1. Introduction
The K-Gill propeller vane (k vane) is an anemometer for measuring turbulent fluxes as well as mean flow properties. The k vane consists of two propellers—one oriented 45° upward, the other 45° downward—that are aligned into the mean wind direction by a vane. From the angular velocities of the propellers, horizontal and vertical wind speed components can be calculated. From the instantaneous values of the horizontal and vertical wind speed, momentum fluxes can be calculated using the eddy correlation method. Advantages of this design above its precursors, for example, the Gill UVW system (Gill 1975) or twin propeller-vane anemometers with a horizontal and a downward-looking propeller used earlier (Large and Pond 1981, 1982; Ataktürk and Katsaros 1987), are (a) its symmetry for up- and downdrafts, (b) the propellers are operating at moderate angles of attack, so the cosine response is well defined and no stalling of the propellers occurs, and (c) there is no need to align the instrument in the mean wind direction. Extension of the main shaft above the pivot of the vane and propeller mounting has improved symmetry even more. A similar design has been presented by Desjardins et al. (1986), but the k-vane anemometer in its present form was first introduced by Ataktürk and Katsaros (1989). So far the k vane has been used mainly to measure momentum fluxes over sea (Katsaros et al. 1987; Katsaros et al. 1993).
The k vanes discussed in this paper are used in a research project concerning turbulent fluxes of momentum and sensible heat in the atmospheric boundary layer over heterogeneous terrain (Verkaik 1997). Six k vanes have been installed at three levels (20, 100, and 180 m) at the 213-m meteorological mast of the Royal Netherlands Meteorological Institute (KNMI) at Cabauw (van Ulden and Wieringa 1996). In this project k vanes were preferred above, for example, sonic anemometers because the latter fail to operate in rain, wet snow, and heavy fog (Wyngaard 1981). Since our purpose was to operate continuously at Cabauw for at least a year, we could not consider using fair-weather instruments. Another operational advantage of the k vane is that it does not need to be pointed into the wind, as is the case with many sonics.
One copy of model 35301 and seven copies of a special model 35301DTX (manufactured by R. M. Young Co., United States) have been tested in the wind tunnel. The 35301 model has also been tested in a field comparison experiment. The 35301 will be referred to as the “old” k vane and the 35301DTX as the “DTX.”
The propellers and vane all have limited response times, so measurement errors can be expected when the k vane is exposed to high-frequency turbulence. In the present article the magnitude of the errors due to k-vane inertia is assessed by perturbation theory, wind tunnel, and field experiments. Based on the spectral behavior of the k vane, simple methods are presented to correct for instrument inertia.
2. Instrument description
The total height of the k vane (see Fig. 1), including the base and extension tube, is 1.01 m; the distance from the top of the base (diameter 0.16 m) to the vane arm and propeller mounting is 0.60 m. The diameter of the main shaft is 29 mm, while the minimum distance from the main shaft to the propellers is 0.18 m. The vane arm extends 0.40 m from the main shaft and the vane dimensions are 0.30 m × 0.36 m. Sturdy carbon fiber thermoplastic (CFT) propellers are used (diameter 0.20 m, pitch 0.30 m).
Model 35301DTX is a slightly different version of model 35301. The original shafts of the propellers and vane have been replaced by stronger ones, the mounting of the propellers has been modified to ensure a 90° angle between the two propellers, and additional electric wires have been inserted to enable us to place thermocouple electronics in the extension tube above the pivot of the vane. We extended our k vanes with electrolytic level sensors (Inclinometer NB3, AE Sensors, the Netherlands) and thermocouples. The level sensor can be used to correct for alignment errors, and the thermocouple enables the k vane to measure sensible heat fluxes. We first intended to mount the level sensor at the beginning of the vane arm. This resulted in an erroneous reading of the level sensor since vane movements generate centripetal accelerations. A solution to this problem was found in mounting the level sensor inside the top of the extension tube on the axis of the main shaft.
