A primer of turbulence at the wind turbine rotor

Solar Energy Vol. 41, No. 3, pp, 281-293. 1988 Printed in the U.S.A,

0038-092X/88 $3.00 + .00 Copyright ~ 1988 Pergamon Press pie

A PRIMER OF TURBULENCE AT THE WIND TURBINE ROTOR J. R. CONNELL Pacific Northwest Laboratory, P. O. Box 999, Richland, WA 99352, U.S.A. Abstract--Some wind turbine fatigue problems are caused by the turbulent wind, and there is an engineering need to understand both qualitatively and quantitatively the characteristics of the wind that are causing untimely damage to wind turbine components. To help meet this need, this article describes, in simple terms, turbulent motion: where it is generated and how it changes with differences in terrain, height above the ground, meteorological conditions, time, and distance from its source. For quantitative information on conventional atmospheric turbulence, the reader is referred to the classic literature, and • for current information specialized for wind turbine dynamics the reader is guided to the modern literature of the American and European wind energy programs. The article concludes with examples of several computational methods and several measurement methods of modeling the turbulent wind that loads and stresses turbine rotors.

1. INTRODUCTION Atmospheric turbulence causes important fluctuating aerodynamic forces on wind turbines[I]. The stress and power output fluctuations of a wind turbine are strongly a function of both the rotation of the rotor and the moment-by-moment variations of turbulent wind velocity in both space and time (see Fig. 1). The major factors in quantifying the turbulence effects on turbines are blade rotation through the wind field that varies in space and time and the different character of the turbulence with different atmospheric conditions and local terrain complexity. Several comprehensive papers, monographs, and books are available from which to study the current state of knowledge about the atmospheric boundary layer[e.g., 2-5]. The reader who wishes either a substantial tutorial or precise values is referred to those works. This article schematically summarizes relevant features of the wind and their relation to the response of a wind turbine rotor for practitioners of wind turbine design, construction, siting, and operation. The reader is encouraged to use it as a guide and stepping-stone to the necessarily more extensive articles referenced above. Similarly, it is a guide to the specialized literature developed in the Department of Energy Wind Turbine Technology Program to characterize the wind as it is experienced from the rotating frame of reference of wind turbine rotors. This article briefly describes the features of the turbulent atmosphere that are important in understanding the fluctuating forces on a wind turbine. First, where and how turbulence occurs are indicated in a description of the atmospheric boundary layer. This overview is followed by brief definitions of some basic propertie.s and descriptors of turbulence as observed at a single nonmoving point and some basic properties and descriptors that represent the most important turbulent spatial variations. Inclusion of the properties of turbulence into a description of wind turbine response requires transformation into a rotat-

ing system from which turbulence properties are experienced. The resulting significant modification of turbulence properties is described. Models of the rotationally sampled turbulent wind are briefly described and discussed. The article concludes with a discussion of the values and limitations of current turbulence models and efforts to achieve useful improvements in turbulence models. The overview begins by discussing the relation of the turbulent boundary layer to the region of the atmosphere in which a wind turbine turns, the "wind turbine layer."

2. THE ATMOSPHERICBOUNDARYLAYER The atmospheric boundary layer is that region next to the surface of the earth in which turbulence exists. The atmospheric boundary layer has a distinct top, above which air is not turbulent (see Fig. 2). Within the boundary layer, the turbulent surface layer[3], up to the first 150 meters or so in which a simple mathematical-physical wind speed and turbulence relation holds, is of greatest interest. The thinner sublayer, below the turbulent surface layer, is not of direct concern in wind energy. Typically, the wind turbine rotor is in the turbulent surface layer, often extending into the top parts of the boundary layer or even into the nonturbulent layer above. The nonturbulent flow is called the ~free stream."* The depth, or thickness, of the boundary layer, as well as the character of the turbulence, depend on two main factors. One is energy generation by shearing of wind either dragging on the surface of the earth, or dragging on adjacent streams of air that are moving with different relative velocities ("shear layers").

* However, "free stream" is also used to describe turbulent boundary layer flow that is outside of any influence of the wind turbine.

281

J. R.CONNELL

282

Ilk

Profile~

~.

.<..--..,111.:

useav

Fig. 1. A turbine in a turbulent wind. The other factor is the vertical distribution of temperature, the thermal stratification or layering that produces the static stability. The importance of static stability is that the atmosphere is either indifferent to, suppresses, or enhances turbulence motion in the boundary layer. The static stability is usually clas-

sifted as neutral, stable, and unstable. Static stability in the turbine layer usually changes strength gradually within one of the categories; however, at times it is in rapid transition between major categories. The effect of the temperature layering on the turbulence is shown in Fig. 3. Turbulence mixes through

Nonturbulent Free Stream Flow Instantaneous Boundary Height

o >=

, Intermittent Turbulence

j z= ]

ill

z-< [

~

T"~"m eA~e"-" Boundary Layer

/ /

~--

Height, 6

L'

Fully Turbulent Flow

J-

111IIIIIIIIII]1111111111111111111111111111111]11111111111~ Y///////////1/1/1////Ill Alongwind Distance.--~ Fig. 2. Atmospheric boundary layer. The surface layer is some variable fraction of the boundary layer, perhaps as small as 0.1&

283

Turbulence at the wind turbine rotor I

• Wtnd Olrectlon

!

