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A statistical model for wake meandering behind wind turbines
Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
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A statistical model for wake meandering behind wind turbines Robert Braunbehrens a, Antonio Segalini b, * a b
Innogy SE, Wind Onshore, Kapstadtring 7, 22297, Hamburg, Germany Linne Flow Centre, STandUP for Wind, KTH Mechanics, Royal Institute of Technology, S-100 44, Stockholm, Sweden
A R T I C L E I N F O
A B S T R A C T
Keywords: Numerical wake model Linearised RANS Wake meandering Atmospheric dispersion
A new wake model is proposed to account for wake meandering in simulations of wind-turbine wakes performed on steady solvers, through a wake-meandering description based on the dispersion theory of Taylor (1921, P. Lond. Math Soc., vol. 20, pp. 196–211). Single-turbine simulations were performed by means of the linearised solver ORFEUS. By analysing the steady wake behind a turbine, a set of parameters describing the wake was first obtained and synthesised into a look-up table. The proposed meandering model extended the simulation results by superimposing the lateral and vertical meandering motions to the steady wake. As a result, the time-averaged velocity distribution of the wake was increased in width and reduced in intensity. Through this combination, the model provides rationale for the wake-deficit decrease and for the power underestimation effects of several wake models. The new wake model is validated against the Lillgrund and Horns Rev data sets.
1. Introduction As offshore wind farms are built with increasing size and investments, an accurate estimation of the wake losses is important in the design phase. However, for a computational simulation of a turbine wake, a wide range of turbulent length scales needs to be accounted for. This is why full RANS simulations of wind farms are still too computationally costly in an iterative design process. The industry relies therefore heavily on wake models, which do not solve for the flow field with the turbines, but rather in a post-processing manner: They are usually semi-empirical formulations that require the introduction of model parameters, that resemble physical conditions like the ambient turbulence or ground roughness. Remarkably, most wake models used in industry do not explicitly account for the phenomenon of wake meandering, and this could be a reason for the tendency to under-predict the power outputs of wind farms (Vermeer et al., 2003). Wake meandering was observed in full-scale measurements by Baker and Walker (1984) and Taylor (1990). It refers to a fluctuating movement of the whole wake region in the lateral and vertical directions. As a consequence of the wake being displaced, the power output of downstream turbines increases. Since these first studies, several attempts to model and verify experimentally the proposed dynamics have been made. Ainslie (1988) created a description in his wake model based on the thin-shear layer equations. He suggested that the meandering motion was due to the variation in the wind direction,
excluding however short timescale fluctuations. Further research was not pursued until the first wind-tunnel measurements were carried out by Medici and Alfredsson (2006, 2008) with a small-scale wind-turbine model. They observed a periodically-fluctuating helical movement of the wake, which made them suggest that an instability of the wake itself was causing the meandering, much like the vortex shedding behind a bluff body. Although the experimental setup had no free-stream large-scale turbulence, it became a reference case in the wind community. In contrast to Medici and Alfredsson, Larsen et al. (2008) introduced the Dynamic Wake Meandering model (DWM) with the fundamental assumption that the wake behaves like a passive tracer, so that no wake instability is accounted for. The large-scale atmospheric turbulence is responsible for the displacement of the wake as a whole and thus for the wake meandering. Larsen et al. defined that the threshold in characteristic length scales influencing the wake is two wake diameters, which means that only eddies with a length scale larger than that will induce wake meandering. This is based on the consideration that, in order for a wake-displacing eddy to move all points of a cross section, it has to be twice the size, since half of the displacement is going to be in the positive direction and the other half in the negative direction. In the final DWM model, the velocity-deficit shape is based on the wake model from Ainslie (1988) that assumes a steady eddy viscosity. The meandering motion is obtained through the low-pass filtering of a synthetic turbulent field to obtain the effect of the large-scale turbulence. The DWM purpose is mainly to provide a rapid option to calculate blade loadings.