3. Interaction between propeller and vane dynamics
Propellers and vanes have been used for many years in meteorology, and many articles have been devoted to their dynamical properties. Propeller dynamics (appendix A) and vane dynamics (appendix B) relevant to the k vane are briefly summarized. In this section the interaction between propeller and vane dynamics of the k vane will be discussed.
a. K-vane response to a turbulent wind field
An excellent analysis of the interaction of propeller and vane dynamics for a propeller vane in a turbulent wind field was given by Zhang (1988). From perturbation theory he found an expression for the over- or underspeeding of the propeller vane in terms of propeller and vane parameters and turbulent wind velocity spectra. In this section the analysis will be extended so that it can also be applied to the k vane. The overspeeding error, artificial vertical wind speed, and measured (co-)variances will be expressed in terms of the k vane’s propeller and vane parameters and spectra of atmospheric turbulence.
The inclination of the wind vector Δψ∞ is calculated from measured
For fast propeller response (ωτ ≪ 1) this equation yields Δψ∞ = w′/U.
The first term and the first part of the second term in Eq. (3.24) represent the propeller overspeeding; the second part of the second term is the result of the discrepancy between the measured and real (instantaneous) inclination angle. The real wind inclination is usually larger than the measured inclination. Inclination of the wind vector will reduce the angle of attack on one propeller while increasing the angle of attack on the other. However, the increase in response of the former is larger than the decrease in response of the latter. So, the net effect will lead to an increase in the jointly measured horizontal wind speed. The correction to the total wind speed, which is applied using the measured inclination angle, is largest at zero inclination. When the inclination angle is underestimated, propeller responses are corrected using a too-large correction, resulting in an overspeeding error. The third term in Eq. (3.24) represents the total υ error. It is smaller than that derived by Zhang (1988) since decomposition of wind speed into horizontal components is done before averaging. In case of an infinitely fast propeller vane response, so that ωτ ≪ 1 and β′ = ϕ′, δ equals zero.
b. Gyroscopic stability propellers
Wieringa (1967) and Busch et al. (1980) mention the possibility of the angular momentum of the propeller (L) to be responsible for the gyroscopic stability of the vane. This applies, however, only to propeller vanes that can swivel in two directions, which are called trivanes. For vanes that can rotate only about a single axis, gyroscopic stability of propellers is not possible, as will be explained below.
Vane movements will alter the direction of L, so dL/dt is in the horizontal plane. Therefore, forces that are induced by azimuthal movements act in the elevation direction on the propeller axis. The propeller vane or k-vane axis cannot be elevated. Azimuthal movements of trivanes, however, can change elevation angles and vice versa.
The only way gyroscopic stability could possibly influence vane dynamics is by increased friction, as a result of the torque, on the bearings that support the vane. However, these torques will be small compared to other torques on the vane. For U = 12.5 m s−1 the k-vane propellers will rotate at 150 rad s−1. To assess the moment of inertia of the propeller, a tiny load has been attached to the tip of one of the blades and then the period of oscillation has been determined. The moment of inertia found this way equals 8.6 × 10−5 kg m2. The angular momentum of the two propellers Lprop1 + Lprop2 = 1.8 × 10−2 kg m2 s−1. Typical angular velocity of the vane equals 0.75 rad s−1. So the torque on the propeller axis is 1.4 × 10−2 N m. The torque on the vane blade at 3° from equilibrium equals 0.4 N m at this wind speed. So, in general, torques from gyroscopic stability are very small compared to torques on the vane blade; however, those torques are perpendicular. In the case of the k vane, the torque by drag on the extension tube is probably much larger.
4. Wind tunnel experiments
a. Propeller tests
The author has tested the CFT propellers (model 08254) in the wind tunnel of the Department of Meteorology of the Wageningen Agricultural University (WAU). This wind tunnel has an octagonal working section with a length of 0.4 m and a radius of 0.2 m (Monna 1983). So it is just large enough to do propeller tests (radius 0.1 m). Step changes in wind speed were used to determine the propeller’s response length. To perform step-down tests without significantly disturbing the mean flow, a fine cotton wire was wound round the propeller shaft. By pulling the wire the propeller was sped up like a top. This way propeller velocities of 4 m s−1 could be achieved. When the propeller is speeded up in reverse direction, the same procedure can be used for step-up tests. Equation (A5) has been fitted to the measured response to determine the response time. Only the tail of the response curve, after 60% adaption, has been used.
Results are summarized in Fig. 2. The response time τ is plotted as function of U∞. The solid line corresponds to τ = D/U∞ with D = 3.0 m, the overall average. From this figure it is clear that for small U∞’s, τ is less than would be expected from D = 3 m for both the step-up and step-down tests. The dashed curve gives the relative decrease of D for step-up tests in percentages. For U∞ < 4 m s−1 D decreases with 30%, so at low wind speeds the propeller responds quicker.
