Prediction of the wind speed probabilities in the atmospheric surface layer G. C. Efthimiou1a, P. Kumarb, S. G. Giannissia, A. A. Feizb, S. Andronopoulosa a
Environmental Research Laboratory, INRASTES, NCSR Demokritos, Patriarchou Grigoriou & Neapoleos Str., 15310, Aghia Paraskevi, Greece. LMEE, Université d’Evry Val-d’Essonne, 40 Rue du Pelvoux 91080 Courcouronnes, France. b
Abstract Accurate prediction of the wind speed probabilities in the atmospheric surface layer is very important for wind energy assessment studies and many other practical applications such as the design and operation of wind turbines and human exposure to wind extremes. In a recent study, an optimized beta distribution was developed for the prediction of the wind speed probabilities in the atmospheric surface layer. Various uncertainties arise in real scenarios due to the composite atmospheric variability, the topography of the terrain, nearby obstacles, orographical features, and other synoptic conditions. Measurements from field experiments are essential and useful for validation of a theory or a model in realistic atmospheric conditions. Thus, in the first part of this study, the beta distribution is validated further with the wind speed database of the FUSION Field Trial 2007 (FFT-07) tracer field experiment for various atmospheric stability conditions. The model is applied without any change in its constants and a high degree of agreement with the field experiment is achieved. The advantages of the proposed beta distribution are that it can predict successfully all the range of the real cumulative distribution function (from the 1st to the 100th percentile) and it can be incorporated in computational models that are able to predict the mean, the variance and the integral time scale of the wind speed. The second part of the paper includes the incorporation of the beta distribution in the Reynolds Averaged Navier Stokes (RANS) methodology. Initially, the “RANS-beta” model is validated against wind speed measurements performed in a wind tunnel over a rough ground. The wind speed skewness and kurtosis were found to be highly dependent on the height and below 50 m the model gave comparable results with the experiment. Then, the wind speed database of the field experiment JU2003 is used to examine the “RANS-beta” model’s performance. The 25th, 50th, 75th and 95th model percentiles at 20 sensors located inside the complex urban area were found to be in good agreement with the experimental ones (FAC2=0.8).
Keywords beta distribution; wind speed; RANS; atmospheric surface layer.
1
Corresponding author. Tel: +30 2106503405. E-mail address:
[email protected] (G. C. Efthimiou).
1. Introduction The prediction of the wind speed in the atmospheric surface layer is a very interesting research field. During a short period (e.g. 15 minutes) and at all locations of the atmospheric surface layer, it is essential to know the range of the wind speed values as well as the probability of the wind speed to exceed a threshold. These have practical applications in the design and the operation of wind turbines (e.g. Seshaiah and Sukkiramathi, 2016). A reliable estimation of the extreme wind speeds is significant for wind energy applications as the wind speed exceeding from a threshold value is undesirable, because the wind farms provide negligible power for the wind values greater than their cut-off thresholds. The extreme values of the wind speed can also seriously compromise the mechanical safety of an installation (Chiodo and De Falco, 2016). Statistical models are usually used for the prediction of wind speed probabilities. In engineering practice, the probability distribution functions (pdf) of the wind speeds are associated with the average wind turbine power (Masseran et al., 2012). The wind speed probability distribution functions are also useful to downscale the General Circulation Models (GCMs) output to the small-scale because the needs of user communities require a description of the wind speed pdf (e.g. Pryor et al., 2005; Devis et al., 2013). Also, even, after selecting a suitable probability distribution for a particular application, the estimation of the corresponding parameters is not exempt from issues. In the literature, the most common selection is the Weibull distribution (e.g. Ramenah and Tanougast, 2016; Dabbaghiyan et al., 2016; Gómez-Lázaro et al., 2016, etc.). However, the downside of the Weibull distribution is that it cannot predict the finite extreme wind speed (i.e. the 100th percentile). For the prediction of extreme wind speeds one seeks distributions of the extreme value theory that can model the upper tail of the real pdf (Perrin et al., 2006; D' Amico et al., 2015). It should be noted that to overcome the drawbacks of parametric models, some researchers introduced non-parametric methods as an alternative to estimate wind speed probability distributions. For example, Wang et al. (2016) used four sites in central China to review and compare the popular parametric and non-parametric models for wind speed probability distribution and the estimation methods for these models’ parameters. The simulations revealed that the non-parametric model outperformed all of the selected parametric models in terms of the fitting accuracy and the operational simplicity, and the stochastic heuristic optimization algorithm was superior to the widely used estimation methods. In quantitative terms, we would like to be able to predict the time-averaged wind speed V within the time interval Δτ that is encountered at an arbitrary receptor location: V
1
v(t )dt
(1)
where v(t) is the instantaneous wind speed. The time interval Δτ can represent the typical durations of gust events or can be understood as an exposure time, e.g. based on the time an individual is expected to stay at a location. Especially inside the street networks, wind speeds can be very high and associated timescales are as low as the typical pedestrian walking time
(i.e. a few seconds), resulting in gusts that can cause discomfort to pedestrians and create the potential for accidents. The derivation of a new pdf for V(Δτ) and the associated maximum expected wind speed Vmax(Δτ) was the main objective of a recent study by Efthimiou et al. (2017). Efthimiou et al. (2017) proposed an optimized beta distribution function that satisfies these criteria i.e. (i) it has a finite upper extreme, and (ii) it can predict all the percentiles. The model has been successfully validated with wind tunnel and field measurements and direct numerical simulation (DNS) and can be incorporated in computational fluid dynamics (CFD) models that use the Reynolds averaged Navier Stokes methodology (RANS). The high degree of agreement of the Efthimiou et al. (2017) statistical model with the wind tunnel and field experiments and with the DNS simulations generates further thoughts about the applicability of the model in real cases. Due to the composite atmospheric variability, the topography of the terrain, nearby obstacles, orographical features, and other synoptic conditions various uncertainties arise in real scenarios. To examine the performance of the statistical model, which is based on an optimized beta distribution function, and for its further validation, a real field experiment was used in this study. A comprehensive dataset of the meteorological, turbulence and concentrations observations from the recently conducted FUSION Field Trial 2007 (FFT-07) field experiment (Storwold, 2007) provides a useful dataset for this evaluation in various atmospheric stability conditions. Furthermore, in the present study, incorporation of the beta distribution in the RANS methodology is performed. The “RANS-beta” performance is assessed by comparing the model with experimental wind speeds measured over a rough surface in a wind tunnel as well as measurements inside a complex urban area based on the JU2003 field experiment (GarcíaSanchez, 2018). The basic equations of the statistical model and the steps for incorporation of the beta distribution in RANS methodology are given in Section 2. In Section 3, the experiments and the selected wind speed databases are described. The results of the model are presented and discussed in Section 4 and the conclusions are given in Section 5. 2. Methodology 2.1. The statistical model When adopting any finite range pdf for the time-averaged wind speed V(Δτ), the prerequisite is the ability to estimate the extreme value Vmax(Δτ). In an earlier study (Efthimiou et al., 2017), an approach from another field of research (i.e. the release of airborne pollutants from point sources) introduced by Bartzis et al. (2008) was used as a basis. Bartzis et al. (2008) analyzed concentration time series from a field experiment and observed that the maximum value of each time series depends on the fluctuation intensity and the integral time scale:
V V max b I V TV
(2)
