1. 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

2. 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)

3. 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

4. GOLDSTEIN MODEL Vortex sheet constructed as perfect helix with variable pitch

5. 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

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

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

8. APPLICATION OF BIOT-SAVART LAW

9. BLADE ELEMENT FLOW CONDITIONS

10. 2-D VISCOUS POLAR S809 profile at Re=500,000 using XFOIL + linear extrapolation to

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

12. TOWER SHADOW MODEL DOWNWIND CONFIGURATION

13. TOWER SHADOW MODEL Model includes Wake Width and Velocity Deficit Profile, Ref: Coton et Al. 2002 Model Based on Wind Tunnel Measurements Ref: Snyder and Wentz ’81 Parameters selected: Wake Width 2.5 Tower Radius, Velocity Deficit 30%

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

15. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

34. 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

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

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

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

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

46. APPENDIX C Homogeneous blade; First mode

47. APPENDIX C Homogeneous blade; Second mode

48. APPENDIX C Homogeneous blade; Third mode

49. APPENDIX C Nonhomogeneous blade; M’ distribution

50. APPENDIX C Nonhomog. blade; EIx distribution

51. APPENDIX C Nonhomogeneous blade; First mode

52. APPENDIX C Nonhomogeneous blade; Second mode

53. APPENDIX C Nonhomogeneous blade; Third mode

54. APPENDIX D KUTTA-JOUKOWSKI LIFT THEOREM

55. APPENDIX D NONLINEAR TREATMENT Discrete equations: If Where

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

57. 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

58. APPENDIX E CONVECTION IN THE WAKE (continued)

59. APPENDIX F Blade working conditions: attached/stalled

60. APPENDIX G STEADY FLOW Power output comparison

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