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Air flow behind wind turbines
Journal of Wind Engineering and Industrial Aerodynamics 80 (1999) 169—189
Air flow behind wind turbines M. Magnusson*, A.-S. Smedman Department of Earth Sciences, Meteorology Uppsala University, Villava( gen 16, S-752 36 Uppsala, Sweden Received 18 March 1996; accepted 17 April 1998
Abstract Wind turbine wakes have been studied by analysing a large set of atomospheric data, from a wind farm with four turbines sited on a flat coastal area. The results obtained have been generalized to allow tests against data from other full scale wind turbines as well as wind tunnel simulations. These comparisons are found to give very satisfactory results. Most results refer to single turbine wakes. The thrust coefficient, C , is found to be a better parameter for description 2 of wake characteristics than wind speed, because it implicitly includes the effect of regulation. It is also found that down wind travel time is more convenient to use than down wind distance in this context. The travel time, t , to the end of the near-wake region, i.e. to the point where a single velocity deficit peak first appears, is found to be inversely proportional to the rotational frequency of the turbine and to the turbulence intensity of the ambient air flow and proportional to the ratio of the wake radius and the hub height. For larger travel times, i.e. for the far-wake region, it is found that the centreline relative velocity deficit decreases with the logarithm of the time travelled and is parametrically dependent on t and C . The combined 2 wake of two turbines aligned with the wind direction is found to have a lower relative velocity deficit than the wake of a single turbine for the same travel time. This is explained by the inverse relation between t and turbulence intensity, the latter being much higher in the double wake, initially. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction 1.1. Background For large-scale exploitation of wind energy, it will be necessary to put wind turbines together in clusters or wind farms, as areas with suitable wind resources are limited. But in wind farms the turbines will always interfere with each other. The flow field
* Corresponding author. E-mail: [email protected]. 0167-6105/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 1 2 6 - 3
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behind a wind turbine is characterised by low wind speed, strong wind shear and a high degree of turbulence, and thus a second wind turbine placed behind the first one along the wind direction is likely to produce less energy than the undisturbed one, by an amount that will decrease with increasing distance. As wind shear and turbulence is recognised as two causes for dynamic loads on wind turbines, it implies that detailed knowledge of the flow field behind wind turbines is needed for planning of wind farms. The flow characteristics behind single wind turbines have been studied extensively during the last decade. Among the first investigators is the report by Templin [1]. He modelled theoretically the effect of an infinite wind farm in terms of its integrated roughness effect on the flow. Lissaman [2] constructed a model of a single wind turbine wake, based on the analogy of jets in turbulent flows (see below). The Lissaman model was first tested on full-scale data from Kalkugnen in Sweden [3]. Early wind tunnel simulations were performed at TNO in Holland [4,5] and at FFA in Sweden [6]. Vermeulen [7] summarised the laboratory data and suggested some modifications to the original Lissaman model, now renamed MILLY. Later measurements were performed in a wind tunnel in United Kingdom [8]. Single-wake studies in the atmosphere were reported by Faxe´n et al. [3], Vermeulen et al. [9], Taylor [10], Swift-Hook et al. [11], Baker and Walker [12], Haines et al. [13] and Ho¨gstro¨m et al. [14]. In the early studies the velocity deficit was the main objective but later most investigations also include turbulence characteristics. A large number of wind tunnel data have accumulated since 1980 but also full-scale experiments in the atmosphere. In Denmark several investigations of wind farms have been performed. First, the wind and turbulence field behind the two Nibe turbines were thoroughly investigated [11,15—17]) and later the two large wind farms Tændpibe [18] and Norrekær Enge [19,17] were studied. In Holland a research wind farm, consisting of 18 300 kW wind turbines, has produced a large amount of data of the flow structure in a wind turbine cluster [20—22]. In Sweden data have been taken at the Alsvik wind farm since 1990 [23]. Data from Alsvik are also analysed in the present investigation. The abovementioned sites can all be described as flat and homogenous but there are also studies from some mountainous areas. In USA wind farms have been studied at Castello Ranch in Altamont Pass [24] and Alta Mesa [25,26] and in Greece on the Samos Island [27]. Since the first attempt to model the flow field behind a turbine [2] a number of rather sophisticated models have been developed. Apart form the MILLY model and other models of the Lissaman type e.g. the Park model from Ris+ [28], several models of the k—e type have been developed [29—32]. 1.2. Flow structure behind a turbine A wind turbine extracts energy from the mean flow, which results in a region behind the machine where the wind speed is reduced, with the largest reduction at the centreline at hub height, which decreases with downstream distance. This is usually looked upon as the ‘wake’ behind the wind turbine. Classical boundary-free shear flows that occur in nature or in engineering applications are wakes, jets and mixing layers (shear layers). In wakes the cross stream
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advection term is very small compared to the longitudinal advection but in jets and mixing layers the two advection terms are of the same order [33]. With the notation used in the atmospheric boundary layer: º
*º *º +¼ *x *z
(1)
where º is the longitudinal mean wind speed, ¼ is the mean vertical wind speed, x and z are the downwind and vertical directions, respectively. It has been shown from turbulence measurements (see Section 2) that Eq. (1) is fulfilled for the velocity field behind the Alsvik turbines: º
*º *º +0.04!0.05, ¼ +0.01!0.04. *z *x
(2)
In a jet or mixing layer as much momentum is carried by the transverse flow as by the downstream flow, which also seems to be the case for the velocity field behind a wind turbine. A consequence of Eqs. (1) and (2) is that the flow field behind a wind turbine is, strictly speaking, either a negative jet or a three-dimensional mixing layer but not a wake. However, hereafter the flow field behind a wind turbine will be called a wake, as being the most common notation. In this investigation wakes behind the wind turbines in the Alsvik wind farm on the island of Gotland in the Baltic Sea are described. In Section 2 the site and the intrumentation are presented and in Section 3 some earlier results are discussed. The velocity deficit behind the turbine is analysed in Section 4 and finally in Section 5 the wake measured behind two wind turbines is presented.
2. Site and instrumentation The Alsvik wind farm consists of four 180 kW Danwind turbines and is located close to the shore line on the West coast of Gotland. The site, which is presented in Fig. 1, is a flat coastal strip covered with grazed grass and low herbs. Only data with wind coming from the sea (200—290°) is used in this investigation, with the exception of the analysis of the double wake (see below). The four turbines are stall regulated, three bladed with a diameter of 23 m and a rotation speed of 42 r.p.m. The hub height is 35 m. Cut-in speed is 5 m s\ and the rated wind speed is 12 m s\. Meteorological measurements have been performed at Alsvik since June 1990. There are two masts (M1 and M2 in Fig. 1) which at the beginning had a height of 42 m, but in June 1991 a new top section was erected giving a total height of 54 m. Wind speed and direction are measured at 7(8) heights on both masts and temperature at 5 heights on mast M1. Wind speed and direction are measured with a type of sensor developed at the Department of Meteorology in Uppsala (MIUU). The instrument is a combination of a small three cup anemometer and a very light wind vane. Wind
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Fig. 1. The Alsvik site. T1—T4 are the wind turbines, M1 and M2 are the meteorological masts.
speed and direction are sampled with 1 Hz. The instrument is described in Refs. [34,35]. Temperature is measured with 500 W platinum sensors. The four turbines are marked T1—T4 in Fig. 1. Wake profiles from turbines T1—T3 are measured on mast M2, while the undisturbed wind profile is recorded on mast M1. The distance from T1, T2 and T3 to mast M2 is 9.6D, 4.2D and 6.1D, where D is the rotor diameter (23 m). During a concentrated field effort in June 1991 turbulence measurements were performed on mast M2 at three levels. The turbulence instrument used was the MIUU instrument, which is described in Refs. [36,37]. It is basically a wind-vane-based three-axial hot film system supplemented with dry and wet bulb temperature sensors. The sampling rate was 20 Hz. Measurements from this field campaign were used to estimate the two terms in expression (2). Here data from June 1991 to November 1993 have been analysed. The averaging time is usually 1 min. 2.1. Internal boundary layer When the wind is blowing from the sea an internal boundary layer is building-up over the land surface with its different roughness and surface heat flux. A simple model [38] has been used to calculate the height of the internal boundary layer at the position of mast M2. For unstable stratification the new internal boundary layer will reach up to 20 m, but the height will decrease with increasing atmospheric stability. The same result is found when comparing simultaneous measurements on the two masts, when the turbines are not working. An example is given in Fig. 2, which shows
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Fig. 2. Examples of the effects of the internal boundary layer: (a) velocity profile; (b) turbulence intensity profile; (;) mast M1, (*) mast M2.
wind speed in (a) and turbulence intensity profiles in (b). The growth of a new internal boundary layer will affect the wind measurements at the lowest heights on mast M2.
