with the communication of those marine animals which are sensitive to this noise frequency bands. The direct victims of turbine noise are those marine animals which are sensitive to turbine noise frequency bands in which they communicate. As biology is not the focus of this study, we just review the previous research to survey the hearing ability of some marine animals. The description below is a comprehensive review of papers by marine biologists such as Au (1993), Payne and Webb (1971), Stafford et al. (1998), Wahlberg and Westerberg (2005) and Madsen et al. (2006). In particular, Wahlberg and Westerberg (2005) and Madsen et al. (2006) reviewed fish and marine mammal hearing ability with an emphasis on those sensitive to the band of noise generated by offshore wind turbines. The main frequency of the noise generated from tidal current turbines is relatively lower than the frequency of the noise generated by wind turbines. Thus, in studying the interference of tidal current turbine noise with marine animal communication, we focus on those animals which are sensitive to low frequency noise. - 87 -  Among the marine mammals, large baleen whales generate low frequency, long-duration and powerful calls that in some cases have ocean-traversing potential (Payne and Webb 1971; Stafford et al. 1998). Toothed whales generate short ultrasonic transients (clicks) for navigation and echolocation of prey within the range of tens to hundreds of meters (Au, 1993). These whales use low frequency underwater signals (10 ~10,000 Hz) for communication and navigation, whereas toothed whales produce sounds for echolocation and communication in the frequency range of 1 ~ 150,000 Hz (Richardson et al. 1995). Pinnipeds communicate by vocalizing with a frequency of about 50 ~ 60,000 Hz (Richardson et al. 1995). In general, Baleen whales are more sensitive to low frequency signals while tooth whales are very insensitive to low frequency signals although most marine animals can be affected by the low frequency noise (Madsen 2007). Froese and Pauly (2008) summarized the fish acoustic information, part of which is shown in Table 3-24. Being affected by the acoustic emission of a tidal current turbine farm, marine animal may have to increase its pitch or change the frequency to communicate with each other (Pauly 2008). Table 3-24 Fish acoustic information  Fish name  Sensitive frequency (Hz)  American eel  75  Atlantic herring  30 -1200  Atlantic thread herring  1200  Banded rudderfish  60-1000  Bar jack  30  Bigeye mojarra  75-1000  Bigeye scad  50-3000  Bluefish  60-1200  Coney  75  Cownose ray  150  - 88 -  3.6.2 Conclusions Based on the research in this chapter, we draw the following conclusions: y  The DVM-UBC is able to simulate turbine rotating both in clockwise and counter-clockwise rotation and both in two-dimensional and three-dimensional representation. Reasonable agreements (in both dynamics and kinematics) are obtained for unsteady flow with both the UBC experimental results, published experimental results, and the results generated by using other numerical methods  y  The DVM-UBC shows a significant improvement compared with traditional DVM overall in predicting the performance of turbines; the results obtained with DVM-UBC are about 50% more accurate than those obtained with the traditional DVM in the cases that we study  y  DVM-UBC is a cost-effectiveness method; it can predict turbine performance almost as accurate as a commercial RANS CFD, but takes much less time (less than 1% of the time that RANS CFD takes)  y  The stand-alone tidal current turbine model can be served as the base for a two-turbine system model, which is presented in Chapter 4  y  The stand-alone turbine model can be used to test the performance of the turbines with new configuration, and to provide information for other disciplines (such as predicting the acoustics emission as shown in Section 3.5 and assessing the impact of turbines on ocean flow).  In general, the DVM-UBC is a reasonably accurate method for simulating the behavior of a stand-alone tidal current turbine and works relatively fast and efficiently. However, the DVM-UBC highly relies on the availability of hydrodynamic characteristics data so that errors can not be avoided when the turbine is not simulated at its design TSR. Besides, arms and shafts are not simulated in this model, which means that their effects are neglected. Therefore, the real turbine efficiency is expected to be less than the numerical results as suggested. According to recent towing tank experimental test results (UBC Naval Architecture Laboratory 2007), the arm effects are observed to reduce the overall efficiency  - 89 -  by over 10% when TSR is around 2.75 for a NACA 63(4)-021 blade turbine. That implies the numerical results could be higher than the real values.  3.6.3 Future Work Based on the experience gained from this study, we identify a few relevant issues worthy of further investigation besides the quantification of the shaft and arm effects and conducting experimental test in unrestricted water area: y  Investigation of turbine fatigue and reliability. In this study, the fatigue and reliability information are obtained from empirical data and reported results for related offshore structures. Considering the unexpected sea conditions and advance of the turbine materials, new fatigue and reliability data are needed. One can obtain them either by conducting finite element analysis or experiments  y  Quantification of sea bottom and free surface effects. The turbine investigated in this paper is assumed to work in an ideal flow where the turbine is far from the water surface and the sea bottom. In the real world, both of them are expected to affect the turbine performance considering the change of tidal range. Both analytical and experimental investigations are necessary  y  Analysis of the unsteady torque and its effect on the electric power generator (the voltage and current). Although an optimally-designed two-turbine system(See Section 4.3) is able to reduce the torque ripple, considering the unsteady angular velocity in the experimental test, the electrical power system still have to be modified (See Sections 2.2.3 and 5.3.1 for more discussion regarding electrical system of the turbine). A systematic dynamic modeling may be required  y  Based on hydrodynamic analysis in this chapter, the important factors affecting turbine power output are identified here for further investigation for design purpose in the future. They are blade arm angle (or Pitch angle), turbine radius, blade numbers, solidity and tip speed ratio.  - 90 -  Chapter 4 Hydrodynamic Analysis of a Two-Turbine System The hydrodynamic interactions between turbines in a tidal current turbine farm affect the overall performance (torque, force and power output) of the turbine systems. In this chapter, these hydrodynamic interactions are studied. Firstly, previous research on aerodynamic interactions between multiple wind turbines in a wind turbine farm, and hydrodynamic interactions between multiple tidal current turbines in a conceptual tidal current turbine farm, is reviewed. A numerical model for simulating a two-turbine system is developed, based on the DVM-UBC for a stand-alone tidal current turbine as described in Chapter 3. Power output and torque fluctuation of the two-turbine system are extensively analyzed for different system configurations. Also, the acoustic emission from the two-turbine system is analyzed. Then a numerical procedure is developed to study the hydrodynamic interactions between tidal current turbines in an N-turbine system, and an example is developed to predict the power output from an N-turbine system.  4.1 Review of Previous Research In this section, we review previous research on the hydrodynamic interactions between two turbines: the incompressible aerodynamic interactions between wind turbines and the marine hydrodynamic interactions between tidal current turbines.  4.1.1 Incompressible Aerodynamic Interactions between Two Wind Turbines There are two types of wind turbine farms: on-land wind turbine farms (Figure 4-1) and offshore wind turbine farms (Figure 4-2). On-land wind turbine farms have been operating for thousands of years since the first one was constructed in Persia, and offshore wind turbine farms have been operating since 1971 in Denmark (CFE 2006). Research on incompressible aerodynamic interactions between wind turbines in a wind turbine farm, however, has barely - 91 -  been reported, because wind farms have often been constructed in open areas where the geological conditions do not restrict farm planning. As a result, the turbines are too far apart to interact with each other. Only recently (during the past three decades), with the increase in the scale and number of wind turbine farms as well as the increase in concerns regarding land use, the aerodynamic interactions between wind turbines in wind turbine farms started to receive attention (Templin 1974; Hahm and Kröning 2002). Nonetheless, the turbine density in a wind farm is still expected to be lower than that in a tidal current turbine farm.  Figure 4-1  Figure 4-2  The Cronalaght wind farm in Donegal, Ireland (EMD 2006)  An offshore wind farm in Denmark (Sandia National Lab 2003) - 92 -  4.1.2 Marine Hydrodynamic Interactions between Two Tidal Current Turbines Tidal current turbine farms are still in their infancy. To date, no commercial farm has been built yet, although conceptual tidal current turbine schemes have been proposed by some tidal current turbine companies. Figure 4-3 shows a conceptual horizontal axis tidal current turbine farm proposed by Marine Current Turbine Ltd. in the United Kingdom. Figure 4-4 shows a conceptual vertical axis tidal current turbine farm proposed by Kobold in Italy. A tidal current turbine farm has to be constructed in a restricted space like a channel, a narrows or a straight, where high flow-velocity currents exist, for cost-effectiveness concerns. In such a restricted area, it is difficult, if not impossible, to avoid severe hydrodynamic interactions between tidal current turbines. In previous research, investigators either use the results of aerodynamic interactions between wind turbines to represent hydrodynamic interactions between tidal current turbines (Bedard 2002) or use commercial RANS CFD packages to estimate hydrodynamic interactions between tidal current turbines (MacLeod et al. 2002). Using the results of aerodynamic interactions between wind turbines to represent hydrodynamic interactions between tidal current turbines neglects the differences between wind turbines and tidal current turbines in turbine design, turbine distribution and their working environment (air vs. ocean flow). By using commercial RANS CFD packages, it is difficult to obtain a family of accurate solutions for the hydrodynamic interactions with current computational capacity. The inaccurate estimation of hydrodynamic interactions could lead to inaccurate power output prediction for a tidal current turbine farm. A method which provide get a family of accurate estimates of these interactions (at reasonable computational cost) is needed. For such a purpose, a numerical model is developed in this chapter by using the DVM-UBC to study the hydrodynamic interactions between tidal current turbines. Part of the research has been published in IEEE/MTS Oceans07 (Li and Calisal 2007b).  - 93 -  Figure 4-3  A conceptual horizontal axis tidal current turbine farm (Courtesy of Marine Current Turbine Ltd.)  Figure 4-4  A conceptual vertical axis tidal current turbine farm (Courtesy of Professor Coiro at the University of Naples) - 94 -  4.2 Numerical Model for a Two-Turbine System In Chapter 3, a numerical model is developed using DVM-UBC to analyze and simulate a stand-alone turbine and predict its performance (torque, force and power) and wake structure. Good agreements have been obtained between the results generated by using DVM-UBC and the results from published experimental tests and other numerical methods. In this section, we develop a numerical model using DVM-UBC to simulate a two-turbine system and predict its performance. In modeling the two-turbine system, we use the same formulations and equations as we use in Chapter 3 to represent the blades of the two-turbine system. Given that a two-turbine system is composed of two individual turbines, the procedure developed for studying a stand-alone turbine can serve as a reference for studying the two-turbine system and as a sub-module of the two-turbine model.  4.2.1 Review of the Stand-alone Turbine Model For those readers who read this dissertation from Chapter 4 directly, we review the model developed in Chapter 3. In Section 3.2, we formulate a new three-dimensional Discrete Vortex Method with free wake structure (DVM-UBC) to represent a turbine in unsteady flow. In the DVM-UBC, a turbine is represented by a set of blades only (no shaft and no arm). The incoming flow is assumed to be uniform, blades are represented by bounded vortices, and wake flow is represented by free vortices and uniform flow. Viscosity is included in DVM-UBC by introducing blade drag, vortex decay, vortex shedding frequency and nascent vortex, and by using lamb vortices. Then, by using DVM-UBC, a numerical model is developed to analyze and simulate a stand-alone turbine and predict its performance (torque, force and power) and wake structure. Specifically, by using the relationship between lift and blade bounded vortex, Eq. (3.7), bounded vortex strength can be obtained. With this calculated vortex strength, the induced velocities can be calculated by Eq. (3.11). Then, a new blade local relative velocity is obtained by Eq. (3.12). With this induced velocity, the new bounded vortex strength can be predicted by Eq. (3.7) again. This process forms a computational loop, which ends when the convergence criteria are satisfied, Eq.(3.44) or Eq.(3.45). Then, blade forces and turbine performance can be calculated by using the basic governing equations, Eqs. (3.5) to (3.7), and the hydrodynamic equations, Eq. (3.35) and Eq. - 95 -  (3.36). The wake structure can be calculated by using the wake vortices system relationship , Eq. (3.13).  4.2.2 Assumptions for the Two-turbine Model As the stand-alone turbine model is used as a sub-module of the two-turbine model, all the assumptions of Section 3.2.2 apply to the two-turbine model (e.g., shafts and arms are not simulated in the two-turbine system). Besides those assumptions, we make the following assumptions, particularly for the two-turbine model: z  As arm effect and shaft effect are not simulated in the stand-alone turbine model, the effects of interactions between arms and shafts of the two turbines are not simulated in the two-turbine model  z  There are no other auxiliary structures (such as mounting frame and anchors) and the third turbine around the two-turbine system.  4.2.3 Program Flowchart In modeling a stand-alone turbine, we use bounded vortices to represent turbine blades and use free discrete vortices to describe the wake of the turbine. Similarly, in modeling a two-turbine system, we use two sets of bounded vortices to represent the blades of the two turbines and two sets of free discrete vortices to describe the wakes of the two turbines. We also obtain the lift and drag coefficient according to the Reynolds number. The stand-alone turbine model serves as a reference for the two-turbine system model. In the computational work, the single turbine model is a sub-module of the two-turbine system. The computational flowchart for the two-turbine model is presented in Figure 4-5. In this procedure for predicting the performance and estimating the wake of the two-turbine system, the initialization of the calculation is the same as that for the stand-alone turbine model, and the strengths of bound vortices are set as zero in the beginning. In the meantime, the main program of the two-turbine model checks the relative positions and relative rotating directions of the two turbines to make sure that there is no physical overlap between the  - 96 -  mechanical components of these two turbines29. After the positional check, the main program calls the model for a stand-alone turbine (Figure 3-5) to predict the bounded vortices and the wake vortices of each of the turbines. Then, wake vortices generated from each of the turbines are used to calculate the new blade-bounded vortices strengths. With the new strength of blade-bounded vortices and the free vortices in the wake, the main program calls the stand-alone turbine model again to calculate the performance of each turbine, and a loop is formed. This loop process is the calculation of the hydrodynamic interactions between turbines. The loop ends based on the given criteria, which are similar to the criteria for the loop in the single-turbine model. In the single turbine model, we suggested two criteria, which are the critical converge deviation value, Eq.(3.44), and the critical number of loops, Eq.(3.45), and used the latter. For the two-turbine model, because the hydrodynamic interactions are significant, we suggest to use the first one (i.e., the deviation of the current value of the strength of the blade-bounded vortex from the value in the last loop is less than the critical convergence deviation value), and we set the critical convergence deviation value ( ε ) as 0.02.  29  In this model, the arm and shaft of the turbines are not simulated. Thus, the program still works even if there is an overlap between the some of the components (e.g., arm) of the two turbines. In order to avoid this overlap, we check the possible overlap between the components of the two turbines at the beginning. - 97 -  Figure 4-5  Flowchart of the computation program for the two-turbine model  4.2.4 Parameters of a Two-turbine System To implement the two-turbine model systematically, we identify the main parameters that affect the performance of the two-turbine system, which are 1) the relative rotating direction of the two turbines, 2) the relative position of the two turbines as shown in Figures 4-6 to 4-9, - 98 -  and 3) the phase shift ( Δφ , See Figure 4-10). The relative rotating direction of a two-turbine system can be either co-rotating (both turbines rotate in the same direction, either clockwise or counterclockwise) or counter-rotating (both turbines rotate in the opposite direction: one clockwise and the other counterclockwise). The relative position is the main factor affecting the hydrodynamic interactions between two turbines, and thus the performance of the two-turbine system. It has two vectors: the layout of the two-turbine system and the relative distance between the two turbines. The relative distance between two turbines (one upstream turbine and one downstream turbine) in the x and y directions can be defined as follows,  xup − xdown ⎧ ⎪Xd = R ⎨ ⎪ Y = yup − ydown ⎩ d R  (4.1)  where X d and Yd denote the relative distance between the two turbines in the x and y directions, respectively, and  ( xup , yup )  and  ( xdown , ydown )  indicate the positions of the  upstream turbine and the downstream turbine, respectively. Here the x direction is the incoming water flow direction (Figure 4-6). In Figure 4-6, we can identify three typical layout for two turbines: 1) canard (turbine A and turbine C, or turbine B and turbine D), 2) tandem (turbine A and turbine B, or turbine C and turbine D), and 3) diagonal (turbine A and turbine D, or turbine B and turbine C).  - 99 -  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)  Canard layout A two-turbine system is said to be in canard layout if the two turbines are located side by side and their shafts are in a straight line that is perpendicular to the incoming flow direction (Figure 4-7). That is to say, if the flow is coming along the x direction, then X d = 0 and Yd ≠ 0 . - 100 -  Figure 4-7  A sketch of two turbines in canard layout  Tandem layout A two-turbine system is said to be in tandem layout if one turbine is located downstream of the other and the shafts of both turbines are in a straight line that is parallel with the incoming flow (Figure 4-8). That is to say, if the flow is coming along the x direction, then X d ≠ 0 and Yd = 0 .  - 101 -  Figure 4-8  A sketch of two turbines in tandem layout  Diagonal layout A two-turbine system in diagonal layout means that the two turbines are neither in canard layout nor in tandem layout. Rather, this layout is a combination of the canard layout and tandem layout (Figure 4-9). That is to say, if the incoming flow is along the x direction, then X d ≠ 0 and Yd ≠ 0 .  - 102 -  Figure 4-9  A sketch of two turbines in diagonal layout  - 103 -  The phase shift is defined as the difference between the azimuth angles of the two turbines as shown in Figure 4-10. It is noted that the possible range of phase shift is between 0 o and 60 o. Considering the low vortex shedding frequency, which implies that the difference between azimuth angles of two continuous shed vortices from one blade is relatively large (e.g., 15o), there are not too many possibilities of phase shifts between two turbines. Additionally, based on our calculation30, it won’t affect the performance of the system too much; therefore, it is only briefly presented here. Monte Carlo method is used to conduct this investigation. Monte Carlo method is a well know computational algorithm in solving the properties of a system by generating random values. It has been widely used in battle system analysis (Sotak, et al., 2006) and engineering and science (Amar 2006) including hydrodynamics (Yoshizaki and Yamakawa 1996). Particularly, Li and Calisal(2007d) shown how to use Monte Carlo method to study the hydrodynamic interaction of a turbine. In this two-turbine system case, we use the Monte Carlo method to randomly choose a variety of combinations of turbine geometry and flow velocity and fix the phase shift, and then calculate the power output of the two-turbine system for each combination. We find that the optimal phase shift for a counter-rotating two-turbine system is 60o and the optimal phase shift for a co-rotating two-turbine system is 0o if we are to maximize the efficiency and reduce the torque fluctuation. Additionally, the phase shift can be adjusted in real time. In this case, we can call the two-turbine system as a variable phase shift two-turbine system, which is similar to the variable pitch turbine. Thus, in the following analysis, from now on, when parameters of the two-turbine system are mentioned, they are referred to relative distance and incoming flow angle.  