where V is the mean wind speed, I is the wind speed fluctuation intensity, and TV is the wind speed integral time scale derived from the wind speed autocorrelation function RV(τ). b and ν are the empirical parameters and need to be determined by the observations. These variables in Eq. (2) are defined as follows:
I
V2 V
2
, V V , TV RV d and RV 2
2
0
V t V t
(3)
V 2
where V2 is the wind speed variance and V is the wind speed fluctuation. The parameters b and ν in Eq. (2) are derived empirically. Any variability in their value reflects a combination of the Bartzis et al. (2008) model limitations, the experimental errors, insufficient stationarity of the time series and the finite duration of the signal in the data that was used to derive these values. Our previous study suggested indicative values of b = 6 and ν = 0.3 (Efthimiou et al., 2017). However, since we are interested in the upper bound of the pdf, the central point of interest is to go beyond the indicative maximum values (measured during the experiments) and focusing on the extreme value Vmax(Δτ) reached within a time interval of infinite duration. It is noted that such an extreme value naturally cannot be verified meas experimentally. When comparing the measured peak value Vmax with the expected meas extreme value Vmax(Δτ) for a specific receptor location, in theory, the relation Vmax ≤
Vmax(Δτ) will always hold due to the ultimately finite length of the measured signal. Given the indicative values of the parameters b and ν, it is proposed again that the extreme value Vmax(Δτ) can be approximated based on Eq. (2). It is assumed that the pdf for the timeaveraged wind speed V(Δτ) at a certain location is given by the beta function (Gupta and Nadarajah, 2004): 1 pdf x x 1 1 x ; 0 ≤ x ≤ 1
x
V Vmax
(4a) (4b)
in which α and ζ can be defined as:
1 1 1
(5a) (5b)
The parameters η and I in Eqs. (5a) and (5b) are defined in Eqs. (2) and (3), respectively. The values of V and V2 at the receptor under consideration are assumed to be known. These quantities are routinely available from several types of atmospheric wind flow models by solving the relevant equations for V and V2 (Hertwig et al., 2012; Koutsourakis et al., 2012)
or from measurements. As mentioned above, the estimate of the extreme value Vmax(Δτ) required in Eq. (4b) can be provided by Eq. (2). The value of the parameter b was estimated equal to 6. This value will be kept the same also in this study. 2.2. The “RANS-beta” model The “RANS-beta” modeling methodology for the prediction of wind speed probabilities includes the following three steps: i.
Prediction of the wind speed mean, variance and time scale using the RANS methodology.
ii.
Prediction of the wind speed maximum (Eq. 2) as well as the parameters of the beta model (Eqs. 5a and 5b).
iii.
Prediction of the wind speed percentiles as well as the skewness and kurtosis.
2.2.1. The RANS wind speed mean, variance and time scale The mean wind speed is derived by the components of the mean wind velocity using the 2
2
2
relation u v w . Thus, we have first to predict the mean wind velocity components. The RANS methodology can predict the velocity components by solving the corresponding Navier Stokes equations (u, v, w momentum) along with a turbulence model. The standard k-ε model (Launder and Spalding 1974) can be used as a first simple choice. However, we must keep in mind that previous studies have shown that k-ε is not the ideal model for atmospheric applications because it fails to predict the height of the atmospheric boundary layer (Bartzis, 2005). An alternative solution can be the k-l model (Bartzis, 1989) or the k-ζ model (Bartzis, 2005; Bartzis, 1989; Hertwig et al., 2012). However, the purpose of this study is not to find the ideal turbulence model for the specific application but to propose a general method to predict wind speed probabilities using the RANS methodology. For this reason, the standard k-ε model is used. Other models are expected to provide better results. The mean speed variance is actually the Reynolds stress V 2 , which according to the gradient hypothesis can be derived by:
V 2 2 t
V 2 k x 3
(6)
where νt is the eddy viscosity. The first term can be assumed negligible compared to the second term and thus the equation becomes, V 2
2 k 3
(7)
In case of k-ε model, the turbulent time scale can be calculated by (Efthimiou and Bartzis, 2011): TV 0.5
k
(8)