] Neutral

~

(SurfaceShear Or,yen)

Unstable (Surface and Buoyancy Drwen)

• Wind Direction ~

I

~

~

=

~

=•I~

P

~

~

Stable (Laver Oecouphng and • Shear

Fig. 3. Atmospheric stability classifications. The boundary layer height, ~, may be as small as 10 m and as large as 500 m.

a deep layer, ~, (say, 100 to 500 m deep) when the wind is blowing strongly. The unstable boundary layer has an additional source of turbulence--buoyant air bubbles that usually are caused by solar heating at the surface of the earth. A statically stable atmosphere is one in which turbulence is suppressed and there is often a decoupling of one layer of wind from

I

T

T

i

I I

A

a A

I I

I

N

N S

I I

I

T

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Boundary Layer Height, 6 ///,f

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another. If the wind is sufrtciently strong, turbulence still may be generated near the surface of the earth, or at a shallow elevated layer where waves form and break much as they do on water. Some layers may actually be calm in stable stratification. Figure 4 shows that the thickness of the turbulence layer depends very much on the combination of generation and suppression of turbulence that associates with the static stability of the atmosphere. A boundary layer having nearly neutral stability, caused by strong enough wind and not too much heating or cooling, has an intermediate depth. At least a 10 m / s wind and an overcast sky during the day assure neutral stability in the mixing layer at a typical noncoastal country location. If the cloud cover is partial, wind speeds greater than about 10 m / s may be required to produce strong enough mechanical mixing forces to assure near neutral stability. As night approaches, especially under clear skies, the air cools near the ground, turbulence is suppressed, and the boundary layer becomes shallower. As morning begins under clear skies, the boundary layer begins a transition from stable through neutral to unstable layering in which bubbles and plumes rise and mix the air. The boundary layer is mixed to its greatest depth during unstable stratification and the energy of the mean wind is spread more uniformly over a deeper layer. Vertical mixing, caused by turbulence, changes the vertical distributions so that vertical profiles of both mean wind velocity and the strength (variance) of turbulence are characteristic of the stability category. Examples of wind profiles for each stability category are shown in Fig. 5. Simple terrain is flat and horizontal or nearly so. The surface is uniform, or homogeneous. Its uniform roughness would be caused by brush, tall or short

R

S I

///////////~ Unstable

T I 0 N

/ / / / / / / ~ , . /~

Neutral

T % ( / ////// I

i

I

I I I

I I

"~::....-,,..,,$'~y~ Day

,/I

Stable

Night Time of Day

Day

Fig. 4. Schematic diagram of changes in thickness of mixing (boundary) layer. Thickness may vary. as indicated in Fig. 3.

284

J.R. CONSELL Neutral

U(Z)

%(z)

T

Ij

I

///////////////////11////

Turbulence

Wind Speed Unstabl,

U(Z)

I=

au(Z)

I

T

I

_J

./i

I/////////////////////// Wind Speed I,

,,////////////////////////

Turbulence Stable

U(Z)

I

I

1=

%(Z)

I I I

! ( I/

Wind Speed

Turbulence

Fig. 5. Vertical profiles of mean wind speed and of RMS turbulence speed.

grass, or plowed fields, for example. Its boundary layer is well developed after the wind has been blowing over the uniform surface for at least a kilometer, and the flow characteristics are steady. In the simplest version of this flow, there is no heating or cooling so the stability is neutral. Then, for a layer from about a meter above the surface of the earth to above the top of the turbine layer, a physically sound simple logarithmic mathematical formula for the wind speed profile applies. This ~logadthmic" wind profile is sometimes approximated by the " 1 / 7 power law" profile. The power law profile depends on the arbitrary choice of the height chosen for the reference wind speed. The ratio of the logarithmic wind speed to the power law wind speed at any height, z, is given by eqn (3). This relation may be used to estimate the deviation of a power law from a log law representation of the mean wind profile.

Ue (z) = "-~ In

(logarithmic wind profile equation)

(1)

where u* is the friction velocity, z0 is the roughness length, and k is the yon Karman constant.

(power-law wind profile equation)

(2)

where U~ is the wind speed at a reference height, z~ is the reference height, and p is the power law exponent. A measure of the discrepancy between wind speed calculated using the power law and the wind speed calculated using the log law may be made by

Turbulence at the wind turbine rotor forming the ratio of these two expressions for wind speed at the same height, z. as shown in relation (3).

Up (z)

(ratio of wind speeds by two relations).