3. Data 3.1. Earlier analyses In Ref. [23], the relative velocity deficit and the turbulence in the wake were studied as function of distance and stability. The relative velocity deficit is defined as *º/º "(º !º )/º , where ‘a’ denotes the ambient wind speed and ‘w’ the wind speed in the wake and the increase of turbulence was expressed as ‘added turbulence’, ((pº) "(pº) !(pº)) where rº is the standard deviation of the mean wind speed. The stability parameter used was the Richardson number, Ri, g *¹/*z Ri" , ¹ (*º/*z)
(3)
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where ¹ is a reference temperature, g is the acceleration of gravity and ¹ is the mean temperature. It was found that the relative velocity deficit decreased with increasing distance from the turbine and increased with increasing stratification. The added turbulence in the wake showed, close to the turbine, two separate peaks at the positions of the blade tips. The two maxima disappeared downwind the turbine more rapidly in unstable air than in stable. In the present investigation atmospheric stratification is divided into three stability classes: unstable (Ri( !0.05), neutral (!0.05(Ri(0.05) and stable (Ri'0.05). Richardson number was calculated using temperature and wind speed at 18 and 31 m.
3.2. Wind speed corrections The difference in wind speed between the two masts was calculated for all wind directions, when the wind turbines were working. Fig. 3 shows (º !º )/º as + + + a function of wind direction at a height of 31 m. As indicated in the figure all the different wakes can be identified. For wind directions when both masts are undisturbed there should be zero difference in wind speed, but according to Fig. 3 there is always a slight difference, which can be both positive (220—260°) and negative (280—310°). Dahlberg [39] has calculated the tower disturbance on the wind field for the towers used in Alsvik with a two-dimensional model based on potential flow theory, including solid body effects and wake flow. The dotted line in Fig. 3 shows the theoretical calculations, which agree very well with the measured differences. All wind speed measurements have been corrected according to the calculated curve.
Fig. 3. Boom and mast effects on the measurements: (—) mean value for the measurements; (- -) calculated boom and mast effect, after Ref. [39].
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3.3. The thrust coefficient and power production In Ref. [40] the effect on the wake of the thrust coefficient, C , was discussed. The 2 thrust coefficient is defined [41] as F C " , 2 0.5oºA
(4)
where F is the axial thrust, o is the air density and A is the rotor area. The thrust coefficient varies with wind speed and Fig. 4 shows the calculated function for the Alsvik wind turbine, taken from Ref. [40]. When C is large, close to one, the wind 2 turbine works with a high degree of efficiency, but this occurs at low wind speeds and thus the power production is low. For higher wind speeds the turbine is less effective, but the power production is large. Power production can thus be expressed as a function of C . But produced power can also be written as 2 P"oC ºA, (5) N where P is produced power, o is air density and C is efficiency coefficient of the N turbine. Fig. 5 shows the power wind speed curve for the Alsvik turbines. For a given wind speed the variation of produced power is large. This is mainly so because the turbine is very often not correctly aligned with the mean wind direction.
Fig. 4. Calculated wind-C curve for the Alsvik wind turbines. According to Ref. [40]. 2
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Fig. 5. Wind-effect curve for the Alsvik wind turbines.
An alternative method is to use the thrust coefficient instead of wind speed to describe the conditions of the turbine. The assumption that the turbine operates according to the wind effect curve does not need to be fulfilled. This method also yields data taken above rated wind speed can be used. 4. Relative velocity deficit 4.1. Dependence of transport time and stability Fig. 6a shows measurements of relative velocity deficit for unstable stratification as a function of distance and for C " 0.84. This is the usual way to plot relative velocity 2 deficit but the data points are very scattered at each distance (4.2D, 6.1D and 9.6D). A more physical way to represent the variation of deficit is to use transport time behind the turbine instead of distance. The near wake is determined by the properties of the turbine but the diffusion of the wake downstream is settled by the ambient turbulence and the longer transport time the stronger influence of atmospheric turbulence.
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Fig. 6. The relative velocity deficit as function of (a) normalised distance, (b) time downstream of the turbine. C " 0.84 and unstable stratification. 2
The transport time is calculated using Taylors hypothesis [42] B(x)"B(ut).
(6)
This hypothesis assumes that the eddies are transported with the mean velocity, without distortion. This is not completely the truth, but assuming this evolution to be relatively slow, the errors introduced are small. The free stream velocity is used as transport velocity of the eddies. The transport time, t, at a fixed height, z, is calculated: x t(z)" . u(z)
(7)
Fig. 6b shows the same relative velocity deficit values as in Fig. 6a but as a function of transport time. There is still a rather large scatter, but the data points form a more continuous band. Taking the data at x"6.1D with the standard deviation p + 0.075, and dividing it in data intervals of 1 s, it will range from 17 to 23 s. The mean of the standard deviations is in this case reduced to p + 0.06. The mean of the relative velocity deficit at this distance is of the order 0.25.
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By using time scale instead of length scale, yields a more continuous pattern and also reduces the scatter. We conclude that the relevant parameter is the transport time instead of the distance. In Fig. 7 isolines of relative velocity deficit in unstable stratification are given as function of height above ground and transport time for C + 0.85 in Fig. 7a and 2 C + 0.6 in Fig. 7b. The relative velocity deficit is much larger for large C -values 2 2 (low wind speeds) than for small (high wind speeds), because the turbine is working much more effectively at low wind speed, although the power production is less. In Fig. 7c and 7d the corresponding relative velocity deficit values are given but for stable air. It is quite clear that the relative velocity deficit increases with increasing stability. The ambient turbulence is less effective to diffuse the wake during stable conditions, and the stability tends to suppress vertical motions. The same result was found in Ref. [23]. In Fig. 8 the relative velocity deficit is given as a function of Ri for C " 0.74 and 2 t " 10 s. For Ri-values below 0.25 (the ‘critical value’) the relative velocity deficit increases at first with increasing stratification, as expected. But for larger Ri there is a tendency for the deficit to decrease, which would indicate more effective mixing. When Ri'0.25 the flow is considered not to be fully turbulent [43,44]. But breaking waves and other kinds of bursts can stir the atmospheric boundary layer and create large temperature and wind speed variations [45], and hence more effective mixing. 4.2. Near wake In the region closely behind the turbine the wake is mainly determined by the characteristics of the turbine, although atmospheric stratification also comes into play. In Ref. [40] theoretical calculations, supported by measurements, show that just behind the rotor the relative velocity deficit has two peaks, situated at the middle section of the blades. With increasing transport time behind the turbine momentum transport towards the centre gradually wipes out the two peaks and one maximum at the centre will occur. Generally speaking, the time needed to get one maximum at the centreline, t , depends on the rotation frequency, f, C and atmospheric conditions. 2 A large C will give a large deficit which will increase the peak values. But at the same 2 time the shear within the wake, and thus the momentum transport towards the centre, will also be large which will increase the transport of momentum towards the centre. Thus as a first approximation we will assume that t is independent of C . As shown in 2 Ref. [40] atmospheric turbulence influences the transport time in spite of the very short times involved, &5 s. For neutral stratification turbulence intensity in the undisturbed atmospheric surface layer can be written as [46] p 2.4k S" º ln z/z
(8)
where k is von Karmans constant equal to 0.4 and z is the surface roughness. Eq. (8) is valid for 30 min data.