3. Datasets and test cases 3.1. The FFT-07 field experiment The wind dataset from a highly instrumented tracer field experiment, referred to as the Fusing Sensor Information from Observing Networks (FUSION) Field Trial 2007 (FFT-07) is utilized for the validation of the described statistical model. The FFT-07 was a short range (~500 m) comprehensive tracer field experiment that was conducted to provide high spatial and temporal resolution dispersion and meteorological measurements at the U.S. Army's Dugway Proving Ground (DPG), Utah in September, 2007 (Storwold, 2007). The terrain of the experimental site was uniform and homogeneous consisting primarily of short grass interspersed with low shrubs with a height between about 0.25 and 0.75 m with the momentum roughness length, 𝑧0=1.3±0.2cm (Yee, 2012). The FFT-07 experiment acquires a comprehensive meteorological and dispersion dataset and it involves various instantaneous, continuous, single as well as multiple point releases in various atmospheric stability conditions varying from neutral to stable, and unstable conditions. In this experiment, a tracer propylene (C3H6) was released at 2 m height above the ground surface and the concentrations were measured at 100 fast response digital Photo Ionization Detector (digiPID) samplers arranged in a rectangular staggered grid of area 475 m × 450 m in 10 rows and 10 columns. The digiPID samplers were deployed at 2 m height above the ground surface.
Figure 1. The layout of the FFT-07 tracer field experiment. The black filled circles denote the position of 100 receptors. The stars denote the source locations in selected four trials of the single releases.
Extensive meteorological and turbulence measurements from many Portable Weather Instrumentation Data system (PWIDs) and ultrasonic anemometer/thermometer (sonic) and other instruments were acquired during this experiment (Fig. 1). Three-dimensional sonics were mounted at five levels (2, 4, 8, 16, and 32 m) on three towers located at grid center, 750 m north-northwest of grid center, and 750 m south-southwest of grid center. The sonic data from these three towers is processed to produce wind and turbulence statistics and fluxes of heat and momentum and evaluation of the described optimized beta distribution. In this study, the observations from four FFT-07 trials 07, 15, 46, and 64 in different atmospheric stability conditions are considered and utilized to evaluate the optimized beta distribution for the prediction of the wind speed probabilities in the atmospheric surface layer. Based on the sign of the Obukhov length (𝐿) computed by the eddy covariance method from the sonic data measured at 4 m level of the south tower, trials 07, 46, and 64 were in stable conditions and trial 15 was categorized in convective atmospheric stability (Kumar et al., 2017). 3.2. Wind tunnel experiment To validate the proposed “RANS-beta” modeling methodology, boundary-layer wind-tunnel measurements are used. The experiments were performed by the Environmental Wind Tunnel Laboratory (EWTL) of University of Hamburg, which is specialized in this field. More details can be found in Efthimiou et al. (2016). In this study, the case of the boundary-layer flow over a very rough surface is chosen for the validation. The flow was physically modeled at a geometric scale of 1:225 and point-wise velocity measurements were conducted by means of a Laser Doppler Anemometry (LDA). The LDA was operated both in UV-mode and UW-mode. The vertical profiles perpendicular to the mean inflow direction are available. The velocity data are scaled to a full-scale reference height of zref = 100 m with reference wind speeds Uref ranging between 4.75 and 6 ms-1. The experimental roughness length on the ground was estimated equal to z0 = 1.53 m. Fig. 2 shows a picture of the wind-tunnel test section.
Figure 2. Test section showing the setup of floor roughness elements and vortex generators for the boundary-layer flow over a very rough surface, representative of the agglomerated roughness effect of an urban environment (source: EWTL; Bernd Leitl).
3.3. The JU2003 experimental data One test case from the JU2003 campaign, , the IOP3, is selected as the study case in this paper. The variability of the meteorological conditions, wind speed and direction outside the central city area and atmospheric stability, is considered to be sufficiently small during the IOP test case. The measured wind speed and direction from the Portable Weather Information Data System (PWIDS) No. 15 (located at about 1 km upwind of the central city area at 40 m height above ground) for IOP3 are shown in Fig. 3. Fig. 4 shows the position of the meteorological sensors and the direction of the incoming wind.
