The document discusses modeling of wind turbine wakes and wind farm layout optimization. It begins with an overview of the physics of wind power extraction and wake effects between turbines. It then describes several wake modeling approaches, from simple analytical models like Jensen's model to more complex Reynolds-averaged Navier-Stokes (RANS) models like Ainslie's model. The goal of wake modeling is to optimize wind farm layouts to maximize energy production while minimizing wake interference between turbines.

1. Wind Energy Meteorology
Modeling of wind turbine wakes
Gerald Steinfeld
(Slides partly taken from presentations of Oliver Bleich
(Uni Oldenburg) and John Prospathopoulos (CRES))
Carl-von-Ossietzky Universität Oldenburg
Summer term 2011, June 21st, 2011

2. Power production of a wind turbine I
How much power is in the wind?
A ds
v
Mass flow rate:
dm dV ds
  A  Av
dt dt dt
Kinetic 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 II
Power 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 coefficient
To maximize the total power production we
have to put the wind turbine in a place with
maximum wind speed
But which layout is best for a wind farm to
guarantee 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 land
However, the situation is more complex
in large wind farms.
Adjacent wind turbines will interact with
each other, i.e. a wind turbine will often
be influenced by the wake of an
upstream wind turbine

5. Offshore wind farms
The same statements hold for offshore
wind farms.
However, there are also differences
between inland and offshore wind
farms. While there is no impact of a
complex topography on the flow, the
atmospheric flow is the result of a
complex interaction between the ocean
and the atmosphere.

6. Wind farm layout
Finding the “best” location of every single
wind turbine with respect to each other,
i.e. finding the “best” wind farm layout is
a 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 area
For reaching these targets we have to be
able to calculate wake effects properly.

7. Definition of wind turbine wakes – wake effects
The rotor of the wind turbine extracts energy
from the wind. This leads to a deceleration of
the flow downstream of the wind turbine. At
the same time, the flow gets more turbulent.
The region showing a wind speed deficit and
an increased turbulence intensity is called
wake of the wind turbine.
As the flow proceeds downstream there is a
spreading of the wake and the flow gradually
recovers to free stream conditions.
The reduced wind speed and increased
turbulence intensity in a wind turbine wake
influences an adjacent wind turbine in
downstream direction.
This is called wake effect.

8. Impacts of the wake effect
Wake effects have two significant
impacts on downstream wind
turbines:
a) Reduction of their power
production due to a reduced
wind speed
b) Reduction of their lifetime due
to increased structural
loading.
These impacts have to be taken
into account, when the layout of a
new wind farm is planned.
Source: b

9. Wake effects in a real offshore wind farm
Nysted 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
Rrotor
u ( 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 velocity
u ( 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 ein
gaußförmiger Verlauf des Geschwindigkeitsdefizits im Nachlauf einer
Anlage beobachtet;
Jensen schlug Multiplikation mit Kosinusfunktion vor
  Rrotor   1  cos9   
 
2
u ( 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 possible
b. The flow is considered to be incompressible
c. There are no external forces or pressure gradients
d. Gradients of the standard deviation of u are neglected
e. Viscous terms are neglected
Momentum equation: u
u
v
u


1  r uv 
x r r r
u 1 r v
Continuity equation: 
x r r
2 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
uv  
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 εa
2. 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 distances
rw: 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 
4u0  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 9.761  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 region
b) Boundary-layer approximation methods
c) Parabolic approximation of Navier-Stokes equations
d) Axisymmetric Navier-Stokes equations simplified for the far wake using a
given initial velocity profile
e) Combination of vortex methods for the calculation of near wake profiles with
axisymmetric Navier-Stokes equations for the far wake
f) 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 effects
b) To simulate atmospheric stability
c) To simulate complex terrain
d) To better simulate the wind turbine effect
e) To simulate the interference between the wind turbine wakes
f) To produce guidelines for a better calibration of the simple engineering
models
• RANS (Reynolds Averaged Navier-Stokes) solvers with 2-equation
turbulence models
a) They have been widely used for the last two decades  enough
experience in flow field applications is available
b) 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:
k
t 

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 model
Modification of these coefficients for atmospheric conditions results from the
measurements of the friction velocity:
2 2
u*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 an
incompressible 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  , tGx   , t  tdtd 
~ 

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 dissipation
S
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 in
the wake => ACL model

93. Results of wake flows simulated with an
actuator 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 an
actuator line model (II)

95. Velocity profiles in dependency on the distance
from the WT
land surface: sea surface:

96. Wake extension in vertical direction
land 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 data
Dashed line: Uniformly loaded actuator disk model
Solid line: Non-uniformly loaded actuator disk model
Black dots: actuator line model
Red 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 by
a. Oliver Bleich in the seminar “Aktuelle Forschungsthemen der
Energiemeteorologie”, Carl von Ossietzky Universität Oldenburg, summer
term 2011. Title: Wake modelling
b. John Prospathopoulos during the WAUDIT summer school 2010 in
Pamplona. Title: Wind turbines wakes modeling

102. Thank you for your attention!