Different models to simulate wakes in wind farms are reviewed [1], [2]. Special emphasis is put in those elaborated by the authors. First, kinematic models, based on self-similar velocity deficit profiles and global momentum conservation, are briefly revisited.  Then, field models of a different degree of complexity, which solve numerically the flow equations with several types of simplifications and turbulence closure models, are analyzed and compared. Models like Ainslie´s that uses a simple algebraic expression for the eddy viscosity, and are still widely used, are compared to other more complete models like UPMWAKE, which uses a k-ε turbulence closure.  Ground effect, simulation of atmospheric conditions, atmospheric stability, simulation of turbulence characteristics, wake superposition and computer time consumed, will be used as terms of comparison.  Results of an algebraic Reynolds Stress Model will also be presented, and compared with those of the k-ε turbulence closure. Then a comparison will follow of elliptic and parabolic models, and how boundary conditions at the disk are imposed. A discussion on the displacement of the wake origin for parabolic models will be made. The simplicity and computer time consumed to simulate wake superposition in whole large wind farms of the elliptic and parabolic models will be analyzed and compared. Results of a large eddy simulation model will also be presented and compared to those of previously mentioned models [3]. Wake meandering, and its role in wake turbulence generation when using simpler eddy viscosity models will be analyzed, and compared with other mechanisms of turbulence generation. The possibility and convenience of introducing unsteady terms in parabolic field models of wind farms, like UPMPARK, and thus simulate wake meandering will be discussed. Results of UPMPARK with and without meandering will be compared with available experimental data

Crespo Martínez, Antonio

García García, Javier

Jiménez Alvaro, Ángel

Migoya Valor, Emilio

Prieto Ortiz, Juan Luis

Archivo Digital UPM

Study of isolated wakes and their superposition in wind farms, using different turbulence models

