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Dr. Detlev Heinemann

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- 1. Wind Energy Meteorology Modeling of wind turbine wakesGerald Steinfeld(Slides partly taken from presentations of Oliver Bleich(Uni Oldenburg) and John Prospathopoulos (CRES))Carl-von-Ossietzky Universität OldenburgSummer term 2011, June 21st, 2011
- 2. Power production of a wind turbine IHow much power is in the wind? A ds vMass flow rate: dm dV ds A Av dt dt dtKinetic energy per unit time, or power, of the flow: 1 dm 2 1 P v Av 3 2 dt 2
- 3. Power production of a wind turbine IIPower production of a single wind turbine 1 dm 2 1 PWT C p v C p Av 3 A: 2 dt 2 Rotor swept area C p : Power coefficientTo maximize the total power production wehave to put the wind turbine in a place withmaximum wind speedBut which layout is best for a wind farm toguarantee that a single wind turbine in it „sees“the highest possible wind speed?
- 4. Inland wind farms Installation of wind turbines in a row This can be efficient when: a) a prevailing wind direction occurs b) the number of wind turbines is small in relation to the extent of the exploitable landHowever, the situation is more complexin large wind farms.Adjacent wind turbines will interact witheach other, i.e. a wind turbine will oftenbe influenced by the wake of anupstream wind turbine
- 5. Offshore wind farms The same statements hold for offshore wind farms.However, there are also differencesbetween inland and offshore windfarms. While there is no impact of acomplex topography on the flow, theatmospheric flow is the result of acomplex interaction between the oceanand the atmosphere.
- 6. Wind farm layoutFinding the “best” location of every singlewind turbine with respect to each other,i.e. finding the “best” wind farm layout isa subject of continuous research.The main targets are:a) a maximization of the energy yield of the wind farm.b) a minimization of the interaction between the wind turbines while taking into account the limited size of the installation areaFor reaching these targets we have to beable to calculate wake effects properly.
- 7. Definition of wind turbine wakes – wake effectsThe rotor of the wind turbine extracts energyfrom the wind. This leads to a deceleration ofthe flow downstream of the wind turbine. Atthe same time, the flow gets more turbulent.The region showing a wind speed deficit andan increased turbulence intensity is calledwake of the wind turbine.As the flow proceeds downstream there is aspreading of the wake and the flow graduallyrecovers to free stream conditions.The reduced wind speed and increasedturbulence intensity in a wind turbine wakeinfluences an adjacent wind turbine indownstream direction.This is called wake effect.
- 8. Impacts of the wake effectWake effects have two significantimpacts on downstream windturbines:a) Reduction of their power production due to a reduced wind speedb) Reduction of their lifetime due to increased structural loading.These impacts have to be takeninto account, when the layout of anew wind farm is planned. Source: b
- 9. Wake effects in a real offshore wind farmNysted offshore wind farm Source: a
- 10. Wake effects in a real offshore wind farm Difference from the farm- averaged energy production per wind turbine in % Source: a
- 11. Single wake in real conditions Source: b