30  Only results of the relationship between incoming flow angle, relative distance and the performance of the two-turbine system are presented(refer to Section 4.3) - 104 -  Figure 4-10 An illustration of the phase shift between two turbines  4.3 Performance of the Two-turbine System in Different Layouts In this section, we apply the approach developed in this chapter to predict the performance of a two-turbine system. In order to systematically investigate the efficiency of the system31, we define the relative efficiency of a two-turbine system (η2T ) as the ratio of the efficiency of a two-turbine system ( η2T ) to the stand-alone turbine’s efficiency32 ( η S ) under the same operation conditions, given as follows,  η2T = ηup + ηdown  (4.2)  η2 T =  η2 T ηS  (4.3)  ηup =  ηup ηS  (4.4)  ηdown =  ηdown ηS  (4.5)  31  Torque fluctuation coefficient is a nonlinear and logarithm relative value by its definition; therefore, we still use the torque fluctuation coefficient as that used in Chapter 3 (see Eq.3.47) to analyze the torque fluctuation of a two-turbine system. 32 In Chapter 3, CP is used to represent the power coefficient of a stand-alone turbine. In this chapter, we use the efficiency of a stand-alone turbine, η S , to refer to its power coefficient to keep the symbols consistent. - 105 -  where ηup and η down denote the efficiency of the upstream turbine and the downstream turbine, respectively, while ηup and ηdown denote the relative efficiency of the upstream turbine and the downstream turbine, respectively. Also, in order to quantify the effect of the layouts of the two turbines on the performance of the two-turbine system, we introduce incoming flow angle (ψ ) and relative distance ( Dr ) (Figure 4- 11), as follows,  ψ = tan −1  Yd Xd  (4.6)  Dr = ( X d2 + Yd2 )  (4.7)  With the definition on the incoming flow angle, the three typical layouts (See Figures 4-7 to 4-9) are summarized in Table 4-1. Table 4-1  The relationship between the incoming flow angle and the layout of the two turbines Incoming flow angle  ψ =±  Layout  π  Canard  2 33  ψ =0 −  33  π 2  Tandem  < ψ < 0 and 0 < ψ <  π 2  Diagonal  π  π  and the wake at − for a 2 2 co-rotating canard layout two-turbine system. For a counter-rotating canard layout system, we define that the Due to the symmetry of the wake, there is no difference between the wake at  π  π  when the two turbines are inner rotating and the incoming flow angle is − 2 2 when the two turbines are outer rotating (See Figure 4-12).  incoming flow angle is  - 106 -  Figure 4-11 An illustration of the incoming flow angle (ψ ) and the relative distance ( Dr ) of a two-turbine system - 107 -  4.3.1 The Performance of an Example Two-turbine System In this section, we use the two-turbine model presented in Section 4.2.2 to investigate the relative efficiency and torque fluctuation coefficient of a two-turbine system.  Relative Efficiency We calculate the relative efficiency of a two-turbine system. The basic specification of the system is given Table 4-2. Three cases are developed according to the layout of the system, as shown in Table 4-3. Table 4-2  Basic specification of the two-turbine system  Parameters  Values  Number of blades  3  Blades type  NACA 0015  Solidity  0.375  TSR  4.75  Reynolds number  160,000  Efficiency of a stand-alone turbine  36.7%  Torque fluctuation coefficient of a stand-alone turbine  38dBS  Table 4-3  1.  The layout of the two-turbine system  Case 1  Case 2  Case 3  Canard layout  Tandem layout  Diagonal layout  Canard layout  According to its relative rotating direction and the symmetry of the wake of the two-turbine system, a canard layout two-turbine system can be studied with three scenarios (Table 4-4 and Figure 4-12).  - 108 -  Table 4-4 Scenario 1  Scenario description for a canard layout two-turbine system Incoming flow angle (ψ )  Relative rotating direction  π  Counter-rotating  2  2  −  3  ±  π  Counter-rotating  2  π  Co-rotating  2  Figure 4-12 An illustration of the three scenarios of the canard layout two-turbine system Figure 4- 13 shows the relative efficiencies of the two-turbine systems in different rotation scenarios. We note that the relative efficiency of the counter-rotating two-turbine system (i.e., scenario 1 and scenario 2) is relatively high when these two turbines are very close (i.e., the distance between the two turbines is 1.5 (times the turbine radius)). Their trends are similar with respect to the relative distance while the relative efficiency of Scenario 1 is higher than that of Scenario 2 at the same relative distance, which indicate the hydrodynamic interaction of the inner counter rotating system (Scenario 1). In order to avoid physical overlap, the relative distance of the co-rotating system can not be - 109 -  less than 2.25. The maximum efficiency of such a system is less than 2 when the relative distance is equal to 2.25, and the efficiency drops to its minimum value when the relative distance is equal to 3, after which the relative efficiency of the system slowly increase. In general, when the system is a canard layout system, the efficiency of the counter-rotating system is higher than that of a co-rotating system. 3.00  Relative efficency  2.50 2.00 1.50 1.00 Scenario 1: two counter-rotating turbines (ψ= π/2) Scenario 2: two counter-rotating turbines (ψ= −π/2) Scenario 3: two co-rotating turbines  0.50 0.00 1  1.5  2  2.5  3  3.5  4  4.5  Relative distance (Dr)  Figure 4-13 The relative efficiency of a canard layout two-turbine system under different rotation scenarios 2.  Tandem layout  According to its relative rotating direction, the tandem layout two-turbine system can be either co-rotating (scenario 1) or counter-rotating (scenario 2). The relationship between the relative efficiency of the two-turbine system and the relative distance between the two turbines is shown in Figure 4-14. It is noted that the relative efficiency of the tandem layout two-turbine system is always less than 2. The relative efficiency of two co-rotating turbines increases smoothly, with some fluctuation, as the relative distance between the two turbines increases. On the other hand, the relative efficiency of two counter-rotating two turbines is - 110 -  relatively high when the relative distance between the two turbines is very short (e.g., less than 4 times the turbine radius). When the relative distance is greater than 4 times the turbine radius, the relative efficiency of a counter-rotating two-turbine system is always less than that of a co-rotating two-turbine system at the same relative distance.  2.50  Relative efficency  2.00  1.50  1.00 Scenario 1:two co-rotating turbines 0.50  Scenario 2:two counter-rotating turbines  0.00 0  5  10  15  20  Relative distance (Dr)  Figure 4-14 The relative efficiency of a tandem layout two-turbine system 3.  45o Diagonal layout  The diagonal layout two-turbine system is studied with four different scenarios according to its relative rotating direction and the incoming flow angle, as shown in Table 4-5.  - 111 -  Table 4-5  Scenario description for a diagonal layout two-turbine system  Scenario  Incoming flow angle (ψ )  Relative rotating direction  1  π  Counter-rotating  2  -π  3  π  4  -π  4  Counter-rotating  4  Co-rotating  4  Co-rotating  4  Figure 4- 15 shows the relationship between the relative efficiency of the diagonal layout two-turbine system and the relative distance between the two turbines in different turbine rotating scenarios. It can be seen that the relative efficiency of a counter-rotating two-turbine system is always lower than that of a co-rotating two-turbine system when the relative distance between the two turbines is greater than 2 (times the turbine radius).  Figure 4-15 The relative efficiency of a diagonal layout two-turbine system - 112 -  4.  Relative efficiency of an individual turbine in a two-turbine system  Besides the total relative efficiency of a system, we also investigate the relative efficiency of the upstream turbine (ηup ) and the relative efficiency of the downstream turbine (ηdown ), respectively. We use an example to show these two relative efficiencies in a co-rotating tandem layout two-turbine system (Figure 4-16). One may notice that the relative efficiency of the upstream turbine is almost always around 1 (except when the relative distance is less than 5 time the turbine radius), while that of the downstream turbine slowly increases and finally reaches 1. In this study, we focus more on the relative efficiency of a two-turbine system (η2T ) rather than the individual relative efficiencies of the two turbines. Thus, the latter will not be shown in the following discussions although the model is able to predict these efficiencies.  1.4  Relative efficiency  1.2 1 0.8 0.6 0.4 Relative efficiency of the downstream turbine 0.2  Relative efficiency of the upstream turbine  0 0  5  10 Relative distance (Dr)  15  20  Figure 4-16 The relative efficiency of individual turbines in a tandem layout co-rotating two-turbine system  - 113 -  From the results of Figures 4-13 to 4-16, we notice that the relative efficiency of the system can be higher than two times the efficiency of a stand-alone turbine under same operation condition when the system is optimally configured. In some distance, it seems that the hydrodynamic interactions of the canard-layout and 45o diagonal-layout system are constructive. How the interaction varies with the distance and the incoming flow angle will be extensively investigated in Section 4.3.2.  Torque Fluctuation One of the problems with a stand-alone tidal current turbine is that the torque varies significantly during one revolution (i.e., torque fluctuation), which increases the fatigue of the shaft and induces reliability issues, and can lead to severe electricity intermittency when integrating into the electricity grid, as shown in Figures 3-18 and 3-19. The torque fluctuation of a two-turbine system may be reduced by selecting an appropriate combination of the phase shift and the layout. As shown in Chapter 3, we use the Welch method to estimate the torque fluctuation coefficient here. In this case, the two turbines are assumed to be in a diagonal layout. The basic specification of the example two-turbine system is shown in Table 4-6. Table 4-6  Basic specification of the two-turbine system for torque comparison Parameters  Values  Turbine blade  NACA0015  Blades number  3  Solidity  0.375  TSR  4.75  Rotating direction  Co-rotating  ψ  π  Relative distance  2.25  Reynolds number  160,000  4  The power spectrum of the torque for the case specified in Table 4-6 is shown in Figure 4-17, which indicates that the second peak (i.e., the secondary torque of the system) is much lower - 114 -  than the first peak (i.e., the main torque of the system). On the other hand, the magnitude of the second peak is similar to the first one in the stand-alone turbine case (Figure 3-19). Additionally, the third and fourth peaks in this case are much lower than those in the stand-alone turbine case. By using the definition of the torque fluctuation coefficient, Eq.(3.47), we obtain the torque fluctuation coefficient for this case34, which is CTF = 69dBS . Compared with the torque fluctuation coefficient of a stand-alone turbine under the same conditions, which is 38dBS , the torque generated by a typical two-turbine system is much more stable. Like the discussion on relative efficiency of individual turbines in a two-turbine system above, the torque fluctuation of individual turbines in a two-turbine system is also discussed. Figure 4-18 shows the torque spectrum of the two individual turbines in the case specified in Table 4-6, which indicates that torque fluctuation coefficient of either the downstream turbine, 72dBS , or upstream turbine, 65dBS , is greater than that of the stand-alone turbine, 38dBS . That is to say, both of them are more stable than that of the stand-alone turbine.  However, we focus on the torque fluctuation coefficient of the two-turbine system other than the individual turbines in the system. Thus, the latter will not be shown in following discussions although the model is able to predict these coefficients.  34  Torque fluctuation coefficient is a coefficient for measuring the level of torque fluctuation in frequency domain as defined in Chapter 3. Readers skipped Chapter 3 may refer to Section 3.4.4 and Eq.(3.47) for detailed discussion on torque fluctuation coefficient. In short, the greater the torque fluctuation coefficient is, the less the torque fluctuates. - 115 -  Figure 4-17 Torque of a two-turbine system in the frequency domain  Figure 4-18 Torque of the upstream and the downstream turbines in a two-turbine system in the frequency domain - 116 -  4.3.2 Generalized Results for the Performance of a Two-turbine System We have discussed the performance of two-turbine systems according to their specific layouts (canard, tandem and 45° diagonal). In this section, we investigate how the performance of a two-turbine system changes with the relative positions (as characterized by using relative distance and incoming flow angle) of the turbines in general. Thus, unlike Figures 4-17 and 4-18 in Section 4.3.1, which only present the relationship between the relative efficiency of the two-turbine system and relative distance of the two turbines, here we plan to describe the relationship between the performance (both relative efficiency and torque fluctuation coefficient) and the relative position of the two-turbine system. In order to generate this general result, we need to calculate the performance of the two-turbine system for each combination of relative distance and incoming flow angle. It is obvious that the finer the grid (the step of relative distance or incoming flow angle), the more accurate the results are and the more costly the computation is. One practical way to reduce the computational cost while maintain the accuracy of the results is to reduce the computational domain. For example, one can only calculate the scenarios where the incoming flow angle is positive; this requires that the results of the positive plane is able to represent the results of the entire computational domain. That is to say, one has to make sure that the results of the positive plane and negative plane of a two-turbine system are symmetric. Considering the wake symmetry of a co-rotating two-turbine system, there is no difference between the results of two clockwise and two counter-clockwise turbines as shown in Section 4.4.1. Thus, one can assume that the results of the co-rotating two-turbine system are essentially symmetric with respect to the middle plane where the incoming flow angle is zero. We develop a case (Case 1) for a counter-rotating two-turbine system to check the symmetry of the results produced by this system.  Case 1 A counter-rotating two-turbine system is studied and its basic specification is shown in Table 4-7.  - 117 -  Table 4-7  The basic specification for Case 1  Parameters  Values  Number of blades  3  Blades type  NACA 0015  Solidity  0.375  TSR  4.75  Relative rotating direction  Counter-rotating  ψ  -  Reynolds number  160,000  π 2  ~  π 2  The results in Figure 4-19 show the relationship between the relative efficiency and the relative position of the two turbines35,36. The results in Figure 4-20 show the relationship between the torque fluctuation coefficient and the relative position of the two turbines. It is noted that the impacts of the hydrodynamic interactions on the torque fluctuation coefficient are mainly constructive37 while the impacts of the hydrodynamic interactions on relative efficiency are partly destructive and partly constructive. For relative efficiency, it is noted that the hydrodynamic interactions are destructive when incoming flow angle is between −π  4  and π  4  and the relative distance is between 2.5 and 6, and the hydrodynamic  interactions are always destructive when the relative distance is between 2.5 and 4 no matter how much the incoming flow angle is. The maximum relative efficiency is 2.49 when the relative distance is 1.5 (time the turbine radius) and the incoming flow angle is π  2  , and the  minimum relative efficiency is 1.28 when the relative distance is 5 and the incoming flow angle is 0o. The maximum torque fluctuation coefficient is 78 dBS when the relative distance is 1.5 and the incoming flow angle is π  2  , and the minimum torque fluctuation coefficient is  35  One may notice that Figure 4-20 is a summary of the relative efficiency of the counter-rotating two-turbine system depicted in Figures 4-13 to 4-15. 36 The relative efficiencies and torque fluctuation coefficients in Section 4.3.2 (i.e., Figures 4-19 to 4-30) are depicted in both surface and contour diagram. 37 If the torque fluctuation coefficient of the two-turbine system is higher than that of the corresponding stand-alone turbine, one can claim that the hydrodynamic interaction of this system is constructive; otherwise, the hydrodynamic interaction is destructive. - 118 -  44 dBS when the relative distance is 6 and the incoming flow angle is - π  2  .  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 To systematically investigate the symmetry of the results, we quantify the relative deviation of the results in the negative planes from that in the positive plane with respect to the incoming flow angle. For example, when the relative distance is equal to 1.5, the relative efficiency is 2.35 when the incoming flow angle is equal to π is 2.47 when the incoming flow angle is equal to − π  4  4  , and the relative efficiency  ; thus, the relative deviation is 4.8%  when relative distance is equal to 1.5 and the incoming flow is equal to ± π . Table 4-8 4 shows the results (both the relative efficiency and the torque fluctuation coefficient) in the whole domain and it is noted the maximum relative deviation is less than 14%. Thus, one can assume the result of a counter-rotating two-turbine system are quasi symmetric with respect to the middle plane.  - 120 -  Table 4-8  The relative deviation of the value in the negative plane from the corresponding value in the positive plane in Case 1  Relative efficiency Torque fluctuation coefficient Dr  ψ  ±  π 4  ±  π 2  ±  π 4  ±  π 2  1.5  4.8%  10.4%  8.2%  3.9%  2  8.5%  13.5%  1.0%  3.3%  2.5  15.9%  11.9%  1.0%  3.1%  3  13.0%  4.3%  3.6%  3.2%  4  12.2%  2%  3.4%  3.1%  5  9.6%  2.5%  3.9%  12.7%  6  1.3%  1%  4.2%  11%  In short, the performance of a counter-rotating two-turbine system is quasi symmetric and the performance of a co-rotating two-turbine system is symmetric with respect to the middle plane. Thus, once we get the performance of the two-turbine system when 0 ≤ ψ ≤ π can extrapolate it to − π  2  2  , we  ≤ψ ≤ π . 2  Then, in testing the relationship between the performances of a two-turbine system and the relative positions of the two turbines under different conditions (e.g., different blade types and different TSRs), we only show the results when 0 ≤ ψ ≤ π  2  , but with finer steps (i.e.,  15o per step). Table 4-9 shows the specifications of five different cases, studied for relative efficiency and torque fluctuation coefficient. In these cases, the number of blades of each turbine remains 3. Two solidities, two relative rotation situations and three TSRs are studied at a Reynolds number of 160,000.  - 121 -  Table 4-9 Cases  Basic specifications of a set of two-turbine systems  Case 2  Case 3  Case 4  Case 5  Case 6  TSR  4.25  4.25  5.25  2.75  2.25  Rotating direction  Counter-rotating  Co-rotating  Counter-rotating  Co-rotating  Co-rotating  Blades type  NACA 0015  NACA 0015  NACA 0015  63(4)-021  63(4)-021  Number of blades  3  3  3  3  3  Solidity  0.375  0.375  0.375  0.425  0.425  Incoming flow angle  0°~90°  0°~90°  0°~90°  0°~90°  0°~90°  Reynolds number  160,000  160,000  160,000  160,000  160,000  Parameters  - 122 -  Case 2 In this case, we investigate the performance of the counter-rotating two-turbine system when the TSR (TSR=4.25) is lower than that in Case 1 (TSR=4.75). This TSR is close to the design TSR (the TSR at which a corresponding stand-alone turbine achieves its maximum efficiency). Figures 4-21 and 4-22 show how the relative efficiency and torque fluctuation coefficient of the two-turbine system change with the relative position of the two turbines, respectively. It is noted that the relative efficiency of the two-turbine system hardly exceeds 2. That is to say, the hydrodynamic interactions are always destructive in terms of generating power. In most scenarios (combinations of the relative distance between two turbines and the incoming flow angle), the torque fluctuation coefficient of this two-turbine system is greater than that of the stand-alone turbine. This means that in most scenarios, the hydrodynamic interactions between the two turbines is constructive for torque fluctuation. Comparing with the results in the positive plane of Case 1, it is noted that both relative efficiency and torque fluctuation coefficient are less than those in Case 1. The maximum relative efficiency is 1.98 when the relative distance is 3 and the incoming flow angle is π  2  , and the minimum  relative efficiency is 1.26 when the relative distance is 5 and the incoming flow angle is  π  12  . The maximum torque fluctuation coefficient is 69 dBS when the relative distance is  1.5 and the incoming flow angle is π  2  , and the minimum torque fluctuation coefficient 42  dBS when the relative distance is 3.5 and the incoming flow angle is π  - 123 -  6  .  