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Fig. 7. Vertical distribution of the relative velocity deficit as function of time: (a) C + 0.85 and unstable 2 stratification, (b) C + 0.6 and unstable stratification, (c) C + 0.85 and stable stratification, (d) C + 0.6 2 2 2 and stable stratification.
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Fig. 7. Continued.
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Fig. 8. The centreline relative velocity deficit as function of Richardson number in stable air (Ri ' 0), for C " 0.74 at t + 10 s. 2
The radius of the rotor will determine the size of the wake just behind the turbine, and with a larger rotor it will take longer time for the momentum transport to create one peak value at the centreline. As turbulent eddies scale with the height in near neutral conditions, it is plausible to assume that with increasing hub height larger turbulent eddies will twist the wake and make the momentum transport more effective. For neutral stratification t can tentatively thus be expressed as
1 H R t "C ln , f z H
(9)
where C is a constant. f is the rotational frequency. H the hub height, z the roughness height and R the rotor radius. For the Alsvik turbines t is estimated from both calculations and measurements to be approximately 5.2 s [40]. With a z -value of 0.0005 for the sea sector at Alsvik and H"35 m, R"11.5 and f"0.7 Hz, the constant C is determined to C "1. To test the generality of Eq. (9), data from wind tunnel experiments and from 4 wind turbine sites, besides Alsvik, are used. The wind tunnel data performed at TNO
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M. Magnusson, A.-S. Smedman/J. Wind Eng. Ind. Aerodyn. 80 (1999) 169–189 Table 1 Data for the other sites used in the investigation (for notations, see text) Site
f (sU)
z (m)
H (m)
R (m)
t (s)
TNO TNO TNO TNO TNO Alta Nurra Nibe Sexbierum Na¨sudden
29.7 38.2 56.0 72.1 78.9 1.51 0.57 0.50 0.42
0.0006 0.0006 0.0006 0.0006 0.0006 0.12 0.07 0.06 0.04
0.36 0.36 0.36 0.36 0.36 17 45 35 77
0.18 0.18 0.18 0.18 0.18 6.75 20 15 37.5
0.1077 0.0837 0.0571 0.0444 0.0405 1.35 5.04 5.46 8.77
in Holland are published in Ref. [47]. The additional full-scale turbines are Nibe in Denmark [16], Sexbierum in Holland [21], Alta Nurra in Italy [48] and Na¨sudden in Sweden [14]. In Table 1 the relevant parameters for the turbines are given. Except for Na¨sudden there is no stability information from the sites in Table 1. Considering the latitudes of the sites and the fact that turbines will normally work at rather high wind speeds, it is justified to assume the stability to be near neutral on the unstable side. In Fig. 9 t -values calculated from Eq. (9) are plotted against rotational frequency of the turbines in a log—log representation. As can be seen from the figure, the different turbines fall on a straight line with the slope-1. Regarding the fact that the very different turbines, both full-scale machines and wind tunnel structures, used in the comparison, obey Eq. (9) make us believe in the assumption behind the expression. 4.3. Far wake Beyond t"t the relative velocity deficit decreases with downstream travel time. As discussed above, t is more or less independent of C , but of course at time t the 2 relative velocity deficit is larger for larger C . This can be seen in Fig. 10, where the 2 relative velocity deficit is plotted as a function of transport time t, for t't , for the three different C -classes. Fitting the curve 2
*º º
t "C ln #C , t't 2 t
(10)
to the data points gives C "0.4. The three curves drawn in Fig. 10 represent Eq. (10) with three different C -values. The data scatter reasonably around the, respectively, 2 curves. Note that Eq. (10) implies that (*º/º)"C at t"t . In Fig. 11 (*º/º) is 2 given as function of transport time for the sites: (a) Nibe, (b) Sexbierum, (c) Na¨sudden, (d) Alta Nurra and (e) wind tunnel measurements at TNO. There is some uncertainty
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Fig. 9. Time constant, t as function of the rotational frequency for different sites: ( ) Na¨sudden, (;) Sexbierum, (#) Nibe, (*) Alsvik, (*) Alta Nurra and ( ) TNO-wind tunnel.
of the C -value used for Alta Nurra. The solid curves in all figures are Eq. (10). The 2 overall impression is that this equation fairly well describes the deficit decrease as a function of transport time. The site with largest difference between measurements and calculation is Na¨sudden. This is due to the fact that the averaging time is 30 min. The averaging time for the other full-scale sites is 1 min (Nibe, Alta Nurra) and 3 min (Sexbierum). Case studies at Alsvik show that longer averaging times gives lower relative velocity deficit, because of meandering of the centre. The difference of the relative velocity deficit measured with a sampling time of 1 min and 30 min, is calculated to be 20%, which is just about what is needed to get full agreement between measurements and the prediction obtained with Eq. (10). So far only near neutral conditions have been treated. Fig. 12 shows the influence of stability for the Alsvik turbines for C "0.76. The two curves have been adjusted to fit 2 the measurements through changing the value of t . In unstable air with higher degree of turbulence and larger turbulent eddies t is shorter than the neutral value 5.25 s and during stable stratification turbulence is suppressed and t is larger. The relative velocity deficit is largest and the decay with time is slowest during stable stratification. The time constant in the figure is t " 4.5 s and t " 6 s, for unstable and stable stratification, respectively.
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Fig. 10. Comparison between measured and calculated relative velocity deficit for three C -values at 2 Alsvik: (#) (- -) C " 0.88, (*) (—) C " 0.74, (*) ( ) ) ) ) C " 0.61. 2 2 2
5. Double wake When the wind is blowing from 165°, the wake of turbine T3 will affect turbine T2 (cf. Fig. 1) and the double wake behind the two turbines can be recorded on mast M1. As turbine T2 is working in the wake of turbine T3 the incoming wind is lower than the ambient wind speed. This means that the thrust coefficient and thus the relative velocity deficit should be larger for the second turbine, according to the above analyse. But both measurements at Alsvik and several other investigations e.g. [16], show that this is not the case. The wake behind the second turbine is less pronounced than behind the first at the same time downstream. This is probably a result of higher turbulence level upstream the second turbine and hence a more effective mixing of the flow. One case study at Alsvik gives the following picture. The undisturbed wind speed at turbine T3 is 9.1 m s\, which corresponds to a C of 0.73. At turbine T2 the wind 2 speed has been reduced to 5.9 m s\, giving a thrust coefficient of 0.82. Using Eq. (8) will give a relative velocity deficit of 0.43 after 15 s (at mast M1), while measurements
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Fig. 11. Comparison between measured and calculated relative velocity deficit for different wind turbines. (#) Tower measurements, (*) Sodar measurements, (—) Calculated, using Eq. (9). (a) Nibe, C " 0.82, (b) 2 Sexbierum, C " 0.75, (c) Na¨sudden, C " 0.8, (d) Alta Nurra, C " 0.85, (e) TNO-wind tunnel, C 2 2 2 2 " 0.87.
only give 0.36. It was discussed, earlier in Section 4.3, that atmospheric stratification will affect the transport time t . This time will be shorter in unstable air when the degree of turbulence is higher and longer for stable stratification. However, if we make the assumption that the higher turbulence intensity in the wake of the first turbine (T3) will create a shorter transport time t for the second turbine, it is possible in an empirical way to fit a new t -value to the measurements taken on mast M1. Fig. 13 shows the velocity deficit in the double wake behind turbine T2, measured on mast M1, for a C -value of 0.82. The solid curve is Eq. (10) with t "3.8 instead of 5.25 as 2 valid for the single-wake cases (see Section 4.3). In the literature two cases with double wakes, with the C -values given, are reported 2 from the Nibe site [16]. The distance between the turbines is 200 m and measurements were performed 100 m downstream the second turbine. The measured relative velocity deficit was 0.42 as the first turbine operated at C "0.82. In both cases the second 2 turbine was operated at C +0.9. Assuming that the Eq. (10) is valid and the time 2 constant in the single wake is 5.04, gives the time constants 3.9 s, respectively 3.8 s, for the double wake. This modification of the time scale is of the same order as was found for the Alsvik case. These results support the above assumption, but further work has to be done, in order to validate the above theory.