* Corresponding author. E-mail address: [email protected] (A. Segalini). https://doi.org/10.1016/j.jweia.2019.103954 Received 4 January 2019; Received in revised form 9 June 2019; Accepted 10 July 2019 Available online 29 August 2019 0167-6105/© 2019 Elsevier Ltd. All rights reserved.
R. Braunbehrens, A. Segalini
Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
To verify the model, Bing€ ol et al. (2010) and Trujillo et al. (2011) measured the wake velocity with a LIDAR system mounted on the back of a full-scale turbine. They were able to detect a meandering motion of the wake due to large-scale turbulence and no periodic vortex shedding was observed. Wind-tunnel investigations were continued by Espa~ na et al. (2012) who used a porous disk submerged by an atmospheric boundary layer with length scale 10 times larger than the disk model. They were able to detect a meandering motion, more prominent in the horizontal than in the vertical direction. The assumption of the wake as a passive tracer was further supported by Muller et al. (2015), although they found that the characteristic length scale of the governing eddies was larger than assumed by Larsen et al. (2008). Finally, Coudou et al. (2017) used a setup of small wind-turbine models to investigate whether the wake meandering persisted inside a wind farm. With an incoming flow characterised by a turbulence length scale larger than three turbine diameters, they were able to assess that the meandering occurred throughout the whole farm model. Furthermore, in contrast to Espa~ na et al. (2012) and Muller et al. (2015), they observed a periodic movement of the wake similar to Medici and Alfredsson (2006) and concluded that intrinsic instability is amplified by large-scale turbulence. Several works assume that the success of the Jensen (also known as PARK) model (Katic et al., 1986) is associated to the fact that the empirical wake-expansion parameter already accounts for the wake meandering (Thøgersen et al., 2017). To avoid this implicit consideration of the wake-meandering mechanism, a simple model for wake meandering model based on Lagrangian particle dispersion (Taylor, 1921) is presented in this paper. It is implemented into a wake model for power prediction that was derived from the linearised code ORFEUS, which has been introduced by Segalini (2017) and Ebenhoch et al. (2017). The paper is structured as follows. Section 2 describes how the original theory of Taylor can be used to characterise wake meandering. It is followed by section 3 describing how the simulations of the linearised code ORFEUS were combined to create look-up tables to construct the wake. The proposed wake model is a combination of sections 2 and 3. Thereafter, section 4 assesses the performance of the new wake model for the two reference cases given by the Lillgrund and Horn Rev wind farms, while some final conclusions are stated in section 5.
Fig. 1. Schematic representation of the approach proposed in this paper. The “instantaneous” field is obtained from steady ORFEUS simulations (or LUT method) without wake meandering. The “time-averaged” field results from the statistical ensemble of the steady ORFEUS fields averaged according to the assumption that the wake centre behaves as a passive tracer.