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

STUDY OF ISOLATED WAKES AND THEIR 
SUPERPOSITION IN WIND FARMS, USING
DIFFERENT TURBULENCE MODELS
V European Conference on Computational Fluid Dynamics 
ECCOMAS CFD 2010
Mini-symposium on "Computational Wind-Farm-Wake-
Aerodynamics” Lisbon, Portugal, 14–17 June 2010
A. Crespo,  E. Migoya,  J.L. Prieto, A. Garcia-Uceda ,  A. Jiménez 
and J. García
Grupo de Investigación  de Mecánica de Fluidos Aplicado a la  Ingeniería 
Industrial
ETSI Industriales, 
Universidad Politécnica de Madrid
-Motivation of the study
-Kinematic Models
-Field models
-Wake meandering
-Parabolic model for wind farms UPMPARK
-Unsteady version of UPMPARK to simulate wake meandering
CONTENTS
-Simple kinematic model to estimate turbulence generated by wake
meandering. Comparison with unsteady UPMPARK
-Final Conclusions
IMPORTANCE OF WIND ENERGY
08/07/2011 3
MOTIVATION OF THE STUDY OF WIND TURBINE WAKES
-Typical wind farms consist of several wind turbines of about 1 MW
power
- Due to the cost of land and civil works, wind turbines tend to be built as
closely as possible to each other
-It is necessary to locate them adequately to minimize interference effects
-The momentum deficit and the increased level of turbulence created by
turbines in a wind farm may cause a reduction in power output and
08/07/2011 4
unsteady loads on other machines
-There are significant energy losses in arrays spaced at less than 7 turbine
diameters, although the most importante effect is that turbulence may
increase in arrays, sufficiently to cause measurable damage due to fatigue
and dynamic loading
-Also the vertical velocity profile that causes unsteady loads on the blades
may be affected by the wake. loads.
KINEMATIC MODELS
-These methods require the solution of ordinary differential equations, and
are valid both for single wakes and wind farms.
-In spite of their simplicity they may retain many of the physics of the
problem
-They are based on imposed velocity deficit profiles, that are self-similar
in the far wake
- Use global momentum conservation to relate the total velocity defect to
08/07/2011 5
the thrust force over the wind-turbine.
-Wake growth is due to both ambient turbulence and turbulence generated
in the wake itself
-Ground effect is simulated by imaging techniques
-Superposition of wakes in wind farms are both linear (Lissaman et al.)
and non-linear (Katic et al.)
-There are also analytical solutions (Larsen et al.) based on the classical
solution of an axisymmetric wake .
FIELD MODELS
-Solve the flow partial differential equations with different degrees of
approximation to calculate flow conditions at grid points
-Some of them make the equations linear
- They may use either elliptic or parabolic modelization of flow
- Use different models for turbulence closure:
* Ainslie uses an analytical model for the eddy viscosity which 
also includes a contribution from ambient turbulence.
08/07/2011 6
* UPMPARK uses a k-ε model 
* Cabezón et al. Mason etc, use different modifications of k-ε
* Gómez et al. use a Reynolds Stress  algebraic model
* More recently Large Eddy Simulation is used to calculate the 
eddy viscosity.  
- In this presentation we will include a non-steady version of  UPMPARK 
to study wake meandering contribution to wake turbulence generation 
shear stresses are described using an
eddy viscosity closure scheme in which the eddy viscosity is 
represented by a simple analytical form based
FIELD MODELS PRESENTED RECENTLY AT THE EUROMECH 
COLLOQUIUM 508, MADRID, OCTOBER 2009 
- 4 PAPERS ON RANS CLOSURE MODELS:
Rethore et al., Cabezón et al., Migoya et al. and Crasto et al.
-8 PAPERS ON LES CLOSURE MODELS
Jiménez et al,, Calaf et al., Hahm et al., Steifeld et al., Sanderse et al., 
Troldborg et al., Larsen et al., Ivanell et al. 
08/07/2011 7
- 15 PAPERS ON WAKE MEANDERING
shear stresses are described using an
eddy viscosity closure scheme in which the eddy viscosity is 
represented by a simple analytical form based
AINSLIE’S MODEL
- Parabolic eddy viscosity model (EVMOD) which assumes     
axis-symmetric wake flow.
- Only the continuity and the axial momentum equations have 
to be solved. 
- Eddy viscosity is an average value over a cross-section, and 
variations in turbulent properties across the wake cannot be estimated 
from the model.
- The model is fairly simple and gives reasonable results when 
08/07/2011 8
compared with wind tunnel experiments.
- For large-scale experiments the results are corrected by taking into 
account meandering effects
- Velocity defects are predicted, but the model does not 
provide information about turbulence characteristics
WAKE MEANDERING AND TURBULENCE GENERATION
-Wake meandering is the large scale movement of the entire wake due to 
eddies that are large in comparison with the size of the wake itself. 
-According to the Dynamic Wake Meandering Model, Larsen et al.,  this 
wake meandering mechanism will not only increase the (apparent) 
turbulence intensity, but may also significantly change the structure of the 
(apparent) turbulence as seen by downstream turbines. 
-They calculate velocity deficit profiles using Ainsliés model, that are 
moved as a passive tracers by ambient low frequency turbulence
08/07/2011 9
- Is the wake meandering taken into account by conventional closure 
models? 
-In our LES model it was barely observed. 
-In the following we will retain unsteady terms in the UPMPARK k-ε
model to include oscillations of low frequency of the incident wind 
turbulence as seen by downstream turbines,
UPMPARK TO MODEL WAKES IN WINDFARMS
The wind turbine is supposed to be immersed in a non-uniform basic flow 
corresponding to the surface layer of the atmospheric boundary layer
-UPMPARK model is a CFD code that describes the diffusion of multiple
wakes in the atmospheric surface layer parameterized by Monin-Obukhov
scaling,
-The equations describing the flow are the conservation equations of mass,
momentum, energy, turbulence kinetic energy, and the dissipation rate of the
turbulence kinetic energy
-The modelling of the turbulent transport terms is based on the k-ε method for
UPMPARK TO MODEL WAKES IN WINDFARMS
the closure of the turbulent flow equations.
-A parabolic approximation was made. Turbulent diffusion and pressure
variation in the main flow direction are neglected.
-The developed wake model is three-dimensional and pressure variations in
the cross-section have to be retained in order to calculate transverse
velocities.
-Finite-difference methods were used in the discretization of the
equations.
-This set of equations has been solved numerically using the
SIMPLE algorithm proposed by Patankar and Spalding (1972).
-The equations were solved numerically by using an alternate-
UPMPARK TO MODEL WAKES IN WINDFARMS
direction implicit (ADI) method.
UPMPARK TO MODEL WAKES IN WINDFARMS
-Consistently with the parabolic approximation, boundary
conditions are applied in the expanded wake, where wake
pressure is approximately the ambient pressure
-UPMPARK is less accurate than LES models, but consume less computing
time, and can be used, with moderate computing resources to study the
behavior of a whole wind farm.
-The code has two options; there is a simplified option in which only the
EXTENSION OF UPMPARK TO MODEL WAKES IN WINDFARMS
convective terms containing the main flow velocity are retained; in the more
complicated version the convective terms containing the velocity
components perpendicular to the main flow direction are also retained.
-Running downwind in the numerical marching procedure associated with
the parabolic model, each turbine found at any cross-section of the farm
acts as a source (or sink) of the three velocity components, k, and ε.
0,0 m/s
UPMPARK TO MODEL WAKES IN WINDFARMS
-4,5 m/s
Velocity defect distribution in a wind farm. 
Horizontal view at hub height
Wind is blowing from the left
Typical distribution of k in a wind farm. 
0 m2/s2
12,87 m2/s2
-Unsteady terms have been incorporated using a simplified version of
UPMPARK that considers convection only in the instantaneous flow
direction, whose average is in the direction in which the wind turbines are
aligned
x direction
UNSTEADY UPMPARK TO MODEL MEANDERING WAKES IN 
WINDFARMS
-The incident flow oscillates around the x-axis in the horizontal direction.
ε,,
...... 000
kVJ
y
J
V
x
J
V
t
J
JV
t
J
x
yx
=
+
∂
∂
+
∂
∂
+
∂
∂
=+∇+
∂
∂ r
θ
V0=Vmeandering
Vhub=Vcalculation
-A stochastic simulation, based in Shinozuka (1971), has been implemented to
take into account the horizontal oscillations of the incident wind.
-Kaimal and Von Karman expression could be employed to obtain spectra for
x and y directions, IEC-61400.
Li4 2σ
( )
6
11
2
2
8.701
4