Figure 3. Measured wind speed and direction from the anemometer PWIDS15 upwind of the central city area for IOP3.
Figure 4. Top view of part of the computational domain, showing position sensors that measured wind, direction of the incoming wind and the building heights for IOP3. The x- and y-coordinates are model coordinates. 4. Results and Discussion 4.1. Validation with the FFT-07 experiment meas At each sensor location, an experimental peak Vmax was identified and a bmeas value was meas estimated from Eq. (2) using Vmax . Based on the analysis of all sensors, the frequency
distribution of bmeas values is shown in Fig. 5.
0.3
0.25
f
b
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
12
bmeas
Figure 5. Frequency distribution of bmeas (total number of data points analyzed: 60). The values of bmeas range from 0.18 to 11.14. The value of bmeas, however, is exceeded by 6 only one time out of 60 (i.e. 1.7%) and seems to represent the marginal part of the distribution’s upper tail pointing more to ‘outliers’ behavior rather than to regular upper tail data. It is worth highlighting that it is rather impressive that three completely different kinds of datasets, the wind tunnel measurements, the DNS and the present field measurements, corresponding to very different near-surface flow scenarios show such a similar behavior in the upper tail of the bmeas frequency distribution. This provides strong support for the hypothesis that a representative value of the b parameter is equal to 6 for various types of near-surface atmospheric flows. Fig. 6 shows scatter plots comparing V(Δτ) as obtained from the proposed method and the equivalents derived from the field experiment corresponding to cumulative probabilities of cdf(V) = 0.25, 0.50 and 0.75. For these simulations, the figures show considerably good agreement between the statistical model and the field experiment.
Figure 6. Trials: 07, 15, 46 and 64 (15 sensors per trial): modeled versus measured wind speeds (V = (u2 + v2+w2)1/2) corresponding to (a) cdf(V) = 0.25, (b) cdf(V) = 0.50 and (c) cdf(V) = 0.75. One critical question is how the derived individual cdf compares with the experimental ones at each sensor. As an example, and for the results illustration, three random sensors per trial have been selected. Generally, the results are acceptable taking into account the complex physics of the atmosphere. We can notice that in almost all cases the 100th percentile is overestimated by the model which is a positive sign about the selection of the parameter b (Figure 7). Trial 07
Trial 15
Trial 46
Trial 64
Figure 7. Modeled versus measured cumulative distribution function results for random sensors of the selected trials. 4.2. Validation with the wind tunnel experiment 4.2.1. The numerical simulations In this test case, the turbulent flow is parallel to the ground and thus only u component is accounted for, while the other two components are assumed to be zero. This implies onedimensional problem. The wind flow computation was performed using a CFD model ADREA-HF (Bartzis 1991, Andronopoulos et al. 1994, Andronopoulos et al. 2002, Venetsanos et al. 2010). This is an Eulerian model, solving the unsteady RANS equations. The code uses finite volumes numerical techniques with rectangular cells for the discretization of the partial differential equations. To describe the complex geometry, the volume porosity concept is used with solid surfaces of any orientation allowed to cross the computational cells. One-dimensional equations for the streamwise velocity U, the turbulent kinetic energy k, and the dissipation ε were solved. The computational grid consisted of 72 hexahedral equidistant cells with size 3.1 m. The height of the domain is 223.45 m. On the ground (at z = 0 m) a fully rough wall boundary condition was imposed, while on the top of the domain (at z = 223.45 m) a constant value equal to 6.359 ms-1 was set based on the measurements. Moreover, the experimental value of the roughness length was used (z0 = 1.53 m), to avoid detailed simulation of all the roughness elements of the wind tunnel presented in Fig. 2. Special attention should be given, in order, the height of the center of the adjacent to the ground computational cell (zP) to be greater than the roughness length, i.e. zP > z0, for wall function approach to be employed. In the present study, zP was set equal to 1.55 m. At the adjacent to the ground cell, the logarithmic law profile for the velocity was used,