- 12. Physical processes in wind turbine wakes• A velocity deficit appears downstream of each wind turbine corresponding to the kinetic energy extract• The shear layers produced expand through convection and diffusion forming the wakes of the wind turbine• The atmospheric boundary layer interacts with the wind turbine wakes through turbulent mixing• Turbulent mixing also occurs between the wakes of the different wind turbines• Combination of the effects above physical processes leads to a complex flow field in the wind farm characterized by varying velocity and turbulence Source: b
- 13. Physical processes in wind turbine wakes Source: a
- 14. Tip vortices
- 15. Single wake in real conditions Source: a
- 16. Single wake in real conditions Source: a
- 17. Single wake in real conditions Source: NREL
- 18. Physical mechanisms to be modelled• Generation of a circular shear layer directly behind the rotor disk due to the extracted wind energy• Development of the circular shear layer downstream in the three- dimensional turbulent atmospheric boundary layer (over a complex terrain or offshore)Dominant factors that need to be modeled:• Atmospheric boundary layer• Rotor thrust• Turbulence (atmospheric and wind turbine induced)• (Complex terrain)• (Air-sea interaction) Solution of full 3D Navier-Stokes equations ? Source: b
- 19. Simple analytical wake model (Jensen model)• Based on the equation of continuity:Rrotoru2 Rwake ( x) Rrotor u0 Rwake ( x)u ( x) 2 2 2 2 2 Rrotor u ( x) 2 u2 2 Rwake ( x) Rrotor 2 2 Rrotor u0 u0 2 u2 u0 2 Rwake ( x) Rwake ( x) Rwake ( x) u0 u0 Rrotor u2 u(x) Rwake
- 20. Simple analytical wake model (Jensen model) 2 Rrotoru ( x) 2 u2 Rwake ( x) Rrotor u u Rrotor u u 2 2 2 2 0 0 2 2 0 Rwake ( x) Rwake ( x) Rwake ( x) 2 Rrotor R kx u2 u0 mit Rwake ( x) Rrotor kx u ( x) u0 wake experimentally ? u0 obtained factor u0 Rrotor u2 u(x) Rwake
- 21. Simple analytical wake model (Jensen model)u2 1 2a u0 1 ct u0 Obtained from 1D-momentum theory (although to use Rrotor is not really correct) Axial induction factor Thrust coefficient Wind turbine specific data, dependent on the 1 c u 2 Rrotor free stream velocityu ( x ) u0 R kx t 0 u0 wake Rrotor 2 1 1 ct 1 u0 R kx wake
- 22. Simple analytical wake model (Jensen model) 1 c u 2 Rrotor u ( x ) u0 R kx t 0 u0 wake Rrotor 2 1 1 ct 1 u0 R kx wake Das Ergebnis wäre eine Kastenfunktion, tatsächlich aber wird eher eingaußförmiger Verlauf des Geschwindigkeitsdefizits im Nachlauf einerAnlage beobachtet;Jensen schlug Multiplikation mit Kosinusfunktion vor Rrotor 1 cos9 2u ( x, ) 1 1 ct 1 R kx u0 , 20 wake 2
- 23. Simple analytical wake model (Jensen model)u(x)/u0 Source: Jensen
- 24. Simple analytical wake model (Jensen model)u(x)/u0 Source: Jensen
- 25. Simple analytical wake model (Jensen model)u(x)/u0 Source: Jensen
- 26. Superposition of wakes (PARK model)• Aim: Calculation of the effect of wakes on the power production of downwind wind turbines• Velocity of the weakened air stream downwind of a wind turbine is calculated according to the Jensen model• Adding the energy effects of the upwind converters, the effect on the downwind converters is calculated• uj-k,j denotes the velocity in the wake of the wind turbine j-k at the position of the wind turbine j. In case of partial shading the individual velocity deficits have to be multiplied by the respective weighting factor βk. βk is defined as the ratio of the affected rotor area in relation to the total rotor area. Upwind velocity acting on rotor: j 1 2 u j k , j u j u0 1 k 1 k 1 u0 • Used for a single wake calc. with Jensen model in seq. of downw. pos.