Figure 4-21 The relative efficiency of the two-turbine system in Case 2  Figure 4-22 The torque fluctuation coefficient of the two-turbine system in Case 2 - 124 -  Case 3 In order to identify the impacts of turbine rotating direction on the performance of the two-turbine system, we change the relative rotating direction of the two turbines from counter-rotating in Case 2 to co-rotating in Case 3 while keeping other parameters unchanged. Figures 4-23 and 4-24 show how the relative efficiency and torque fluctuation coefficient of the co-rotating two-turbine system change with the relative position of the two turbines, respectively. We compare the performance of the co-rotating two-turbine system in this case with that of the counter-rotating two-turbine system in Case 2. It is noted that the relative efficiency of the co-rotating two-turbine system is greater than that of the counter-rotating two-turbine system except when the incoming flow angle is 90°. The torque fluctuation coefficients in this case are all around 60dBS . The maximum relative efficiency is 2.29 when the relative distance is 3.5 and the incoming flow angle is π  4  , and the minimum  relative efficiency is 1.61 when the relative distance is 2.5 and the incoming flow angle is 0. The maximum torque fluctuation coefficient is 66 dBS when relative distance is 3 and the incoming flow angle is π  3  , and the minimum torque fluctuation coefficient 54 dBS when  the relative distance is 3 and the incoming flow angle is π  - 125 -  6  .  Figure 4-23 The relative efficiency of the two-turbine system in Case 3  Figure 4-24 The torque fluctuation coefficient of the two-turbine system in Case 3 - 126 -  Case 4 The TSR in Case 1 and Case 2 are both less than 5 and they are close to the design TSR. In order to investigate the impact of TSR on the performance of the two-turbine system, we increase the TSR to 5.25 and see how the performance of the two-turbine system changes with the relative position of the two turbines in Case 4. Figures 4-25 and 4-26 show the relationship between the performance of a counter-rotating two-turbine system and the relative positions of the two turbines. It is noted that the results in Case 4 (relative efficiency and torque fluctuation coefficient) fluctuate much less than those in Case 1 and Case 2. Its relative efficiency is higher than that in Case 2 and lower than that in Case 1 and its torque fluctuation coefficient is similar to those in Case 1 and Case 2. The maximum relative efficiency is 2.51 when the relative distance is 4 and the incoming flow angle is π  2  , and the  minimum relative efficiency is 1.53 when the relative distance is 4 and the incoming flow angle is π  6  . The maximum torque fluctuation coefficient is 66 dBS when the relative  distance is 2 and the incoming flow angle is π  2  , and the minimum torque fluctuation  coefficient is 54 dBS when the relative distance is 5 and the incoming flow angle is 0.  - 127 -  Figure 4-25 The relative efficiency of the two-turbine system in Case 4  Figure 4-26 The torque fluctuation coefficient of the two-turbine system in Case 4 - 128 -  Summary of Cases 1 to 4 (NACA 0015 blade two-turbine system) We presented the performance of four different NACA 0015 blade two-turbine systems (Cases 1 to 4). Table 4-10 shows a comparison of the maximum and minimum values of the performance in each case with the corresponding relative position of the two turbines. The best performances can be obtained when the TSR is around 4.75 where both relative efficiency and torque fluctuation coefficients are high although these two values fluctuate more than those when TSR is around 5.25. The minimum performance of a canard-layout counter-rotating two-turbine system (i.e., when the incoming flow angle is π  2  ) can be  obtained when the relative distance is around 2. This relative distance for achieving the minimum performance of a canard-layout counter-rotating two-turbine system will increase as the incoming flow angle decreases. The impacts of the hydrodynamic interactions on the torque fluctuations coefficients are almost constructive in all conditions while those on the relative efficiencies are partly constructive and partly destructive and the portion of destructive impacts increase when TSR decreases. For a co-rotating system, the impacts of the hydrodynamic interactions on the relative efficiency are all destructive when the system is in canard or tandem layouts; if the two turbines are in diagonal layout, these impacts are partly constructive and partly destructive depending on the incoming flow angle. In these four cases, it can be observed that the maximum efficiency of a co-rotating two-turbine system is achieved when the incoming flow angle is about π  4  , while the  maximum efficiency of a counter-rotating two-turbine system is achieved when the incoming flow angle is about π  2  (i.e., the system is in a canard layout two-turbine system). It is  noted that the higher the TSR, the better the performance, although the performance when TSR is equal to 5.25 is almost the same as the performance when TSR is equal to 4.75. At the same TSR, we notice that 1) the performance of a co-rotating two-turbine system fluctuates less than that of a counter-rotating two-turbine system, 2) the maximum values of performance of a co-rotating system is larger than that of a counter-rotating system, unless the system is a canard-layout system which may be attributed to the fact that it is easy for a  - 129 -  canard layout counter-rotating system to destruct the vortices shedding which will reduce the fluctuation. The relative efficiency of the NACA 0015 blade two-turbine system is hardly over 2 and the torque fluctuation coefficients are close to those of a stand-alone turbine when the TSR is low (i.e., TSR≤4.25). In order to investigate if the performance of a two-turbine system at low TSR can be better than that of a stand-alone turbine, we develop two new cases by using NACA 63(4)-021 blade turbine: Case 5 and Case 6. Only co-rotating two-turbine system is studied here, due to lack of hydrodynamic characteristics data on NACA 63(4)-021 blades. The performance of the corresponding stand-alone turbine has been extensively discussed in Section 3.4.2.  - 130 -  Table 4-10 The performance of the NACA 0015 blade two-turbine systems in different cases Case  038  1  TSR and relative rotation  Maximum η2T  direction  ( Dr ,ψ )  4.75, co-rotating  4.75, counter-rotating  2  4.25, counter-rotating  3  4.25, co-rotating  4  5.25, counter-rotating  38  2.57 (3,  π 4  1.98 (3,  2  π 2  2.29 (3.5,  2.51 (4,  2  ( Dr ,ψ )  ( Dr ,ψ )  )  1.28 (5, 0)  π  1.26 (3,  π  1.61 (2.5, 0o)  )  )  1.53 (4,  12  π 6  76 dBS (3,  π 3  )  )  2  π  69 dBS (1.5,  66 dBS (3,  66 dBS (2,  2  π 3  π 2  We define the co-rotating two-turbine system in Section 4.3.1 as Case 0 for comparison purpose. - 131 -  )  π  78 dBS (1.5,  )  4  π  Maximum CTF  1.41 (2.5,0o)  )  π  2.49 (1.5,  Minimumη2T  )  )  Minimum CTF  CTF of a stand-alone  ( Dr ,ψ )  turbine  π  58 dBS (2.5,  π  )  44 dBS (6, -  )  42 dBS (3.5,  54 dBS (3,  6  )  38 dBS  )  2  π 6  π 6  54 dBS (5, 0o)  )  38 dBS  )  36 dBS  36 dBS  42 dBS  Case 5 Similar to Case 2, in this case, the two-turbine system works at the design TSR (TSR =2.75). Figures 4-27 and 4-28 show the relationship between the relative efficiency and the torque fluctuation coefficient of this co-rotating system and the relative positions of the two turbines, respectively. It is found that the maximum relative efficiency and the maximum torque fluctuation coefficient are obtained when the incoming flow angle is equal to 5π  12  and  their minimum values are obtained when incoming flow angle is around 0o (i.e., the system is a tandem layout system). The relative efficiencies are partly higher than 2 and partly lower than 2. The torque fluctuation coefficient is always higher than that of a corresponding stand-alone turbine. The maximum relative efficiency is 2.44 when the relative distance is 3.5 and the incoming flow angle is 5π  12  , and the minimum relative efficiency is 1.19 when  the relative distance is 2.5 and the incoming flow angle is 0. The maximum torque fluctuation coefficient is 80 dBS when the relative distance is 3.5 and the incoming flow angle is 5π  12  , and the minimum torque fluctuation coefficient is 52 dBS when the relative distance  is 2.5 and the incoming flow angle is 0.  - 132 -  Figure 4-27 The relative efficiency of the two-turbine system in Case 5  Figure 4-28 The torque fluctuation coefficient of the two-turbine system in Case 5 - 133 -  Case 6 To systematically investigate the performance of the NACA 63(4)-021 blade two-turbine systems, especially when they operate at low TSR, we develop a case by setting the TSR to be 2.25. Figures 4-29 and 4-30 show the relationship between the relative efficiency and the torque fluctuation coefficient of this co-rotating system and the relative positions of the two turbines, respectively. It is found that the relative efficiency is mostly less than 2 which indicates that the hydrodynamic interactions are mostly destructive for the efficiency. The minimum torque fluctuation coefficient is still a little bit higher than that of the stand-alone turbine, which indicates that the impacts of hydrodynamic interactions on torque fluctuation coefficient are all constructive. Unlike the relationship between the incoming flow and the torque fluctuation coefficient in other cases, the torque fluctuation coefficient decreases as the incoming flow angle increases in this case. The maximum relative efficiency is 2.07 when the relative distance is 3 and the incoming flow angle is π  3  , and the minimum relative efficiency is 1.41 when the relative distance is  2.5 and the incoming flow angle is 0. The maximum torque fluctuation coefficient is 66 dBS when relative distance is 5 and the incoming flow angle is 0, and the minimum torque  fluctuation coefficient is 46 dBS when the relative distance is 5 and the incoming flow angle is 5π  12  .  - 134 -  Figure 4-29 The relative efficiency of the two-turbine system in Case 6  Figure 4-30 The torque fluctuation coefficient of the two-turbine system in Case 6 - 135 -  Summary of Case 5 and Case 6 (NACA 63(4)-021 blade two-turbine system) In Case 5 and Case 6, we investigate the performance of NACA 63(4)-021 blade co-rotating two-turbine systems at two different TSRs (2.75 and 2.25). A comparison between the results of these two systems is given in Table 4-11. It is noted that the maximum relative efficiency and torque fluctuation coefficient can be obtained when incoming flow angle is around π to 5π  12  3  and the relative distance is around 3 to 3.5. As the NACA0015 blade system, we  find that the higher the TSR, the better the performance of a NACA 63(4)-021 blade system. Table 4-11 Comparison of the performance of the NACA 63(4)-021two-turbine system Maximum  Minimum  Maximum  Minimum  CTF of a  rotating  η 2T  η 2T  CTF  CTF  stand-alone  direction )  ( Dr ,ψ )  ( Dr ,ψ )  ( Dr ,ψ )  ( Dr ,ψ )  turbine  2.75  2.44  1.19  80 dBS  52 dBS  46 dBS  Case TSR(Relative  5  (co-rotating) 6  2.25 (co-rotating)  5π ) 12  (3.5, 2.07 (3,  (2.5, 0o) 1.41  π 3  o  )  (2.5, 0 )  (3.5,  5π ) 12  (2.5, 0o)  66 dBS (5,  46 dBS (5,  0o)  5π ) 12  42 dBS  4.3.3 Summary In section 4.3.2, we present the results of two-turbine systems with two different blades (i.e., NACA 0015 and NACA 63(4)-021). Table 4-10 summarizes the results of NACA 0015 blade two-turbine systems and Table 4-11 summarizes the NACA 63(4)-021 blade two-turbine systems. In this section, a comparison of the two-turbine systems with different blade types is provided with a focus on the co-rotating two-turbine system because we only investigate the NACA 63(4)-021 blade co-rotating system in this study. Table 4-12 compares the maximum and minimum values of the performances between NACA 0015 blade and NACA 63(4)-021 blade two-turbine systems. It is noted that the - 136 -  performance of the NACA 0015 blade co-rotating two-turbine systems may achieve its maximum value when the incoming flow angle is around π  4  , while the performance of the  NACA 63(4)-021 blade co-rotating systems achieves its maximum value when the incoming flow angle is around 5π  12  . The performance of the systems with both blade types may  achieve its minimum value when the incoming flow angle is 0o. Table 4-12 Comparison of the performance of two-turbine systems with two different blade types Parameters  NACA 0015  NACA 63(4)-021  Relative deviation  Maximum η2T  2.57 (4.75, 3,  (TSR, Dr ,ψ )  π 4  )39  2.44 (2.75, 3.5,  5π ) 12  6.1%  1.41 (4.75, 2.5, 0o)  1.19 (2.75, 2.5, 0o)  18.3%  Maximum CTF  78 dBS  80 dBS  -2.25%  (TSR, Dr ,ψ )  (4.75 , 1.5,  Minimum CTF  46 dBS  (TSR, Dr ,ψ )  (4.25, 1.5, 0o)  Minimum η2T (TSR, Dr ,ψ )  π 2  )40  (2.75, 3.5,  5π ) 12 0%  46 dBS (2.25, 5,  5π ) 12  It is noted that the maximum relative efficiency of the NACA 0015 blade system is 6.5% higher than that of the NACA 63(4)-021 blade system and the minimum relative efficiency of the former system is about 18.3% higher than that of the latter system. The maximum and minimum torque fluctuation coefficients of both systems are quite similar, although the results of the NACA 0015 blade system are from the counter-rotating system. However, as presented in Table 4-11, the torque fluctuation coefficient of a NACA 0015 counter-rotating system is higher than that of a co-rotating system. That is to say, if we just compare the 39 40  Both maximum and minimum relative efficiency results are extracted from Section 4.3.1. Both maximum and minimum torque fluctuation coefficient results are for counter-rotating systems. - 137 -  co-rotating system, the torque fluctuation coefficient of the NACA 0015 system is lower than that of the NACA 63(4)-021 blade system based on the cases investigated in this study. In general, one can say that the NACA 0015 blade two-turbine system is similar to the NACA 63(4)-021 blade two-turbine system in terms of the performance although the latter system works better at lower TSR. A turbine farm planner may focus on other characteristics of turbines such as blade cost and blade reliability, but more investigations on NACA 63(4)-021 blade system and other blades (e.g., NACA 0018 and NACA 0021) systems are desired so as to choose an appropriate type of blades for a new turbine.  4.4 Wake and Acoustic Impacts 4.4.1 Wake Like the stand-alone turbine model developed in Chapter 3, the two-turbine system model is also able to describe the wake of the system, which can provide necessary flow information for ocean scientists and environmental scientists as well as maritime engineers ( for navigation purposes). Here we develop an example case for predicting the wake of a two-turbine system. The basic specification of the two-turbine system is shown in Table 4-13. Unlike the wake of a stand-alone turbine which is separated along the two sides of the turbine (Figure 3- 9), the wake of this two-turbine system (Figure 4-31) comes close to the middle plane while the boundary becomes fuzzy, which may suggest that this system have less impact on local flow and the vibration of the system could be less.  - 138 -  Table 4-13 Basic specification of the two-turbine system for wake prediction Parameters  Values  Turbine blade  NACA0015  Blades number  3  Solidity  0.375  TSR  4.75  Rotating direction  Counter-rotating  Layout  Canard  Relative distance  3  Reynolds number  160,000  Figure 4-31 The wake of a canard layout two-turbine system  4.4.2 Acoustic Emission Similar to the analysis in Section 3.5, we can also use the two-turbine system model to estimate the noise intensity, and the numerical procedure for predicting the acoustic emission is the same as that explained in Section 3.5.4. Thus, we develop a case to study the noise intensity of a two-turbine system, the basic specification of which is given in Table 4-13 and - 139 -  the locations of three receivers are given in Table 4-14 and the locations of turbines and receivers are illustrated in Figure 4-32. Table 4-14 Location of three receivers Parameters  Values  Receiver 1  (4,0)  Receiver 2  (3,1)  Receiver 3  (3,-1)  Figure 4-32 Locations of three receivers Figures 4-33 to 4-35 show the noise intensity at the three receiver locations of the two-turbine system. The same as the acoustic emission of a stand-alone turbine (See Section 3.5), the main noise frequencies of this two-turbine system are all around 4Hz too.  - 140 -  Figure 4-33 Power spectrum of the noise intensity of the two-turbine system at Reciever 1  Figure 4-34 Power spectrum of the noise intensity of the two-turbine system at Reciever 2 - 141 -  Figure 4-35 Power spectrum of the noise intensity of the two-turbine system at Reciever 3  4.5 A Procedure for Simulating an N-Turbine System We develop a numerical model for predicting the performance (i.e., the efficiency and the torque fluctuation) of a two-turbine system in Section 4.2 and Section 4.3. In this section, as an extension, we develop a procedure for predicting the performance of an N-turbine system (i.e., a tidal current turbine farm) with a focus on the efficiency.  4.5.1 N-Turbine System Formulation The total efficiency of an N-turbine system can be written as follows, N  ηtotal = ∑ηi  (4.8)  i =1  Considering the hydrodynamic interactions between turbines, the efficiency of any individual - 142 -  turbine is affected by other turbines in the system. Thus, the efficiency of turbine i ( ηi ) in an N-turbine system can be written as follows,  ηi = f (η1, η2 ⋅ ⋅⋅, ηi −1, ηi +1, ⋅ ⋅ ⋅, ηN −1, ηN ) + ηS  (4.9)  ηi = ηr ,i ⋅ ηS , i = 1, 2 … N  (4.10)  where ηr ,i denotes the relative efficiency of turbine i in the N-turbine system, which is the ratio of the efficiency of turbine i ( ηi ) to the efficiency of a stand-alone turbine ( ηS ). To systematically investigate the total efficiency of the N-turbine system, we use the relative efficiency ( ηr ,i ) to represent the efficiency ( ηi ) of turbine i because ηS is independent of ηr ,i . Then, Eq.(4.7) and Eq.(4.8) can be rewritten as follows, N  ηr −total = ∑ηr ,i  (4.11)  ηr ,i = f (ηr ,1 , ηr ,2 ⋅⋅⋅, ηr ,i−1 , ηr ,i+1 ,⋅⋅⋅, ηr , N −1 , ηr , N ) + 1 i = 1, ⋅⋅⋅, N  (4.12)  i =1  To solve Eq. (4.11), one has to solve Eq.(4.12) for all turbines (totally N equations). As modeling a stand-alone turbine and the two-turbine system, we use DVM-UBC to model all N tidal current turbines. Obviously, we can follow the procedure for simulating the two-turbine system to simulate an N-turbine system. However, the related computational cost is expected to be extremely high due to the large number of turbines and the nonlinearity of the hydrodynamic interactions between turbines. Table 4-15 shows the typical computational time of simulating an N-turbine system by using a standard PC (CPU: Intel-Core 2 Duo 2.4 with 4Mb Cache and Memory 2Gb). Li and Calisal (2007 c) simplified an N-turbine system by using the vortex decay formulation, Eq.(3.15) to represent the effect of the hydrodynamic interactions on power output. To reduce the computational cost while approximating the physics of the interactions, we suggest using a linear theory to simplify the N-turbine system.  - 143 -  Table 4-15 Typical computation time of simulating an N-turbine system by using DVM-UBC Parameter  Stand-alone turbine N-turbine system41 N=2(two-turbine system) N=3  Computational time 1 to 2 minutes  15 to 20 minutes  1 to 1.5 hour  4.5.2 System Simplification The linear theory that we employ in this study is a classical theory in the marine hydrodynamics and aerodynamics community - the linear unsteady foil theory. It is a theory for simplifying the nonlinear characteristics in the airfoils and the hydrofoils. Linear theory had been used for such a purpose during World War I, which afterward was shown in two world-widely known papers (Von Karman and Sears 1938; Sears 1941). The original theory is subject to the restrictions that the flow is assumed to be two-dimensional and inviscid, the Kutta condition of the smooth flow at the trailing edge holds, the deviations from a uniform flow are small enough to allow the boundary conditions on a foil to be projected onto its mean position, and the wake is represented as a flat vortex sheet convecting with the free stream. Several decades later, a three-dimensional analysis for this problem was given by Streitlien (1994). In the modeling process, each turbine is modeled by using a three-blade system. The N-turbine system can be expressed as a 3N-blade system, which is a system with 3N blade bounded vortices by using DVM-UBC. Therefore, the turbine system and the flow can be expressed as the sum of uniform flow and vortices as depicted in Figure 4-36. By using perturbation theory and keeping only the 1st order term, we can treat such a system as a linear system. That is to say, the hydrodynamic impact from all upstream turbines on a certain downstream turbine is quasi equal to the sum of hydrodynamic impact from each of them on the downstream one. Therefore, Eq.(4.11) can be rewritten as follows42, 41  The computational time highly depends on the relative position of the system, i.e., the longer the relative distance is, the longer the computational time is. 42 Some terms on the right hand side can be removed since their values are all quasi equal to zero because they - 144 -  ηr ,i = f (ηr ,1 ) + f (ηr ,2 ) +⋅⋅⋅, + f (ηr ,i−1 ) + f (ηr ,i+1 ) +⋅⋅⋅, + f (ηr , N ) + 1 (4.13)  Figure 4-36 An illustration of a linearlized turbine system After being expressed as a linear system, the total relative efficiency of an N-turbine system can be extrapolated from the relative efficiencies of a set of two-turbine systems.  4.5.3 A Case Study – the Relative Efficiency of an N-turbine System When planning a tidal current turbine farm, one can choose an optimal turbine distribution based on the procedure for predicting the power output from the farm as discussed above, given the site conditions (e.g., topography and channel curvature). In this section, we use a hypothetical case to illustrate the procedure. In this case, a group of turbines are distributed in a rectangular channel as shown in Figure 4-37 and the basic farm specification for this case is given in Table 4-16. The objective here is to maximize the total relative efficiency of the system, ηr −total , which is given in Eq.(4.11); this objective is subject to the constraint of Eq.