density function of the position yp ðtN Þ tends to be Gaussian as N → ∞ pffiffiffiffi N Δt σ v . If one expffiffiffiffi presses the total time as T ¼ NΔt, it is clear that σ yp ∝ T , so that the standard deviation does not grow linearly with time, but at a slower rate. This is true regardless of the velocity probability density function. At this point, one should remove the assumption of independence of the velocity displacements and consider a continuous velocity field correlated in time. The position of the particle is now given by with mean yp ¼ NΔtv and standard deviation σ yp ¼
Z
t
xp ðtÞ ¼ 2. Statistical description of wake meandering
N X
vðti Þ :
(2)
0
The mean position is simply obtained as xp ðtÞ ¼ vt, while the variance of the generic component of the position is given by
The fundamental assumptions of the present model were stated by Larsen et al. (2008) with the hypothesis that the wake behaves as a passive tracer and that the entire wake preserves its shape when laterally displaced. Under these two hypotheses, the instantaneous wake characteristics are defined by the wake centre position and by the undisturbed geometry of the wake. This splits the characterisation of wake meandering into two independent problems as schematically represented in Fig. 1: One unsteady, where the wake centre must be determined with a Lagrangian approach in the turbulent field and a time-averaged one, where the wake properties are determined in statistically-steady (but still turbulent) conditions. The first problem is tackled by following the same formulation of Taylor (1921) for the diffusion of a passive scalar. Since only a statistical model is desired, the statistical properties of the wake centre are the target here. Let us consider the idea of a simplified problem where a particle, p, with no inertia in a one-dimensional space is subjected to a random velocity field vi ¼ vðti Þ during the time interval ½ti ; ti þΔt with position yp ðt0 ¼ 0Þ ¼ 0. The velocity field has mean v and variance σ 2v . Here and in the following the over-bar will indicate the average operator assuming ergodicity. The position of the particle after N time intervals is given by yp ðtN Þ ¼ v1 Δt þ v2 Δt þ … ¼ Δt
v xp ðτÞ; τ dτ :
σ 2xp ðtÞ ¼
Z
t
Z
0
t
v’ xp ðτÞ; τ v’ xp ðζÞ; ζ dτdζ ;
(3)
0
with v’ ¼ v v as the velocity fluctuation in the same direction pointed by x. Here one introduces the assumption that the two-point two-time correlation v’ðxp ðτÞ; τÞv’ðxp ðζÞ; ζÞ depends exclusively on the separation time ξ ¼ ζ τ since the turbulent velocity field is homogeneous in space and time and their respective positions also depend on time. Indeed, one can introduce the correlation function as Rðξ ¼ ζ τÞ ¼
1
σ 2v
v’ xp ðτÞ; τ v’ xp ðζÞ; ζ ;
(4)
simplifying equation (3) to
σ 2xp ðtÞ ¼ 2σ 2v
Z
t
ðt ξÞRðξÞdξ :
(5)
0
Equation (5) is of utmost importance as it gives the variance of the particle position for given characteristics of the incoming velocity field, like the velocity variance and the correlation function. The latter can be modelled by an exponential function of the form RðξÞ ¼ expð ξ=ΛÞ, allowing for the analytical integration of (5) to
(1)
i¼1
By using the central limit theorem, one obtains that the probability 2
R. Braunbehrens, A. Segalini
σ 2xp ðtÞ ¼ 2σ 2v Λ2
ht Λ
Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
i þ et=Λ 1 :
"
(6)
Equation (6) is one of the key results of the present analysis as it relates the variance of the position to the convection time of the wake, t ¼ Δx=Uc (where Δx is the streamwise distance from the turbine and Uc is some unknown convection velocity). The integral time scale Λ is also unknown a priori but it depends on the characteristics of the turbulent field where the wake evolves. This is dependent on the atmospheric turbulence and the operating conditions of the turbine: However, one can neglect the contribution from wake-induced turbulence since the atmospheric turbulence is dominating being characterised by large energetic eddies. Equation (6) indicates also that, at small distances from the rotor, the standard deviation of the wake centre goes as σ xp ∝ Δx, while instead pffiffiffiffiffiffiffi at large distances it goes as σ xp ∝ Δx, similarly to the smoke of a chimney (Taylor, 1921). If we assume that the wake deficit is a passive scalar, it makes sense that only the characteristics of the atmospheric turbulence, minimally synthesised by the parameters σ v and Λ, play a key role in the wake-centre transport. This could be true even when considering the convective instability of the wake that acts as a noise amplifier. Now, according to the central limit theorem, since the wake is given by sequential correlated displacements, it is expected that the wakecentre position should be a Gaussian random variable with mean given by xp ¼ vΔx=Uc and variance given by (6). Since the transport in the streamwise