Response times for step-down tests seem to be smaller than for step-up tests; D for step-down tests equals about half the value of D for step-up tests when U < 2 m s−1. A possible explanation is the friction of the bearings. This will increase the deceleration of the propeller and decrease its acceleration. However, the step-down response at low wind speeds is not very well described by Eq. (A5), and the scatter of individual measurements is considerable.
b. Vane tests
1) Determination of λN and ζ
The author has tested the k vanes in the wind tunnel of Delft University of Technology, Faculty of Mechanical Engineering and Marine Technology, Laboratory for Aero- and Hydrodynamics. The open working section of this wind tunnel is 0.7 m in height, 0.9 m in width, and 1.6 m long. Vane tests were done with the propellers mounted on the k vane. At several wind speeds the vane was given a deviation (less than 15°) of its equilibrium position and then released. Vane and propeller responses were recorded using a Campbell 21X datalogger. This procedure was repeated twice for both back and veer wind deviations and for every k vane used at Cabauw. Special care was given to the symmetry of the experimental setup since some k-vane tests suggested different response characteristics for back and veer wind.
Results are shown in Tables 1 and 2; standard deviations are given in parenthesis. The results have been evaluated by two methods. First, overshoot ratios and the time between successive overshoots have been determined. From Eq. (B6), the damping ratio ζ can be calculated, and the time between two successive overshoots multiplied by the wind speed equals the half-damped wavelength (λd). From this the natural wavelength (λn = λd
2) Torque on the vane as function of attack angle
The description of vane response as a damped harmonic oscillator is based on the assumption M = Nβ, where the torque M increases linearly with the angle of attack β. The validity of this assumption has been tested in a wind tunnel experiment. A fine cotton wire was attached to the end of the vane arm. Using a pulley and some little weights, a force could be applied to the vane arm. With the wind tunnel running at constant speed more weights were added. This procedure was repeated for two wind tunnel speeds (6.5 and 10.1 m s−1) and for two k vanes. Results are shown in Fig. 3. The ordinate is M/U2 and the abscissa is β. Two important features are clear from Fig. 3. First, M/U2 does not increase linearly with β; rather, a parabolic increase seems to fit the data. Second, in veer wind deviations (k vane is turned in a back wind direction), M/U2 increases faster compared to back wind deviations for these k vanes. Dynamic tests of the same k vane revealed slightly stronger damping for back wind deviations, which suggests the opposite.
From Fig. 3 it seems there is a little offset in vane response for β > 0. If so, the vane would have an equilibrium position with β ≠ 0. Regression results indicated only insignificant offsets, however. For β < 0 least squares fitting yields M/U2 = −(0.061 ± 0.004)β;for β > 0, M/U2 = (0.045 ± 0.003)β.
3) Influence of propeller rotation on vane dynamics
To check empirically the theoretical considerations in section 3b, vane response tests have been repeated with fixed propellers. In spite of the conclusion that no effect could be expected, there was a clear difference in vane response. Both λn and ζ decreased to 9 (1) m and 0.43 (0.06), respectively, so the vane is indeed better damped when propellers are rotating. The same effect was observed earlier by Wieringa (1967). Scatter in λn is considerably less when propellers are fixed, especially using the least squares method. In Fig. 4 the difference in response between fixed and rotating propellers can clearly be seen. When propellers are rotating, the vane is usually critically damped after the first overshoot. This behavior cannot be described by Eq. (B2), which assumes equal overshoot ratios for successive overshoots. With propellers fixed the vane behaves much more in agreement with Eq. (B2) so the least squares method will be much more successful.
Flow distortion is certainly different with the propellers fixed than with propellers rotating or removed. The latter experiment was not carried out, unfortunately. It is difficult to understand, however, how flow distortion by the propellers can have such a remarkable effect on the vane dynamics.