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Fig. 12. Alsvik. C " 0.76. Relative velocity deficit as function of time, (#) measured in unstable air, (;) 2 measured in stable air. (—) 0.4 ln 4.5/t # 0.76, (- -) 0.4 ln 6/t # 0.76.
Fig. 13. Alsvik double wake, downstream turbine with C " 0.86, (—) 0.4 ln 3.8/t # 0.86, (*) measurements. 2
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6. Conclusions A large data set of wind speed, turbulence and temperature measured on two 54 m masts within the Alsvik wind farm on the island of Gotland, have been analysed to obtain characteristics of wind turbine wakes. The results of the analysis have been tested on data from several full-scale wind turbines at other sites as well as on wind tunnel simulation data. The relations obtained are found to have very wide applicability. Most of the results refer to single turbine wakes, but tentative results for two superimposed wakes have also been obtained.
6.1. Single turbine wakes (i) The thrust coefficient, C of the turbine is found to be a better variable for 2 description of turbine wake characteristics than wind speed, because it implicitly includes the effect of regulation on power output from the turbine. (ii) The characteristics of the wake is found to be primarily a function of downwind travel time as opposed to travel distance as conventionally assumed. The scatter of the data points is found to be reduced when time instead of distance is used in the various expressions describing downwind development of the wake. This behaviour is thought to be due to the time of exposure of the developing wake to the turbulence in the ambient air flow being of decisive importance. (iii) In the near wake the crosswind profile has two peaks, which gradually merge into a single centreline peak. The downwind travel time for this merge to appear, t , is found to be inversely proportional to the rotational frequency of the turbine and to the turbulence intensity of the ambient atmospheric flow and proportional to the ratio of the wake radius to the hub height. At travel time t it is found that the centreline relative velocity deficit, *º/º is equal to C . 2 (iv) For travel times t't , i.e. for the far-wake region, the relative centreline velocity deficit is found to be linearly related to ln (t /t). The data show that the centreline relative velocity deficit decreases faster with travel time during unstable atmospheric conditions compared to stable conditions. This is in agreement with the scaling of the parameter t with atmospheric turbulence intensity noted under (iii).
6.2. Double turbine wakes The centreline relative velocity deficit in the double wake behind two turbines is found to be lower than at the same travel time behind a single turbine. This is contradictory to the prediction from the relation between C and wind speed: the 2 second turbine is working in lower than ambient wind speed, which would give a larger C and thus larger relative deficit. But turbulence intensity upstream the 2 second turbine is much higher than upstream the first turbine, and this reduces the value of t , which according to (iv) above would give lower relative velocity deficit.
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Acknowledgements This project was sponsered by NUTEK (contract no. 506 226-8). The authors want to thank Professor U. Ho¨gstro¨m and Dr. H. Bergstro¨m for fruitfull discussions. The instruments used at Alsvik were developed by Mr. K. Lundin.
References [1] R.J. Templin, An estimate of the interaction of windmills in widespread arrays, National Establishment laboratory, Technical Report LTR-LH 171, Ottawa Canada, 1974. [2] P.B.S. Lissaman, Energy effectiveness of arbitrary arrays of wind turbines. Proc. 17th Aerospace Science Meeting, New Orleans, 1979. [3] T. Faxe´n, A.-S. Smedman-Ho¨gstro¨m, U. Ho¨gstro¨m, The Meteorological Field Project at the Wind Energy Test Site Kalkugnen, Sweden. Report no. 51, Meteorologiska Institutionen Uppsala Universitet, 1978. [4] P.E.J. Vermeulen, Wind tunnel study of the wake of a horizontal axis wind turbine, TNO Report 78-09674, 1978. [5] P.J.H. Builtjes, Wind turbine wake effects, TNO-Report 79-08374, 1979. [6] P.-H. Alfredsson, J.-A> . Dahlberg, Measurements of wake interaction effects on the power output from small wind turbine models, Technical note FFA HU-2189, Stockholm, 1981. [7] P.E.J. Vermeulen, An experimental analysis of wind turbine wakes, Proc. Third Int. Symp. on Wind Energy Systems, Copenhagen, 1980, pp. 431—450. [8] D. Smith, Multiple wake measurements and analysis, National Power Technology and Environmental Centre Leatherheads Surrey, 1991. [9] P.E.J. Vermeulen, P.J.H. Builtjes, Mathematical modelling of Wake interaction in wind turbine arrays, Part 1. Description and evaluation of the mathematical model, TNO Report 81-01473, Appeldorn, The Netherlands, 1981. [10] G.J. Taylor, Wake and performance measurements on the Lawson—Tancred 17 m horizontal-axis windmill, IEE Proc. Part A 130 (9) (1983). [11] D.T. Swift-Hook, J. H+jstrup, D.N. McIntosch, D.J. Milborrow, G.J. Taylor, Nibe wake measurements project in Denmark, European Wind Energy Conference 22—28 October 1984, pp. 47—55. [12] R.W. Baker, S.N. Walker, Wake velocity deficit measured at the Goodnoe Hills MOD-2 Site, Oregon State University, Dept. of Atmospheric Science, Report no. BPA 84-15, 1985. [13] R.S. Haines, D.J. Milborrow, D.I. Page, Wake Interaction Between the Howden HWP-300 and the WEG MS-1 Wind Turbine Generators on Orkney, UK, European Wind Energy Conf. and Exhibition, Rome, Italy, 1986, pp. 453—455. [14] U. Ho¨gstro¨m, D.N. Asimakopolos, H. Kabezidis, C.G. Helmis, A.-S. Smedman, A field study of the wake behind a 2 MW wind turbine, Atmos. Environ. 22 (4) (1988) pp. 803—820. [15] E. Lukken, Nibe wake measurements data analysis and model validation, TNO Report, Ref. no. 88-301, 1988. [16] G.J. Taylor, Wake measurements on the Nibe wind-turbines in Denmark, Part 2: data collection and analysis, ESTBL/L/0158/R80, National Power, UK, 1990. [17] J. H+jstrup, M.S. Coutney, C.J. Christensen, P.Sanderdorf, Full scale measurements in wind-turbine arrays, N+rrekær Enge II. CEC/JOULE, Ris+ National Laboratory, Roskilde, Denmark, 1993. [18] J. H+jstrup, P. N+rga rd, Tændpibe wind Farm Measurements 1988, Ris+-M-2894, 1990. [19] J. H+jstrup, M.S. Courtney, P. Sanderhoff, Measurements of structural loads in the N+rrekær Enge Windfarm, Amsterdam, EWEC’91, 1991, pp. 382—384. [20] J.W. Cleijne, Results of Sexbierum wind farm; double wake measurements. TNO Report Ref. no. 92-288, 1992. [21] J.W. Cleijne, Results of Sexbierum wind farm; single wake measurements. TNO Report Ref. no. 92-082, 1992.