(4.13). We make the following assumptions for this case, z  All turbines are identical and are operating at the same TSR  z  All turbines are constructed at the same water level (i.e., we do not have to consider the vertical layout of the turbines)  z  Because the site is an ideal rectangular site (i.e., no obstacles in the site), the N-turbine system is uniformly distributed as an Nr -row by Nc -column array as depicted in Figure 4-37. Mathematically, the total number of the turbines can be obtained by multiplying the number of rows with the number of columns, given as follows,  are downstream turbines with respect to turbine i (see Figure 4-16). - 145 -  N = Nr × Nc  (4.14)  where Nr and Nc denote the number of the rows and number of the columns, respectively z  Since the system is distributed as an array, the distance between the adjacent columns is assumed to be optimized based on the results in Section 4.3. In this case, we set this distance as 6 (turbine radiuses) so as to avoid hydrodynamic interactions (there are totally 5 columns). Table 4-16 Basic specification of the case for estimating the efficiency of the farm Parameters  Values  Site length  400R  Site width  30R  Turbine blades  NACA0015  Number of blades  3  TSR  4.75  Solidity  0.375  - 146 -  Figure 4-37 A hypothetical rectangular turbine farm site Based on assumptions above, Eq.(4.11) can be simplified as to maximize the total relative efficiency in a column ηr −column as given in Eq.(4.15) by selecting an optimal number of rows ( Nr ) or distance between two adjacent rows ( X d ): Nr  ηr −column = ∑ηr ,i  (4.15)  Site Length Xd  (4.16)  i =1  where Nr =  Then, we use the procedure developed in this chapter to exhaustively search the tandem distance between two adjacent rows for achieving the maximal total relative efficiency of this - 147 -  farm. We search X d from 5 to 100. By plugging a certain X d in Eq.(4.16), we obtain the corresponding Nr . Then, with Nr and X d , by using the two-turbine interaction results in Section 4.3, we can solve Eq.(4.12) and finally use Eq. (4.15) to obtain the total relative efficiency in a column. The results are shown in Figure 4-38 and summarized in Table 4-17. The maximum total relative efficiency of the turbines in a column, 14.4, can be achieved when the tandem distance between two adjacent turbines is about 23 turbine radiuses and the number of turbines in a row is about 18. If we use the previous research methods that avoid the hydrodynamic interactions by setting the tandem distance between two adjacent turbines as more than 35 turbine radiuses(See Section 4.3.1), the maximum total relative efficiency of the turbines in a column is about 12, i.e., totally 12 turbines (Li et. al. 2007b). That is to say, the relative efficiency of the farm with an optimal distribution is 20% more than that of efficiency of the farm where the hydrodynamic interactions between two adjacent turbines in a column is avoided (i.e., no hydrodynamic interaction between two tandem turbines). 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 Circumstances  Maximum  X r corresponding to  Nr corresponding to  ηcolumn  the maximum ηcolumn  the maximum ηcolumn  14.4  23~24  18  Avoid interactions 12  35  12  Relative deviation  -  -  With interactions  20%  - 148 -  (a)  (b) 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 -  These results may not represent the theoretical maximum power output from a turbine farm in a large rectangular channel (especially if the shape of channel is a square) since we have made a few ideal assumptions as discussed above. More discussion on this is given in Section 4.6.1.  4.6 Discussion and Conclusions This chapter presents a numerical model for predicting the performance of a two-turbine system and then extrapolates the numerical model for predicting the power output of an N-turbine system. In this section, an extended discussion on the results is given, and the conclusions of this chapter are drawn.  4.6.1 Discussion In Sections 4.3 to 4.5, we conduct solid analysis and make a few judgments on the impacts of turbine distribution on the performance of a multiple turbine system. In this section, an extended discussion is given to explore possible reasons for some of the results and to make suggestions for future research.  Performance of a two-turbine system In order to simplify some terminologies, we define an “effective distance” for the hydrodynamic interactions between the two turbines. It is the distance at which the hydrodynamic interactions affect either the relative efficiency or the torque fluctuation coefficients and beyond this distance, the hydrodynamic interactions do not affect either the relative efficiency or the torque fluctuation coefficients. At the same incoming flow angle, the relative distance at which the torque fluctuation coefficient of the two-turbine system is equal to the torque fluctuation coefficient of a stand-alone turbine is much longer than the relative distance at which the relative efficiency is equal to 2. At the distance where the relative efficiency is equal to 2, there are still hydrodynamic interactions between two turbines and it can be observed in torque fluctuation. Table 4-18 summaries the effective distance for Case 4 in Section 4.3.2. It is noted that the effective distance for torque  - 150 -  fluctuation coefficient is almost two times the effective distance for the relative efficiency except when the incoming flow angle is equal to 0o (i.e., the system is a tandem layout counter-rotating two-turbine system). By using such a table, a turbine farm designer can easily choose the distribution of the turbine system. Table 4-18 Effective distance between the two turbines in Case 4, Section 4.3.2 at different incoming flow angles Incoming flow angle Effective distance  Effective distance  Relative deviation  for η2T  for CTF  0o  14  16  12.5%  π  8  20  60%  10  18  44.4%  8  16  50%  π  6  14  57.1%  3 5π 12  7  18  61.1%  6  22  59.1%  12  π  6  π  4  π  2 Among the three layouts (canard, tandem, and diagonal), canard layout has been widely used in studying marine propellers and researchers found that the performance of a counter-rotating two-propeller system is better than the performance of a corresponding co-rotating two-propeller system. This is because some wake vortices are destructed in the counter-rotating two-propeller system; so that power is increased and the vibration is reduced (Baquero et al. 1997). This is also observed in studying the performance of a canard-layout two-turbine system (See section 4.3.1). When one plans to design a two-turbine system, the key to find the optimal location of the downstream turbine is to study the wake of the upstream turbine, which is the grey zone in Figure 4-39. Two scenarios should be considered in this process: - 151 -  When the downstream turbine is not in the wake of the upstream turbine. Then, the  z  downstream turbine can easily achieve the maximum efficiency as if it is a stand-alone turbine since there is no hydrodynamic interaction between the two turbines. If the planners do not want a high turbine density, they can distribute turbines in this way When the downstream turbine is in the wake of the upstream turbine. Then, one can  z  analyze this scenario according to the relative rotation directions of the two turbines: A) the two turbines are counter-rotating, and B) the two turbines are co-rotating. The wakes of the two turbines in a co-rotating two-turbine system are quasi symmetric, while the wakes of the two turbines in a counter-rotating two-turbine system are not. One may find that it is relatively easier to find the optimal location for the downstream turbine in a co-rotating system (and thus to receive the constructive impacts from the upstream turbine) than in a counter-rotating system. The wake vortices of the upstream turbine is quasi symmetric with respect to the x axis when the incoming flow is along x axis. Although the wake is symmetric, the signs of the strength of the wake vortices is opposite with respect to the x axis (i.e., half positive and half negative). For example, in Figure 4-39, the downstream turbine is in the negative wake vortices region of the upstream turbine, and the part of this downstream turbine in this wake generates positive wake vortices; in this case, it will easily benefit from the upstream wake vortices.  . Figure 4-39 An illustration of the upstream turbine wake (not representative of the real structure of the wake) and the downstream turbine - 152 -  Additionally, it seems that the higher the TSR is, the better the performance of a NACA 0015 blade two-turbine system is (i.e., higher efficiency and higher torque fluctuation coefficient), compared with a similar NACA 63(4)-021 blade two-turbine system. The difference in their performances may have originated from the difference in their blade hydrodynamic characteristics.  Acoustic emission from the two-turbine system As far as the acoustics emission is concerned, it is noted that the receiver at the middle plane (Receiver 1) receives a signal with multiple peaks (See Figure 4-33) in the two-turbine system, which is different from the signal received by Receiver 1 in the stand-alone turbine case (See Figure 3-25). This implies that there is a harmonic acoustic emission at the middle plane of this two-turbine system. It is known that turbine’s torque fluctuation affects the vibration of the turbine which is related to the noise intensity. As shown in Section 4.3, one can find an optimal relative position for a two-turbine system to reduce the torque fluctuation of the system, which is equivalent to increasing the torque fluctuation coefficient. That is to say, an optimally-designed two-turbine system may reduce the noise intensity of the system as well. In order to evaluate that, a comparison of the noise intensity of the first peak and the second peak are presented in Table 4-19 and Table 4-20, respectively. In general, one can say that the canard layout counter-rotating two-turbine system can reduce the noise intensity except the noise intensity at location 1. Also, except the noise at location 1, the bandwidth of the noise of the two-turbine system is wider than that of the corresponding stand-alone turbine. The effect of the relative position of the two-turbine system on the noise intensity is significantly location-sensitive.  - 153 -  Table 4-19 Results of the first peak in the two cases Cases  Location 1  Location 2  Location 3  Power/Frequency Bandwidth(Hz)/ Power Power/Frequency Bandwidth(Hz) Power Power/Frequency Bandwidth(Hz) Power (dB/Hz)  Frequency  of (dB)  /Frequency of (dB)  (dB/Hz)  (dB/Hz)  the peak(Hz)  the peak(Hz)  the peak(Hz)  /Frequency of (dB)  Case 1  -17.5  0~220/4  -300  -14  0~24/4  -250  -7  0~18/4  -250  Case 2  -13  0~18/4  -475  -19  0~24/4  -350  -8  0~20/4  -500  Deviation  -26.4%  92%  58%  35.7%  Same  40%  14.2%  11%  100%  Table 4-20 Results of the second peak in the two cases Cases  Location 1  Location 2  Location 3  Power/Frequency Bandwidth(Hz) Power Power/Frequency Bandwidth(Hz) Power Power/Frequency Bandwidth(Hz) Power (dB/Hz)  /Frequency of (dB)  (dB/Hz)  /Frequency of (dB)  the peak(Hz)  (dB/Hz)  the peak(Hz)  /Frequency of (dB) the peak(Hz)  Case 1  --  --  --  -30  24~48/36  -80  -22  18~42/30  -180  Case 2  -32  36~68/46  -300  -35  24~74/45  -320  --  --  --  --  --  16.7%  108%  300%  --  --  --  Deviation --  - 154 -  An N-turbine system As shown in the example case in Section 4.4.3, in which the incoming flow direction is perpendicular to the row of the turbines, the distance between two adjacent rows, X d , is the most important variable. This is because X d is the very variable representing the vortices decay rate (see Eq.(3.15)), which affects the strength of the vortex, and it is identified as the main factor affecting the turbine efficiency. The results of this example case just show how to distribute turbines based on those assumptions in an ideal rectangular site and a linear N-turbine system. Practically, it may be hard to achieve such a result for, but not limited to the following reasons: z  The water flow may be disturbed by islands in the channel or channel bottom topographical conditions  z  Requirements on channels from local ship transportation regulation might affect turbine distribution  z  It may not be possible to distribute the farm as an ideal array.  Additionally, the uniform array may not be the best distribution. In this linear system, the maximum total relative efficiency from the farm can be obtained by searching all possible turbine distributions and individual turbine configurations, which requires advance searching algorithm and more computational efforts. The procedure for predicting the total relative efficiency of the N-turbine system is, however an approximation to the physics of the hydrodynamic interactions between turbines by linearizing the system since the relative distance between turbines are far; when the turbine density is relatively high (e.g. X d <3 or Yd <3), the results may not be precise enough to approximate the physics of the system due to the nonlinearity of the hydrodynamic interactions. Other procedures or numerical methods should be developed. These topics are all beyond the scope of this study.  - 155 -  4.6.2 Conclusions Based on the research in this chapter, we conclude as follows: z  The two-turbine system model can predict the total relative efficiency of a two-turbine system under different conditions (i.e., relative position, relation rotating direction and blade type), although arm and shaft effects are neglected. The results suggest that the efficiency of a two-turbine system could be 25% higher than two times the efficiency of a stand-alone turbine although arm and shaft effects may reduce the power output. This requires a careful selection of the parameter of the system such as relative distance, incoming flow angle and relative rotating direction  z  The two-turbine system model also suggests that an optimally-designed two-turbine system can significantly reduce the torque fluctuation during each revolution (i.e., increase the torque fluctuation coefficient). For example, the torque fluctuation coefficient of a stand-alone turbine is around 38 dBS , while the torque fluctuation coefficient of a two-turbine system can be up to 80 dBS  z  The noise emission and signal interference from a two-turbine system may be reduced by up to 35% by optimally designing the system, i.e., carefully selecting the relative distance, incoming flow angle and relative rotating direction of the system  z  This procedure for predicting the performance of an N-turbine system can be used to predict and optimize the power output from a tidal current turbine farm. The accuracy of the prediction may be affected by linearization procedure and assumptions in the two-turbine model  z  The challenge in the two-turbine system mainly lies in the gearing or belting system design.  In general, if one needs to put two turbines close to each other, i.e., when the relative distance of two turbines is less than 4, the counter-rotating two-turbine system should be used because a counter-rotation two-turbine system can generate more power and have higher torque fluctuation coefficient than a co-rotation two-turbine system. On the other hand, if one need to put two turbines far from each other, i.e., when the relative distance of two turbines is more than 4, the co-rotating two turbine system should be used. One example is given in - 156 -  Chapter 6.  4.6.3 Future Work The two-turbine model and the procedure for predicting the efficiency of an N-turbine system are based on the stand-alone turbine model developed in Chapter 3; Therefore, the limitation of the stand-alone turbine model also applies here. The obvious limitation in this method is that it neglects turbulence effect, free surface effects and bottom effects as well as the arm and shaft effects. These effects may affect the performance of a turbine system more considering the effects of the hydrodynamic interactions, and the relationship between these effects and the turbine performance could be highly nonlinear. These effects need to be numerically investigated. Some particular important factors for a two-turbine system are summarized here for further investigation for design purpose in the future. They are, the relative distance (two turbine position configuration), relative rotational direction, incoming flow angle, and phase angle difference (although it can be offset by relative distance). Although the basics of the two-turbine system model, i.e., the stand-alone turbine model, has been validated by experimental tests, the experimental tests for the two-turbine system need to be carried out to validate and calibrate the two-turbine system model. Additionally, future nonlinear analysis on the N-turbine system is expected.  - 157 -  Chapter 5 Tidal Current Turbine Farm System Modeling In Chapter 3 and Chapter 4, we analyze the hydrodynamic interactions between turbines in a tidal current turbine farm and develop a procedure to predict power output from such a farm. It is shown that the total power output from a tidal current turbine farm is significantly affected by turbine distribution (i.e., the layout of the turbines). In this chapter, we develop a model, which integrates the procedure for predicting power output from a tidal current turbine farm (and thus energy output) with an approach for estimating the cost that may be incurred in producing the energy, to predict the energy cost from a tidal current turbine farm. This cost is important for governments to make subsidy decisions and investors to make investment decisions related to the construction of tidal current turbine farms. We minimize the energy cost by minimizing the total cost and maximizing the total power output under constraints related to the local conditions (e.g., geological and labor information) and the turbine specifications using a scenarios-based analysis.  5.1 Introduction The construction of a commercial tidal current turbine farm is expected to require substantial investment. Expected energy cost is one of the important factors for the justification of constructing such a farm. Estimating energy cost requires the information regarding the expected energy output from a tidal current turbine farm and the cost (both capital cost and O&M cost) that might be incurred. However, it is not practical to conduct experiments regarding unit cost estimation by building and operating a fairly large scale tidal current turbine farm to obtain information regarding energy output and total cost. Normally, designers turn to modeling the farm system to obtain this information. A few models have been proposed for estimating potential energy output from (Myers and Bahaj 2005; Fraenkel 2002; Bedard 2006) and energy cost of (Fraenkel 2002; Bedard 2006) a tidal current turbine farm. The former papers, which proposed models for estimating energy - 158 -  output from a tidal current turbine farm, all use the efficiency of a stand-alone turbine to represent the efficiency of individual turbines in the farm and neglect the hydrodynamic interactions between turbines. The hydrodynamic interactions between turbines have significant impact on turbine efficiency and thus power output from a tidal current turbine farm, as shown in Chapter 4. The latter papers, which estimated the energy cost, all assume that the 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 (the 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), which is considered as one of the major barriers to the industrialization of tidal current turbine farms (Bregman et al. 1995; Eaton and Harmony 2003). In this chapter, we develop a model to estimate energy cost of a tidal current turbine farm. The energy cost is defined as the ratio of the total cost to the total energy output over the lifetime of such a farm. Mathematically, the energy cost can be estimated by using Eq. (5.1).  ∑∑levco ∑∑ Energy  i, j  cenergy  =  i  j  (5.1)  i, j  i  j  where levcoi , j and Energyi , j denote the levelized cost43 and energy output of turbine i in the year j , respectively. We conduct a cost-effectiveness analysis to calculate the energy cost, which is widely used in military operations, and health care and energy system analysis. It is a typical economic analytical approach that compares the relative expenditure (costs) and outcomes (effects) of two or more courses of action. Specifically, we minimize the energy cost by strategically planning the turbine distribution in a farm with full consideration of the hydrodynamic interactions between turbines (thus to maximize the energy output from the farm) and strategically selecting O&M plans (thus to minimize the total cost).  43  Present value of the total cost of building and operating a power plant over its economic life time. - 159 -  In the next section, we state the assumptions that we have made in formulating the model for estimating energy cost. In Section 5.3, we then introduce the integrated model (including an integrating module, a hydrodynamic module and an O&M module) that we use to estimate the energy cost and discuss the integrating module, which is the engine of the integrated model. In Section 5.4, we explain the hydrodynamic module, which is used to calculate energy output for a given set of turbines and turbine configuration. In Section 5.5, we present the O&M module with different O&M strategies. We then apply the integrated model to estimate the energy cost for an example turbine farm, as a case study.  