(3)

We express U~ in terms of relation (1) and pick two specific reference heights for the power law, z~ = 30 m and 10 m; then, relation (3) becomes (4) and (5) below.

E = - -

z=60m

= 1.3

p = 1/7

zj = 3 0 m

zo = 0 . 0 3 m

(4)

and E =

2 + In 3.33

(6) j/7 = 1.12

(5)

3+1n2 where the values have been obtained from z~ = 10 m, z = 60 m, p = 1/7, and z0 = 0.03 m. Comparing the logarithmic profile wind and the 1/7 power law wind at 60 m for a power-law reference height of 30 m (eqn 4), and 10% difference is found. If the power law reference height is set at 10 m (eqn 5), a 30% difference is found at the 60-m height of comparison. The logarithmic profile of wind speed may be specified by three parameters. The roughness of the ground is included by the use of roughness length, z0. The mixing of momentum, or velocity, caused by the turbulence may be characterized by the rms speed of the turbulence, tr, and a mixing factor, Ks, which are related to the friction velocity, u*, and the yon Karman constant, k; or to some reference height, z~, and the roughness length, z0, by relation (6). cr K, = u * / k = u (z0/ln (zt/zo).

(6)

The logarithmic profile equation in these three forms may be written as U(z) = (r K, In (Z/Zo), = u * / k In (Z/Zo), and = [u(zO/ln (zl/zo)] In (Z/Zo).

(7)

If z0 has a value of about 0.5 m (a large value), the equation represents the wind profile for deep mixing (see Fig. 6a). If zo is about 0.05 m (modest roughness, but not smooth), the equation represents shallow mixing (see Fig. 6b). Typical values of Zo range from about 0.01 m to about 1 m. The l-m value

285

is for very rough terrain for which, it is important to realize, the wind profile close to the height of the elements of roughness is not modeled by the logarithmic profile. The roughness elements that correspond to a profile with Zo = 1 m may be as high as large buildings. The logarithmic wind profile does not apply down into the height range of the buildings that create the turbulence. Typical growth patterns of sagebrush bushes about 1 m high have values of roughness length of from 0.03 m to 0.06 m. A relation between the height, shape, and spacing of roughness elements and a value of roughness length is very specific to the terrain and the wind speed and direction. Such relations have only been derived successfully by empirical methods that incorporate measurement of wind profiles to derive values of -'0. Tables matching values of z0 with geometry of terrain and direction of wind for specific sites are available in several references (e.g., [31). The discussion has been for very homogeneous terrain of not too great roughness, for which a simple equation, the logarithmic law, represents the vertical variation of the mean wind speed from above the roughness elements up to about 100 m above the ground in moderate neutral wind conditions. The logarithmic equation may also be used in modified form to include some wind cases for both stable and unstable layering of the atmosphere over uniform terrain[3]. Examples of the profiles that are modeled are shown in Fig. 7, a schematic adapted from Figure 6.5, page 137, reference [3]. Many wind cases, usually flows in complex terrain, are not well represented throughout the wind turbine layer by the logarithmic law or, for that matter, by any relation presently known. Empirical graphical descriptions are required for these conditions on a case-by-case basis to account for the additional important physical variables that are not represented adequately by z0. However, on occasion, lower layers of a flow can still be described by a logarithmic law. Figure 8 contains two examples of flows in complex terrain. Regions of excess speed, low speed, wakelike situations, and extreme turbulence may exist in complex flow. Usually, these complex flows are over complex terrain, but not always. Furthermore, these conditions may or may not exist in specific terrain depending on wind speed, wind direction, and static stability. It is not a simple matter to model mean velocity or turbulence of complex flow. The length scales of turbulence are of obvious importance in complex terrain, because some turbulence is generated on the large scale by the peaks and valleys. Although the existence of eddies of different sizes may be most obvious in complex terrain, the distribution of length scales is variable in simple terrain also and variably influences wind turbine response there also[l 1,12]. The strength of turbulence, an absolute quantity represented by the variance, is a useful measure when the absolute forces on a rotor blade are to be estimated. The intensity of turbulence, a relative quan-

286

J. R. CONNELL

(a)

lTI

Height

I Zo = O.5m Deep Mixing

1 I

Zo = 0.05 m Shallow Mixing

I I

/l L/'

(b)

I I

.,~ i. • ¢ t~k Wind Speed ~

Trees

/i

~

.....

