1. Wind Energy Systems MASE 5705
Spring 2017, Feb , L7 + L8

2. 2.1 Atmospheric boundary layer (ABL) and surface layer turbulence
L7 and L8
1. Key points of L5+L6
2. Turbulence ( pp. 39 – 52)
2.1 Atmospheric boundary layer (ABL) and surface layer turbulence
2.2 Turbulence intensity (TI)
2.3 Autocorrelation and Power Spectral Density (psd)
2.4 Preview of later lectures on turbulence
2.5 Recommended reading

3. 3. Ch. 3 (Aerodynamics of Wind Turbines) 3
3. Ch. 3 (Aerodynamics of Wind Turbines) 3.1 One-dimensional momentum theory and Betz limit 4. Project 2

4. 1. Key points of L5+L6

5. Key Points of L5+L6
Bin width, bin probability or frequency and the corresponding pdf
For arbitrary values of bin width ΔU,
2. Estimation of AEO (Annual Energy Output) of a proposed site
a) When bin database is available;
That is, (Umax, Umin, no. of hours in each bin)
b) When only is known

6. For case (2a), we use the method of bins.
For case (2b) , we create our own bins; then we
estimate the number of hours in each bin by the Rayleigh estimation.
3. Estimation of AEO of a specific wind turbine in a
proposed site
a) Specific wind turbine ≡ a wind turbine with
specified Uci, Uco and power curve.
b) power curve ≡ an official document that gives
measured power output vs wind speed (other
details are similar to (2a) and (2b))
For example see next table and figure, and also consider the method of bins:

7. Measurements of site Bin Data and WT power
output data for Mod MW WT *
* Reasons for such a wide bin width are not known. Lower values are more likely than higher values. Take Ū18= m/s 7

8. (Measured) Power Curve for Mod-2 2. 5 MW HAWT (Uci = 6
(Measured) Power Curve for Mod MW HAWT (Uci = 6.25 m/s , Uco = 22.4 m/s) 8

9. Method of Bins

10. 2. Turbulence (pp )

11. "Turbulence is a dangerous topic which is often at the origin of serious fights in the scientific meetings devoted to it since it represents extremely different points of view, all of which have in common their complexity, as well as an inability to solve the problem. It is even difficult to agree on what exactly is the problem to be solved.” Marcel Lesieur, Turbulence in Fluids , Springer 2008, The Netherlands

12. One is usually at a loss to calculate with greater accuracy, say ±50%" Lumley, AIAA Journal, Valid even today

13. 2.1 Atmospheric boundary layer (ABL) and surface layer turbulence

14. 2.1 (Ambient Boundary Layer (ABL) and Surface Layer Turbulence
Of concern to wind turbines is turbulence in the lowest levels of the ABL, also referred to as surface-layer turbulence. The ABL thickness is not a precisely defined quantity; it varies “from a few hundred meters to several kilometers.” For practical purposes, surface layer encompasses operational heights up to 200 ft, say 10% of the boundary layer, and the rest is called outside layer.

15. Surface turbulence has been an actively researched area of the past 30 years by meteorologists, and engineers associated with dynamic loading on exposed structures and wind turbines. As for modeling, “ classical turbulence theory” is keyed to surface layer conditions on the basis both of phenomenological and analytical considerations as well as other guidelines such as continually updated Engineering Science Data Unit (ESDU) series.

16. Basically, the von Karman turbulence model with empirically adjusted parameters correlate well with test data and it is widely used. Therefore in this lecture, we follow the text and briefly describe turbulence intensity and von Karman power spectral density (PSD), which is a common frequency-domain description of turbulence, Equation (2.27), p. 43). This PSD is usually referred to as von Karman turbulence model.

17. 2.2 Turbulence intensity (TI)

18. In this lecture, we follow the text (Ch
In this lecture, we follow the text (Ch. 2): Just mention the basic equations and apply them. We address turbulence in some detail, well beyond the text, after covering Ch. 3 (Aerodynamics of Wind Turbines) and Ch. 4 ( Mechanics and Dynamics).

19. U = Ū + u Ū = wind speed (x-direction)
u = fluctuating (random) component in the x- direction (at present we neglect lateral and vertical components)

20. z(vertical) x
lateral or
y-direction
disk plane x U

21. In General, For the present, we follow the text:
Wind speed in the x direction, which is perpendicular to the disk (γ=0)