5. Field comparison experiment
a. Experimental setup
A field comparison experiment was carried out in June and July 1994 at the meteorological site of WAU. The site has a free fetch of more than 20 obstacle heights in most directions (Bottema 1995). A sonic anemometer (Solent A1012R2, Gill Instruments, United Kingdom) was used as reference instrument. The k-vane model 35301 and sonic were place on top of a 20-m mast (diameter 0.15 m, open lattice structure), each on either side of a 1.5-m-long boom. The gap in the potentiometer of the k vane was oriented toward the sonic (150°). Nearly 300 28-min runs of raw data have been collected at a sampling rate of 10.4 Hz and spectra were computed. Both the finite response of the sonic as well as the separation between the sensors are insignificant when compared to the distance constant of the k vane (Bottema 1995).
No instrument is free of error and neither is the Solent sonic anemometer. Flow distortion by the sonic probe may cause an overestimation of 4%–6% in mean wind speed and 20% in momentum flux, according to Grelle and Lindroth (1994). Mortensen and Højstrup (1995), on the other hand, report a too-low response of the Solent for all wind speed components. However, most effects of flow distortion by the Solent show periodic behavior (period 120°). In the data selection used in the present analysis no such periodic effects were found. So the effect of flow distortion by the Solent on the results is expected to be small, and no corrections were applied to the Solent data.
b. Statistical results
In total 139 h of data were collected. Situations with weak wind were dominant; only 20% satisfied U > 4 m s−1. About 60% of the time unstable situations occurred, and almost 65% of the time the wind did not have a very disturbed fetch. From every 28-min file averages and (co-)variances have been calculated in three approximately 10-min blocks. No detrending was done. The 28-min averages of
The absolute accuracy of wind direction by the vane was not determined since the absolute alignment of both sonic and k vane is rather difficult. The overall average wind direction difference was put at zero. The standard deviation of all 10-min averages differences equaled 0.8°, so the accuracy of the vane is better than 1°. Large differences in wind direction were restricted to low wind speeds. Maximum differences in the selections U ≥ 1, 2, and 4 m s−1 were 5°, 3°, and 2°, respectively.
The k-vane-measured
Scatterplots of the data selected on wind direction and stationarity are shown in Fig. 5. There seems to be no minimum wind speed to ensure reliable measurements. The selection on stationarity, however, tends to reject low wind speed situations. The minimum wind speed in this selection is 0.3 m s−1.
c. Determination of k-vane properties from spectra
After selection of minimum wind speed (2 m s−1), stationarity, and wind direction (undisturbed fetch), average
Transfer functions can be calculated by dividing the k-vane spectra by the sonic spectra. Transfer of
Equations (A6) and (B3) have been fitted using least squares method to the observed transfer functions calculated from the selected data; D45° was found to be 2.9 m (±0.5 m), λn = 7.8 m (±0.9 m), and ζ = 0.49 (±0.05). The fitted transfer function of
6. Evaluation of instrument response errors
The k-vane overspeeding and the ratio of measured to actual (co-)variances can be estimated from the k-vane parameters and spectra of atmospheric turbulence. Standard spectra for stable stratification were taken from Olesen et al. (1984); for unstable stratification, spectra from Højstrup (1982) were used. Cospectra of
a. K-vane overspeeding
The k-vane parameters derived from the field experiment (D = 2.9 m, λn = 7.8 m, ζ = 0.49) were used to estimate the overspeeding. In Figs. 8 and 9 the total overspeeding is plotted for heights from 10 to 200 m as a function of stability (L is Obukhov length). Calculations for stable stratification are only meant for estimation of the order of magnitude of the overspeeding since surface layer scaling does certainly not apply over the whole height range in these conditions. Except for very unstable conditions, when the turbulence intensities become very large, k-vane overspeeding or underspeeding is less than a few percent. Note, however, that here again the parameterizations used for the turbulence intensities and spectra are out of their range of validity.
From Eq. (3.26) note that
b. Correction of variances and momentum flux
The measured fractions of second-order moments have been estimated by integrating the product of the k-vane transfer functions and the relevant spectra. In Fig. 10 the results are plotted for different values of z/D and D/L. Here, D, λn, and ζ were taken from the field experiment, where zi = 1000 m and z0 = 0.1 m. For
Although D from the field experiments is significantly smaller than that from the wind tunnel tests, the expected loss in (co-)variance is not significantly different when D45° = 3 m/
When z/L and zi are available, the measured variances and momentum flux can be corrected using the estimated losses from the standard spectra. This has been done for the data selected on stationarity and undisturbed fetch. Again regression coefficients were calculated. The results are summarized on the left-hand side of Table 4. When compared to the results of the uncorrected data (Table 3), it can be seen that part of the lost variances and momentum flux can be restored without increasing scatter.