uP
z ln P z0
u*
(11)
where uP is the velocity of the adjacent to the ground cell, u* is the friction velocity and κ = 0.4 is the von Karman constant. The turbulent kinetic energy and the dissipation rate at the adjacent to the ground cell are given by:
kP
P
u*2 C
(12)
C3 4 k P3 2
(13)
z P
4.2.2. Mean and maximum wind speed The results of the 1-D problem for U and Umax and the corresponding experimental results are presented in Fig. 8 up to the height of 223.45 m. For the mean wind speed, the model agrees well with the measurements. We have also observed in previous simulations that near the wall there is a deviation of the model from the measurements due to the wall function. The maximum wind speed is overestimated by the model. These results are in agreement with the results obtained by Efthimiou et al. (2016). It is also observed that the model computed mean and maximum wind speed profiles similar to a power law profile contrast to the experiment where the profiles of the maximum wind speed and the mean wind speed are different to each other. The experimental maximum wind speed follows approximately a constant profile around the value of 8 ms-1. 250
Height (m)
200
150 Experiment
100
Model
50
0 0
1
2
3
4
Mean wind speed (m/s)
5
6
7
250
Height (m)
200
150 Experiment
100
Model
50
0 0
2
4
6
8
10
12
Maximum wind speed (m/s)
Figure 8. Profiles of the mean and maximum wind speed as measured from the experiment and as calculated by the 1-D simulation. 4.2.3. Wind speed skewness, kurtosis, and percentiles The results of the 1-D calculation of the model wind speed skewness, kurtosis and percentiles and the corresponding experimental results are presented in Fig. 9. As far as the skewness is concerned, the model is in very good agreement with the experiment at the heights below 50 m, while at higher heights the model gives an overprediction. At this point, it should be reminded from statistics that skewness is a measure of the asymmetry of the probability distribution. Based on Fig. 9 the skewness is positive below 50 m and negative above this height. This indicates that at low heights the right tails are longer, while at higher heights the left tails are longer. The model is skewed in the same directions but with shorter tails. As far as the kurtosis is concerned, based on Fig. 9 the model is in very good agreement with the experiment at the heights below 150 m. Kurtosis is a measure of the ‘‘tailedness’’ of the probability distribution. In the experimental data, kurtosis is positive indicating that the distribution is leptokurtic. The model underestimates the kurtosis at the heights above 150 m and thus it is flatter. Finally, all the experimental percentiles 25%, 50% and 75% are predicted very well by the model in all heights. The profiles of the percentiles follow a power law profile in both the experiment and the model.
250
Height (m)
200
150 Experiment
100
Model
50
0 -1.5
-1
-0.5
0
0.5
Wind speed skewness
250
Height (m)
200
150 Experiment
100
Model
50
0 0
1
2
3
4
5
6
Wind speed kurtosis
250
Height (m)
200 Experiment 25%
150
Experiment 50% Experiment (75%)
100
Model (25%) Model (50%)
50
Model (75%) 0 0
2
4
Wind speed percentiles (m/s)
6
8
Figure 9. Profiles of the wind speed skewness, kurtosis, and percentiles as measured from the experiment and as calculated by the 1-D simulation.
4.3. Validation with the JU2003 experiment 4.3.1. The numerical simulations The geometry of the buildings of the Oklahoma City center has been provided as “shape” files by the UDINEE project coordinators. These files were imported in the code ADREA-HF to construct the solid boundary surfaces of the domain. The computational domain was cut out from the total area covered by the shape files according to the UDINEE project requirements. In Fig. 10, the city center geometry included in the shape files and the computational domain are presented in a view from the top. The buildings’ heights that are included in the computational domain range between 2 and 113 m.