direction should be identical, xp ¼ Δx, from which follows that Uc ¼ v and the wake velocity can be used as convection velocity. In the lateral directions, the mean transport velocity should be zero, leaving dispersion as the only mechanism to provide meandering. The streamwise dispersion is however complex since the wake must be present continuously downstream of the rotor: Therefore, the streamwise dispersion of the wake is here neglected when compared to the mean advection. However, this is not the case for the lateral dispersion that can be statistically described with a Gaussian distribution with known mean and variance. It is possible at this point to move the focus to the wake geometry to solve the second problem posed at the beginning of this section. Similar to the model proposed by Bastankhah and Porte-Agel (2014), the wake deficit is assumed to be Gaussian and given by the formula " ΔU ¼ C exp
# ðy yc Þ2 ; 2σ 2y
σ yc σy
"
2 #1=2
exp
# ðy yoc Þ2 ; 2σ 2y þ 2σ 2yc
(10)
indicating that the wake meandering has the effect to increase the wake 1=2
size (from σ y to ðσ 2y þ σ 2yc Þ ) and to decrease the wake deficit of the same relative amount. To summarise, the complete formula for the mean velocity deficit, including the vertical and lateral wake displacements, is " ΔUðy; zÞ ¼ C 1 þ
σ yc σy
2 #1=2 "
1þ
# ðy yoc Þ2 ðz zoc Þ2 ; exp 2 2σ y þ 2σ 2yc 2σ 2z þ 2σ 2zc "
σ zc σz
2 #1=2
(11)
where the parameters C, σ y and σ z are the wake-deficit maximum and the wake-widths parameter in the lateral and vertical directions (approximately 95% of the velocity deficit is included in a radius from the wake centre of 2σ ) while yoc and zoc represent the wake centre (yoc can be assumed to be zero due to symmetry, although zoc might not be zero allowing for a downward motion of the wake). These four parameters are obtained from steady simulations and are dependent on the turbine operating conditions (like thrust coefficient, hub height, etc.) and on the streamwise distance from the rotor Δx. The other two parameters σ yc and σ zc are obtained from equation (6), necessitating of the measure (or the estimate) of the velocity standard deviations, σ v and σ w , and integral time scales, Λv and Λw . The determination of these parameters with the current setup is described in the next section. 3. Model parameters and setup The basic wake parameters have been here obtained by means of the linearised simulation code ORFEUS developed at KTH (Segalini, 2017). The code solves the linearised Navier-Stokes equations with a spectral approach by using the Fourier transform in the horizontal directions and Chebyshev polynomials in the vertical direction. The turbine is introduced as an actuator disk in the simulation. By analysing the wake region of a single-turbine simulation with ORFEUS, the defining parameters of the velocity deficit distribution can be synthesised into look-up tables (LUTs). These LUTs allow for the rapid wake reconstruction for all individual turbines in a wind farm by means of the proposed wake model. The defining wake parameters C, σ y , σ z and zoc , (since it is assumed that yoc ¼ 0), were obtained from ORFEUS simulations with different free-stream velocities, shear exponents, hub heights and thrust coefficients. However, for a farm with turbines operating with the same thrust coefficient curve and hub height, only few pre-run simulations over all the relevant free-stream velocities are required. To illustrate this process, the velocity deficits in the wake of a single
(7)
where C indicates the wake-deficit intensity, yc the wake centre and σ y the size of the wake. The inclusion of a displacement in the other lateral direction is trivial since it is assumed that the displacements in the two directions are uncorrelated, as expected from symmetry arguments (this might not be the case if the wind turbine is yawed). The only stochastic entry in (7) is given by the wake centre, yc , while the wake-deficit intensity and the wake size can be determined from steady non-meandering simulations. The probability density function of the wake centre is assumed to be given by " # 1 ðyc yoc Þ2 f ðyc Þ ¼ pffiffiffiffiffi exp ; 2σ 2yc 2π σ yc
ΔUðyÞ ¼ C 1 þ
(8)
where the offset centre of the wake, yoc , is obtained from the steady simulation and the wake centre standard deviation from (6). The average wake deficit profile can then be obtained from the integral Z ΔUðyÞ ¼ C ¼ pffiffiffiffiffi 2π σ yc
Z
∞
∞
"
∞
∞
ΔUðy; yc Þf ðyc Þdyc ¼
# " # ðy yc Þ2 ðyc yoc Þ2 exp exp dyc : 2σ 2y 2σ 2yc
(9) Fig. 2. Evolution of the velocity deficit at hub height in the horizontal plane z ¼ zhub (a) and in the vertical plane y ¼ 0 (b) for x =D ¼ 1; 2; 4; 7 (sorted according to the arrows). The dashed line in (b) indicates the hub height zhub ¼ 0:875D. The present ORFEUS simulation was performed with: U∞ ¼ 8 m=s, cT ¼ 0:82, and a shear exponent of α ¼ 0:09.