- 27. Simple RANS model – Ainslie model• Ainslie model uses a parabolic approximation of the Reynolds-averaged Navier-Stokes-equations (the thin shear layer approximation) and the continuity equation, field model: calculates complete flow field• Assumptions:a. The wake is considered to be axisymmetric 2d-formulation in cylindrical co-ordinates possibleb. The flow is considered to be incompressiblec. There are no external forces or pressure gradientsd. Gradients of the standard deviation of u are neglectede. Viscous terms are neglectedMomentum equation: u u v u 1 r uv x r r r u 1 r vContinuity equation: x r r2 equations, but three unknowns turbulence closure required
- 28. Simple RANS model – Ainslie model• Turbulence closure: The Reynolds stress, which indicates momentum transport across the flow, is modelled with the eddy viscosity approach v uv r• ε is the eddy viscosity• Ainslie suggested to split up the total eddy viscosity ε into two components:1. The ambient eddy viscosity of the atmospheric flow εa2. The eddy viscosity generated by the wind shear in the wake εw u* z I au0 zh a Km m z / L 2.4 1 x 4.5 3 krw u0 uc ( x) 0.65 für x 5.5D w Km * F ( x) F (x) 23.32 m z / L k: determined empirically (0.015), 1 for all other distancesrw: width of the wake, u0-uc: centreline deficit
- 29. Simple RANS model – Ainslie model• Main simplification of the Ainslie model: Separation between wind shear and related eddy viscosity of wake and ambient flow• This allows the two-dimensional description of the wake flow, which leads to a very fast-running model• No height dependence of the ambient eddy viscosity!• No calculation of an eddy viscosity from the local wind shear!
- 30. Simple RANS model – Ainslie model• The model requires an inflow boundary condition, as the near wake cannot be calculated with the model, as pressure gradients cannot be neglected in that region. Actually, pressure gradients dominate the flow in that region.• Calculation of the wake starts at the end of the near wake, which is assumed to be 2D behind the wind turbine• An empirical, Gaussian shaped profile is used as boundary condition. The profile has a centreline velocity deficit u0-uc and a wake width rw (2.83% of the wake deficit in the centre): u0 uc ct 0.05 16ct 0.5 I 10 3.56ct rw 4u0 uc 2 u0 uc r 2 u (r ) u0 uc exp 3.56 r w
- 31. Simple RANS model – Ainslie model• In a modified version of the Ainslie model the near wake length is not longer fixed to 2D, but it is calculated with an empirical approach.• The near wake is divided into two parts, of which the first has the length xH, which is modelled to be dependent on ambient turbulence, rotor-generated turbulence and shear-generated turbulence 1 dr dr dr 2 2 2 2 xH r0 dx a dx dx m D m 1 1• r0 is the effective radius of the fully expanded rotor disc r0 m 2 2 1 ct dr 2 2.5I 0.05 for I 0.02 dx a 5I for I 0.02 2 dr 0.012 B dx dr 2 1 m 1.49 m dx m 9.761 m
- 32. Simple RANS model – Ainslie model• Finally, the length of the near wake region is calculated from the length of the first region by 0.212 0.145m 1 0.134 0.124m xn xH 1 0.212 0.145m 0.134 0.124m
- 33. Simple RANS model – Ainslie model• Wind farm model: In order to estimate the average wind speed over the rotor, the momentum deficit is averaged over the rotor swept area: u0 urotor 2 1 uo uw dA 2 A Rotor• The influence of multiple wakes on the wind speed of the rotor area is calculated by adding the momentum deficits of all incident wakes and integrating over the rotor area: u0 urotor 2 1 u i, all wakes rotori uwi dA A Rotor 2• From the mean wind speed of the rotor area, with all wakes taken into account, the power output of a turbine is estimated from its power curve
- 34. Simple RANS model – Ainslie model Source: Lange et al.
- 35. Simple RANS model – Ainslie model Source: Lange et al.
- 36. Simple RANS model – Ainslie model Source: Lange et al.
- 37. Simple RANS model – Ainslie model Source: Lange et al.
- 38. Simple RANS model – Ainslie model Source: Lange et al.
- 39. Simple RANS model – Ainslie model Source: Lange et al.