5.2 Assumptions In formulating the integrated model for estimating the energy cost, we make the following assumptions in addition to those assumptions for hydrodynamic analysis given in Chapters 3 and 4: z  Fees such as licensing and permission are ignored at the construction stage  z  Energy losses that are increased from year to year due to equipment degradation are offset by energy gains from improved management strategies and monitoring technologies  z  No electricity transmission cable is shared by two or more turbines. That is to say, one turbine is assigned to one cable  z  Turbines, during their life time, will not be replaced with turbines having higher efficiency, which may be available in the market due to technological developments, which means that turbine efficiency will not increase over time  z  The traveling distance of maintenance vessels and helicopters from turbine to turbine for routine maintenance is neglected  z  Routine maintenance frequency and its effect on emergency maintenance are assumed constant over time  z  The labor and maintenance materials costs are functions of the farm information (e.g., the size of the farm, and offshore distance of the farm). For example, the larger the farm size, the cheaper the unit cost of the maintenance material  z  The time needed to acquire replacement parts is assumed constant. That is, logistics will - 160 -  not be affected by the weather and types of the failures z  The performance of the maintenance vessels, helicopters and labor force are assumed perfect so that there are no additional costs due to vessel failures and labor behavior.  5.3 Description of the Integrated Tidal Current Turbine Farm System Model The integrated tidal current turbine farm system model consists of three sub-modules, which are the hydrodynamic module, the O&M module, and the integrating module, as shown in Figure 5-1. The main inputs of these modules are given in Table 5-1. In this model, we use scenario-based analysis to identify the minimum energy cost. The hydrodynamic module calculates hydrodynamic power output ( Pout , see Eq.(5.2)) for a given combination of turbine configuration, total number of turbines, and turbine distribution in a farm, which is called a scenario. Given a set of different scenarios, the hydrodynamic module will calculate the power output for each scenario, and identify the one that achieves the module objective (e.g., provides the maximum power output). The inputs for this module include turbine geometry (e.g., blade span and chord length), farm information (e.g., turbine distribution), local geological conditions (e.g., current velocity) and turbine material reliability. The O&M module calculates the O&M cost for different scenarios, which are combinations of weather condition, farm specifications, labor cost, other facility cost (e.g., transportation vehicle cost and maintenance equipment cost) and O&M strategies, and identifies the one that achieves the module objective (e.g., provides the minimum O&M cost). The inputs for the O&M module include most of the inputs of the hydrodynamic module and local labor and weather conditions and material cost information. The integrating module is used firstly to convert the total hydrodynamic power to total energy output, then to calculate the total cost by adding the capital cost and the O&M cost together, and finally to estimate an energy cost and identify the scenario that achieves the model objective (e.g., provides the minimum energy cost).  - 161 -  Figure 5-1  Tidal turbine farm system model (the same as Figure 1-2) Table 5-1  Inputs for each of the sub-modules  Module  Integrating  Hydrodynamic O&M  Input  module  module  module  Geometry of individual turbines  –  Yes  Yes  Turbine material cost  Yes  –  Yes  Farm information, e.g., total number  Yes  Yes  Yes  –  –  Yes  of turbines, turbine reliability, farm offshore distance, turbine distribution and bathymetry Local conditions, e.g., weather and labor cost  - 162 -  5.3.1 Energy Output The energy output here refers to the amount of energy in the load center which is ready to be delivered to the existing electricity grid. Total energy output can be calculated by using Eq.(5.2). T  Energy = ∫ Pout dt  (5.2)  0  where Pout denotes the final electrical power output from the tidal current turbine farm, given the total number of turbines and turbine distribution, which is ready to be delivered to the electricity grid, t and T denote the time annual increment and the lifetime of the tidal current turbine farm, respectively. The power generation and transmission process can be modeled using three systems, including 1) the hydrodynamic system, which generates hydrodynamic power from tidal current using turbines, 2) the mechanical system, which converts the hydrodynamic power from the hydrodynamic system to mechanical power using a generator (e.g., a gearbox and flywheel), and 3) the electrical system, which converts the mechanical power to electrical power and then transmits the electrical power to the local load center.  Electrical system For a given tidal current turbine farm, the power output Pout from the electrical system can be expressed as follows, Pout = f ( Pe ) ≈ ft Pe  (5.3)  Pe = f ( Pm ) ≈ f e Pm  (5.4)  where ft denotes the electrical power transmission efficiency of the tidal current turbine farm, Pe denotes the total electrical power of the tidal current turbine farm, f e denotes the - 163 -  electrical power conversion efficiency of the tidal current turbine farm, and Pm denotes the total mechanical power of the tidal current turbine farm. By substituting Pe in Eq.(5.3) with Eq.(5.4), we can summarize the final power output as follows, Pout ≈ f t f e Pm  (5.5)  Technically, the turbine configuration and turbine distribution in a farm have little influence on the conversion efficiency, f e , and the transmission efficiency, ft . Thus, we generally focus on mechanical power output, Pm , instead of electrical power output , Pout .  Mechanical system The mechanical power is converted from hydrodynamic power and it can be written as follows, Pm = f m Phydro  (5.6)  where f m denotes the mechanical power conversion efficiency and Phydro denotes the hydrodynamic power from a tidal current turbine farm.  Hydrodynamic system Hydrodynamic power ( Phydro ) can be estimated by using Eq.(5.7). N  Phydro = ∑ηi Pideal − s  (5.7)  i =1  where ηi denotes the hydrodynamic power efficiency of turbine i , N denotes the total number of turbines in the farm, and Pideal − s denotes the ideal hydrodynamic power that a stand-alone turbine can generate, given as follows (White 2002), - 164 -  Pideal − s =  1 ρ AU ∞3 2  (5.8)  where ρ denotes the density of sea water, A denotes the turbine frontal area as explained in Chapter 2, and U ∞ denotes the free stream incoming flow velocity.  Total energy output Based on the relationship among the set of variables in Eqs.(5.5) to (5.8), we can rewrite Eq.(5.3) as follows, Pout ≈ ft f e f c  N 1 ρ AU ∞3 ∑ηi 2 i =1  (5.9)  Similar to the electrical power conversion efficiency, the mechanical power conversion efficiency is independent of the turbine distribution in a farm and the turbine design too. In other words, the hydrodynamic interactions among turbines have no impact on the mechanical power conversion efficiency. Given that the velocity of tidal current and the cycle of tides are highly predictable44, the total energy output from a tidal current turbine farm can be estimated using Eq.(5.10). T  Energy = ∫ Pout dt = Eideal − Edown  (5.10)  0  where Edown denotes the downtime energy loss during the maintenance when the turbines have to be shutdown, which is affected by maintenance strategies and weather only, and Eideal denotes the energy output from a tidal current turbine farm when no maintenance is needed (i.e., there is no energy loss due to downtime). The ideal energy output can be written as follows, Eideal = PoutT 44  (5.11)  Interested readers are referred to Pond (1983). - 165 -  where Pout denotes average final power output over time. To simplify the relationship between variables in Eq.(5.10), we define a tidal coefficient ( ftidal ) as the ratio of the average final power output to the final electrical power output as follows, ftidal =  Pout Pout  (5.12)  Additionally, we define the downtime coefficient ( f down ) as the ratio of the downtime energy loss to the ideal energy output as follows, f down =  Edown Eideal  (5.13)  By substituting Eqs. (5.11) to (5.13) in Eq.(5.10), the equation for estimating the total energy output can be written as follows, Energy = Pout f tidal (1 − f down )T  (5.14)  Then, by substituting Eqs.(5.6) to (5.9) into Eq.(5.14), the total energy output can be rewritten as follows, N  Energy ≈ f e ft f c f tidal (1 − f down )T ∑ηi Pideal − s  (5.15)  i =1  As mentioned earlier in this chapter, the electrical efficiency coefficient, f e , the conversion coefficient,  f m , the transmission efficiency coefficient, ft , and the downtime  coefficient, f down , are not affected by the turbine distribution and turbine configuration, and the tidal coefficient, ftidal , can be treated as a constant in certain periods (detailed reasons are given in Section 3.3.7). Thus, we define a new coefficient, f , to account for all the non-hydrodynamic energy losses as follows, - 166 -  f = f e ft f m f tidal (1 − f down )  (5.16)  Finally, by substituting Eq.(5.16) into Eq.(5.15), the total energy output can be rewritten as follows, N  Energy = f TPideal − s ∑ηi  (5.17)  i =1  As f and Pideal − s are all deterministic and fixed values, only the total hydrodynamic efficiency and the turbine farm life expectancy are variables. Thus, when we use a scenario-based analysis to maximize the total energy output, we shall focus on the total hydrodynamic efficiency and turbine farm life expectancy. Specifically, the total hydrodynamic efficiency is affected by the turbine distribution which is discussed in Section 5.4 in detail. The turbine farm life expectancy is affected by the turbine component material, weather and O&M strategies, which all are discussed in Section 5.5 in detail.  5.3.2 Total Cost The total cost is the sum of the capital cost and the O&M cost, so that the levelized cost in Eq.(5.1) can be calculated by summing these two components as follows, levcoi , j =capi , j + O & M i , j  (5.18)  where O & M i , j and capi , j denote the levelized O&M cost and the levelized capital cost of turbine i in the year j , respectively. The capital cost here refers to the cost that is incurred in purchasing turbines and in constructing the turbine farm. In this study, the life time capital cost of the tidal current turbine farm is estimated by multiplying the unit capital cost (i.e., the sum of the cost of one turbine, the cost of the cable for electricity transmission and the cost of the turbine and cable installation and decommissioning) with the total turbine number in a farm as follows,  - 167 -  caplife = ∑∑ capi , j = N ⋅ capunit  (5.19)  capunit = capturbine −u + Doffshore ⋅ cableunit  (5.20)  i  j  where capunit denotes the unit capital cost, capturbine−u denotes the capital cost of one turbine including the cost of manufacturing, installing, and decommissioning a turbine, Doffshore denotes the offshore distance from the farm to the local load center, and cableunit  denotes the unit cable cost which is the sum of the per meter cost of manufacturing, installing, and decommissioning a cable. The O&M cost includes all the costs except those incurred in purchasing turbines and constructing the turbine farm. The capital cost is a deterministic and fixed value, while the O&M cost of an offshore structure is uncertain and variable because of the unexpected factors such as weather and sea states which will lead to uncertain O&M needs. For example, Hurricane Katrina led to unexpected O&M needs, which are responsible for more than a billion US dollars in losses incurred in the offshore industry (Buckley 2005). Surprisingly, little attention has been paid to the unexpected factors in offshore turbine farm industry. Considering the similarity between offshore wind turbine farms and on-land wind turbine farms, the system modeling of offshore wind turbine farms focused on modeling the entire system (e.g., planning, manufacturing, and integrating electricity) following the experience from on-land wind turbine farm research (South 1978; Wake et al. 1979; Daniel 1981; Harrison et al. 2001; Gipe 2004; Heier 2006). Failures of what have happened in offshore wind farms (Sasse 2006) were mainly caused by the lack of knowledge related to maintenance vessel transportation and weather offshore. Then, in order to accurately predict energy cost, research on modeling offshore wind turbine farms shifted from modeling the cost of the entire system towards giving special attention to different O&M strategies (Rademaker 2003; van Bussel and Bierbooms 2003 a; van Bussel and Bierbooms 2003 b). Given the documented experience of modeling the cost of offshore wind farms, when estimating total cost of a tidal current turbine farm here, we give special attention to the systematical analysis of its O&M cost, and this is discussed in Section 5.5 in detail.  - 168 -  5.4 Hydrodynamic Module The hydrodynamic module is designed to estimate the hydrodynamic power output from a tidal current turbine farm; this module is based on the analytical and numerical investigations of turbine-farm hydrodynamics discussed in Chapters 3 and 4. In Chapter 3, we formulate a new discrete vortex method to describe the vortex shedding unsteady flow, i.e., DVM-UBC; by using DVM-UBC, we then present a model to estimate turbine performance (hydrodynamic power and torque fluctuation) of a stand-alone turbine according to the turbine configuration (turbine height, turbine radius, and blade geometry). In Chapter 4, we present a model to estimate the performance of an N-turbine system with an emphasis on a two-turbine system according to the turbine configuration and turbine distribution, and this model is based on the stand-alone turbine model developed in Chapter 3. The basic structure of the hydrodynamic module is depicted in Figure 5- 2. The inputs of the hydrodynamic module are turbine configuration and turbine distribution. The output of this module is the hydrodynamic power of a tidal current turbine farm based on Eq.(4.8). Given the specification of turbine configuration, the power output of each turbine in the farm as a stand-alone turbine can be predicted; with the turbine distribution information, the effect of hydrodynamic interactions between turbines on the power output of each turbine can be calculated and thus the power output of the farm can be obtained. The detailed calculation procedure is presented in Section 4.5. By choosing appropriate inputs, we can maximize the power output. Specifically, this objective is subjective to the constraints of the hydrodynamic interactions between turbines, which is the relationship between individual turbine efficiency in a farm, given as follows45 (see Section 4.5 for more details), ηi = f (η1 ) + f (η2 ) + ⋅ ⋅ ⋅, + f (ηi−1 ) + f (ηi +1 ) + ⋅ ⋅ ⋅, + f (ηN ) + ηS (5.21)  Section 4.5.3 presents the important factors affecting the total hydrodynamic efficiency (total hydrodynamic power output) of a tidal current turbine farm. They are turbine configuration (e.g., blade type and blade geometry), total turbine number and turbine distribution.  45  The same as Eq.(4.12). - 169 -  Figure 5-2  The structure of the hydrodynamic module  5.5 Operation and Maintenance (O&M) Module The O&M module calculates the O&M cost for a given combination of farm information, local information and maintenance strategy (a scenario). In this section, we present the major structure and formulation of the O&M module, details of which are given in Appendix B. The main structure of the O&M module is shown on the left hand side of Figure 5-346 which includes an emergency maintenance cost sub-module, a routine maintenance cost sub-module, a service sub-module and a farm attribute sub-module, and these four sub-modules are presented in Section 5.5.1 to Section 5.5.3 in detail. The inputs of the O&M module are services and farm attributes information from the service sub-module and the farm attribute sub-module, respectively. The services and the farm attribute sub-modules provide inputs for the emergency and routine maintenance sub-modules. The output of the O&M module (i.e., the O&M cost) can be obtained by summing the routine maintenance cost and the emergency maintenance cost which are calculated in the routine and emergency maintenance sub-modules. That is to say, the levelized O&M cost can be written as follows, 46  The structure showed on the right hand side is the emergency sub-module which will be discussed in Section 5.5.1 in detail. - 170 -  O & M i , j = ECi , j + RCi , j  (5.22)  where ECi , j and RCi , j denote the levelized emergency maintenance cost and routine maintenance cost of turbine i in the year j, respectively.  Figure 5-3  The structure of the O&M module (Left) and an expansion of the emergency maintenance sub-module (Right)  5.5.1 Emergency Maintenance Cost The emergency maintenance cost is the sum of the material, equipment, transportation and labor cost for the emergency maintenance, Eq.(5.23), and these costs are related in a way as shown on the right hand side of Figure 5-3. ECi , j = ELCi , j + ETCi , j + EECi , j + EMCi , j  (5.23)  where ELCi , j , ETCi , j , EECi , j and EMCi , j denote the levelized emergency labor, transportation, equipment and material cost incurred for the emergency maintenance of - 171 -  turbine i in the year j , respectively. The structure of the emergency maintenance sub-module is the most complicated one among all the four sub-modules because the preparedness of the emergency situation and the optimization of the emergency operation are complicated. Emergency maintenance entails fixing an existing or pending failure of one or more devices. The major components affecting the emergency maintenance cost are the failure rates, the replacement cost for the broken components, and the turbine downtime. The type of equipments needed for the emergency maintenance affects both the equipment cost and the transportation costs. Larger equipment such as cranes might be required for some severe emergency maintenance depending on the level of failure severity47, and these equipments require special vessels. Labor cost as well as the type and cost of the materials used in maintenance also depend on the level of the failure severity. The labor, equipment and transportation costs are proportional to the required maintenance time, and some emergency maintenance requires the turbine to be shut down for a relatively long time depending on the failure situation, accessibility of the turbine and availability of the materials. More details are given in Appendix B.  5.5.2 Routine Maintenance Cost Figure 5-4 shows the structure of the routine maintenance sub-module and Eq.(5.24) shows the mathematical expression for estimating the routine maintenance cost. Similar to the emergency maintenance cost, the routine maintenance cost also consists of material, equipment, transportation, and labor costs. RCi , j = RLCi , j + RTCi , j + RECi , j + RMCi , j  where RLCi , j , RTCi , j , RECi , j  (5.24)  and RMCi , j denotes the levelized routine local,  transportation, equipment and labor cost of turbine i in the year j , respectively. More details are given in Appendix B. Routine maintenance is conducted once or twice a year (BBV 2001). Routine maintenance 47  In this study, there are three level of severities defined: minimal, mid-level and severe. - 172 -  includes both maintenance and monitoring. Some tasks can only be performed on site such as device vibration tests and seal checks, while some other tasks, such as connection and stability test, can be self-checked where these results are sent to the control center remotely via data cable and can be accessed through internet (Gore 2005). Factors affecting the costs of routine maintenance include the number of turbines, the magnitude of labor-hours per turbine that are needed to perform a maintenance operation (i.e., routine inspection time), labor skill, transport cost, and the cost of diagnostic equipment. The duration of the routine maintenance each year will increase as the life of the turbine farm increases because older turbines need more attention, and the increment of the duration is determined by the routine inspection time increase rate (see Figure 5-4). Onsite routine maintenance only needs vessels (mainly tug), and the vessel operation cost is determined by the vessel speed and the farm offshore distance. In addition, a few routine maintenance procedures require the turbine to be temporarily taken off-line although most procedures are conducted when the turbine is shut down because of low flow velocity. The mathematical relationships between these factors are given in Appendix B in detail.  Figure 5-4  The routine maintenance sub-module - 173 -  5.5.3 Service Sub-module and Farm Attribute Sub-module Service sub-module and farm attribute sub-module are relatively simpler than the two maintenance sub-modules. There are no lower level sub-sub modules in these two sub-modules. The inputs of the service sub-module are all the information related to service such as labor performance, unit material cost, material reliability, and facility availability, and the inputs of the farm attribute sub-module are the information related to the farm such as turbine geometry, weather, geological condition and flow velocity. The outputs of these two sub-modules are the inputs of those two maintenance sub-modules. The basic function of these two modules is to transform their inputs into outputs by using different transformation techniques. For example, the weather data and material reliability data are transformed from discrete format (the input) to continued probability functions (the output)48. Particularly, compared with the inputs of previous energy cost prediction models, the inputs here are treated in greater detail via the service sub-module and the farm attribute sub-module. For example, in previous models, researchers discuss the turbine fatigue by considering the turbine as one component; in this study, the turbine is decomposed into several components (e.g., brake, blade, shaft and belt) when the fatigue is concerned in the farm attribute sub-module.  