TOWER SHADOW MODELIZATION WITH HELICOIDAL VORTEX METHOD Jean-Jacques Chattot University of California Davis OUTLINE Motivations Review of Vortex Model Tower Shadow Model Conclusion 45 th AIAA Aerospace Sciences Meeting and Exhibit 26 th ASME Wind Energy Symposium, Reno, NV, Jan.8-11, 2007

MOTIVATIONS Take Advantage of Model Simplicity and Efficiency for Analysis of Unsteady Effects with Impact on Blade Fatigue Life and Acoustic Signature - Include Tower Interference Model (Upwind 2006) - Include Tower Shadow Model (Downwind 2007)

REVIEW OF VORTEX MODEL Goldstein Model Simplified Treatment of Wake -Rigid Wake Model -“Ultimate Wake” Equilibrium Condition -Base Helix Geometry Used for Steady and Unsteady Flows Application of Biot-Savart Law Blade Element Flow Conditions 2-D Viscous Polar

GOLDSTEIN MODEL Vortex sheet constructed as perfect helix with variable pitch

SIMPLIFIED TREATMENT OF WAKE - No stream tube expansion, no sheet edge roll-up (second-order effects) -Vortex sheet constructed as perfect helix called the “base helix” corresponding to zero yaw

“ULTIMATE WAKE” EQUILIBRIUM CONDITION Induced axial velocity from average power:

BASE HELIX GEOMETRY USED FOR STEADY AND UNSTEADY FLOWS Vorticity is convected along the base helix, not the displaced helix, a first-order approximation

APPLICATION OF BIOT-SAVART LAW

BLADE ELEMENT FLOW CONDITIONS

2-D VISCOUS POLAR S809 profile at Re=500,000 using XFOIL + linear extrapolation to

FLEXIBLE BLADE MODEL Blade Treated as a Nonhomogeneous Beam Modal Decomposition (Bending and Torsion) NREL Blades Structural Properties Damping Estimated

TOWER SHADOW MODEL DOWNWIND CONFIGURATION

TOWER SHADOW MODEL Model includes Wake Width and Velocity Deficit Profile, Ref: Coton et Al Model Based on Wind Tunnel Measurements Ref: Snyder and Wentz ’81 Parameters selected: Wake Width 2.5 Tower Radius, Velocity Deficit 30%

SIMPLIFIED MODEL LINE OF DOUBLETS PERTURBATION POTENTIAL If |Y’|>2.5 a, Outside Wake, Use Where: If |Y’|<2.5 a, Inside Wake:

RESULTS V=5 m/s, Yaw=0, 5, 10, 20 and 30 deg V=7 m/s, Yaw=0, 5, 10 and 20 deg V=10 m/s, Yaw=0, 5, 10 and 20 deg V=12 m/s, Yaw=0, 10 and 30 deg Comparison With NREL Sequence B Data

RESULTS FOR ROOT FLAP BENDING MOMENT V=5 m/s, yaw=0 deg

RESULTS FOR ROOT FLAP BENDING MOMENT V=5 m/s, yaw=5 deg

RESULTS FOR ROOT FLAP BENDING MOMENT V=5 m/s, yaw=10 deg

RESULTS FOR ROOT FLAP BENDING MOMENT V=5 m/s, yaw=20 deg

RESULTS FOR ROOT FLAP BENDING MOMENT V=5 m/s, yaw=30 deg

EFFECT OF ROTOR INDUCED VELOCITY ON WAKE V=5 m/s, yaw=30 deg

RESULTS FOR ROOT FLAP BENDING MOMENT V=5 m/s, yaw=30 deg

NREL ROOT FLAP BENDING MOMENT COMPARISON V=7 m/s, yaw=0 deg

NREL ROOT FLAP BENDING MOMENT COMPARISON V=7 m/s, yaw=5 deg

NREL ROOT FLAP BENDING MOMENT COMPARISON V=7 m/s, yaw=10 deg

NREL ROOT FLAP BENDING MOMENT COMPARISON V=7 m/s, yaw=20 deg

NREL ROOT FLAP BENDING MOMENT COMPARISON V=10 m/s, yaw=0 deg

NREL ROOT FLAP BENDING MOMENT COMPARISON V=10 m/s, yaw=5 deg

NREL ROOT FLAP BENDING MOMENT COMPARISON V=10 m/s, yaw=10 deg

NREL ROOT FLAP BENDING MOMENT COMPARISON V=10 m/s, yaw=20 deg

NREL ROOT FLAP BENDING MOMENT COMPARISON V=12 m/s, yaw=0 deg

NREL ROOT FLAP BENDING MOMENT COMPARISON V=12 m/s, yaw=10 deg

NREL ROOT FLAP BENDING MOMENT COMPARISON V=12 m/s, yaw=30 deg

CONCLUSIONS Simple model for tower shadow easy to implement Good results obtained for “downwind” configuration Some remaining unsteady effects possibly due to tower motion Vortex Model proves very efficient and versatile

APPENDIX A UAE Sequence Q V=8 m/s  pitch=18 deg CN at 80%

APPENDIX A UAE Sequence Q V=8 m/s  pitch=18 deg CT at 80%

APPENDIX A UAE Sequence Q V=8 m/s  pitch=18 deg

APPENDIX B Optimum Rotor R=63 m P=2 MW

APPENDIX C Homogeneous blade; First mode

APPENDIX C Homogeneous blade; Second mode

APPENDIX C Homogeneous blade; Third mode

APPENDIX C Nonhomogeneous blade; M’ distribution

APPENDIX C Nonhomog. blade; EIx distribution

APPENDIX C Nonhomogeneous blade; First mode

APPENDIX C Nonhomogeneous blade; Second mode

APPENDIX C Nonhomogeneous blade; Third mode

APPENDIX D KUTTA-JOUKOWSKI LIFT THEOREM

APPENDIX D NONLINEAR TREATMENT Discrete equations: If Where

APPENDIX D NONLINEAR TREATMENT (continued) If is the coefficient of artificial viscosity Solved using Newton’s method

APPENDIX E CONVECTION IN THE WAKE Mesh system: stretched mesh from blade To x=1 where Then constant steps to Convection equation along vortex filament j: Boundary condition

APPENDIX E CONVECTION IN THE WAKE (continued)

APPENDIX F Blade working conditions: attached/stalled

APPENDIX G STEADY FLOW Power output comparison

APPENDIX H YAWED FLOW Time-averaged power versus velocity at different yaw angles =5 deg =10 deg =20 deg=30 deg