+
=
x
hub
x
x
xKarmanVon
V
fL
V
L
fS
σ
NON-STEADY UPMPARK TO MODEL WAKE MEANDERING
-Input data: the average incident wind direction, the average and variance
incident wind speed and the integral scale.
( ) yxi
V
fL
V
fS
hub
i
hub
i
iKaimal ,
61
3
5
=






+
=
  hub
( )
6
11
2
2
2
8.701
18912














+














+
=
hub
y
hub
y
hub
y
y
yKarmanVon
V
fL
V
fL
V
L
fS
σ
Model and Component Kaimal x Kaimal y Von Karman x Von Karman y
Standard deviation, σi σx 0.8σx σx σx
Integral scale, Li Lx 2.7Lx/8.1 Lx Lx
-Stochastic time series generators are based on the integration of velocity
spectrum. Therefore, summations of harmonics, with a random phase and
amplitudes which follow one of the previously mentioned spectral density
function, are used:
yxiktffSfV
upjj
jj
jjiji ,)2cos()( =+∆= ∑
=
=
pi
1=
hub
highest
V
Wf
NON-STEADY UPMPARK TO MODEL WAKE MEANDERING
k is a random phase distributed between 0 and 2π,
jup and jdown mean the index of higher and lower filtered scales and
W is a distance of the order the maximum wake width
down
22
yxmeandering VVV += 







=
y
x
V
V
arctanθ
Instantaneous profiles
Distribution of kDistribution of velocity deficit
NON-STEADY UPMPARK TO MODEL WAKE MEANDERING
Distribution of kDistribution of velocity deficit
-In the following figures are compared the averaged results with and without
meandering.
-Typical non-linear profiles that show a maximum are expected to be softened
because of the meandering effect.
NON-STEADY UPMPARK TO MODEL WAKE MEANDERING
-Comparison with experiments will be made in following works.
∑
=>
∑
=>
Without meandering
Distribution of velocity deficit
Meandering stochastic simulation
Distribution of velocity deficit
NON-STEADY UPMPARK TO MODEL WAKE MEANDERING
Distribution of k
Without meandering
Distribution of k
Meandering stochastic simulation
Without meandering Meandering stochastic simulation
NON-STEADY UPMPARK TO MODEL WAKE MEANDERING
Distribution of velocity deficit Distribution of velocity deficit
Distribution of k
Without meandering
Distribution of k
Meandering stochastic simulation
COMPARISON WITH EXPERIMENTS
Take part in inter-comparisons exercises, Danish Noordtank, Vindeby and
Tjareborg sites.
Initially, the intention is to compare speed deficits and turbulent kinetic
energy increases.
COMPARISON WITH EXPERIMENTS
 Preliminary results to Vindeby:11x Bonus 450kW, h=37m D=35m
COMPARISON WITH EXPERIMENTS
 Preliminary results to Vindeby to be compared later with experimental results:
θ=140º, V=10.4m/s, Lx=65m, σx=2.7m2/s2
Velocity deficit
-6
-5
-4
-3
-2
-1
0
0 20 40 60 80 100 120 140 160 180
Wind direction (m)
V
e
l
o
c
i
t
y
 
d
e
f
i
c
i
t
 
(
m
/
s
)
Without meandering Meandering stochastic simulation
-5.6 m/s                                                                   0  m/s          -5.4   m/s                      0 m/s  
Without meandering              Meandering stochastic simulation
-6
-5
-4
-3
-2
-1
0
0 20 40 60 80 100 120 140 160 180
Wind direction (m)
V
e
l
o
c
i
t
y
 