When standard spectra do not apply, spectra measured by the k vane itself may be used to correct for loss of variance. Variance spectra can be divided by the appropriate transfer function, and the resulting spectra can be integrated to obtain corrected variances. Integration has to be truncated at the high-frequency end where the signal-to-noise ratio or the transfer function is very low. In this analysis, integration was truncated when the transfer function was below 0.04. The results are summarized on the right-hand side of Table 4.
Except for
7. Discussion
a. Minimum wind speed
The threshold wind speed of a propeller with well-maintained bearings is of the order of 0.1–0.2 m s−1. From the scatterplots (Fig. 5), including many runs with U between 0.3 and 1 m s−1, it can be concluded that the minimum wind speed the k vane needs for reliable measurements is of the same order of magnitude. When bearings wear during long-term field experiments, however, the threshold wind speed will increase and the sensitivity of the propellers will decrease. To exclude any influence of friction at low rotation speed, situations with U below 1–2 m s−1 should not be considered. Note that the propeller response deviates in the wind tunnel from its regular response when U is below 4 m s−1.
b. Bottema’s results
Bottema (1995) tested the k vane’s propellers (model 08254) in the wind tunnel of WAU before the field comparison experiment took place. He found the calibration of propellers was in agreement with their pitch and no significant deviations of k from unity [see Eq. (A2)] could be measured. The threshold wind speed Uthr and correction ΔUf both equaled 0.2 m s−1. The best fit of measured cosine response was expressed in goniometric functions, inspired by the expansion formulation in Busch et al. (1980), and was given in Eq. (A4).
Bottema determined the distance constant at 2.5 m and claims that the dependence on angle of attack agreed with D(ψ) = D(0°) /
Because of the size of the k vane (length of arm and blade 0.7 m, working section wind tunnel 0.40 m × 0.40 m), vane tests could hardly be done in the WAU wind tunnel. For want of something better, Bottema still evaluated vane properties from experiments in this wind tunnel. His reported values of the natural wavelength λn and the damping ratio ζ vary considerably with wind speed. Bottema argues that the most reliable estimates of λn and ζ were made at low wind speeds because of undesirable oscillation phenomena at high wind speeds. At U = 2 m s−1 he found λn = 4 m and ζ = 0.4.
c. Propeller response at low wind speeds
The smaller D for both step-up and step-down changes (Fig. 2) can be the result of the size of the step change. Doing tests at low wind speeds usually means applying small wind speed changes as well, especially for the step-down tests. Hicks (1972) found that “the time required for a propeller to respond to sudden increases in wind speed increases with the magnitude of the fluctuation.” In other words, at low wind speeds, applying small wind speed changes, the propeller responds quicker.
This also explains the smaller D and larger D45° found by Bottema. He used wind tunnel speeds of 2, 4, and 6 m s−1 when doing step response tests and found a D of 2.4, 2.7, and 2.8, respectively (M. Bottema 1997, personal communication). This is in close agreement with Fig. 2. The reported average value for D is biased because the wind speeds used were too low. Since the propeller response is less when it is inclined to the flow, Bottema probably used a larger wind tunnel speed when assessing D45°. The resulting response length will be larger because of this larger wind tunnel speed.
The faster propeller response for wind speed decreases compared to wind speed increases will reduce the overspeeding of the propeller. If the difference between step-up and step-down response times as well as the magnitude of the wind speed fluctuations was to be large, the step-down response time could even be smaller than for a step-up time. In that case the propeller could underspeed.
d. Field versus laboratory response
For both propeller and vane it seems that the field response is faster than the tunnel response, resulting in a smaller D and λn. For the propellers discussed in this report D = 3.0 m (wind tunnel) and D45° = 2.9 m (field experiment), which do not correspond at all to the observed increase of D with angle of attack. The parameters found for the vane are λn = 7.8 m, ζ = 0.49 (field comparison) and λn = 13 m, ζ = 0.54 (wind tunnel). A reason for the difference may be that the wind tunnel used is too small for the present propellers. However, other researchers also found a faster response in the field than in the laboratory (Fichtl and Kumar 1974; Pond et al. 1979). Because in a turbulent wind field there are no step changes, the propeller will usually be closer to its equilibrium response. Hicks’ (1972) results suggest that the faster field response may be caused by the smaller wind speed changes that are applied. On the contrary, Horst (1973) explained a larger D found from field experiments as the result of the increase of D with angle of attack. Since the field comparison results show less scatter and the field performance is thought to be of major importance, the author recommends using these results only when assessing instrument response parameters.