22. Turbulence Intensity TI (2.23 p. 40)
static loads in the system.

23. Text,
p. 41 u Ū Ū u
Sample wind data (fig, 2.14 , 41)

24. U

25. (y, z) ≡ rotor plane
x ≡ +ve in the downwind direction

27. Wind Fluctuations and Average Power
U = Ū + u ?

30. E(u) = 0 E(un) = 0 for n odd (why?) E(u2) = σU2

35. (2.54) p. 58
(2.64) p. 60
(2.54) p. 58 and (2.63) p. 60 in 2nd Edition

36. (2.23) p. 40 2nd Edition
(2.23), p. 40
No U-bar^2

38. (p. 61)
p. 61, 2nd Edition

39. Variation of parameters with
Weibull k shape factor
Table 2.4 P. 61 k Ke
1.2
0.837
3.99 2
0.523
1.91 3
0.363
1.40 5
0.229
1.15
Compare Ke = 1+3(TI)2 = 1+3(.837)2 = (.523)2 = (.363)2 = (.229)2 = 1.16

41. For details, see
Random Data, Analysis of Measurement Procedures (3rd Edition, 2000), Wiley J.S. Bendat and A.G. Piersol A thorough and easy-to-read account of random processes.

42. Each Ui(t) represents a unique set of measurements (not likely to be repeated) Ui(t) = sample function or a random signal Ui(t) = wind speed (in our case) {Ui(t)} ≡ {U(t)} The ensemble {U(t)} describes a random or stochastic process of wind speed, and the sample function Ui(t) belongs to this process.

43. UN(t)
U3(t)
U2(t)
U1(t)

44. 2.3 Autocorrelation and Power Spectral Density

45. Autocorrelation
We consider a stochastic process {U(t)} and consider two time instances ‘t’ and ‘t + ’. Given {U(t)}, we cannot predict {U(t + )}. But {U(t)} and {U(t + )} are ‘related’ or correlated. This correlation is described by the autocorrelation function RUU(t,) RUU(t, t + ) = E[U(t) U(t + )]

46. Wind Speed measurements belong to a Stationary random process
Wind Speed measurements belong to a Stationary random process. That is, Ū(t) = constant, E[U2(t)] = constant Autocorrelation function t2 – t1 depends only on time difference: RUU(t1 , t2) = RUU(t2 – t1) RUU() = E[U(t)U(t+)]

47. While the autocorrelation function characterizes turbulence as a function of time or time lag  in the time domain, the corresponding description in the frequency domain leads us to the power spectral density function. That is, the power spectral density function characterizes turbulence as a function of frequency in the frequency domain.

48. It is proved that (Wiener-Khinchine relation) autocorrelation function and power spectral density function are a pair of Fourier transform. (N. Wiener in the United States and A.I. Khinchine in the USSR).

49. UU UU UU UU

51. Turbulent Wind. Velocity about Mean
For future reference we also define the Gaussian pdf (2.25), p. 42 2nd Edition
(2.25) p. 41 p
Fig. 2.15
p. 41
pdf -1 1 -4 4
Turbulent Wind. Velocity about Mean

53. Turbulence Modeling
Two important parameters in turbulence modeling

54. (p. 43)
p. 43, 2nd Edition

55. (2.27) p. 43
(2.27) p. 43, 2nd Edition

59. , Fig. 2.16, p. 42

60. Power spectral density functions, Fig. 2.`17, p. 44

61. von Karman Turbulence Model
Based on first principles;
No empirical constants;
For above about 150 m (height z > 150 m); good correlation with test data;
For z < 150 m, some terrain-sensitive deficiencies.

62. B. 2.13 (p. 620) Power Spectral Density Estimation
Similar to Equation 2.27 in the Text (p. 43) , the following empirical expression has been used to determine the power spectral density (psd) of the wind speed at a wind turbine site with a hub height of z. The frequency is f(Hz), and n (n = f z / Ū ) is a non-dimensional frequency. Where Plot the power spectral density of the wind at a site where the surface roughness is 0.05 m (z0) and the hub height is 30 m (z), and the mean wind speed Ū is 7.5 m/s.

63. Values of surface roughness length for various types of terrain
Table 2.2 (p. 46)
Values of surface roughness length for various types of terrain
Terrain Description
Zo (mm)
Very smooth, ice or mud
0.01
Calm open sea
0.20
Blown sea
0.50
Snow surface
3.00
Lawn grass
8.00
Rough pasture
10.00
hallow field
30.00
Crops
50.00
Few trees
100.00
Many trees, hedges, few buildings
250.00
Forest and woodlands
500.00
Suburbs
Centers of cities with tall buildings

65. next:

66. b) Plot of S(f) vs f S(f) ms2 /hz f(hz)10 1
0.1
0.01
0.001
f(hz)
S(f) ms2 /hz
(Observe the rapid decrease in turbulence energy with increasing frequency.)

67. bending

68. An example:
bending

69. Around 4 Hz The turbulence has little energy!
100 10 1
0.1
0.01
0.001
f(Hz)
S(f) m/s2 /Hz
fn= 4 Hz

70. b ) Why is it that blades are sensitive to gust
(turbulence) excitation?
The answer is keyed to the difference between turbulence PSD in space-fixed non-rotating coordinates (hub element) and turbulence psd in blade-fixed rotating coordinates (blade element). Hub element and blade element feel turbulence differently.
See next section on preview of later lectures on turbulence. We will study this difference in detail under rotational sampling. pp. 143, 330.)