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[22] F.J. Verheij, J.W. Cleijne, Wake and wind data analysis, 25th meeting of Experts, Increased Loads in Wind Power Stations, Wind Farms, 1993, pp. 65—71. [23] M. Magnusson, A.-S. Smedman, Influence of Atmospheric Stability on Wind Turbine Wakes, Wind Eng. 18 (3) (1994) 139—152. [24] R.L. Simon, D.F. Matson, J.M. Fuchs, Wake effects in a Fayette 95-IIS Wind Turbine Array, SERI/STR-217-3186A, 1988. [25] T.F. Pedersen, S.M. Petersen, K. Thomsen, P.H. Madsen, J. H+jstrup, Loads for wind turbines in inhomogeneous terrain measurements, Report Ris+-M-2922, 1991. [26] K. Thomsen, Loads on wind turbines in complex terrain wind farms, 25th Meeting of Experts, Increased Loads in Wind Power Stations, Wind Farms, 1993, pp. 15—23. [27] C.G. Helmis, J. Whale, K.H. Papadopoulos, C.G. Andersson, D.N. Asimakopoulos, D.J. Skyner, A comparative laboratory and full-scale study of the near wake structure of a wind turbine, EWEC’94, Thessaloniki, 1994, pp. 465—469. [28] I. Katic, J. H+jstrup, N.O. Jensen, A simple model for cluster efficiency, EWEC’86, Rome, 1986, pp. 407—410. [29] J.F. Ainslie, Calculating the flow field in the wake of wind turbines, J. Wind Eng. Ind. Aerodyn. 27 (1988) 213—224. [30] A. Crespo, J. Herna´ndez, E. Frafa, C. Andrau, Experimental validation of the UPM computor code to calculate wind turbine wakes and comparison with other models, J. Wind Eng. Ind. Aerodyn. 27 (1988) 77—88. [31] S.G. Voutsinas, J.P. Glekas, A. Zervos, Investigation of the effect of the initial velocity profile on the wake development of a wind turbine, J. Wind Eng. Ind. Aerodyn. 39 (1992) 293—301. [32] A. Crespo, L. Chacn, J. Herna´ndez, F. Manuel, J.C. Grau, UPMARK a Parabolic 3D Code to Model Wind Farms, EWEC’94, Thesanoliki, 1994, pp. 454—459. [33] H. Tennekes, J.L. Lumley, A first course in turbulence, The Massachusetts Institute of Technology, 1972. [34] K. Lundin, A. Smedman, U. Ho¨gstro¨m, A system for wind and turbulence measurements in a wind farm, Proc. European Community Wind Energy Conf. 1990, pp. 110—113. [35] H. Bergstro¨m, K. Lundin, Climatological mean wind and turbulence measurements using a high quality cup anemometer system, Proc. ECWEC’93, Travemu¨nde, 1993, pp. 697—700. [36] U. Ho¨gstro¨m, A critical evaluation of the aerodynamical error of a turbulence instrument, J. Appl. Meterol. 21 (1982) 1838—1844. [37] H. Bergstro¨m, U. Ho¨gstro¨m, Calibration of a three-axial fiber film system for meteorological turbulence measurements, Dantec Information No. 5 September 1987. [38] H. Bergstro¨m, A simplified boundary layer wind model for practical applications, J. Clim. Appl. Meterol. 25 (1986) 813—834. [39] J-A> . Dahlberg, Mast, boom & velocity gradient effects on cup anemometers, FFAP-V-009, 1996. [40] M. Magnusson, Near-wake behaviour of wind turbines, J. Wind Eng. Ind. Aerodyn. 80 (1999) 147—167. [41] L.L. Freris, Wind Energy Conversion System, Prentice-Hall International Ltd., Englewood Cliffs, NJ, 1990. [42] A.S. Monin, A.M. Yaglom, Statistical Fluid Dynamics, 3 Ed., MIT Press, Cambridge, 1987. [43] R.B. Stull, An Introduction to Boundary Layer Meteorology, Kluwer Academic Publisher, Dordrecht, 1988. [44] B. Grisogono, A curvature effect of the critical Richardson number, Hrvatski meteorolos\ ki c\ asopis 29 (1994) 43—46. [45] C.J. Nappo, Sporadic breakdowns of stability in the PBL over simple and complex terrain, Bound. Layer Meteorol. 54 (1991) 69—87. [46] H.A. Panofsky, J.A. Dutton, Atmospheric Turbulence, Wiley, New York, 1984. [47] A.M. Talmon, The wake of a horizontal-axis wind turbine model; measurements in uniform approach flow and in a simulated atmospheric boundary layer, TNO Report 85-010121, 1985. [48] G. Botta, V.A. Giraldi, Results of wake measurements at the Alta Nurra Wind Power Station, ENEL direnzione e richerch centro di ricera elettrica, 1990.
Air flow behind wind turbines M. Magnusson*, A.-S. Smedman Department of Earth Sciences, Meteorology Uppsala University, Villava( gen 16, S-752 36 Uppsala, Sweden Received 18 March 1996; accepted 17 April 1998
Abstract Wind turbine wakes have been studied by analysing a large set of atomospheric data, from a wind farm with four turbines sited on a flat coastal area. The results obtained have been generalized to allow tests against data from other full scale wind turbines as well as wind tunnel simulations. These comparisons are found to give very satisfactory results. Most results refer to single turbine wakes. The thrust coefficient, C , is found to be a better parameter for description 2 of wake characteristics than wind speed, because it implicitly includes the effect of regulation. It is also found that down wind travel time is more convenient to use than down wind distance in this context. The travel time, t , to the end of the near-wake region, i.e. to the point where a single velocity deficit peak first appears, is found to be inversely proportional to the rotational frequency of the turbine and to the turbulence intensity of the ambient air flow and proportional to the ratio of the wake radius and the hub height. For larger travel times, i.e. for the far-wake region, it is found that the centreline relative velocity deficit decreases with the logarithm of the time travelled and is parametrically dependent on t and C . The combined 2 wake of two turbines aligned with the wind direction is found to have a lower relative velocity deficit than the wake of a single turbine for the same travel time. This is explained by the inverse relation between t and turbulence intensity, the latter being much higher in the double wake, initially. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction 1.1. Background For large-scale exploitation of wind energy, it will be necessary to put wind turbines together in clusters or wind farms, as areas with suitable wind resources are limited. But in wind farms the turbines will always interfere with each other. The flow field
* Corresponding author. E-mail: [email protected]. 0167-6105/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 1 2 6 - 3
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behind a wind turbine is characterised by low wind speed, strong wind shear and a high degree of turbulence, and thus a second wind turbine placed behind the first one along the wind direction is likely to produce less energy than the undisturbed one, by an amount that will decrease with increasing distance. As wind shear and turbulence is recognised as two causes for dynamic loads on wind turbines, it implies that detailed knowledge of the flow field behind wind turbines is needed for planning of wind farms. The flow characteristics behind single wind turbines have been studied extensively during the last decade. Among the first investigators is the report by Templin [1]. He modelled theoretically the effect of an infinite wind farm in terms of its integrated roughness effect on the flow. Lissaman [2] constructed a model of a single wind turbine wake, based on the analogy of jets in turbulent flows (see below). The Lissaman model was first tested on full-scale data from Kalkugnen in Sweden [3]. Early wind tunnel simulations were performed at TNO in Holland [4,5] and at FFA in Sweden [6]. Vermeulen [7] summarised the laboratory data and suggested some modifications to the original Lissaman model, now renamed MILLY. Later measurements were performed in a wind tunnel in United Kingdom [8]. Single-wake studies in the atmosphere were reported by Faxe´n et al. [3], Vermeulen et al. [9], Taylor [10], Swift-Hook et al. [11], Baker and Walker [12], Haines et al. [13] and Ho¨gstro¨m et al. [14]. In the early studies the velocity deficit was the main objective but later most investigations also include turbulence characteristics. A large number of wind tunnel data have accumulated since 1980 but also full-scale experiments in the atmosphere. In Denmark several investigations of wind farms have been performed. First, the wind and turbulence field behind the two Nibe turbines were thoroughly investigated [11,15—17]) and later the two large wind farms Tændpibe [18] and Norrekær Enge [19,17] were studied. In Holland a research wind farm, consisting of 18 300 kW wind turbines, has produced a large amount of data of the flow structure in a wind turbine cluster [20—22]. In Sweden data have been taken at the Alsvik wind farm since 1990 [23]. Data from Alsvik are also analysed in the present investigation. The abovementioned sites can all be described as flat and homogenous but there are also studies from some mountainous areas. In USA wind farms have been studied at Castello Ranch in Altamont Pass [24] and Alta Mesa [25,26] and in Greece on the Samos Island [27]. Since the first attempt to model the flow field behind a turbine [2] a number of rather sophisticated models have been developed. Apart form the MILLY model and other models of the Lissaman type e.g. the Park model from Ris+ [28], several models of the k—e type have been developed [29—32]. 1.2. Flow structure behind a turbine A wind turbine extracts energy from the mean flow, which results in a region behind the machine where the wind speed is reduced, with the largest reduction at the centreline at hub height, which decreases with downstream distance. This is usually looked upon as the ‘wake’ behind the wind turbine. Classical boundary-free shear flows that occur in nature or in engineering applications are wakes, jets and mixing layers (shear layers). In wakes the cross stream