I Brush

Wind Speed --~

Fig. 6. Logarithmic profiles for two values of mixing length, zo (schematic). tity, is more useful as a measure when the relation of terrain complexity to the turbulence level is to be modeled. Typical approximate values of intensity for, say, a 20-rain period of time series with the nonstationary trend removed (the square root of turbulence variance divided by mean wind speed) (see the next section for a more detailed definition) are listed as a function of terrain complexity in Table 1. The intensity of turbulence in complex terrain may double or triple that found in simple terrain.

an irregular motion of fluid that appears when fluids flow past solid surfaces or when streams of fluid flow past or over each other. The generation mechanisms discussed earlier--drag on the earth, buoyancy motion, and breaking waves--are all encompassed by this description of turbulence. 3.1 Turbulence at a single p o i n t

Turbulence velocity at any given single point and time has a value that seems not to be determinable from other information or to be controllable or to have a predictable character. Unlike the predictable pressure in a balloon, which is measured on a scale much larger than the scale of random motion of individual molecule motions that cause the macroscopic pressure, turbulence velocity in an air volume varies with the same scale as the random processes that produce

3. DESCRIPTION OF TURBULENCE PROPERTIES

Turbulence is an important source of aerodynamic forces on wind turbine rotors. Some commonly used turbulence terms that refer to the physical descriptions of the wind are defined below. Turbulence is

Schematic f

p,®

~;

~'~

u = Ks ~ Lr , n -Z-

/

3

0

Stable

/"

0

¢

/ -Z ~1

/

1

1

2

Neutral

Unstable

~

3

Wind Speed in m/s

4I

5I

Fig. 7. Logarithmic wind speed model (schematic). Adapted from [3].

Turbulence at the wind turbine rotor Multiple Hills

t hi

l

d2

l

.,e

Topography for Large Scales. Z* for Smallest Scales

dl

liolated Hill

Z~for Small Scales. h, d for Large Scales

9

,,d

287

One thing is certain about a field of flow having such an irregularity and interaction of scales. No single point of measurement can adequately describe the flow. Rather, a number of simultaneous and continuous measurements must be made at a number of positions in the flow. A simpler description starting with parameters calculated from a time history of turbulence velocity measured at a single nonmoving point is useful, however. The time history is usually called a "time seties." In order to compute a parameter that has a nonrandom aspect equivalent to a macroscopic property, the time average is used. The mean of the u component of wind velocity, U, and the corresponding 2 variance, cr,, derived from the time series u(t) as follows, are commonly used single-point parameters of turbulent flow[3,4].

,,, _U

= -1 f0T u ( t ) d t ,

T

Fig. 8. Turbulent wind in complex terrain (schematic).

(8)

where T is a long enough time that the value of U is a stable one. the turbulence. Thus, turbulence velocity is not deterministic at a point moment by moment. This feature is described by terms such as irregular motion, lack of periodicity, random motion, stochastic nature, turbulent eddies, and turbulent vortical elements. These turbulent flows are in contrast to periodic flows or vortex flows, which can also pass by a point and create wind fluctuations but are not part of turbulence until they become dynamically unstable and degenerate. An important feature of turbulence is the size, the distribution of size, and the rotational nature of turbulent flow, calling to mind a flow field of eddies. There is a distribution of scales of turbulence: time scales, space scales, and energy scales caused by generation in different scales of shear and by degeneration of larger scales to smaller scales. A wide and continuous distribution of scales is readily observed in turbulence measured with an anemometer on a tower. Rhyming reminders, such as the following one by Lewis F. Richardson[10], say the same thing more poetically: Big whirls have little whirls, Which feed upon their velocity. Little whirls have smaller whirls, And so on to viscosity.

lff

0"5 = T

[u' (t)] ~ d t ,

where u ' (t) = u (t) - -U is the fluctuating part of the total turbulent wind velocity, u ( t ) . If a stable value of either parameter is not achieved by the computation, there is a ~trend" in the data. For typical winds, the averaging period. T, must be at least 10 to 20 rain, so that the values of mean wind speed and variance are independent of the starting time of the average. If a time series has this regularity property, it is called "statistically stationary." Then it has some macroscopic character. Shorter time averages can be used for many purposes if it is understood that (1) they will be unsteady and (2) correlation with other measurements will only make physical sense if the other measurements are made very close by. The strength of turbulence is indicated by the variance, ¢r2, which has the units of energy per unit mass of air. The intensity* of turbulence, i, is a dimensionless parameter indicating the strenph of turbulence relative to the mean wind velocit2,'. i = (r.IU.

.... Table 1. Approximate turbulence intensity (rms longitudinal turbulence/mean wind speed) Terrain type

Stable

Neutral

Unstable

Simple Gentle complex Moderate Severe complex

0.08 0.12 0.18 0.25

0.15 0.2 0.3 0.35

up to 0.4 0.4 0.5 0.5

(9)

(10)

Useful but incomplete information about the distribution of scales of turbulence may also be derived from a time series of wind velocity at a single point. In the material that follows, descriptors of turbulence are discussed as theoretical concepts. One of the

* It is common to find the word "intensity.- of turbulence also used loosely to mean the root mean square value of turbulence, or what is the same, the square root of the variance, sigma.