5.6 Computational Procedure of the Farm System Model In Sections 5.2 to 5.5, we presented the assumptions, the structure and the main equations of the tidal current turbine farm system model for estimating its energy cost. We name the model Tidal Energy-UBC (TE-UBC). In this section, we show the computational procedure of TE-UBC. Figure 5-5 shows the flow chart of TE-UBC for estimating energy cost. This program (model) starts with the inputs given in Table 5-1. As a scenario-based analysis model, the program will check if the scenario is a “reasonable” scenario according to certain criteria (e.g., whether the number of turbines or current velocity is within certain range). In this study, we 48  One can also just keep these data as in the discrete data format. To conduct sensitivity analysis, we transfer them into continuous functions. - 174 -  suggest that as long as the maximum current velocity is between 2m/s and 6m/s, the scenario is a reasonable scenario due to the economic concerns49. If the scenario is reasonable, the program will start the simulation. By using the hydrodynamic module, the power output from the turbine farm is obtained. By using the O&M module (mainly Eqs.(5.22) to (5.24)), the O&M cost is obtained. Then, in the integrating module, with the power output calculated in the hydrodynamic module, the energy output can be calculated by using the relationship between power output and the energy output, Eq.(5.17). On the other hand, the capital cost of the farm is calculated by using Eq. (5.19) and Eq.(5.20). Then, by adding the capital cost and the O&M cost together, Eq.(5.18), the total cost (i.e., the sum of the levelized costs) can be obtained. The energy cost can be calculated as the ratio of the total cost to the total energy output, Eq.(5.1). The program will end if there are no more scenarios to be simulated. Otherwise, the program will check whether the new scenario is reasonable again and repeat the procedure. This forms a loop and the program continually saves the lowest cost scenario. Finally, the program will identify the scenario which achieves the objective (e.g., minimum energy cost).  49  When the current velocity is too low(e.g., lower than 2m/s), the energy output is low while the operation cost is not low; when the current velocity is too high(e.g., higher than 6m/s), the turbine may have a significant reliability issue. - 175 -  Figure 5-5  The flowchart of TE-UBC - 176 -  5.7 Optimal Design of a Tidal Current Turbine Farm in the Quatsino Narrow Using data for potential turbine farms near Vancouver, BC, Canada, we demonstrate the use of the TE-UBC for finding the minimum energy cost with a scenario-based analysis. Two cases are examined: in Case 1, turbines in the farm are distributed far away from each other (i.e., there are no hydrodynamic interactions between turbines), and in Case 2, turbines in the farm are distributed with constructive hydrodynamic interactions (See Figure 4-37). Basic turbine specifications are assumed in Table 5-2, and this turbine is well discussed in Sections 3.5 and 4.5. The farm site is in Quatsino Narrows, Vancouver, BC, Canada, where the average current velocity is 2m/s and the detailed hydrographic data are obtained from Canadian Hydrographic Service (CHS). Cost and material information (manufacturing, labor, and maintenance material costs and device fatigue information) are adapted from BBV (2001) and Rademakers (2003). Weather information, e.g., foggy and wind, is obtained from Environmental Canada. Major farm specifications from the references above are given in Table 5-3. More detailed farm specifications are given when they are used in Appendix B. Table 5-2  Basic turbine specifications  Parameter  Value  Turbine blades  NACA0015  Turbine Blade number 3 Turbine height (m)  12.5  Turbine radius (m)  5  Solidity  0.375  - 177 -  Table 5-3  Farm specifications  Parameter  Value  Labor cost (Technician salary $/hr)  80  Labor (Technician) workload(Hr/day)  10  Vessel speed (kn)  12  Foggy time offshore (day/year)50  5  Extreme wind and wave condition offshore (day/year)51 8 Offshore distance(km)  0.5  Current velocity (m/s)  2  Capital cost including installation ($/turbine)  150,000  Routine maintenance frequency (time/year)  2  Given the basic farm specifications for both cases, several scenarios are investigated for different farm sizes (number of turbines) and different farm lifetimes. In both cases, three different farm sizes: a small farm (10 turbines), a medium-sized farm (30 turbines), and a large farm (100 turbines), and six different lifetimes, which are 5, 8, 11, 14, 17 and 20 years52, are analyzed. The expected lifetime of a turbine is extrapolated from that of offshore platforms, which typically extends to 25 years or so (Buckley 2005). Designers of tidal current turbines in the UK and Canada have projected a 30-year lifetime for their designs (Pearson 2005). Although the actual operational lifetime of tidal current turbines will not be available until more experiences is gained from sea tests on full-scale devices, some causes for optimism lie in the fact that pre-commercial testing of near-shore turbines has resulted in turbines operating without failure over a five-year period (Gulli 2005), despite the lack of a systematic maintenance program. The data of Figure 5-6 (a) and Figure 5-6 (b) show the energy costs for Case 1 and Case 2, respectively. It is noted that the energy cost reduces significantly when the farm lifetime increases. However, the lifetime of the device will significantly affect the O&M cost, because 50  Technician working time and vessel travel time will be doubled during the foggy day. Extreme wind and wave condition offshore contribute the downtime delay in the cost calculation, which is analyzed in Appendix B in detail. 52 As a new offshore system, the life time is hard to be predicted given the offshore conditions such as extreme wave. 51  - 178 -  older equipment requires more attention. For a given lifetime, the larger the farm size is, the lower the energy cost will be, which is mainly due to the difference in O&M cost for farms of different sizes, but not due to the difference in capital cost (i.e., no economies of scale)53. Additionally, for two farms of the same size and with the same lifetime and maintenance strategies, the one of Case 2 is more competitive than the one of Case 1 economically since the energy cost of Case 2 is lower than that of Case 1. To quantify this difference, we calculate the relative deviation of energy cost of Case 2 with respect to that of Case 1 (Table 5-4). For example, the energy cost of the 5-year lifetime farm in Case 1 is 42.5 cents/KWh while that in Case 2 is 35.5 cents/KWh; the relative deviation is 16.5%. We find that the energy cost of Case 2 is at least 14% lower than that of Case 1. This is mainly because the energy output of Case 2 is more than that of Case 1 while the total costs of both cases are almost the same. Table 5-4  The relative deviation of the energy cost of Case 2 with respect to the energy cost of Case 1 Lifetime  5 years  10 years  15 years  20 years  Small farm  -16.5%  -17.3%  -16.7%  -17.9%  Medium farm  -14.7%  -15.5%  -14.2%  -17.8%  Large farm  -15.2%  -15.4%  -14.0%  -16.8%  Farm scale  53  More discussion on this topic is included given in Section 5.8.1. - 179 -  45  Energy cost (cent/Kwh)  40 35 30 25 20 15 10  Small farm (10 turbines) Medium sized farm (30 turbines)  5  Large farm (100 turbines) 0 0  5  10  15  20  15  20  Life time (years)  (a) 45  Energy cost (cent/Kwh)  40 35 30 25 20 15 10  Small farm (10 turbines) Medium sized farm (30 turbines)  5  Large farm (100 turbines) 0 0  5  10 Life time (years)  (b) Figure 5-6  Energy cost: (a) no hydrodynamic interaction; (b) with constructive hydrodynamic interactions - 180 -  Figure 5-7 shows the 2005-2006 average electricity prices in New York city, New York, USA, San Francisco, California, USA and British Columbia, Canada54 in four different demand sectors, which are residential, small commercial, medium commercial and industrial sectors. It is noted that the energy costs for all scenarios in both cases are higher than the maximum market price in British Columbia which is the price for small commercial usage, 7.5 cents/KWh. In such a situation, governmental subsidies can be applied to help the new technology penetrate into the market. Mathematically, the ideal governmental subsidy can be given as at least greater than the difference between the energy cost and the market price as follows, sG = cenergy − price  (5.25)  The energy cost of the 20-year lifetime large farm (100 turbines) in Case 2 is the lowest among the energy costs of all scenarios, and we use it as the energy cost in Eq.(5.25). Then, by substituting the BC electricity price (the values in Figure 5-7) into Eq.(5.25), we calculate the ideal government subsidies, and the results are shown in Table 5-5. These estimates suggest that the required subsidy would approximately be 2.4 cents/KWh on average for a large farm if the demands from all four demand sectors are uniform assuming world price holds and thus local prices don’t change. If we only use tidal current turbine farms to support small commercial activities, the ideal governmental subsidy may only need to be as low as 0.4cent/KWh. These results are based on the assumption that there is no cost for transmitting the electricity from the local load center to the end users.  54  The electricity market in BC is a regulated market where the electricity price is flat in all cities. - 181 -  2005-2006 Electricity Price 25 British Columibia, Canada New York, USA San Francisco, USA  Cent/Kwh  20  15  10  5  0 Residential  Figure 5-7  Small Commercial  Medium Commercial  Industrial  2005-2006 Electricity price in several regions in North America (BCHydro 2007)  Table 5-5  The ideal governmental subsidy in British Columbia, Canada Lifetime  5 years  10 years  15 years  20 years  Residential (cent//KWh)  23  8.1  3.0  0.9  Small commercial (cent//KWh)  22.5  7.6  2.5  0.4  5  2.9  Sector  Medium  commercial 25  (cent//KWh)  10.1  Industrial (cent//KWh)  26.8  11.9  6.8  4.7  Average55 (cent//KWh)  24.3  9.4  4.5  2.4  It is noted that the energy costs shown in Figure 5-6 is the levelized energy cost. In this study, 55  Assume that demands from all sectors are uniform. - 182 -  we also investigate the energy cost in a particular year. Data in Figure 5-8 show the energy cost for a certain year for a medium sized farm (50 turbines) with a life time of 20 years in which the levelized energy cost is 9.66 cents/KWh. It is noted that the energy cost in a certain year does not significantly increase during its life time: the energy cost increase from 8 cents/KWh in the year 5 to 9.9 cents/KWh in the year 20, with a net increase of about 1.8 cents/KWh in 15 years. Any increase is mainly due to the turbine fatigue and maintenance strategies, resulting O&M cost increases during the life time of the turbines.  12  Cents/Kwh  10 8 6 4 2 0 5  Figure 5-8  8  11 14 Working Year  17  20  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 )56  56  The energy cost at year 5 serves as a reference value (blue) and the increment in a certain year is shown in red. - 183 -  5.8 Discussion and Conclusions 5.8.1 Discussion In section 5.7, we conduct sensitivity analysis on the effects of three major components in TE-UBC on energy cost, which are the size of the farm, the lifetime of the turbines and the hydrodynamic interactions between turbines (whether the hydrodynamic interactions between turbines are avoided). To understand how the energy cost respond to the changes in other factors, we conduct a set of sensitivity analysis for the specific situation when the farm is a medium-sized farm, and the lifetime of the turbines is 20 years life (Case 2 in Section 5.7), under which the energy cost is 9.66 cents/KWh57. Data in Figure 5-9 shows the effects of five factors on energy cost. These five factors are such as technician salary, technician workload58, farm offshore distance, wind and wave condition (accessibility) and fog condition (visibility). It is noted that 1) the energy cost increases from 8 cents/KWh to 10 cents/KWh as the technician salary increases from 60$/hour to 90$/hour; 2) the energy cost increases from 9.4 cents/KWh to 9.9 cents/KWh as the workload decreases from 14 hours/day to 8 hours/day; 3) the energy cost increases from 8.99 cents/KWh to 12.05 cents/KWh as the farm offshore distance increases from 0.25km to 3km; 4) the energy cost increases from 8.97 cents/KWh to 14.05 cents/KWh as the number of the extreme wind and wave days increases from 4 days/year to 24 days/year; 5) and the energy cost increases from 9.23 cents/KWh to 11.98 cents/KWh as the number of foggy days increases from 2.5 days/year to 15 days/year. The results suggest that keeping all the other factors unchanged, within plausible ranges, the effects of the offshore distance and the number of extreme wind and wave days have more effects on energy cost than technician salary, technician workload and the number of foggy days per year. The length of offshore cable and travel distance of vessels and helicopters both increase as the farm offshore distance increases, which is expected to increase the O&M cost and capital cost significantly. The number of extreme wind and wave days increases the downtime during which the technicians, equipments, and transportation cost remains, while the turbines do not generate power. Technician salary and technician workload both directly affect the labor cost, which is 57  58  This scenario is also used for the energy cost study in a certain year as shown in Figure 5-8. We assume that the technician skill does not change when their salary or their workload change. - 184 -  just a small portion of the O&M cost because transportation and equipment costs are much higher. During foggy days, both the vessel travel time and technician work time are doubled. Thus, the labor cost will be two times the labor cost of the scenario where it is not a foggy day, and the energy cost will increase as well. We expect to have more information regarding O&M practices and turbine reliability after the full scale sea test of such a system is conducted.  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 -  The O&M transportation cost in routine maintenance per turbine is quasi inversely proportional to the total number of turbines in a farm, which means that the larger the farm is, the lower the routine transportation cost per turbine is. In detail, one routine maintenance is conducted on one trip from the harbor to the farm site, no matter how many turbines are at the site. Thus, a farm with a larger number of turbines will result in a smaller routine transportation cost per turbine. On the other hand, we assume the unit capital cost of turbines does not change with the size of turbine farm. In fact, it is understood that the unit manufacturing cost should significantly decrease when the number of manufactured turbines increases. As a new product without too much experience, we suggest, however, using the same capital cost per turbine no matter how many turbines are manufactured (See Eq.(5.19)). It is clear that the electricity market price in New York City and San Francisco is much higher than that in BC. There is a possibility to sell the electricity to such cities and others if transmission system is further utilized or developed. Further investigation on international trading and transmission is required.  5.8.2 Environmental Impact Assessment The construction and operation of a farm are expected to have environmental impacts and social-economic impacts and raise maritime security concerns. Besides exploring the cost of the farm, one also needs to thoroughly investigate these issues. Although they are beyond the scope of this dissertation, as a supplement to the farm system modeling and analysis, possible environmental impacts from a tidal current turbine farm are briefly discussed here. Environmental impact assessment is required either by law or regulations at different jurisdictional levels in most countries before a major construction project is carried out, to make sure that the decision maker considers the associated environmental impacts. This is of course applicable to the construction of tidal current turbine farms. The development of tidal current turbine farms is still in its infancy. Accordingly, only a few studies related to the environmental impact of individual tidal current turbine have been reported. Wang et al. (2007) presented an experimental study on cavitation noise from a horizontal axis tidal current turbine. Dolman et al. (2007) discussed the impact of ocean power technologies on - 186 -  cetaceans. There is, however, no systematic analysis being conducted on the environmental impacts from tidal current turbine farms. In this study, we review previous research on possible environmental impacts and survey environmental impacts from other ocean energy technologies (e.g., barrage and offshore wind) that might have implications on environmental impacts from tidal current turbine farms. We identified a set of major attributes that can be used to characterize environmental impacts during the life cycle (construction, operation and decommissioning) of a tidal current turbine farm, which include 1) marine animal mortality; 2) impact on ocean circulation; 3) impact on marine ecosystem; 4) wetland damage; and 5) acoustic emission59. These attributes are by no means exhaustive or mutually exclusive, and we believe significant future research on the environmental impact could improve this list.  Marine animal mortality In the tidal power industry, the threat of the barrage technology to the lives of marine animal’s raised concerns since the 1960’s and is one of the main reasons that the the advancement of the barrage technology has been restrained since the 1980’s (Pelc and Fujita 2002). Although tidal current turbines are considered to be more friendly to marine animals than the barrages (Lang 2003), they still have the potential to kill marine animals (Rulifson and Dadswell 1987), like wind turbines killing birds (Barrios and Rodríguez 2004; Smallwood and Thelander 2004; Percival 2004; Chamberlain et al. 2006). The operation of a tidal current turbine may harm or and kill marine animals through the contact with the rotation of the turbine blades. It has been observed that some fishes were killed in this way by turbines in barrages, which happens when the flow going in and out of the sluice is fast, and the tip speed of the turbine blade tip is much higher than the fish swimming speed. In this case, fish might not be able to escape (Pelc and Fujita 2002). Although tip speed of a full scale tidal current turbine is much lower than that of a turbine in a barrage, those slow-reacting marine animals still have difficulty maneuvering an escape, especially when they meet a strong pressure gradient. Also, fish usually swim around the sea bottom or other structures where a boundary layer exists (Farrell 2005). Tidal current  59  Acoustic emission has been extensively discussed in Chapter 3 and Chapter 4, respectively. - 187 -  turbines are usually attached to some auxiliary structures such as ducts, anchors or platforms, which create thick boundary layer and attract fish60.  Impact on ocean circulation and iceberg The rotation of a turbine is driven by the tidal or ocean current. Inevitably, there are hydrodynamic interactions between turbines and the flow. These interactions may change the ocean flow to some degree. In previous research, Miles (1981) showed that the operation of turbines in the Severn Barrage significantly changed the local current. Although more evidence is needed to substantiate the impact of tidal current turbines on the local current, the large-scale change of the ocean flow induced by the hydrodynamic interactions between turbines and the ocean flow may change ocean circulation in the long run. Additionally, some of the potential abundant sites for constructing turbine farms are in icy areas, which are very close to glaciers (e.g., the sites in northern Canada as shown in Figure 2-1 and Figure 2-2). The interactions between turbines and icebergs may directly or indirectly affect the glacier’s motion and erosion. This interaction has not been studied until recently when Sanders (2007) conducted research on hydrodynamic and solid mechanic interactions between icebergs and turbines.  Impact on marine ecosystem A tidal current turbine farm may affect the marine ecosystem during its construction, operation and decommissioning by disturbing the communication between marine animals, damaging their habitat, and changing their living environment (e.g., increasing the temperature of their environment). The impacts from the construction and decommissioning of an offshore structure on the marine ecosystem have received wide attention in the community of the offshore structure industry, especially in the offshore oil and gas (O&G) industry (Aramco 2007). Klima et al.  60  There are two studies focused on ducted turbine design in the Naval Architecture Laboratory at UBC, Nabavi (2008) and Rawlings (2008) using a numerical method and an experimental method, respectively. These studies explored the boundary layer created by various duct shapes. - 188 -  (1988) reported that over 50 sea turtles and over 40 dolphins were killed and countless fishes were damaged during the decommissioning of an O&G platform in the Gulf of Mexico. Burns  and  Codi  (1999)  reported  that  sediment  concentration  decreased  from  2.45 ± 1.29μ g ⋅ g −1 to 0.86 ± 0.45μ g ⋅ g −1 after the construction of an O&G platform in offshore Northwest Australia, while the marine ecosystem is significantly affected by sediment concentration (Yamashiki et al. 2006; Kiirikki et al. 2006). During its operation, a turbine farm may induce energy diffusion in the ocean. Large scale energy diffusion in the ocean may increase sea temperature, which then affects the marine ecosystem. As an example, Maury et al. (2007) stated that a 2 Co temperature change caused by energy diffusion can lead to a 20~40% decrease in ecosystem biomass in oligotrophic regions and a 15~32% decrease in biomass in eutrophic regions.  