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advection term is very small compared to the longitudinal advection but in jets and mixing layers the two advection terms are of the same order [33]. With the notation used in the atmospheric boundary layer: º
*º *º +¼ *x *z
(1)
where º is the longitudinal mean wind speed, ¼ is the mean vertical wind speed, x and z are the downwind and vertical directions, respectively. It has been shown from turbulence measurements (see Section 2) that Eq. (1) is fulfilled for the velocity field behind the Alsvik turbines: º
*º *º +0.04!0.05, ¼ +0.01!0.04. *z *x
(2)
In a jet or mixing layer as much momentum is carried by the transverse flow as by the downstream flow, which also seems to be the case for the velocity field behind a wind turbine. A consequence of Eqs. (1) and (2) is that the flow field behind a wind turbine is, strictly speaking, either a negative jet or a three-dimensional mixing layer but not a wake. However, hereafter the flow field behind a wind turbine will be called a wake, as being the most common notation. In this investigation wakes behind the wind turbines in the Alsvik wind farm on the island of Gotland in the Baltic Sea are described. In Section 2 the site and the intrumentation are presented and in Section 3 some earlier results are discussed. The velocity deficit behind the turbine is analysed in Section 4 and finally in Section 5 the wake measured behind two wind turbines is presented.
2. Site and instrumentation The Alsvik wind farm consists of four 180 kW Danwind turbines and is located close to the shore line on the West coast of Gotland. The site, which is presented in Fig. 1, is a flat coastal strip covered with grazed grass and low herbs. Only data with wind coming from the sea (200—290°) is used in this investigation, with the exception of the analysis of the double wake (see below). The four turbines are stall regulated, three bladed with a diameter of 23 m and a rotation speed of 42 r.p.m. The hub height is 35 m. Cut-in speed is 5 m s\ and the rated wind speed is 12 m s\. Meteorological measurements have been performed at Alsvik since June 1990. There are two masts (M1 and M2 in Fig. 1) which at the beginning had a height of 42 m, but in June 1991 a new top section was erected giving a total height of 54 m. Wind speed and direction are measured at 7(8) heights on both masts and temperature at 5 heights on mast M1. Wind speed and direction are measured with a type of sensor developed at the Department of Meteorology in Uppsala (MIUU). The instrument is a combination of a small three cup anemometer and a very light wind vane. Wind
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Fig. 1. The Alsvik site. T1—T4 are the wind turbines, M1 and M2 are the meteorological masts.
speed and direction are sampled with 1 Hz. The instrument is described in Refs. [34,35]. Temperature is measured with 500 W platinum sensors. The four turbines are marked T1—T4 in Fig. 1. Wake profiles from turbines T1—T3 are measured on mast M2, while the undisturbed wind profile is recorded on mast M1. The distance from T1, T2 and T3 to mast M2 is 9.6D, 4.2D and 6.1D, where D is the rotor diameter (23 m). During a concentrated field effort in June 1991 turbulence measurements were performed on mast M2 at three levels. The turbulence instrument used was the MIUU instrument, which is described in Refs. [36,37]. It is basically a wind-vane-based three-axial hot film system supplemented with dry and wet bulb temperature sensors. The sampling rate was 20 Hz. Measurements from this field campaign were used to estimate the two terms in expression (2). Here data from June 1991 to November 1993 have been analysed. The averaging time is usually 1 min. 2.1. Internal boundary layer When the wind is blowing from the sea an internal boundary layer is building-up over the land surface with its different roughness and surface heat flux. A simple model [38] has been used to calculate the height of the internal boundary layer at the position of mast M2. For unstable stratification the new internal boundary layer will reach up to 20 m, but the height will decrease with increasing atmospheric stability. The same result is found when comparing simultaneous measurements on the two masts, when the turbines are not working. An example is given in Fig. 2, which shows
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Fig. 2. Examples of the effects of the internal boundary layer: (a) velocity profile; (b) turbulence intensity profile; (;) mast M1, (*) mast M2.
wind speed in (a) and turbulence intensity profiles in (b). The growth of a new internal boundary layer will affect the wind measurements at the lowest heights on mast M2.
3. Data 3.1. Earlier analyses In Ref. [23], the relative velocity deficit and the turbulence in the wake were studied as function of distance and stability. The relative velocity deficit is defined as *º/º "(º !º )/º , where ‘a’ denotes the ambient wind speed and ‘w’ the wind speed in the wake and the increase of turbulence was expressed as ‘added turbulence’, ((pº) "(pº) !(pº)) where rº is the standard deviation of the mean wind speed. The stability parameter used was the Richardson number, Ri, g *¹/*z Ri" , ¹ (*º/*z)
(3)
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where ¹ is a reference temperature, g is the acceleration of gravity and ¹ is the mean temperature. It was found that the relative velocity deficit decreased with increasing distance from the turbine and increased with increasing stratification. The added turbulence in the wake showed, close to the turbine, two separate peaks at the positions of the blade tips. The two maxima disappeared downwind the turbine more rapidly in unstable air than in stable. In the present investigation atmospheric stratification is divided into three stability classes: unstable (Ri( !0.05), neutral (!0.05(Ri(0.05) and stable (Ri'0.05). Richardson number was calculated using temperature and wind speed at 18 and 31 m.
3.2. Wind speed corrections The difference in wind speed between the two masts was calculated for all wind directions, when the wind turbines were working. Fig. 3 shows (º !º )/º as + + + a function of wind direction at a height of 31 m. As indicated in the figure all the different wakes can be identified. For wind directions when both masts are undisturbed there should be zero difference in wind speed, but according to Fig. 3 there is always a slight difference, which can be both positive (220—260°) and negative (280—310°). Dahlberg [39] has calculated the tower disturbance on the wind field for the towers used in Alsvik with a two-dimensional model based on potential flow theory, including solid body effects and wake flow. The dotted line in Fig. 3 shows the theoretical calculations, which agree very well with the measured differences. All wind speed measurements have been corrected according to the calculated curve.
Fig. 3. Boom and mast effects on the measurements: (—) mean value for the measurements; (- -) calculated boom and mast effect, after Ref. [39].
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3.3. The thrust coefficient and power production In Ref. [40] the effect on the wake of the thrust coefficient, C , was discussed. The 2 thrust coefficient is defined [41] as F C " , 2 0.5oºA
(4)
where F is the axial thrust, o is the air density and A is the rotor area. The thrust coefficient varies with wind speed and Fig. 4 shows the calculated function for the Alsvik wind turbine, taken from Ref. [40]. When C is large, close to one, the wind 2 turbine works with a high degree of efficiency, but this occurs at low wind speeds and thus the power production is low. For higher wind speeds the turbine is less effective, but the power production is large. Power production can thus be expressed as a function of C . But produced power can also be written as 2 P"oC ºA, (5) N where P is produced power, o is air density and C is efficiency coefficient of the N turbine. Fig. 5 shows the power wind speed curve for the Alsvik turbines. For a given wind speed the variation of produced power is large. This is mainly so because the turbine is very often not correctly aligned with the mean wind direction.