71. 2.4 Preview of later lectures on turbulence
Ch. 3 (aerodynamics of wind turbines, p. 143) and Ch. 7 (wind turbine design, p. 330)

72. In later lectures on turbulence we cover the following:
Classical theory of turbulence, that is, theory of homogeneous and isotropic turbulence.
Revisiting von Karman model and extensions
Rotational sampling, that is, why turbulence seen by the hub element differs from that seen by a blade element.

73. For later reference: Homogeneous and isotropic turbulence

74. Turbulence is homogenous
Turbulence is homogenous. That is, its statistical properties do not vary form point to point in the field, and thus all the functions described are independent of the location of the origin in the field.

75. The concept of isotropy simplifies the description of turbulence even further. If a turbulence field is isotropic, its statistical properties are independent of direction in the field, and thus they do not change with rotation of the coordinate axes.

76. A Descriptive Account of Rotational Sampling

78. Predicted and Measured Longitudinal Turbulence PSD
Experiment
___ Von Karman
Rotational-coordinates
Fixed-coordinates
(Radius 24.5 m, Rotation Rate = Hz)

79. This figure shows the measured and predicted psd (von Karman) of longitudinal turbulence in both space-fixed and blade-fixed coordinates. The model refers to a 35-m radius HAWT and the blade element has a local radius of 24.5 m, 70% of radial location. Two sets of predictions from von Karman model –with and without rotational effects– are shown.

80. Since the area under the PSD curve represents the turbulence energy, which is the same in both coordinates, the turbulence energy is transferred from the low-frequency region (P<1) to the high-frequency region with PSD peaks at 1P, 2P, etc., where P=0.625 Hz. The correlation in this graph brings out two key points of a much broader significance.

81. First, the classical turbulence theory that is judiciously modified to account for the local conditions provides a means of modeling surface-layer turbulence seen by a turbine blade element.

82. Second, the von Karman model in rotating coordinates correlates qualitatively well with the measurements.

83. (It is available on pass-word-protected Telesys.)
2.5 Recommended Reading
The Nature of Wind
R.I. Harris
(It is available on pass-word-protected Telesys.)
Although some 40 years old, this 25 - page article gives a through description of classical turbulence theory and its adaptation to modeling turbulence for structural applications.

84. 3. Ch. 3 (Aerodynamics of Wind Turbines). 3. 1
3. Ch. 3 (Aerodynamics of Wind Turbines) One-dimensional momentum theory and Betz limit

85. The assumptions are
Fluid flow is incompressible,
No frictional drag,
Uniform thrust over entire disk area,
Nonrotating wake,
Steady-state operation: constant wind speed over the turbine disk and constant rotational speed.

86. Momentum or actuator disk theory of wind turbines

89. +ve direction U - vi U-w U (1) (2) (3) (4) disk F = force on the
fluid
(2)
(3)
(4)
disk
F = force on the
disk = Thrust
= T

90. +ve direction
U-w U
U - vi

92. (1)
(2)
(3)
(4)
Turbine does negative work on the fluid ≡ extracts energy from the fluid
Fluid slows down
 Tube expands

98. Limitations of Betz’s (Momentum) Theory
Windmill or windmill brake state

100. Power is extracted from the fluid by the wind turbine Windmill State
Power is extracted from the fluid by the wind turbine
Windmill State
Velocity slows down (tube expands)  WT brakes the fluid
 Windmill brake state

101. (p. 93)

102. (p. 95)
Operating parameters for a Betz turbine; U, velocity of undisturbed air; U4 , air velocity behind the rotor, Cp power coefficient, CT thrust coefficient

103. In L9 we will consider an extension of the momentum theory with rotational effects on the fluid; p. 96

104. Project 2: Not assigned in 2014

105. B.2.11 Actual Data Analysis and Power Prediction, P. 619
Based on the spreadsheet (MtTomData.xls) which contains one month of data (mph) from Holoyke, MA. determine:
a) The average wind speed for the month
b) The standard deviation
c) A histogram of the velocity data (via the method of bins- suggested bin width of 2 mph)
d) From the histogram data develop a velocity-duration curve
e) From above develop a power-duration curve for a given 25 kW Turbine at the Holyoke site.

106. For the wind turbine,
assume:
P = 0 kW
P = U3/ 625 kW
P = 25 kW
0<U<6mph)
6<U <25 (mph)
25<U<50 (mph)
50 <U(mph)
f) From the power duration curve, determine the energy that would be produced during this month in kWh.

107. Verify and explain:

111. f) The total energy produced can be determined from integrating the product of the turbine power and the numbers of hours of operation at that power level, yielding an annual energy production of 2474 kWh.