Figure 10. Top view of the Oklahoma City center geometry, the horizontal boundaries, and the dimensions of the computational domain. The x-axis is in the west-east direction and the y-axis is in the south-north direction
The spatial discretization of the computational domain is presented in Table 1. The grid is Cartesian. It is uniform in the horizontal directions and stretched in the vertical with a factor approximately equal to 1.2.
Domain dimensions x/y/z (km)
Total number of cells
Number of cells in each axis x
y
Z
1.6 / 1.4 / 0.682 2,419,200 320 280 27
Minimum / maximum cell sizes (m)
dx
dy
dz
5.0 / 5.0
5.0 / 5.0
1.0 / 114.5
Table 1 Dimensions and spatial discretization of the computational domain The 5 m spatial horizontal resolution was a requirement of the UDINEE project to all models to facilitate intercomparison of results. A finer discretization has also been tested using 2.5 m horizontal spatial resolution and keeping the same vertical resolution (9,676,800 cells) to test the grid-independency of the results. The flow field results that were obtained (not shown here) were very similar to those obtained with the grid described in Table 1, confirming the grid independency. For the flow computations, the 3-dimensional Eulerian mass and momentum conservation equations were solved by the code. The standard k-ε turbulence closure scheme (Launder and Spalding, 1974) was used; therefore the conservation equations for the turbulent kinetic energy k and its dissipation rate ε were also solved. Thermal effects were neglected so no temperature or energy equations were solved. Neutral atmospheric stability was assumed by setting and keeping constant an appropriate vertical temperature gradient. The boundary conditions that were used for the solution of the hydrodynamic variables are presented in Table 2.
Plane
Boundary conditions
-x
Inlet: fixed vertical profiles for u, v, k, ε; w = 0
+x
Outlet: 𝜕𝜑⁄𝜕𝑥 = 0, φ = u, v, w, k, ε
-y
Inlet: fixed vertical profiles for u, v, k, ε; w = 0
+y
Outlet: 𝜕𝜑⁄𝜕𝑦 = 0, φ = u, v, w, k, ε
Ground
Standard wall functions, roughness length = 0.1 m
Top
𝜕𝜑⁄𝜕𝑧 = 0, φ = u, v, k, ε; w is calculated from the cell mass balance
Building walls
Standard wall functions, roughness length = 0.05 m
Table 2 Boundary conditions for the hydrodynamic variables (u, v, w: velocity components in the x-, y-, z-axis respectively, k: turbulence kinetic energy, ε: turbulence kinetic energy dissipation rate) The characterization of -x and -y domain planes as the inlet and of +x and +y planes as the outlet is based on the incoming wind direction which was southwest for IOP3. The vertical profiles that were imposed and kept constant at the inlet domain planes, as shown in Table 2, resulted in from solving the 1-dimensional momentum equation, having adjusted the horizontal wind velocity at the top boundary so as to obtain at the height of 40 m above ground the average wind velocity measured by the anemometer PWIDS 15. The 3dimensional flow problem was solved as a transient case. The model was run until a steadystate situation was reached, as revealed by the stabilization of the velocity values at several locations in the domain. A 1st order scheme was used for the time derivative, the Van Leer scheme was used for the approximation of convective terms and a 2nd order central-difference scheme was used for the approximation of diffusion terms. 4.3.2. Wind speed percentiles Fig. 11 presents the FAC2, NMSE and FB (their definition has been described in Efthimiou et al., 2017) of the 25th, 50th, 75th and 95th percentiles of the IOP3. The scatter of the data around the ideal line (1 to 1) is higher for the 25th percentiles (NMSE=0.61) while it is almost the same for the rest percentiles (0.1 for the 75th, 0.13 for the 95th and 0.16 for the 50th). The FAC2 is increased for the higher values of the wind speed with the best value for the 95th percentiles (FAC2=0.95). For all the percentiles the FAC2 is equal to 0.8. According to FB the model slightly underestimates the experiment for the 50th 75th and 95th percentiles while the underestimation for the 25th is high. Generally, it is concluded that the performance of the model is better above the 50th percentile which is a positive indication for the type of studies such as wind comfort and danger. 0.7 0.6