Katsaros et al. (1993) obtained propeller and vane parameters from laboratory tests. The natural wavelength and damping ratio they reported compare well to those found from our field comparison experiment. The vane they used, however, had slightly different dimensions. The distance constant they reported (2.2 m) is small. From their report it is not clear whether this is the distance constant at 0° angle of attack. If so, D45° is approximately 2.6 m, which is close to the value of 2.9 m found from the field comparison experiment.
e. Simple methods for the estimation of vane parameters
For simple vanes Wieringa (1967) derived formulas to estimate their dynamic parameters from the dimensions and weight of the vane (see appendix B). For the k vane S = 0.094 m2 (area of the vane blade), rυ = 0.48 m (distance from the vane pivot to one-quarter of the blade chord), and Jold = 0.086 kg m2 (moment of inertia of the vane). The latter has been assessed by attaching a little weight on the vane and then measuring the period of oscillation. This experiment has been repeated for several weights at different distances from the pivot of the vane. To reduce damping by the vane, the blade was twisted 90°.
When an infinite aspect ratio is assumed, the torque parameter aυ equals 2π, where N/U2 = 0.18 kg, λn = 4.4 m, and ζ = 0.34. These values compare rather well with the values Bottema (1995) found (N/U2 = 0.16 kg, λn = 4 m, and ζ = 0.4 at U = 2 m s−1). Since the blade of the k vane approximates a square, the aspect ratio is not infinite. In fact, aυ = 2.0 (span of the vane blade b = 0.36 m). In that case λn and ζ should equal 6.0 m and 0.25, respectively. When using the results of section 4b on the torque on the vane as function of angle of attack, λn = 8 m and ζ = 0.2.
The presence of the propellers and their mounting can certainly not be neglected in case of the k vanes. The presence of surface before the pivot of the vane will increase λn as well as ζ. The area of the projection of the surface before the pivot on a vertical plane Sw is estimated at 0.02 m2, and the distance of the aerodynamic center to the vertical axis rw is 0.2 m. This results in an increase of almost 5% in λn and of 9% in ζ (Wieringa and van Lindert 1971).
These formulas apply to simple vanes, however. The shape of the propellers and their mounting does not resemble that of a vane. This may explain the large difference, especially in ζ, between the estimated vane parameters and the measured parameters.
8. Conclusions
From the field experiment it can be concluded that the k vane’s measurements of average wind speed and direction are very accurate. Overspeeding or artificial vertical wind speed will generally be very small. Overspeeding could be significant in conditions of very high instability. These occasions usually are accompanied by very low wind speeds. Then the correction Δuf, which is usually neglected, and the different propeller response at these wind speeds may mask any overspeeding.
Both perturbation theory and a field comparison experiment show that the k vane behaves as a first-order sensor. The only relevant instrument parameter for measured variances and fluxes is D45°, the response length at 45° angle of attack (2.9 m for the 35301 model). This parameter can best be determined from a field comparison experiment, not from wind tunnel tests. Wind tunnel tests show much scatter, and the resulting parameters do not correspond very well to those from the field comparison experiment. When used above 20-m height, however, the exact value of D45° is of insignificant importance.
Transfer functions of the (co-)variances can be described by the regular first-order [Eq. (A6),
Instead of standard spectra, spectra measured by the k vane itself can be used to calculate corrections for the measured (co-)variances. In that case no information on atmospheric conditions is necessary. Using this method high-frequency variance may not be fully restored, resulting in lower estimates of
Acknowledgments
The author would like to thank Prof. Jon Wieringa for setting up this project and for good advice at various stages while writing this paper. The author would also like to thank Dr. Marcel Bottema, who prepared and started the field comparison experiment. Arjan van Dijk from the Delft University of Technology helped me doing the wind tunnel tests at his department. In carrying out the experiments many of the technical staff of the meteorology department of WAU were of great help. I would like to especially mention Bert Heusinkveld and Frits Antonysen. Also many thanks to Fred Bosveld from the KNMI, members of the meteorology department of WAU, and the reviewers who carefully read the manuscript and gave lots of valuable comments.