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J. R. CONNELL

challenging tasks in describing turbulence is to compute and interpret these descriptors in a way that is consistent with the theoretical concept. This problem is beyond the scope of this article, but it is of central importance at the moment of computational application of the concepts to a real time series of wind speed. An average large time scale, called the integral scale, is derived by computing a correlation of the fluctuation of velocity with itself measured at all delay times possible in the series. This autocorrelation function, R(r), is used as a weighting function with the delay time to calculate the integral scale, ~ , as follows[3,4]:

1£ r ~ = ~ R(-OdT,

(11)

where -r is the time delay and R('r) is the autocorrelation coefficient defined by

~"R(T) =

lfoT

u' (t)u'(t + z) dt.

(12)

At large time delays, "r --* oe and R('r) ---, 0, and at zero delay "~ = 0 and R('r) = 1. A turbulent (irregularly fluctuating) wind speed time series can be thought of as being the sum of a large set of continuous sine wave time series that each fluctuate at different frequencies. The total strength or variance of all the sine waves must, of course, be equal to the variance of the irregular time series calculated from the original turbulence time series. In the set of sine waves, each small group, j, of sine wave time series carries its share, ,r-'(nj), of the total strength, crz. o"2 = Y. tr2(nj),

(13)

where nj is the center frequency of the jth band of sine waves. The functional relation between the set of values of cr2(n/) and the values of n 2 is directly related to the spectrum of turbulence. A table or graph of the values of (r-'(n2) as a function of the band center frequency, nj, describes the hierarchy of scales in terms of the energy at different frequencies of fluctuation. This is a simple form of the spectrum of turbulence.* It is common to plot an equivalent form that is an energy spectral density function obtained by dividing the value of crZ(ni) by the value of the frequency range in the jth band of the set of sine wave time series. It is also common to show the equations of Fourier transform from the

* Contrary to usage in this paper and in power spectral analysis in general, the word wind "spectrum" is often used to label a probability distribution of frequency of occurrence distribution of wind speed. It is best to reserve the spectral designation for the hierarchy of energy of fluctuations as a function of the frequency of fluctuation.

time domain of the autocorrelation function, R(r), to the spectral frequency domain of the power spectral density function. The author considers such equations to be unnecessary clutter for the purpose of this article, and the interested reader is referred to the citations in the references. This completes the definition of terms for the important features of turbulence velocity at a single, nonmoving point, or Eulerian turbulence at a point. It is well beyond the scope of this article to define in equivalent detail the important features of turbulence that can only be described using multipoint and time parameters of the flow. However, the main terms are named and described briefly below. They are analogous to the single-point parameters described above. A brief explanation of the significance of atmospheric stability in relation to turbulence is appropriate here. The reader is referred to the literature for detailed study[3, e.g.]. We have indicated that turbulence is generated in wind shear. The wind energy is released to turbulence if there is cross-shear mixing in the air. The mixing is caused mainly by mechanical forces caused by flow over surface roughness, buoyancy forces, and other dynamic instabilities. In the absence of these initiating forces, the potential energy for turbulence would remain stored in the static atmosphere (density or temperature layering) and in the nonturbulent kinetic atmosphere (mean wind speed). In using the hydrostatic stability of the atmosphere as the factor in the changing character of turbulence, we are treating the kinetic factor as secondary in importance for simplicity. Both factors are incorporated in the Richardson number, Ri, a nondimensional parameter. The Richardson number is the ratio of the rate of change of buoyancy energy potentially available by mixing in the portion of atmosphere under consideration to the rate of change of kinetic energy potentially available by mixing in the wind shear of that same piece of atmosphere. The sign of the wind shear energy rate factor will always be positive. The buoyancy term may be positive or negative, the latter indicating instability and increasing amounts of turbulence. Instability is commonly thought to occur also for values of Ri that are positive but less than about 0.25. Turbulence in existence would be suppressed in an atmosphere with Ri greater than +0.25. New turbulence would not be expected to form. To write the defining energy equation, or the ratio defining the Richardson number as a mathematical relation, is to get to greater depth than is appropriate here. However, one cautionary comment is vital: the term "Richardson number" is used with about four variations in meaning. The most critical difference is the variation that uses vertical gradient of temperature as opposed to the negative of the vertical gradient of temperature, which is called the "lapse rate." Thus, for example, the condition for suppressed turbulence will be expressed by a different sign of Ri for the two definitions. Another critical difference in

Turbulence at the wind turbine rotor definition concerns the use of gradients and mixing a local point, as opposed to "bulk gradients" and mixing through a finite layer. The "bulk Richardson number" reflects processes that occur on a larger scale of flow whereas the "gradient Richardson number" reflects processes occurring more locally. For more complete information, the reader is referred to the large body of literature in atmospheric science and fluid mechanics from which [3] is a good starting place. 3.2 Turbulence at two points The two-point (two-velocity) correlation coefficient is R,,,,,.. It relates to a deterministic aspect of turbulent flow[3-8].

R .... (r) = ~

ut(t)u,.(t - "r)dt.