Fig. 4. Calculated wind-C curve for the Alsvik wind turbines. According to Ref. [40]. 2
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Fig. 5. Wind-effect curve for the Alsvik wind turbines.
An alternative method is to use the thrust coefficient instead of wind speed to describe the conditions of the turbine. The assumption that the turbine operates according to the wind effect curve does not need to be fulfilled. This method also yields data taken above rated wind speed can be used. 4. Relative velocity deficit 4.1. Dependence of transport time and stability Fig. 6a shows measurements of relative velocity deficit for unstable stratification as a function of distance and for C " 0.84. This is the usual way to plot relative velocity 2 deficit but the data points are very scattered at each distance (4.2D, 6.1D and 9.6D). A more physical way to represent the variation of deficit is to use transport time behind the turbine instead of distance. The near wake is determined by the properties of the turbine but the diffusion of the wake downstream is settled by the ambient turbulence and the longer transport time the stronger influence of atmospheric turbulence.
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Fig. 6. The relative velocity deficit as function of (a) normalised distance, (b) time downstream of the turbine. C " 0.84 and unstable stratification. 2
The transport time is calculated using Taylors hypothesis [42] B(x)"B(ut).
(6)
This hypothesis assumes that the eddies are transported with the mean velocity, without distortion. This is not completely the truth, but assuming this evolution to be relatively slow, the errors introduced are small. The free stream velocity is used as transport velocity of the eddies. The transport time, t, at a fixed height, z, is calculated: x t(z)" . u(z)
(7)
Fig. 6b shows the same relative velocity deficit values as in Fig. 6a but as a function of transport time. There is still a rather large scatter, but the data points form a more continuous band. Taking the data at x"6.1D with the standard deviation p + 0.075, and dividing it in data intervals of 1 s, it will range from 17 to 23 s. The mean of the standard deviations is in this case reduced to p + 0.06. The mean of the relative velocity deficit at this distance is of the order 0.25.
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By using time scale instead of length scale, yields a more continuous pattern and also reduces the scatter. We conclude that the relevant parameter is the transport time instead of the distance. In Fig. 7 isolines of relative velocity deficit in unstable stratification are given as function of height above ground and transport time for C + 0.85 in Fig. 7a and 2 C + 0.6 in Fig. 7b. The relative velocity deficit is much larger for large C -values 2 2 (low wind speeds) than for small (high wind speeds), because the turbine is working much more effectively at low wind speed, although the power production is less. In Fig. 7c and 7d the corresponding relative velocity deficit values are given but for stable air. It is quite clear that the relative velocity deficit increases with increasing stability. The ambient turbulence is less effective to diffuse the wake during stable conditions, and the stability tends to suppress vertical motions. The same result was found in Ref. [23]. In Fig. 8 the relative velocity deficit is given as a function of Ri for C " 0.74 and 2 t " 10 s. For Ri-values below 0.25 (the ‘critical value’) the relative velocity deficit increases at first with increasing stratification, as expected. But for larger Ri there is a tendency for the deficit to decrease, which would indicate more effective mixing. When Ri'0.25 the flow is considered not to be fully turbulent [43,44]. But breaking waves and other kinds of bursts can stir the atmospheric boundary layer and create large temperature and wind speed variations [45], and hence more effective mixing. 4.2. Near wake In the region closely behind the turbine the wake is mainly determined by the characteristics of the turbine, although atmospheric stratification also comes into play. In Ref. [40] theoretical calculations, supported by measurements, show that just behind the rotor the relative velocity deficit has two peaks, situated at the middle section of the blades. With increasing transport time behind the turbine momentum transport towards the centre gradually wipes out the two peaks and one maximum at the centre will occur. Generally speaking, the time needed to get one maximum at the centreline, t , depends on the rotation frequency, f, C and atmospheric conditions. 2 A large C will give a large deficit which will increase the peak values. But at the same 2 time the shear within the wake, and thus the momentum transport towards the centre, will also be large which will increase the transport of momentum towards the centre. Thus as a first approximation we will assume that t is independent of C . As shown in 2 Ref. [40] atmospheric turbulence influences the transport time in spite of the very short times involved, &5 s. For neutral stratification turbulence intensity in the undisturbed atmospheric surface layer can be written as [46] p 2.4k S" º ln z/z
(8)
where k is von Karmans constant equal to 0.4 and z is the surface roughness. Eq. (8) is valid for 30 min data.
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Fig. 7. Vertical distribution of the relative velocity deficit as function of time: (a) C + 0.85 and unstable 2 stratification, (b) C + 0.6 and unstable stratification, (c) C + 0.85 and stable stratification, (d) C + 0.6 2 2 2 and stable stratification.
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Fig. 7. Continued.
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Fig. 8. The centreline relative velocity deficit as function of Richardson number in stable air (Ri ' 0), for C " 0.74 at t + 10 s. 2
The radius of the rotor will determine the size of the wake just behind the turbine, and with a larger rotor it will take longer time for the momentum transport to create one peak value at the centreline. As turbulent eddies scale with the height in near neutral conditions, it is plausible to assume that with increasing hub height larger turbulent eddies will twist the wake and make the momentum transport more effective. For neutral stratification t can tentatively thus be expressed as
1 H R t "C ln , f z H
(9)
where C is a constant. f is the rotational frequency. H the hub height, z the roughness height and R the rotor radius. For the Alsvik turbines t is estimated from both calculations and measurements to be approximately 5.2 s [40]. With a z -value of 0.0005 for the sea sector at Alsvik and H"35 m, R"11.5 and f"0.7 Hz, the constant C is determined to C "1. To test the generality of Eq. (9), data from wind tunnel experiments and from 4 wind turbine sites, besides Alsvik, are used. The wind tunnel data performed at TNO
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M. Magnusson, A.-S. Smedman/J. Wind Eng. Ind. Aerodyn. 80 (1999) 169–189 Table 1 Data for the other sites used in the investigation (for notations, see text) Site
f (sU)
z (m)
H (m)
R (m)
t (s)
TNO TNO TNO TNO TNO Alta Nurra Nibe Sexbierum Na¨sudden
29.7 38.2 56.0 72.1 78.9 1.51 0.57 0.50 0.42
0.0006 0.0006 0.0006 0.0006 0.0006 0.12 0.07 0.06 0.04
0.36 0.36 0.36 0.36 0.36 17 45 35 77
0.18 0.18 0.18 0.18 0.18 6.75 20 15 37.5
0.1077 0.0837 0.0571 0.0444 0.0405 1.35 5.04 5.46 8.77
in Holland are published in Ref. [47]. The additional full-scale turbines are Nibe in Denmark [16], Sexbierum in Holland [21], Alta Nurra in Italy [48] and Na¨sudden in Sweden [14]. In Table 1 the relevant parameters for the turbines are given. Except for Na¨sudden there is no stability information from the sites in Table 1. Considering the latitudes of the sites and the fact that turbines will normally work at rather high wind speeds, it is justified to assume the stability to be near neutral on the unstable side. In Fig. 9 t -values calculated from Eq. (9) are plotted against rotational frequency of the turbines in a log—log representation. As can be seen from the figure, the different turbines fall on a straight line with the slope-1. Regarding the fact that the very different turbines, both full-scale machines and wind tunnel structures, used in the comparison, obey Eq. (9) make us believe in the assumption behind the expression. 4.3. Far wake Beyond t"t the relative velocity deficit decreases with downstream travel time. As discussed above, t is more or less independent of C , but of course at time t the 2 relative velocity deficit is larger for larger C . This can be seen in Fig. 10, where the 2 relative velocity deficit is plotted as a function of transport time t, for t't , for the three different C -classes. Fitting the curve 2
*º º
t "C ln #C , t't 2 t
(10)
to the data points gives C "0.4. The three curves drawn in Fig. 10 represent Eq. (10) with three different C -values. The data scatter reasonably around the, respectively, 2 curves. Note that Eq. (10) implies that (*º/º)"C at t"t . In Fig. 11 (*º/º) is 2 given as function of transport time for the sites: (a) Nibe, (b) Sexbierum, (c) Na¨sudden, (d) Alta Nurra and (e) wind tunnel measurements at TNO. There is some uncertainty
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Fig. 9. Time constant, t as function of the rotational frequency for different sites: ( ) Na¨sudden, (;) Sexbierum, (#) Nibe, (*) Alsvik, (*) Alta Nurra and ( ) TNO-wind tunnel.