Wetland damage Wetlands are defined as “those areas that are inundated or saturated by surface or ground water at a frequency and duration sufficient to support, and that under normal circumstances do support, a prevalence of vegetation typically adapted for life in saturated soil conditions”  (USACE 1994), and the impacts on wetland may induce significant change globally (Nicholls et al. 1999). Thus, wetlands are frequently referred as the “kidney” of the earth. Tidal current turbine farms may affect wetlands in coastal areas either directly or indirectly. Directly, the auxiliary facilities (e.g., services house and cable system) are constructed nearby coastal areas. The activities related to the construction, operation and maintenance of these facilities might damage local wetlands (Findlay and Bourdages 2000). Indirectly, the possible change of near-shore circulation induced by hydrodynamic interactions between turbine and flow has the potential to affect the coastal landscape which then affects the wetland. As an example, significant wetland change caused by the change of ocean circulation in the Pacific Ocean was reported by Hales et al. (1991).  - 189 -  5.8.3 Conclusions Modeling of the tidal current turbine farm planning process, power generation and distribution is complicated problem due to the lack of detailed analysis and understanding of the turbine working principle and the complexity of the ocean natural environment. Given the present knowledge of tides, principles of turbine analysis and computational ability, this chapter presented a systematic framework and a model (TE-UBC) for estimating the energy cost of tidal current turbine farms which integrates different research disciplines and new approaches such as DVM-UBC. In general, some conclusions are drawn here. y  The TE-UBC can be used to estimate the energy cost based on a given turbine configuration and local conditions  y  In the example case, by utilizing constructive hydrodynamic interactions, the energy cost can be reduced by about 15% compared with the case where the hydrodynamic interaction is avoided. The results show that the minimum energy cost for a large farm (100 turbines) with a 20-year life in offshore BC can be about 8 cents/KWh and it is only a few cents more than the local market price  y  In the example case, it is noted that energy cost in a certain year does not significantly increase during its life time  y  The TE-UBC allows the user to test the sensitivity of the results of individual components. Characteristics of every component are inputs in the model. The results of the main example in Section 5.7 show that, besides tidal flow velocity which has been discussed in Chapter 3 and Chapter 4, the two most important control variables for energy cost are farm size and turbine relative distance. With the sensitivity analysis, besides farm lifetime and farm size, it is noted that the farm offshore distance and number of extreme wind and wave days are also two most important factors.  - 190 -  5.8.4 Future Work Some future works are suggested in this section, y  Full scale sea tests are expected to provide more accurate reliability data and operation strategies  y  Investigations of the grid integration of the tidal power are required to make a more practical farm plan  y  Integrating of the government policy information such as taxes and incentives is expected to investigate the precise energy price  - 191 -  Chapter 6 Summary, Conclusions and Future Work In this study, we develop a numerical model to describe the wake of a turbine that sheds vortices in an unsteady flow, which we name DVM-UBC. DVM-UBC is then used to calculate the hydrodynamic interactions between two turbines. The results from the two-turbine system research are then extrapolated to address the hydrodynamic interactions in an N- turbine system and then to predict the performance (power output) of a tidal current turbine farm. To describe the behavior of the turbines, a set of rigorous mathematical models are developed. The objectives were, z  predicting the performance and acoustic impact of a stand-alone turbine with improved accuracy and less computational effort  z  predicting the performance and acoustic impact of a two-turbine system and the power output of an N-turbine system (N = 2,3, 4, ⋅⋅⋅ ), namely, a tidal current turbine farm  z  minimizing the energy cost of a tidal current turbine farm by strategically planning the turbine distribution in the farm and selecting O&M schemes.  In this chapter, the research problems addressed in this study are restated in Section 6.1, the research that we conduct is summarized in Section 6.2, the major conclusions and contributions of this study are summarized in Section 6.3, and future research directions are envisioned in Section 6.4.  6.1 Restatement of the Research Problems Accurate prediction of the power output (and thus energy output) and the lifetime cost of a tidal current farm are important to accurately estimate the energy cost of a tidal current turbine farm. The accurate estimation of energy cost is crucial for the economic justification - 192 -  of constructing such a farm. Understanding the hydrodynamic interactions between turbines is important to accurately predict the performance of an N-turbine system (i.e., a turbine farm) given that the hydrodynamic interactions between turbines are expected to affect the performance of the turbines. However, in previous studies, researchers often neglected these hydrodynamic interactions in predicting power output from a turbine farm and in studying the impact of such a farm on current flows. We develop and use our DVM-UBC to investigate the hydrodynamic interactions between turbines and their impact on the performance of the turbines. Understanding the impact of O&M strategies on the O&M cost is important for accurately estimating the total cost that might be incurred in the lifetime of a turbine farm. However, in previous studies, researchers often used very rough approximations to represent the O&M cost, such as using a fixed percentage of the capital cost. We develop an O&M module for estimating O&M cost with different O&M strategies. By integrating this O&M module with the hydrodynamic module for estimating energy output from a tidal current turbine farm, we develop an integrated model for estimating energy cost, which is important for government to make subsidy decisions and investors to make investment decisions related to the development of renewable ocean energy.  6.2 Summary of the Method and its Applications A brief introduction to the motivation, the research problems and the objectives of this study is given in Chapter 1, and the history and the state-of-the-art of the vertical axis tidal current turbines are described in Chapter 2. In chapters 3 to 5, research problems are solved by using a Discrete Vortex Method with a free-wake structure, a perturbation theory and a linear theory, and a scenario-based cost-effective analytical method; a set of example cases are developed and used to test the validity of the methods, and to demonstrate its utility, robustness and the accuracy. In this section, the methods applied, and the models developed, together with the major findings are summarized61, and a sample application of using the methods and models to help 61  The detailed findings and conclusions are given in each chapter. - 193 -  plan a turbine farm is presented.  6.2.1 Summary of the Work Chapter 3 In Chapter 3, a numerical model is developed for predicting the performance of a stand-alone tidal current turbine by using DVM-UBC which is a method for describing vortex shedding and unsteady flow formulated in that chapter. Similar to the traditional DVM, bounded vortices are used to describe the blade elements and the wake. To approximate the physics of the flow better and to avoid instability in the computations, Lamb vortices are employed. Nascent vortex, vortex decay scheme and vortex shedding frequency are extensively simulated in DVM-UBC. To validate the DVM-UBC in predicting the performance of a stand-alone turbine, we conduct both kinematics and dynamics validation by comparing the wake characteristics, the power output, and torque of a stand-alone turbine obtained with DVM-UBC with those obtained with physical experiments (both published experimental results and results from the UBC towing tank tests) and other numerical methods (e.g., commercial CFD package, traditional DVM and other potential flow methods). The validation results reveal that DVM-UBC predicts the performance with acceptable accuracy while the computational cost of it is relatively low. It is found that the relative deviations of the results obtained with DVM-UBC with respect to the results from other methods are no more than 7% for kinematics validation and are around 10% for dynamics validation. The relative deviation of the results by using DVM-UBC from the results obtained with Fluent is about 5% for kinematics validation and is about 8% for dynamics validation while the computational cost of DVM-UBC is less than 1% of that of Fluent. Furthermore, if the Fluent results are regarded as an approximation to the true results, DVM-UBC can predict the performance 50% more accurately than a traditional DVM method while the computational costs of DVM-UBC and traditional DVM are comparable. We also define a torque fluctuation coefficient to quantify the turbine torque fluctuation.62 62  This coefficient is to measure the increment of the torque magnitude in frequency domain. That is to say, the higher the torque fluctuation coefficient is, the less the torque fluctuates. The torque fluctuation coefficient of an ideal turbine without turbine fluctuation is infinite. - 194 -  By using this model, we quantify the effects of TSR, solidity, Reynolds number and blade type on the turbine performance. Additionally, another numerical model is developed to estimate the acoustic emission from a stand-alone tidal current turbine using DVM-UBC. The results obtained with this acoustic model reveal that the main frequency of the noise of a typical stand-alone turbine63 is around 4Hz. The victims of the noise emission from such a turbine are likely to be Baleen whale and other marine animals using low frequency bands for their communication.  Chapter 4 In Chapter 4, a numerical model is developed to predict the performance of a two-turbine system by using DVM-UBC and based on the stand-alone turbine model developed in Chapter 3. This two-turbine model is able to quantify the effects of important factors (i.e., TSR, relative distance, incoming flow angle and relative rotating direction of the two-turbine system) on the performance of a two-turbine system. We then use this model to calculate the relationship between these important factors and the performance of the system. The results show that by optimizing the relative positions of two turbines, the power output of a two-turbine system can be about 25% higher than two times that of a corresponding stand-alone turbine, and the torque fluctuation coefficient64 of a two-turbine system can be about two times that of a corresponding stand-alone turbine. The results also show that the higher the TSR is, the higher the relative efficiency and torque fluctuation coefficient are. As the towing tank on UBC campus is demolished, we were not able to validate these results. The acoustic model developed in Chapter 3 is modified to predict the noise emission from the two-turbine system. The results show that the main frequency of the noise emission from a two-turbine system is 4Hz, the same as that of the corresponding stand-alone turbine. The results also suggest that, by choosing an optimal relative position for a two-turbine system, the noise intensity of the system is about 5 dB / Hz lower than that of the stand-alone turbine  63  Refer to Section 3.4.4 for the detailed description of such a turbine. In this case, the torque fluctuation coefficient of the two-turbine system is about two times that of a corresponding stand alone turbine. That is to say, the magnitude of the power spectrum density of the torque of the system reduces about half.  64  - 195 -  and the noise band of the system is wider. Additionally, by using a perturbation theory and a linear theory, a procedure is developed to predict the power output (i.e., efficiency) of an N-turbine system by extrapolating the results of a set of two-turbine systems. This procedure is used to estimate the total relative efficiency of an ideal tidal current turbine farm (i.e., a rectangular farm site with a length of 400 times the turbine radiuses and a width of 30 times the turbine radiuses). In this example, turbines are assumed to be uniformly distributed as an array; the objective is simplified as maximizing the total relative efficiency of a column of turbines because the distance between two adjacent columns are fixed to avoid the hydrodynamic interactions. The results from this example show that the optimal number of the turbines in a column is about 18 and the optimal relative distance between two adjacent turbines in a column is 24 times the turbine radiuses. The maximum total relative efficiency in a column is 14.5 times the efficiency of a corresponding stand-alone turbine, which is about 20% more than that of the case where the hydrodynamic interactions between two turbines in the column are avoided (i.e., no hydrodynamic interaction between two tandem turbines).  Chapter 5 In Chapter 5, a tidal current turbine system model is developed to estimate the energy cost of a given tidal current turbine farm, which we name the TE-UBC, by incorporating the turbine model (which we call hydrodynamic module here) developed in Chapters 3 and 4 for predicting the power output and incorporating a scenario-based cost-effectiveness module developed here for estimating the O&M cost (i.e., the O&M module). In this O&M module, turbine components are simulated in greater detail than that in previous models in the literature, and the O&M strategies are modeled as routine and emergency O&M strategies. TE-UBC is then used to predict the energy cost of an ideal turbine farm at a site offshore BC, Canada. The results of this example suggest that the energy cost is about 7.9 cents/KWh if the farm has 100 turbines with a life span of 20 years and 0.5 km offshore distance. This cost is about 16% lower than that of the case in which the hydrodynamic interactions in the farm are avoided. Also, this cost is only a few cents higher than the current market price. In general, TE-UBC can be used to predict the energy cost more accurately than the previous - 196 -  models. Additionally, possible environmental impacts from the tidal current turbine farms are synthesized, which include impacts on marine animals, marine ecosystem, ocean circulation, and wet-land.  6.2.2 Sample Applications Although we predict the energy cost of an N-turbine system in Section 5.6, as the development of tidal current turbines is still in its infancy, the N-turbine system will not be available in the very near future, and only a stand-alone turbine and a two-turbine system are deployed in a site to date. Thus, we apply the methods and models developed in our research to a sample two-turbine system and we shall not only discuss the energy cost, but also the annual energy output, torque fluctuation and noise emission from the two-turbine system. This two-turbine system is assumed to be deployed for the purpose of replacing diesel generators used to supply power to remote communities in BC such as communities around Quatsino Narrows where the average current velocity is 2m/s and the annual electricity usage per household, Ed , is 15MWh(Hawley 2008). The number of household, N h , that the system can serve can be obtained as follows, N h = Energy  (6.1)  Ed  In this application, the turbine is a typical tidal current turbine65, the specifications of which are given in Table 6-1. We develop two cases: in Case 1, two turbines are distributed far away from each other (i.e., there is no hydrodynamic interaction between them), and in Case 2, two turbines are distributed as a canard layout counter-rotating two-turbine system and their relative distance is 2.5 (times the turbine radius).  65  Refer to Sections 3.4.4., 4.3, 4.4, and 5.7 for discussion about this turbine regarding different subjects. - 197 -  Table 6-1  The basic specification included in the cases that we summarize here Parameters  Values  Turbine blades  NACA0015  Blade number  3  Turbine height  3.75m  Turbine radius  2m  Turbine lifetime  20 years  Solidity  0.375  Current velocity  2m/s  By using the TE-UBC and the acoustic model, we calculate the annual energy output, the energy cost and the noise intensity for both cases, which are summarized in Table 6-2. We find that the annual energy output and energy cost of the turbine system in Case 1 are 120MWh and 13.1 cents/KWh, respectively, while those of the turbine system in Case 2 are 140MWh and 10.2 cents/KWh, respectively. The torque fluctuation coefficient in Case 2 is 40dB higher than that in Case 1. The noise intensity in Case 2 is about 35.7% lower than that in Case 1. Table 6-2  Results of the sample application  Parameters  Case 1  Case 2  Relative deviation66  66 67  Annual energy output (MWh)  120  140  16.6%  Number of household powered  8  9.33  16.6%  Torque fluctuation coefficient( dBS )  38  78  40  Cost(cent/KWh)  13.1  10.2  -22%  Noise intensity ( dB / Hz )67  -14  -19  -35.7%  It is defined as the value of Case 2 with respect to the value of Case 1. We pick the first peak of the noise received at Receiver 2(refer to Table 4-22 for detail). - 198 -  6.3 Contributions and Insights This study is expected to make contributions to the ocean engineering field and shed light on research in different disciplines related to ocean energy investigation such as oceanography, environmental sciences, industrial engineering and electrical engineering. For example, the innovation of the two-turbine system may change the focus and procedure of all related research because it suggests that the total efficiency of a two-turbine system could be 25% more than two times that of a corresponding stand-alone turbine while the torque fluctuation is significantly reduced as long as this system is optimally-designed.  Ocean Engineering The DVM-UBC developed in this study can be used for numerical simulation in other marine applications such as rudders, hydrofoils and propellers. The theoretical and numerical investigations of the turbine system conducted in this study can provide insights in designing new turbines, which help designers and experimental hydrodynamists for identifying the important factors affecting turbine performance in improving turbine design and in conducting experimental turbine tests.  Oceanography The wake structure generated by using the method developed in this study can provide information to physical oceanographers in predicting circulation and related local flows and thus for biological and chemical oceanographers in studying the biological and chemical processes in the ocean. Large scale circulation and biological and chemical processes can be predicted by using ocean flow models such as Princeton Ocean Model (POM) and Regional Ocean Modeling System (ROMS).  Industrial engineering and electrical engineering The power output and the energy cost prediction procedures provide a systematical approach for researchers in industrial engineering and electrical engineering in planning energy system such as tidal current turbine farm. - 199 -  Environmental sciences The numerical model for noise intensity estimation can provide information for the environmental scientists and policy analysts in assessing the acoustic emission level and in developing their policies regarding the acoustics impacts.  6.4 Future Research Directions Based on the experience gained from this study, the following areas are worthy of further investigation: z  To analyze the nonlinearity of an N-turbine system in greater details. In this study,  the nonlinear hydrodynamic interactions between N turbines in an N-turbine system are extrapolated from two-turbine systems. In a site with high turbine density, the nonlinearity of an M-turbine system (2<M<N) is more significant and the simplification procedure in this study is not likely suitable for solving the hydrodynamic interactions between these M turbines. z  To investigate turbine fatigue and reliability. In this study, the fatigue and reliability  information are obtained from empirical data and reported results for related offshore structures such as BBV (2001). Considering the unexpected sea conditions and advance of the turbine materials, new fatigue and reliability data are required. One can obtain them either by conducting a finite element analysis or by physical experiments. z  To quantify the arm and shaft effect and identify the way to reduce or utilize these effects. The arm and shaft effects are neglected in this study while this can not be  avoided for the prototype. It significantly reduces the turbine power output, although light arm with appropriate arm profile works better 68 . More investigation and quantification work is needed. z  To quantify free surface and ocean bottom effects. The farm site investigated in this  study is an ideal site, in which the turbines are installed far away from the water surface and the bottom of the ocean. That is to say, free surface and bottom effects are not included in this study. In a real site, these boundaries are expected to affect the turbine 68  Refer to Appendix A for the details. - 200 -  performance considering the change of tidal variables such as velocity and elevation. Both analytical and experimental investigations are necessary. z  To analyze the unsteady torque and its effect on the electric power conversion process (i.e., the voltage and current). Although an optimal two-turbine system is able  to reduce the torque ripple, considering the unsteady RPM in the experimental test, the electrical power system still requires to be modified. A systematic dynamic modeling may be required. z  To investigate governmental subsidy policy for tidal energy. It is noted that the  energy cost of a large scale tidal current turbine farm is only a couple cents higher than the local market price. Experience learned from the positive effects of the subsidy on wind power may be useful; possible new energy policies for tidal power need to be developed. z  To investigate the environmental impacts more extensively. The synthesis of this  study indicates that further experimental investigation is necessary; Hydrophone and more pressure transducers could be installed in the future experimental tests. z  To investigate viscous effect more. Although viscosity was introduced in DVM-UBC  with lamb vortices, vortex decay, nascent vortex and vortex shedding scheme, we still neglected boundary layer and turbulence effects. 