- 40. From simple models to FULL 3D RANS• Starting from simple models other models have been developed• Their complexity ranges according to the approximations of each method:a) Simple engineering models using self-similarity in the far wake regionb) Boundary-layer approximation methodsc) Parabolic approximation of Navier-Stokes equationsd) Axisymmetric Navier-Stokes equations simplified for the far wake using a given initial velocity profilee) Combination of vortex methods for the calculation of near wake profiles with axisymmetric Navier-Stokes equations for the far wakef) Full 3D Navier-Stokes equations (RANS)g) Full 3D Navier-Stokes equations (LES)• Full 3D Navier-Stokes simulations of wind farms are now feasible due to the rapid development of computer systems Source: b
- 41. Advanced CFD wake models• What do we expect from them?a) To better simulate turbulence effectsb) To simulate atmospheric stabilityc) To simulate complex terraind) To better simulate the wind turbine effecte) To simulate the interference between the wind turbine wakesf) To produce guidelines for a better calibration of the simple engineering models• RANS (Reynolds Averaged Navier-Stokes) solvers with 2-equation turbulence modelsa) They have been widely used for the last two decades enough experience in flow field applications is availableb) They have a reasonable computational cost in comparison to more advanced models such as LES (Large Eddy Simulation) Source: b
- 42. Navier-Stokes solvers (RANS)• Governing equations: ui • Continuity equation 0 x x1 , x2 , x3 , u u1 , u2 , u3 , i, j 1,2,3 xi ui, p: time-averaged velocity and pressure ui ui p ui u j ij • Momentum equation u j t x j xi x j x j xi • Reynolds stresses modeled through Boussinesq approximation ui u j 2 ij t k ij x j xi 3 turbulent viscosity turbulent kinetic energy Kronecker delta Source: b
- 43. The k-ω turbulence model (Wilcox) k k u k u j t x j ij i * k x j x j t * x j TKE u u j ij i * 2 t Specific dissipation t x j k x j x j x j The eddy viscosity is given as: kt Closure coefficients of the standard k-ω model: 5 3 * 9 1 1 , , , , * 9 40 100 2 2 Source: b
- 44. Closure coefficients of the k-ω turbulence modelModification of these coefficients for atmospheric conditions results from themeasurements of the friction velocity: 2 2u*2 u* 0.17 * 0.033 k k From experimental observations of Townsend: * 1.2 0.0275 From the momentum and k,ω equations for the limiting case of anincompressible constant pressure boundary layer: * k 2 / * 0.3706 Source: b
- 45. Computational domain Source: b
- 46. Boundary conditions Source: b
- 47. Boundary conditions Source: b
- 48. Computational grid Source: b
- 49. Wind turbine modeling Source: b
- 50. Actuator disk concept Source: b
- 51. Derivation of the uniformly loaded AD model One-dimensional momentum theory (Betz, 1926): Control volume, in which the control volume boundaries are the surface of a stream tube and two cross-sections of the stream tube. The only flow is across the ends of the stream tube. Source: a
- 52. Derivation of the uniformly loaded AD model discontinuity of pressure The thrust force is determined by the change of momentum of the air through the stream tube Assuming a stationary flow and mass flow conservation it follows that Source: a
- 53. Derivation of the uniformly loaded AD model Application of Bernoulli: p2 p4 p1 p3 discontinuity of pressure Assumption: and The thrust force can also be expressed by the pressure gradient force acting on the rotor disc: Source: a
- 54. Derivation of the uniformly loaded AD model Equating equations (9) and (5) leads to: with it follows Source: a
- 55. Derivation of the uniformly loaded AD model Introduction of the axial induction factor a Application of (14) in (12) leads to Problem: what is the reference velocity u1=uref? Source: a
- 56. Definition of the reference velocity Source: b
- 57. Induction factor concept Source: b
- 58. Actuator disk – pressure profiles Source: b
- 59. Actuator disk – velocity profiles Source: b
- 60. Actuator disk – turbulence profiles Source: b
