Real Life Turbulence
and Model
Simplifications
Jørgen Højstrup
Wind Solutions/Højstrup Wind Energy
VindKraftNet
28 May 2015
Contents
• What is turbulence?
• Description of turbulence
• Modelling spectra.
• Wake turbulence – near and far
• Measured turbulence
• Distributions and gusts
What is turbulence?
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What is turbulence?
• In fluid dynamics, turbulence or turbulent flow is a
flow regime characterized by chaotic, stochastic
property changes. This includes low momentum
diffusion, high momentum convection, and rapid
variation of pressure and velocity in space and time.
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Not turbulence
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Turbulence
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Turbulence
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Generating turbulence
• Turbulence is generated by mechanical forces along the wind (Ucomponent).
• Turbulent eddies are generated (or destroyed) by thermal forces in the
vertical component.
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Height [m]
Surface friction decreases wind speed
Windspeed [m/s]
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Height [m]
Windspeed [m/s]
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Height [m]
Mixing of air with different speeds
Windspeed [m/s]
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Wind shear and turbulence?
• Assume increasing wind speed with height
• Assume that we have vertical turbulent
fluctuations
• A positive vertical fluctuation will bring an
air parcel to height “2” where the speed is
higher – therefore we get a negative
variation on the u-component, or in other
words the product uw will be negative
• A negative vertical fluctuation will bring an
air parcel to height “3” where the speed is
lower – therefore we get a positive variation
on the u-component, or in other words
again the product uw will be negative.
• The higher the wind shear, the larger the
absolute value of uw
Description of turbulence
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Taylor Hypothesis
•
•
•
•
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FROZEN TURBULENCE
Turbulent structures are advected past your observation point.
Frequency = propagation speed/wave length
You always see turbulence described as time series, but you
should think of turbulence as spatial structures
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Distribution
14
12
10
8
6
0
200
400
600
800
1000
Time [sec]
180
160
Distribution: We
count how many
times we have a
value in a certain
interval
140
120
100
80
60
40
20
0
7
8
9
10
11
12
13
Standard deviation
14
12
10
8
6
0
200
400
600
800
1000
Time [sec]
Variance = ([a1 – M]2+[a2 – M]2 +.. [an – M]2)/n =1.0
Standard deviation = Variance (also called RMS)
Turbulence intensity = std.dev./mean = 10%
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Spectra
•
•
•
•
Definition: Decomposition of variance on different frequencies (scales)
Length scale = U*f
We plot frequency*power spectra (area preserving in loglin plot)
Area under graph equals the variance
Spectra
Coherence
• “Correlation” of spectra measured at separate points
Modelling Spectra
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Spectra - models
• Kaimal – measurements up to 32 m. Fits neutral data.
• Modified Kaimal (IEC 61400-1), to work higher up than 32m
• Von Karmann – Analytical autocorrelation and coherence functions
(Mann model).
• Does not fit data (shape more pointed than Kaimal)
• Unstable/stable spectra
• High windspeed spectra
Kaimal Spectra
•
•
•
•
Definition: Decomposition of variance on different frequencies (scales)
Length scale = U*f
We plot frequency*power spectra (area preserving in loglin plot)
Area under graph equals the variance
Thermal turbulence
Thermally generated turbulence
appear at scales comparable to the
height of the boundary layer.
Daytime: 1-5 km
We see the opposite effect in stable
conditions (cooling from the
ground) where turbulence with
large length scales are being
suppressed, i.e. the peak moves
towards higher frequencies.
Højstrup: Journal of the Atmospheric Sciences, 1982, 39, pp. 2239-2248
Spectra length scales
Length scales (measurements Vindeby 48m)
• Large range of scales
• IEC 61400-12-1 ed.2
specifies the length scale
as constant 600m,
independent of height
(Kaimal spectrum fixed at
30m height)
• IEC 61400-12-1 ed.3
specifies length scale as
constant 1200 above 60 m
High wind speed data
Additional energy at 3000m length scale. Can be modelled, but gets complicated.
Højstrup, Larsen, Madsen: AMS 9th symposium on turbulence and diffusion, 1990, pp.305-308
Wake turbulence – near and far
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Turbulence in wakes
Strong shear in upper part of wake
creates large amount of turbulence
Turbulence in wakes
Hubheight wake at 2D (Nørrekær Enge II)
• Wakes generate large shears at length
scales comparable with the rotor size
• Consequently wake turbulence is
generated at much smaller scales than
“ordinary” turbulence.
Turbulence in wakes
Turbulence has a long memory
The turbulence “remembers”
upstream conditions much
longer than the average wind
speed.
Low
turbulence
Here exemplified by the island of
Gotland, but might just as well
be an offshore wind farm.
High
turbulence
Measurements by RAF C130.
Low
turbulence
Measured turbulence
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Quick and dirty turbulence intensity
• Scaling velocity u*
• Along wind: σu/u* = 2.5
• Lateral:
σv/u* = 2
• Vertical:
σw/u* = 1.5
• Log profile: U = u*/k * ln(z/z0), k=0.4
=>
• Turbulence intensity: σu/U = 1/ln(z/z0)
• Correlation coefficient <uw>/(σuσw) = -0.3
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Quick and dirty turbulence intensity
• Land, height 70m, grass surface, roughness length = 0.03 m
• => Turbulence intensity: 13%
• Offshore, height 70m, roughness length = 0.0001 m
• => Turbulence intensity: 7%
• Forest, height 70m, grass surface, roughness length = 1 m
• => Turbulence intensity: 24%
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Offshore turb. Intensities. Fetch 100km
Horns Rev 62m
0.11
0.10
0.09
0.08
Turb.int.
0.07
0.06
0.05
Vertical bars denote
+-1 std.err.on mean value
0.04
0.03
0.02
0.01
0.00
5
10
15
20
25
30
Wind speed [m/s]
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Measurement of turbulence
• Because of instrument limitations you do not measure the whole
turbulence intensity.
• Lowest frequency in measurement f=1/T
• Highest frequency in measurement determined by averaging time and
instrument time constant.
• For power curve fluctuations (and some loads) the relevant way to
measure would be to average wind speed over the rotor area.
Therefore measured turbulence will always be higher than the
turbulence relevant for the rotor.
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Measurement of turbulence
• Cup anemometer timecst= L/U. Typical 0.2 sec (L=distance constant,
2m, U wind speed 10m/s)
• Typical timecst for 100m rotor at 10 m/s: 10 sec
At 10 m/s you would measure 93% of the total std.dev. by
cupanemometer (600 sec avg.)
At 10 m/s you would measure 60% of the total std.dev. by a sensor the
size of a 100m rotor (600 sec avg).
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Nice graph showing connection
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Distributions and gusts
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Turbulence is not Gaussian
Wind speed fluctuations are nearly Gaussian
Accelerations are not Gaussian
THEREFORE ATMOSPHERIC TURBULENCE IS NOT A RESULT OF A GAUSSIAN PROCESS
Gust from turbulence intensity
Simulation
Measured
Theory:
Gust from turbulence intensity
3.5
1 hour
3.0
Gust factor: (Umax-Uavg)/u
10 minutes
2.5
2.0
1 minute
1.5
1.0
Height 50m
Solid line: Kaimal spectrum
Dashed line: JH spectrum
0.5
0.0
0
10
20
30
40
Wind speed [m/s]
50
60
Thank you for your attention
Højstrup Wind Energy & Wind Solutions
www.wind-solutions.com
Jorgen@hojstrup.eu