d
e
f
i
c
i
t
 
(
m
/
s
)
Without meandering Meandering stochastic simulation
COMPARISON WITH EXPERIMENTS
 Preliminary results to Vindeby to be compared later with experimental results:
θ=140º, V=10.4m/s, Lx=65m, σx=2.7m2/s2
Turbulent kinetic
energy increase
0
1
2
3
4
5
6
0 20 40 60 80 100 120 140 160 180
Wind direction (m)
T
u
r
b
u
l
e
n
 
k
i
n
e
t
i
c
 
e
n
e
r
g
y
 
i
n
c
r
e
a
s
e
 
(
m
2
/
s
2
)
Without meandering Meandering stochastic simulation
0 m2/s2 5.8 m2/s2 0 m2/s2 4.2 m2/s2
Without meandering              Meandering stochastic simulation
0
1
2
3
4
5
6
0 20 40 60 80 100 120 140 160 180
Wind direction (m)
T
u
r
b
u
l
e
n
 
k
i
n
e
t
i
c
 
e
n
e
r
g
y
 
i
n
c
r
e
a
s
e
 
(
m
2
/
s
2
)
Without meandering Meandering stochastic simulation
TURBULENT KINETIC ENERGY CREATED BY WAKE 
MEANDERING
-Recent wake models use Ainslie´s simple model and postulate that a significant 
part of the turbulence is what is called apparent turbulence intensity, that is 
associated to the oscillating velocity components as a consequence of 
meandering, and will not be present in the meandering frame of reference
-A simple kinematic model is proposed to show that this turbulence can not be 
very significative
KINEMATIC MODEL FOR WAKE MEANDERING
A Gaussian velocity defect profile is assumed, that makes periodic lateral
displacements
b
xx
VV
2
0
0 exp 












 −
−∆=∆
Average velocity defect
Apparent Turbulent Kinetic 
energy dtVV
T
k
Vdt
T
V
T
t
bx
T
T
2
0
0
0
)(
2
1
1
)
2
sin(
∆−∆∫=∆
∫ ∆=∆
=
pi
ε
KINEMATIC MODEL FOR WAKE MEANDERING
 The non-dimensional results turn to depend only on the non-dimensional lateral
displacement ε ε=0, ε=0.5, ε=1, ε=1.5, ε=2,
ε=2.5, ε=3,
Non-dimensional Average 
velocity defect
Non-dimensional Apparent 
Turbulent Kinetic energy, 
maximum value  is 0.09
COMPARISON OF KINEMATIC MODEL AND UPMPARK 
MODEL FOR WAKE MEANDERING
 Now the turbulent kinetic energy associated to wake meandering will be
obtained from UPMPARK calculations and will be extended over the whole
meandering period
Average velocity defect Vdt
T
V
T
T 0
1
1
lim ∫ ∆=∆ ∞→
Apparent Turbulent Kinetic 
energy dtVV
T
k
T
T
2
0
)(lim ∆−∆∫=∆ ∞→
The  results obtained are qualitatively quite similar   to those obtained with the 
simple kinematic model.  This may be an indication that the assumption that the 
velocity deficit profile is moved around like a pasive scalar may be correct
COMPARISON OF KINEMATIC MODEL AND UPMPARK 
MODEL FOR WAKE MEANDERING
ε=0, ε=0.5, ε=1, ε=1.5, ε=2,
ε=2.5, ε=3,
Kinematic modelUnsteady UPMPARK
By comparison of the two figures the  equivalent lateral displacement of the wake 
using UPMPARK  is ε=0,5, half the wake width 
FINAL CONLUSIONS
At the present stage UPMPARK could be a useful tool for optimization of wind
farms. It gives information on both wind velocity and the characteristics of
turbulence generated in the wind farm. So that the energy production of every
wind turbine and the incident turbulence characteristics that produce fatigue loads
can be estimated in a reasonable computer time, for every array configuration.
Unsteady effects have been incorporated in UPMPARK to reproduce
meandering effects. As a consequence of the wake oscillations due to meandering
the average of velocity and turbulent kinetic energy profiles are softened; in
general the profiles are shorter and thicker. This effect is more important for the
turbulent kinetic energy
 But on the other hand additional turbulent kinetic energy created by meandering
should be considered. A kinematic model is proposed based on oscillations of
velocity defect profiles as passive scalars. It gives good agreement with
UPMPARK results, but shows that this additional turbulence is small because the
equivalent amplitude of meandering oscillations is also small.


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Study of isolated wakes and their superposition in wind farms, using different turbulence models

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