- 61. Actuator disk – deficit contours Source: b
- 62. Actuator disk – deficit contours Source: b
- 63. Actuator disk – turbulence contours Source: b
- 64. Blade element approach Non-uniformly loaded AD Source: b
- 65. Actuator line approach Source: b
- 66. Initial single wind turbine predictions – Nibe exp. Source: b
- 67. Initial single wind turbine predictions – Nibe exp. Source: b
- 68. Modifications for near wake correction Source: b
- 69. Turbulence modeling – modification 1 Source: b
- 70. Turbulence modeling – modification 1 Source: b
- 71. Turbulence modeling – modification 2 Source: b
- 72. Turbulence modeling – modification 2 Source: b
- 73. Turbulence modeling – modification 3 Source: b
- 74. Turbulence modeling – modification 3 Source: b
- 75. Comparison with simple models Source: b
- 76. Wind farms – 5 wind turbines in flat terrain Source: b
- 77. Wind farms – 5 wind turbines in flat terrain Source: b
- 78. Wind farms – 5 wind turbines in flat terrain Source: b
- 79. 43 wind turbines in complex terrain Source: b
- 80. 43 wind turbines in complex terrain Source: b
- 81. 43 wind turbines in complex terrain Source: b
- 82. 43 wind turbines in complex terrain Source: b
- 83. 43 wind turbines in complex terrain Source: b
- 84. 43 wind turbines in complex terrain Source: b
- 85. 30 wind turbines in offshore wind farm Source: b
- 86. 30 wind turbines in offshore wind farm Source: b
- 87. 30 wind turbines in offshore wind farm Source: b
- 88. 30 wind turbines in offshore wind farm Source: b
- 89. 30 wind turbines in offshore wind farm Source: b
- 90. Large-eddy simulation• Large eddies are explicitly resolved• The impact of small eddies on large eddies is modeled (SGS-model)• Concept of filtering: Scale separation by application of a filter function u , tGx , t tdtd ~ u x , t 3 • Example for a filter function: box filter 1 / if x / 2 G x 0 otherwise• Application of the filter to the Navier-Stokes equations results in ~ ~~ ~ ~ ui uk ui 1 ~* p ~ f u g ~ T T0 2ui ki ijk f j uk i 3k 3 k g i3 t xk 0 xi T0 xk xk 2 Impact of small eddies on large eddies, term needs to be parameterized
- 91. Comparison between LES and RANS production inertial subrange dissipationS local concentration time series resolved scale subfilter-scale fluctuations (u, c) k x LES: volume average building critical concentration level local concentration time series z smooth result building x RANS, k-ε: ensemble average after Schatzmann and Leitl (2001)
- 92. PALM – simulations with actuator line model figure from Ivanell (2009)Rotating rotor blades => rotating air flow,ACD model does not provide the helical structure of the flow inthe wake => ACL model
- 93. Results of wake flows simulated with anactuator line model (I)• PALM simulations using actuator line method• 1536*512*256 grid points, = 1m, t = 0.01s• cpu-time: 1 week on 1024 PEs of SGI-Altix-ICE turbulent upstream flow laminar upstream flow + tower
- 94. Results of wake flows simulated with anactuator line model (II)
- 95. Velocity profiles in dependency on the distancefrom the WTland surface: sea surface:
- 96. Wake extension in vertical directionland surface: sea surface: x in m ( WT bei x = 150 m ) x in m ( WT bei x = 150 m )
- 97. Validation of LES results by wind tunnel data a: wind tunnel experiment b: LES with non-uniformly loaded actuator disk c: LES with uniformly loaded actuator disk
- 98. Validation of LES results by wind tunnel dataDashed line: Uniformly loaded actuator disk modelSolid line: Non-uniformly loaded actuator disk modelBlack dots: actuator line modelRed dots: wind tunnel data
- 99. Comparison of different wake models
- 100. Some conclusions Source: b
- 101. Many slides have been taken from …• … the presentations given bya. Oliver Bleich in the seminar “Aktuelle Forschungsthemen der Energiemeteorologie”, Carl von Ossietzky Universität Oldenburg, summer term 2011. Title: Wake modellingb. John Prospathopoulos during the WAUDIT summer school 2010 in Pamplona. Title: Wind turbines wakes modeling
- 102. Thank you for your attention!

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