NMSE
0.5 0.4 0.3 0.2 0.1 0 25%
50%
75%
Percentiles
95%
1 0.9 0.8 0.7
FAC2
0.6 0.5 0.4 0.3 0.2 0.1 0 25%
50%
75%
95%
75%
95%
Percentiles
0.45 0.4 0.35
FB
0.3 0.25 0.2 0.15 0.1 0.05 0 25%
50%
Percentiles
Figure 11. Validation metrics of the 25th, 50th, 75th and 95th percentiles for the IOP3. 5. Conclusions In the spirit of addressing the fundamental problem of quantifying the turbulence-related inherent variability of short-time wind speed averages, an existing statistical methodology was validated with real field measurements. The methodology is based on the concept of a finite maximum wind speed at a given time interval and the use of the beta function as the pdf of wind speeds at a specified receptor. The problem itself is quite complex and adequate validation studies require extensive experimental datasets. It is also required to continue to validate the statistical model with several real wind speed experiments in diverse scenarios. The selected reference data for verifying the proposed approach are representative of the turbulent wind flow in the atmospheric surface layer: an atmospheric boundary layer without buildings. The sensor locations of the field experiment cover the boundary layer up to the
height of 32 m. The temporal resolution of the wind speed signals was fine enough (10 Hz). From the results obtained the following main conclusions can be drawn: 1. The second attempt to approximate the statistical behavior of the above-mentioned wind speed variability with a beta probability density function has proved to be successful. 2. The important issue of the extreme value in the beta distribution seems to be properly addressed by the Bartzis et al. (2008) model. 3. The value of the b parameter (equal to 6) in the Bartzis et al. (2008) model was verified also from this field experiment (FFT-07). The new model can broaden the capability of time-averaged computational models such as RANS models to estimate the wind speed cdf provided that reliable predictions of mean wind speeds, wind speed fluctuations, and integral time scales are available from these computations. The “RANS-beta” methodology was examined also in this study. The parameters of the beta distribution for the wind speed were calculated using the predicted mean, variance and time scale of the RANS methodology. A wind tunnel and a field experiment were used for the model validation which involve wind flow over a very rough surface and wind flow inside a complex urban area. The RANS-beta results have been compared with the measurements and an overall good agreement was found. However, in the wind tunnel case, the model tends to overestimate the maximum wind speed similarly to previous studies. The performance of the wind speed skewness and kurtosis was found to be highly dependent on the height. At heights below 50 m the model gave similar results with the experiment. In the field experiment, it is concluded that the performance of the model is better above the 50th percentile which is a positive indication for the type of studies such as wind comfort and danger. Future work can involve simulations using LES type models. Acknowledgments: The authors would like to thank Bernd Leitl of the Environmental Wind Tunnel Laboratory at the University of Hamburg for providing access to the reference database (CEDVAL-LES; https://www.mi.zmaw.de/index.php?id=6339). The FFT-07 dataset is available at https://fft07-dpg.dpg.army.mil/.We would like to thank the Defense Threat Reduction Agency (DTRA) for providing access to the FFT-07 field experiment dataset and a sincere thanks to Nathan Platt for helping to obtain it. The simulations were supported by computational time granted from the Greek Research & Technology Network (GRNET) in the National HPC facility ARIS (http://hpc.grnet.gr) under project CFD-URB (pr004009). We gratefully acknowledge the European Commission Directorate General for Migration and Home Affairs (DG HOME) for their support to the Urban Dispersion International Evaluation Exercise (UDINEE) activity.
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