(14)

O'uI(~u2 Refer to [2,4,6,7] for detail about correlation and coherence in turbulence. The corresponding crossspectral density function, d~,,u:(n), is for two points (two velocities). It is derived from the two-point correlation coefficient by a Fourier transformation by methods to which the reader is referred in the references[3,4,9]. It has an amplitude portion and a phase portion. The cross spectrum is written as the sum of two terms, the cospectrum C,,,,(n) and the quadspectrum Qu~,,(n). cb~,u.,(n) = C,~u:(n) - jQ,,,:(n),

(15)

where j (=V"L-i) denotes the imaginary part of a complex variable. The amplitude cross spectrum becomes, in nondimensional form, a normalized spectral correlation coefficient called the coherence, F,,,.,(n). [ 6.,...(n)12 F.,...(n) = d~.,(n) ~b~,.(n)

(16)

These two-point cross functions of turbulence velocity are of vital importance in determining the wind experienced by rotating turbine blades[l 1-17,23,24]. Crosswind and along-wind differences are both important. There are horizontal and vertical gradients of wind velocity at each moment, even if the time average gradients have zero value. The inclusion of these relations into a theoretical description of the rotationally sampled wind velocity spectrum has been done for a single point and is being worked on for two points. The key aerodynamic point is that forces are generated along each blade of a rotor by turbulent air crossing over the blade as the blade moves rapidly through the field of turbulence. A single point on a blade rotates around a cirele in the crosswind plane as shown from point Uc to Uo in Fig. 9. Figure 10 shows turbulent wind time series for a nonmoving point at the turbine hub location and at single points

289

rotating with the blade at two radial distances and a frequency of about 1 Hz. The effect of the speed and radial sweep of rotational motion of a point is clearly significant. The difference in the three turbulence time series is dramatic. Each of the time series can better be distinguished if it is analyzed into a fluctuation spectrum. Research [ 1,19] has shown that only the rotational time series, such as the examples in Fig. 10b and c, describe the wind that causes the aerodynamic forces on the rotor blade. Furthermore, the amount of turbulence energy that drives the rotor blades at potentially critical response frequencies of oscillation in determining fatigue lifetime is much greater than previously expected from models using time series such as in Fig. 10a plus mean wind shear. Table 2 shows the turbulence variance of or'- (nj), contributed in the bands of fluctuation frequency surrounding each multiple of the frequency of rotation of the rotor. The trend with higher frequency is clear. Figure 11 shows more clearly the magnitude of the rotational effect by plotting for each band the ratio of the rotational energy to the energy in the mean shear plus the non-rotating measurement of turbulence. Thus, the fluctuating aerodynamic forces that cause flapping of the blade are likely to be much greater than expected at frequencies higher than the rotation rate of the rotor. Very little attention has been given by the engineering community to the frequency content in the turbulence in the rotating frame of reference that has a frequency less than no. However, the PNL theory and rotational measurement of the turbulent wind both have shown some potentially important aspects of the spectral content in this "low frequency" region[l, 11]. The effect of observing from the rotating frame of reference appears to be twofold. First, there is a slight general diminishment of the total energy in the low frequency part of the spectrum. Second, there is a large diminishment of the energy, compared to that of the wind spectrum as seen in the nonrotating frame of reference, in the region between 0.5no and no. The loss of energy, appearing as a dip in the spectrum, is about that which would occur if the turbulence in the nonrotating frame of reference were spatially averaged over the whole rotor disk area. This dip shows in each of the spectra in Fig. 12. (Note that the dip does not always appear below the dashed reference spectrum from the nonrotating frame, the Kaimal spectrum. This is because the set of spectra are for different heights on the vertical-axis rotor blade, whereas the reference spectrum is for the equatorial height.) The difference between the dip in the rotational wind spectrum and the disk-averaged wind spectrum is that the disk-averaged spectrum in the nonrotating frame of reference continues to diminish at all higher frequencies, but the rotational spectrum dips and then rises in a set of spikes at the higher frequencies. Although discussion of transforming the wind spectra into aerodynamic force spectra is not the pur-

290

J. R. CONNELL z

Sc.e a,,°

,

~\

u~ ~

\

/

Crosswind Vertical Plane

Mean Wind irection

~D

Fig. 9. Turbulence correlation and convection.

10

~E _~

a) Not Rotating (at Hub Height)

0

m

< ~ -10

0

3

6

9

12

15

18

Time in Rotations

10 bl m/s

0 -10

Rotationally Measured (at r = O.7R) 0

3

6

9 12 t, Rotations

15

18

10 ~ m/s

c)

0 -10

Rotationally Measured (at r = O.2R) 0

9

18

t, Rotations Fig. 10. Comparison of rotationally measured turbulence with turbulence at one hub-height point. The rotational velocity is 1 Hz.

Turbulence at the wind turbine rotor Table 2. Measured variance in each frequency band of rotationally sampled wind speed (neutral stability) Center frequency of band, n,, (no = blade rotation freq.)