After some algebra, equation (2) becomes
3
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Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
height above ground. The two parameters that remain to be determined are the two width parameters in the lateral as well as in the vertical directions, σ y and σ z . If they are equal, the wake will be axisymmetric, otherwise it will be oval. The parameters CðxÞ, zoc , σ y and σ z are determined by least-square fitting a Gaussian curve to the simulation data at every streamwise location. Fig. 3 shows the parameters obtained for the same turbine simulated with ORFEUS and used in Fig. 2. The curve for the maximum velocity deficit, CðxÞ, in Fig. 3a has the expected behaviour and tends towards zero from the beginning of the far wake at x 2D. The comparison of the present result with the wake model proposed by Bastankhah and Porte-Agel (2014) indicates good agreement in the far field region, although the latter has a faster wake decay. The height of the maximum velocity deficit, zoc , on the other hand does not remain constant in 3b and decreases in the downstream direction of 0:15D after 10D. The wake-width parameters, σ y and σ z , develop differently from each other: They both start from 0:5D, but the increase for σ z is stronger making the wake more oval shaped than round after 10D. This could be due to the larger shear in the top of the wake, where there is more turbulent mixing, compared to the lateral direction where only the wake deficit provides shear. The agreement with the model of Bastankhah and Porte-Agel (2014) is reasonably good everywhere in the far wake. Since the thrust coefficient of a wind turbine changes depending on the free-stream velocity, analyses are carried out from simulations ranging from 4 m =s to 20 m =s with steps of 1 m =s and the obtained set of parameters form a turbine-specific LUT. In the wake-model code, the incoming wind speed for a turbine is obtained through averaging the velocity field in front of the rotor. This accounts for partial-wake situations (namely when the wind turbine is partially experiencing the wake of upstream turbines) as the rotor can be considered an integrator over the whole area. For the wake-combination scheme, it was found that the method of adding the square of the velocity deficits, as proposed by Katic et al. (1986), agreed best with the data. The LUT model has now lost some of the features of ORFEUS. Turbines will no longer influence other upstream turbines and the wake-combination scheme usually starts deviating from measurements for long turbine rows. However, the static
turbine in the hub-height plane and in the vertical middle plane are shown in Fig. 2 at several downstream stations. For both planes, the decay of the wake deficit is visible as one moves further downstream. In the literature, the decay process is often separated into two regions: In the near-wake region the flow is still directly influenced by the presence of the rotor and the ring-shaped shear layer has not yet reached the centreline. In the far-wake region the wake should have forgotten about its initial shape, achieving some kind of self-similarity and can therefore be considered to have an axisymmetric Gaussian velocity distribution (Pope, 2000). The starting point for the far wake is suggested by Crespo et al. (1999) to be in the range of 2D-5D downstream of the rotor. In Fig. 2a the profile seems to have reached a Gaussian shape already at the first station x =D ¼ 1, which should still be located in the near-wake region. However, when moving to the profile at x =D ¼ 2, it can be seen that no substantial velocity recovery has occurred. This means that the wake has not yet started to mix with the surrounding flow and the region 0-2D should be still considered as near wake. In the vertical plane, the wake deficit has no symmetric Gaussian shape. It is smoothing out in downstream direction but the shape below hub height differs from the shape above the turbine. This is probably due to the combined effect of wind shear and the confinement of the ground: The incoming boundary layer causes a downwards displacement of the wake and prevents a real axisymmetric