The investigations were (in part) supported by the Netherlands Geosciences Foundation (GOA) with financial aid from the Netherlands Organisation for Scientific Research (NWO), Project No. 750.194.15.
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APPENDIX A
Propeller Dynamics
Note that in this equation, equal to Eq. (43) from Busch et al. (1980), k will be larger than 1 in order to reduce ω at given wind speed U. Usually k is very close to unity (within 1%) and Δuf is very small (less than 0.1 m s−1). The correction Δuf should not be confused with the starting or threshold wind speed Uthr. The former is an offset to be determined from regression of the calibration curve; the latter is the minimum wind speed required to start the propeller from rest. Usually Uthr is larger than Δuf.
Here D is called the distance constant because it is assumed to be independent of wind speed. Since τ is smaller for larger wind speeds, the propeller responds faster to increasing wind speed and slower to decreasing wind speed. Placing the propeller in a turbulent wind field will cause the propeller to overspeed. The magnitude of the overspeeding will increase with D. This feature is similar to cup anemometer overspeeding and has been discussed in detail by Busch and Kristensen (1976). MacCready (1966) called this the u-error of a propeller anemometer. Here D = U∞τ increases slightly as the angle Ψ between wind direction and propeller axis increases. Note that this is only true when U is equal to the total wind, not to the component parallel to the propeller axis [compare Busch et al. (1980) and Bottema (1995)].
APPENDIX B
Vane Dynamics
Experimentally ω and γ or ζ can be found by fitting Eq. (B2) to the measured response or by measuring successive overshoots and the time between them.
Picture of the k vane.
Citation: Journal of Atmospheric and Oceanic Technology 15, 4; 10.1175/1520-0426(1998)015<0901:EOTKGP>2.0.CO;2
Dependence of D on U∞.
Citation: Journal of Atmospheric and Oceanic Technology 15, 4; 10.1175/1520-0426(1998)015<0901:EOTKGP>2.0.CO;2
M/U2 as function of β.
Citation: Journal of Atmospheric and Oceanic Technology 15, 4; 10.1175/1520-0426(1998)015<0901:EOTKGP>2.0.CO;2
Influence of rotation propellers on vane response.
Citation: Journal of Atmospheric and Oceanic Technology 15, 4; 10.1175/1520-0426(1998)015<0901:EOTKGP>2.0.CO;2
Comparison of
Citation: Journal of Atmospheric and Oceanic Technology 15, 4; 10.1175/1520-0426(1998)015<0901:EOTKGP>2.0.CO;2
Measured and fitted transfer functions of
Citation: Journal of Atmospheric and Oceanic Technology 15, 4; 10.1175/1520-0426(1998)015<0901:EOTKGP>2.0.CO;2
Measured and fitted transfer functions of
Citation: Journal of Atmospheric and Oceanic Technology 15, 4; 10.1175/1520-0426(1998)015<0901:EOTKGP>2.0.CO;2
Overspeeding in unstable stratification.
Citation: Journal of Atmospheric and Oceanic Technology 15, 4; 10.1175/1520-0426(1998)015<0901:EOTKGP>2.0.CO;2
Overspeeding in stable stratification.
Citation: Journal of Atmospheric and Oceanic Technology 15, 4; 10.1175/1520-0426(1998)015<0901:EOTKGP>2.0.CO;2
Measured fractions of
Citation: Journal of Atmospheric and Oceanic Technology 15, 4; 10.1175/1520-0426(1998)015<0901:EOTKGP>2.0.CO;2
Measured fraction of
Citation: Journal of Atmospheric and Oceanic Technology 15, 4; 10.1175/1520-0426(1998)015<0901:EOTKGP>2.0.CO;2
Model 35301 vane properties from wind tunnel tests.
Model 35301DTX vane properties from wind tunnel tests.
Results from the field comparison experiment. Regression parameters c0 and c1 from equation k vane = c0 + c1 × sonic; c0 = 0 in all cases.
Regression parameters for the corrected data. Spectra were taken from literature (left-hand side) or spectra measured by the k vane itself were used (right-hand side).