of the C -value used for Alta Nurra. The solid curves in all figures are Eq. (10). The 2 overall impression is that this equation fairly well describes the deficit decrease as a function of transport time. The site with largest difference between measurements and calculation is Na¨sudden. This is due to the fact that the averaging time is 30 min. The averaging time for the other full-scale sites is 1 min (Nibe, Alta Nurra) and 3 min (Sexbierum). Case studies at Alsvik show that longer averaging times gives lower relative velocity deficit, because of meandering of the centre. The difference of the relative velocity deficit measured with a sampling time of 1 min and 30 min, is calculated to be 20%, which is just about what is needed to get full agreement between measurements and the prediction obtained with Eq. (10). So far only near neutral conditions have been treated. Fig. 12 shows the influence of stability for the Alsvik turbines for C "0.76. The two curves have been adjusted to fit 2 the measurements through changing the value of t . In unstable air with higher degree of turbulence and larger turbulent eddies t is shorter than the neutral value 5.25 s and during stable stratification turbulence is suppressed and t is larger. The relative velocity deficit is largest and the decay with time is slowest during stable stratification. The time constant in the figure is t " 4.5 s and t " 6 s, for unstable and stable stratification, respectively.
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Fig. 10. Comparison between measured and calculated relative velocity deficit for three C -values at 2 Alsvik: (#) (- -) C " 0.88, (*) (—) C " 0.74, (*) ( ) ) ) ) C " 0.61. 2 2 2
5. Double wake When the wind is blowing from 165°, the wake of turbine T3 will affect turbine T2 (cf. Fig. 1) and the double wake behind the two turbines can be recorded on mast M1. As turbine T2 is working in the wake of turbine T3 the incoming wind is lower than the ambient wind speed. This means that the thrust coefficient and thus the relative velocity deficit should be larger for the second turbine, according to the above analyse. But both measurements at Alsvik and several other investigations e.g. [16], show that this is not the case. The wake behind the second turbine is less pronounced than behind the first at the same time downstream. This is probably a result of higher turbulence level upstream the second turbine and hence a more effective mixing of the flow. One case study at Alsvik gives the following picture. The undisturbed wind speed at turbine T3 is 9.1 m s\, which corresponds to a C of 0.73. At turbine T2 the wind 2 speed has been reduced to 5.9 m s\, giving a thrust coefficient of 0.82. Using Eq. (8) will give a relative velocity deficit of 0.43 after 15 s (at mast M1), while measurements
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Fig. 11. Comparison between measured and calculated relative velocity deficit for different wind turbines. (#) Tower measurements, (*) Sodar measurements, (—) Calculated, using Eq. (9). (a) Nibe, C " 0.82, (b) 2 Sexbierum, C " 0.75, (c) Na¨sudden, C " 0.8, (d) Alta Nurra, C " 0.85, (e) TNO-wind tunnel, C 2 2 2 2 " 0.87.
only give 0.36. It was discussed, earlier in Section 4.3, that atmospheric stratification will affect the transport time t . This time will be shorter in unstable air when the degree of turbulence is higher and longer for stable stratification. However, if we make the assumption that the higher turbulence intensity in the wake of the first turbine (T3) will create a shorter transport time t for the second turbine, it is possible in an empirical way to fit a new t -value to the measurements taken on mast M1. Fig. 13 shows the velocity deficit in the double wake behind turbine T2, measured on mast M1, for a C -value of 0.82. The solid curve is Eq. (10) with t "3.8 instead of 5.25 as 2 valid for the single-wake cases (see Section 4.3). In the literature two cases with double wakes, with the C -values given, are reported 2 from the Nibe site [16]. The distance between the turbines is 200 m and measurements were performed 100 m downstream the second turbine. The measured relative velocity deficit was 0.42 as the first turbine operated at C "0.82. In both cases the second 2 turbine was operated at C +0.9. Assuming that the Eq. (10) is valid and the time 2 constant in the single wake is 5.04, gives the time constants 3.9 s, respectively 3.8 s, for the double wake. This modification of the time scale is of the same order as was found for the Alsvik case. These results support the above assumption, but further work has to be done, in order to validate the above theory.
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Fig. 12. Alsvik. C " 0.76. Relative velocity deficit as function of time, (#) measured in unstable air, (;) 2 measured in stable air. (—) 0.4 ln 4.5/t # 0.76, (- -) 0.4 ln 6/t # 0.76.
Fig. 13. Alsvik double wake, downstream turbine with C " 0.86, (—) 0.4 ln 3.8/t # 0.86, (*) measurements. 2
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6. Conclusions A large data set of wind speed, turbulence and temperature measured on two 54 m masts within the Alsvik wind farm on the island of Gotland, have been analysed to obtain characteristics of wind turbine wakes. The results of the analysis have been tested on data from several full-scale wind turbines at other sites as well as on wind tunnel simulation data. The relations obtained are found to have very wide applicability. Most of the results refer to single turbine wakes, but tentative results for two superimposed wakes have also been obtained.
6.1. Single turbine wakes (i) The thrust coefficient, C of the turbine is found to be a better variable for 2 description of turbine wake characteristics than wind speed, because it implicitly includes the effect of regulation on power output from the turbine. (ii) The characteristics of the wake is found to be primarily a function of downwind travel time as opposed to travel distance as conventionally assumed. The scatter of the data points is found to be reduced when time instead of distance is used in the various expressions describing downwind development of the wake. This behaviour is thought to be due to the time of exposure of the developing wake to the turbulence in the ambient air flow being of decisive importance. (iii) In the near wake the crosswind profile has two peaks, which gradually merge into a single centreline peak. The downwind travel time for this merge to appear, t , is found to be inversely proportional to the rotational frequency of the turbine and to the turbulence intensity of the ambient atmospheric flow and proportional to the ratio of the wake radius to the hub height. At travel time t it is found that the centreline relative velocity deficit, *º/º is equal to C . 2 (iv) For travel times t't , i.e. for the far-wake region, the relative centreline velocity deficit is found to be linearly related to ln (t /t). The data show that the centreline relative velocity deficit decreases faster with travel time during unstable atmospheric conditions compared to stable conditions. This is in agreement with the scaling of the parameter t with atmospheric turbulence intensity noted under (iii).
6.2. Double turbine wakes The centreline relative velocity deficit in the double wake behind two turbines is found to be lower than at the same travel time behind a single turbine. This is contradictory to the prediction from the relation between C and wind speed: the 2 second turbine is working in lower than ambient wind speed, which would give a larger C and thus larger relative deficit. But turbulence intensity upstream the 2 second turbine is much higher than upstream the first turbine, and this reduces the value of t , which according to (iv) above would give lower relative velocity deficit.
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Acknowledgements This project was sponsered by NUTEK (contract no. 506 226-8). The authors want to thank Professor U. Ho¨gstro¨m and Dr. H. Bergstro¨m for fruitfull discussions. The instruments used at Alsvik were developed by Mr. K. Lundin.
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