Variance, Gr,2(m"/s")

no

0.28

2no 3no 4no 5no

0.085 0.043 0.028 0.024

pose of this article, a relevant point may be made here. In order to completely describe the forces, the wind must be known simultaneously at intermediate radial locations also[ 11,12,14,23,24]. Figure 12 is an example of the relationship between spectra of winds at different radial locations for a vertical-axis wind turbine, derived using the Pacific Northwest Laboratory (PNL) theory. These amplitude spectra are part of the required information. The phase spectra of the turbulence must also be modeled in order to compute the correct radial distribution of aerodynamic forces. The measuring and accurate modeling of the radial distribution of the rotational wind including the phase relation is modeling research yet to be accomplished. The only "nontuned" model of aerodynamic forces that has predicted with believable accuracy the highfrequency response of a turbine rotor used rotationally measured wind from a total of four locations on the two blades of the rotor[21]. It was actually necessary for the data from four radial locations to be interpolated and extrapolated to a number of other locations using other speculations as to what the wind phase distribution might have been. 4. MODELS OF ROTATIONALLYSAMPLED WIND Several models of rotationally sampled wind now exist that help in applying modem rotationally sampled wind characteristics to analysis of wind turbine re-

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sponse[13,15-17,23,241. The first theory of rotationally sampled wind spectrum to be developed is still as accurate as any known model for the wind speed at one point rotating in a circle[11-17]. The theory has now been extended to apply to both horizontal- and vertical-axis turbines[ 12], A key feature is that it includes a wide range of mean wind profiles coupled with the theoretical turbulence and a wide range of turbulence strengths and length scales of turbulence. The length scale of turbulence is an important factor in determining how a wind turbine experiences turbulence. In addition, the theory has attached to it a simulation of a time series for a rotating point, which has the spectrum of rotationally sampled wind identical to the spectrum of the theory. The current objective is to theoretically model two simultaneous time series whose phase relationships are accurate representations of those for real winds. Then the relationship between wind at any number of points around the turbine rotor can be described by the time series and by spectra. This difficult objective is important. The other key feature for the current PNL spectral theory and time series simulation is the economy of its use. It requires only a few tens of seconds of execution time on a digital computer equivalent to a VAX 1170. This contrasts to 10 to 30 rain to some of the multipoint simulations. Another modeling approach has been to interpret simple turbulence data from 3 to 5 levels on a tower spanning the disk of the rotor[18]. The data from the single tower are manipulated to provide a useful time series approximation to a complete crosswind-circle rotationally sampled wind. Although the complete crosswind-circle measurements are not made, those that are taken represent some part of actual site conditions. This is especially valuable for complex terrain and may be the only available realistic way to incorporate wind characteristics of specific complex terrain. The third approach to modeling wind is to make rotational measurements of wind completely around

Ratio of Energy in Bands of Spectra, O'rot 2 / O'hxed 2 10 3

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-

-

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0

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:

4

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Fig. 11. Significance of rotationallly sampled turbulence.

I 8

292

J . R . CONNELL

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I

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Fig. 12. Rotationally sampled spectrum for 17-m Darrieus: Zo = 0.1 m, ~ = m/s, rotor frequency = 1 Hz, calculated using PNL theory. The dashed line is for the Kaimal spectrum modeled for the equator height.

circles of rotation. One method is to place many anemometers on an array of towers and rotationally sample the measurements[11,19]. Another method eliminates the need for many anemometers and towers by measuring with hotfilm anemometers from the blades of a rotor[20,21 ]. Finally, some success has been had using a circle-scanning laser anemometer that requires neither many towers nor attachment of sensors to the rotor blades[22]. Thus, some data tapes are available o f rotationally measured turbulent wind as models of specific wind conditions. A good space and time multipoint description of turbulence velocity that is economical in time to

compute and is accurate is needed. The measurement methods offer the best hope of first success and they are being developed at this time[20,22, e.g.]. Two digital simulations of turbulence throughout a crosswind plane have been formulated: one with a lengthy running time[23] and a more recent simplified one[24]. Evaluation of some features of both models indicates that both require additional attention to make them quantitatively valuable. Efforts to that end are under way in joint effort between PNL and Sandia National Laboratories (SNL) at the same time that the corresponding PNL theory is being developed.

Turbulence at the wind turbine rotor 5. CONCLUDING REMARKS Turbulence characteristics are variable and depend upon the terrain roughness and orientation, largescale complexity, and temperature layering, which tend to suppress or increase turbulence. All of the descriptors of turbulence that we have discussed for fixed points of measurement, Eulerian turbulence, apply also to turbulence at rotating points. Accurate results for multipoint and phase relations await the acquisition of new measurements or much more complete theory. In the interim, a set of tools is available by which to accurately model single-point rotationally sampled wind. Experience with single-point data and some measured multipoint phase information suggests that the currently available numerical multipoint phase relation modeling may not be a particular improvement over single-point rotational modeling. Those who don't have access to computers of sufficient power to exercise each of the theories or models that are available might consider looking at results of the new wind models and simply selecting a new safety or correction factor for the older design criteria. Finally, PNL is currently putting its turbulence models onto floppy disks in a form usable with an IBM-PC type computer. Acknowledgments--This work was performed at the Pacific Northwest Laboratory for the U.S. Department of Energy (DOE) under Contract DE-AC06-76RLO 1830. The Pacific Northwest Laboratory is operated for DOE by Battelle Memorial Institute, Richland, WA. REFERENCES