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(1935). “The rolling up of the surface of discontinuity behind an aerofoil of finite span.” Aeronautic Research Committee Report, Memo No.1692, London  - 215 -  White,F.M.(2002). “Fluid Mechanics” McGraw-Hill Sciences, 2002 Wilson, R. E. (1976). “Vortex sheet analysis of the Giromill.” Journal of Fluid Engineering, 100 (3), 340-342 Wilson, R. E. and Walker, S. N. (1983). “Fixed wake theory for vertical axis wind turbines.” Journal of Fluid Engineering, 105 (4), 389-393  Wong, H. (1990). “A numerical model for vortex shedding from sharp wedges in oscillatory flow.” Master thesis, Department of Mechanical Engineering, the University of British Columbia, Vancouver, BC, Canada Wong, H. (1995). “Slender ship procedures that include the effects of yaw, vortex shedding and density stratification.” Doctoral thesis, Department of Mechanical Engineering, the University of British Columbia, Vancouver, BC, Canada Yamashiki, Y., Nakamura, T., Kurosawa, M., Matsui, S. (2006) “Ecosystem approach to mitigate impacts of sedimentation on the hydrological cycle and aquatic ecosystem.” Hydrological Processes, 20 (6), 1273-1291Younsi, R., El-Batanony, I., Tritsch, J.-B.,  Naji, H., Landjerit, B. (2001). “Dynamic study of a wind turbine blade with horizontal axis.” European Journal of Mechanics, A/Solids, 20 (2), 241-252 Yoshizaki,T. and Yamakawa,H. (1996) “Hydrodynamic and viscosity Radius expansion factors of polymer chains with excluded volume: Monte Carlo Methods” Journal of Chemical Physics, v105, n 13, Oct 1 1996, pp5618-5625 Zhou, Q. and Joseph, P. F. (2005). “A frequency domain method for the prediction of broadband noise from an open rotor.” Collection of Technical Papers-10th AIAA/CEAS Aeroacoustics Conference, Vol. 3, 2694-2709  - 216 -  Appendices A. Extended Hydrodynamic Analysis Fundamentals of the Discrete Vortex Method We formulate a numerical model to simulate turbine rotation and unsteady flow by using a discrete vortex method with a free wake structure (i.e., DVM-UBC) in Chapter 3. In this section, the fundamentals of the discrete vortex method are presented. Mathematically, the vorticity ω is defined as  ω = ∇×u  (A.1)  where u denotes the flow velocity. If we take the curl of the Navier-Stokes equation, Du ∂u 1 = + u ⋅∇u = − ∇p + ν∇ 2u ρ Dt ∂t  (A.2)  We obtain the governing equation of vorticity transport, Dω ∂ω = + u ⋅∇ω = ω ⋅∇u + ν∇ 2ω Dt ∂t  (A.3)  Eq.(A.3) suggests that the rate of vorticity change of a fluid element consists of two components, with one ( ω ⋅∇u ) being the intensification of vorticity due to the stretching of vortex lines caused by the motion of the fluid and the other (ν∇ 2ω ) being viscous diffusion. The direction of the vector ω is perpendicular to the x − y plane while the magnitude of the vector  ω is  ∂v ∂u − . As the mathematical derivation here is complex analysis for a ∂x ∂y  two-dimensional motion, the stretching term ω ⋅∇u = 0 . On the other hand, the flow that we - 217 -  consider here is inviscid flow, which implies that ν = 0 . Therefore, Eq. (A.3) can be rewritten as Dω ∂ω = + u ⋅∇ω = 0 Dt ∂t  (A.4)  Eq.(A.4) implies that the vorticity of a fluid element remains constant and convects with the flow. Knowing the irrotational outer flow, the velocity in the vicinity of the separation point can be found if the vorticity in the flow at a given time is calculated, u − iv =  i 2π  ∫  C  γ ( s ) ds  (A.5)  z − z (s)  where C is the contour, and γ ( s ) is the vortex strength within it, while z ( s ) is a point on the sheet on C . Numerically, Eq.(A.5) can be written as  q( z) =  i 2π  ⎛ Γ ( z − zo ) ⎞ ⎜ ⎟ ⎜ z−z 2 ⎟ o ⎝ ⎠  (A.6)  By using Eq.(A.6), we can calculate induced velocity (See Eq.(3.2)). Additionally, the forces induced due to vortex shedding can be estimated by using theoretical hydrodynamics. It can be found using the unsteady form of the Blasius force equation for infinite small wedge: 2  ⎛ dF ⎞ FV = −i ∫ Fdz − i ∫ ⎜ ⎟ dz 2 C ⎝ dz ⎠ C  ρ  (A.7)  where the integrals are taken around the perimeter of the body, C , on the physical plane z = x + iy , and F is the complex potential.  Schwarz-Christoffel transformation is used to transform the equation from physical plane to the ζ plane. Then Eq.(A.7) can be rewritten as follows, - 218 -  FV = i ρ  ∂ n Γk ∑ ζk +ζ k ∂t k =0 2  (  )  (A.8)  Thus, the force coefficient can be obtained by using the corresponding length scale lz and lζ as follows  CFV =  FV 1 2 ρ LZ Lζ τ −2 2  = 2i  ∂ ∂τ  n  ∑γ k =0  k  Re (ζ k )  (A.9)  Experimental Investigation of the Three-dimensional Effects In Chapter 3, it has been stated that the two-dimensional model is just a simplified version of the three-dimensional model. Section 3.4.3 shows that the three-dimensional effect on the turbine performance is not significant in the numerical model. To understand the physics of the three-dimensional effect more precisely, we conduct an experiment. We use a typical experimental design to quantify the three-dimensional effect, which is by adding end-plates to the turbine blade, so that the test results with end-plates become two-dimensional results. The information of the experiment is given in Table A-1 and two types of end-plates are shown in Figure A-1. Table A-1  Basic information of end-plate experiments Values  Parameter  Scenario 1  Scenario 2  Scenarrio 3  N/A  NACA0012  Circular  Number of blades  3  3  3  Number of arms  2  2  2  End-plate type  Blades type  NACA 63(4)-021  NACA 63(4)-021  NACA 63(4)-021  Solidity  0.435  0.435  0.435  Reynolds number  160,000  160,000  160,000  - 219 -  Figure A- 1 End-plates by using NACA0012 blade (left) and circular shape (right) Figure A-2 and Table A-2 show the comparison between power coefficients of the two-dimensional test (i.e., end-plate test) and the three-dimensional test. It is noted that the relative deviation of the two-dimensional result with respective to the power coefficient of the three-dimensional experiment is less than 7%. This relative deviation is in the same magnitude as that of numerical prediction shown in Section 3.4.3, and it is an acceptable error or noise in the computation. It is known that the computational time of the three-dimensional model is almost twenty times that of the two-dimensional model. Therefore, considering both numerical analysis in Section 3.4.3 and experimental analysis here, one can say that two-dimensional model is more cost-effective than three-dimensional model in this study. However, a three-dimensional model is required for some special purposes such as calculating the vertical flow fluctuation (i.e., the flow fluctuation in the z direction) which is ignored in this study except this section and Section 3.4.3.  - 220 -  Table A-2  Results of the end-plate experiments Scenario 1 Scenario 2  Scenario 3  Maximum Cp  0.13169  0.138  0.139  Relative deviation from scenario 1  ---  5.3%  6.1%  0.16 0.14  Standard Test End Plates: NACA 0012 End Plates: Circular  0.12 0.10  Cp  0.08 0.06 0.04 0.02 0.00 1.00 -0.02  1.50  2.00  2.50  3.00  3.50  -0.04 TSR  Figure A- 2 Comparison of power coefficients between end-plates test and standard test  A.3 Analysis of the Experimental Configuration In Chapter 3, we have shown that the difference between the results generated by using DVM-UBC and the results generated from UBC experiment is larger than the difference between the results generated by using DVM-UBC and the results by using other numerical methods or experimental tests. This is mainly because some experimental configurations were not simulated in the numerical model as summarized in Table 3-22. Besides these 69  One may notice that the result here is much lower than that of Figure 3-14 while they are obtained from the cases with same basic information (Table A-1 and Table 3-10). The reason is discussed in Appendix A.3. - 221 -  reasons, two special issues with the UBC experimental configuration are analyzed here.  Connection between Arm and Blade The arms are not simulated in the numerical method, although it is very important in the experimental tests because they contribute to additional drag on the turbine. One might notice that the maximum efficiency of the experimental results in Figure A-2 is around 13% while that in Figure 3-14 is 35% although both experiments are conducted under the same conditions (see Table 3-10 and Table A-1). The only difference between these two experimental tests is that the experimental test of Table A-1 used heavy arms while the experimental test of Table 3-10 used light arms. Figure A-3 illustrates three arm profiles examined during the test (Type A and Type B are both heavy arms while Type C is a light arm) and Figure A-4 shows the efficiency of the turbine with different arms. It is obvious that the efficiency of the turbine with lighter arms is higher. It is important to know that the clamping mechanism allows for adjustable angle of attack used for Type A and Type B which increases drag at the same time. Upon removing the blades to examine the power absorbed by the arms, a large portion of the clamping mechanism is also removed, which greatly reduces the parasitic drag compared to when the blade is mounted. To quantify the effects of different arms, we investigated the arm effects with the same shaft by testing the turbine without blades, and the results are shown in Figure A-5. It is noted that light arms reduce efficiency less than heavy arms. Although this plot provides insight into what efficiency losses are occurring due to the drag on the arms, simple subtraction of these values from those in Figure A-2 or Figure 3-14 does represent an ideal case without parasitic drag for the following reasons (Rawlings 2008): y  ¼ span mounting of the foils reduces the length of the working span of the blade  y  Positioning of the arms at the ends of the foil will affect tip losses  y  Bolt heads are also removed and thus, in the assembled case the parasitic drag will be larger in those tests with heavy arms.  - 222 -  Figure A- 3 Different types of arms used in the experiments  - 223 -  0.35  Arm A  0.3  Arm B 0.25  Arm C  Cp  0.2 0.15 0.1 0.05 0 -0.05 -0.1 0  1  2  3  4  TSR  Figure A- 4 Different types of arms used in the experiments 0 -0.02 -0.04 -0.06  Cp  -0.08 -0.1 Arm A -0.12 Arm B  -0.14  Arm C  -0.16 -0.18 -0.2 0  0.5  1  1.5  2  2.5  3  3.5  TSR  Figure A- 5 Plausible efficiency reduction induced by arm effects - 224 -  4  Connection between Shaft and Torque Meter When we conduct the dynamic validation for our numerical model in Chapter 3, we show that the locations of maximum torque of the experimental tests appeared shift to a higher azimuth angle than those in the numerical prediction. In all the experimental tests at UBC, there is a connection problem between the shaft and the torque meter which induces a phase shift between these two components. Therefore, the torque remains around its minimum value (0 Nm or even lower) for a while, and then reach its maximum value rapidly afterwards. The average torque during a revolution is almost a fixed value for a constant power output. However, the duration when the torque is at its minimum value is relatively long during a revolution and the value of the minimum torque is lower than that of numerical result (see Figure 3-12(b)). Thus, the maximum torque obtained in experiments is much higher than that generated by using any other numerical methods.  B. Extended Analysis on Operation and Maintenance Module In Section 5.5, we present the basic structure of the O&M module of TE-UBC. In that section, we show more details about this module. In the O&M module, most cost variables are levelized over the lifetime of a tidal current turbine farm. To simplify the discussion, we use cost variables without subscripts to replace those with subscripts in the following description. For example, we use ELC to represent ELCi , j (emergency labor cost of turbine i in the year j , see Eq.(5.23)).  Emergency Maintenance Sub-module The emergency maintenance cost is the sum of emergency transportation, labor, equipment and material cost. Here we explain how these four components are estimated. The simplest one is the emergency material cost (EMC), which is determined by the level of failure severity and the types of failed components. Emergency transportation cost (ETC) can be expressed as either the emergency vessel cost (ETVC, when the failure severity is minimal and mid-level) or the sum of the emergency vessel and helicopter cost (ETVC and ETHC, when the failure is severe), as shown with the following equations: - 225 -  ETVC ⎧ ETC = ⎨ ⎩ ETVC + ETHC  when the level of failure severity is minimal and mid-level when the level of failure severity is severe (A.10)  ETVC = ( ODist / VS × 2 + ( DTime + EWTime ) ⋅ ETNum ) × EVC (A.11) ETHC = ( ODist / HS × 2 + ( DTime + EWTime ) ⋅ ETNum ) × HC (A.12)  where ODist , VS , HS, DTime, ETNum, EVC and HC denote the offshore distance of the farm, vessel speed , helicopter speed, delay time due to weather and related reasons and labor waiting cost, number of turbines that need emergency maintenance, the emergency vessel cost per day, and the emergency helicopter cost per hours. ETNum is determined by component failure rate in regular condition (Table A-3)70, weather  and sea state71. DTime is determined by the type of the failure component and the level of the failure severity as given in Table A-4. Table A-3  Component failure rate in regular condition (times per turbine per year)72 Minimal  Mid-level  Severe  Shaft  0.002  0.007  0.001  Brake  0.0153  0.0325  0.0025  Generator  0.065  0.0545  0.0065  Electrical System (including cable) 0.225  0.09247  0.000002  Blade  0.042  0.0273  0.00007  Gearbox  0.2125  0.0325  0.0005  Control system  0.1  0.1  0.0001  Others  0.03  0.0299  0.00006  70  We assume that two components on one turbine will not fail at the same time. When the weather and sea state become worse, the failure rate will increase. In this study, we assume that effects from weather and sea states are very small. Thus, one can also neglect these effects. 72 Adapted from Rademakers (2003). 71  - 226 -  Table A-4  Downtime of emergency maintenance (Days)73 Minimal  Mid-level  Severe  Shaft  0  1  2  Brake  0  1  1  Generator  1  2  2  Electrical System (including cable) 1  2  3  Blade  1  3  3  Gearbox  1  3  3  Control system  1  1  2  Others  0  1  2  The emergency labor cost (ELC) can be estimated by using Eq.(A.13). ELC = TechS × ETNum × ELNum × EMT + DTime × LWC  Eq. (A.13)  where TechS , ELNum , EMT and LWC denote the technician’s salary, number of technicians required during this emergency maintenance, time required for this maintenance, and labor waiting cost respectively. In this study, technician salary and their waiting costs are constants. ELNum and EMT are both functions of the severity of the failures that an average turbine experiences. The emergency equipment cost (EEC) can be written as follows, EEC = ( DTime × EWaC + EWoC × EWTime ) × ETNum  (A.14)  where EWaC , EWoC and EWTime denote the equipment waiting cost, equipment working cost and equipment working time, respectively. They all are determined by the type of failed component and failure severity level.  73  Adapted from Rademakers (2003) and van Bussel and Bierbooms (2003). - 227 -  Routine Maintenance Sub-module Similar to the emergency maintenance cost, the routine maintenance cost is also the sum of four components, which are the routine labor, transportation, equipment and material costs. Similar to the emergency material cost, the routine material cost is also the simplest one among four routine maintenance costs, and it is determined by the type of the turbine components. The routine labor cost (RLC) can be obtained as follows, RLC = RITime ⋅ TechS × N × LDisc  (A.15)  where RITime and LDisc denote the routine inspection time per turbine and the labor discount rate (i.e., the larger the size of the turbine farm is, the lower the discount rate is. The default value of the discount rate is set as 1). The routine transportation cost (RTC) and routine equipment cost (REC) can be estimated by using Eq.(A.16) and Eq.(A.17), respectively. RTC = ( ODist / VS × 2 + RITime ⋅ N ) × RVC  (A.16)  REC = RITime ⋅ N × EWoC  (A.17)  where RVC denotes the routine vessel cost. The discussion above presents the basic formulation of the routine maintenance cost. In this module, the technician workload is set as 8 hours per day. That is to say, the vessel returns to the harbor and goes to the farm again the next day if the routine inspection cannot be finished in one day. Thus, the program will minimize the routine maintenance cost by choosing the optimal combination of the number of vessels and the number of technicians.  - 228 -  C. Error Analysis In this research, error exists in both numerical and experimental work. In this section, we discuss these errors,  y  Numerical Reasons  1. Quasi steady-state effect: DVM-UBC assumes that the flow is in a quasi steady state, while the flow is actually unsteady. Although it is widely accepted to use the steady state hydrodynamic coefficient, the error is unavoidable because the wake in this study is time-dependent. 2. Viscous effect: Although we introduced vortex decay scheme and nascent vortex into the calculation, DVM-UBC is still an approximation to the unsteady physics of the flow. Boundary layer effects and turbulence effects are not described, which will bring errors into the prediction. 3. No shaft and arm assumptions: The shaft and arm effect is neglected in this study. It significantly affects the turbine performance although the negative effects can be reduced by using new materials to build the components. 4. N-turbine system extrapolation: The procedure for obtaining the efficiency of an N-turbine system is a linearization process (i.e., to extrapolate the efficiency of an N-turbine system from that of several two-turbine systems), which often introduces  errors. 5. Converge criteria: Convergence always leads to errors when we set the convergent criteria, and this is a ubiquitous phenomenon.  y  Experimental Reasons  We use experimental test to validate our numerical method, and to improve the prototype design. Besides the two unique issues mentioned in Appendix A, other reasons that might introduce errors are given here: 1. Design: The design of the turbine is still to be improved by comparing its  - 229 -  performance with that of the turbines with different designs, choosing different materials, adding auxiliary structure (e.g., deflector) and changing location of the arms. 2. Towing tank effect: The towing tank provides an environment mimicking a restricted water area. The results obtained in a towing tank can not totally represent the behavior of the turbine in the open ocean, which is usually an open water area. 3. Measurement and data acquisition: There are always errors caused by the quality of the testing facilities and measuring equipments. 4. Human errors: The experimental test is a complicated procedure. Human errors are often unavoidable especially at the set-up stage (e.g., calibration).  D. Software Description In this study, besides those popular softwares such as Microsoft Office series (Excel, Word, Powerpoint and Visio) and Microsoft Visual Studio Development series (C++, Basic and Fortran), the following professional softwares are used:  MatLab/Simulink Matlab is a well-konwn computational software. Matlab is a high-level language and has an interactive environment that enables people to perform computationally intensive tasks faster than with traditional programming languages such as C, C++, and Fortran. Simulink is a platform for multi-domain simulation and model-based design for dynamic systems. It provides an interactive graphical environment and a customizable set of block libraries, and can be extended for specialized applications.  Analytica Analytica is a popular system modeling software and a visual tool for creating, analyzing, and communicating decision models. Spreadsheet modelers find Analytica a revelation because of its clarity, flexibility, uncertainty scalability and so on.  - 230 -  Fluent The Commercial CFD package Fluent is a popular engineering CFD software that includes geometric tools for pre-processing and post-processing (Gambit) and a CFD solver for numerically modeling complex flows. The flows range from incompressible (low subsonic) and mildly compressible (transonic) to highly compressible (supersonic and hypersonic). Fluent provides several solver options to deal with a wide range of flow regimes. The broad physical modeling capabilities of Fluent have been applied in industrial applications ranging from air flow over an aircraft wing to combustion in a furnace, from bubble columns to glass production, from blood flow to semiconductor manufacturing, from clean room design to wastewater treatment plants as well as general fluid mechanics, aerospace, chemical engineering, biomedical engineering automotive and offshore engineering applications. The ability of the software to model in-cylinder engines, aeroacoustics, turbomachinery, and multiphase systems has served to broaden its reach.  XFOIL XFOIL is used to generate blade hydrodynamic data. XFOIL is an interactive program for the design and analysis of subsonic isolated airfoils. It consists of a collection of menu-driven routines which perform various useful functions such as: y  Viscous (or inviscid) analysis of an existing airfoil  y  Airfoil design and redesign by interactive modification of surface speed distributions  y  Airfoil redesign by interactive modification of geometric parameters  y  Blending of airfoils  y  Writing and reading of airfoil coordinates and polar save files  y  Plotting of geometry, pressure distributions, and multiple polars.  Origin Origin 7.5 is used to processing data and produce numerical graphs together with Excel and MatLab. Origin is developed by OriginLab Corporation. It produces professional data analysis and graphing software for scientists and engineers. - 231 -  

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