1. J. R. Connell and R. L. George. Accurate correlation of wind turbine response with wind speed using a new characterization of turbulent wind. Solar Energy Engineering, 109, 321-329 (1987). 2. J. Counihan, Adiabatic atmospheric boundary layers: A review and analysis of data from the period 18801972. Atmospheric Environment, 9, 871-905 (1975). 3. H. A. Panofsky and J. Dutton, Atmospheric turbulence. Wiley, New York (1984). 4. J. O. Hinze. Turbulence, 2nd ed. McGraw-Hill, New York (1975). 5. E. Simiu and R. H. Scanlon, Wind effects on structures and introduction to wind engineering. Wiley, New York (1978). 6. L. Kristensen and N. O. Jensen, Lateral coherence in isotropic turbulence and in the natural wind. Boundary-Layer Meteorology, 5, 353-363 (1979). 7. L. Kristensen, On longitudinal spectral coherence. Boundary-Layer Meteorology, 7, 309-321 (1979). 8. H. Gustavsson and M. Linde, A gust as a coherent structure in the turbulent boundary layer. Tech. Note AU-1499, Part 5, FFA, The Aeronautical Research Institute of Sweeden, Stockholm (1979).

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9. L. Kristensen, H. A. Panofsky, and S. D. Smith, Lateral coherence of longitudinal wind components in strong winds. Boundary-Layer Meteorology, 21, 199-205 (1981). 10. L. F. Richardson, Weather prediction by numerical process. Cambridge University Press, London (1922). 11. J. R. Connell, The spectrum of wind speed fluctuations encountered by a rotating blade of a wind energy conversion system. Solar Energy, 29, 363-375 (1982). 12. D. C. Powell and J. R. Connell, A model for stimulating rotational data for wind turbine applications. PNL-5857, Pacific Northwest Laboratory, Richland, Washington (1986). 13. D. C. Powell and J. R. Connell, Review of wind simulation methods for horizontal-axis wind turbine analysis. PNL-5903, Pacific Northwest Laboratory, Richland, Washington (1986). l4. D. C. Powell, J. R. Connell, and R. L. George, Verification of theoretically computed spectra f o r a point rotating in a vertical plane. PNL-5440, Pacific Northwest Laboratory, Richland, Washington (1985). 15. L. Kristensen and S. Frandsen, Model for power spectra of the blade of a wind turbine measured from the moving frame of reference. J. Wind Engi. Industrial Aerodynamics. 10, 249-262 (1982). 16. J. B. Dragt, Load fluctuations and response of rotor systems in turbulent wind fields. Report ECN-172, Netherlands Energy Research Foundation, Petten (1985). 17. E. J. Fordham and M. B. Anderson, An analysis of results from an atmospheric experiment to examine the structure of the turbulent wind as seen by a rotating observer. Proc. 4th British wind Energy Association Workshop. University of Cambridge, Dept. of Physics, Paper No. WE4 (1982). 18. J. R. Connell and R. L. George, Scaling wind characteristics for designing small and large wind turbines. Proc. Sixth Biennial Wind Energy Conference and Workshop, American Solar Energy Society, Boulder, Colorado, pp. 513-524 (1983). 19. R. L. George and J. R. Connell, Rotationally sampled wind characteristics and correlations with MOD-OA wind turbine reponses. PNL-5238, Pacific Northwest Laboratory, Richland, Washington (1984). 20. V. A. Samdborn and J. R. Connell, Measurement of turbulent wind velocities using a rotating boom apparatus. PNL-4888, Pacific Northwest Laboratory, Richland, Washington (1984). 21. J. R. Connell, R. L. George, V. R. Morris, and V. A. Sandborn, Rotationally sampled wind and Mod-2 wind turbine response. EPRI Report AP-4335, Electric Power Research Institute, Palo Alto, California (1985). 22. R. M. Hardesty et at. Lidar measurement of wind velocity turbulence encountered by a rotating turbine blade. D O E / E T / 1 0 2 3 6 - 8 1 / I , National Oceanic and Atmospheric Administration, Wave Propagation Laboratory, Boulder, Colorado (1982). 23. R. M. Sundar and J. P. Sullivan, Performance of wind turbines in a turbulent atmosphere. Solar Energy, 31, 567-575 (1983). 24. P. S. Veers, Modeling stochastic wind loads on vertical-axis wind turbines, SAND83-1909, Sandia National Laboratories, Albuquerque, New Mexico (1984).