wake profile. Nevertheless, many authors have reported Gaussian velocity deficits also in the vertical plane, and we will assume as well that the velocity deficit is Gaussian as in the horizontal plane. This is consistent with the observations (not shown here) that the ORFEUS wake becomes increasingly axisymmetric as the turbine hub height increases and as the velocity shear decreases. For the sake of simplicity and usefulness, it was decided to limit the analysis to the far-wake region motivated by the fact that the turbine spacing in wind farms is typically on the order of several diameters. The far wake starting point was decided to be at x =D ¼ 2 and the analysis and fitting process was consequently started from there. The wake can be characterised by the maximum velocity deficit, CðxÞ, defining also the centreline locus. The downwards movement of the maximum velocity deficit is captured by the zoc parameter that describes the centreline
Fig. 3. Distribution of (a) the wake intensity, C, (b) wake-centre height, zoc , and (c) wake widths, σ y (dashed line) and σ z (solid line), for the different downstream positions. The same simulation setup used in Fig. 2 was adopted here. (Dash-dotted line) wake model from Bastankhah and Porte-Agel (2014). 4
R. Braunbehrens, A. Segalini
Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
Fig. 4 shows a comparison of the streamwise velocity field at hub height. The wake in 4a was composed with equation (7). Fig. 4b was composed with the meandering-model superposition given by equation (11). From x =D ¼ 2 the increase in width is visible and it accounted for faster recovery of the centerline deficit when meandering is present.
wake description allows for the combination with the wake-meandering model. The LUT obtained so far did not include wake meandering since that is an unsteady phenomenon associated to the local increase of momentum diffusivity. The proposed approach combines the wake parametrisation discussed above with the wake-meandering theory developed in section 2, as represented in Fig. 1. Assuming some characteristics of atmospheric turbulence like the velocity variances and the integral time scales, it is possible to quickly account for wake meandering statistically, i.e. increasing the wake width and decreasing the wake deficit. Here, it was simply assumed that the fluctuation intensity σ v is related to the momentum transfer to the ground in the atmospheric boundary layer (namely to the friction velocity, u ). This is usually stated as (Stull, 2012)
σ v ¼ 2:5u :
4. Results For validation purposes, the proposed wake model was tested against production data from two different offshore wind farms. Since during the measurement period the wind direction changed continuously, the data had to be binned in sectors of arbitrary width. The sector width can vary from 3o to 30o and include all measurements from the investigated bin. To account for this spatial averaging, multiple flow fields with 1o increment of the model are averaged within the sector. With ORFEUS, or a generic Computational Fluid Dynamics (CFD) code, this procedure would be computationally costly and this is one reason for the use of wake models in industry. In different studies from Gaumond et al. (2012), Wu and Porte-Agel (2015) and Walker et al. (2016) the authors found that wake-model results agree better with data from wider wind sectors (say larger than 10o ). In order to test the model with strong wake effects, inline layout configurations were analysed, namely when the wind direction is aligned with the wind-farm rows. The wakes will directly affect the next turbines and therefore cause significant power losses. However, a large sector width includes most likely situations where turbines are not or just partly covered by wakes, decreasing the wake effect and the error of any wake model. For the Lillgrund wind farm, Dahlberg (2009) claims that a sector width smaller or equal to 5o is necessary to clearly see the wake effects on a turbine from the preceding one. As the goal is to get an insight about the models performance under full-wake conditions, it was decided to compare data from sector widths between 5o to a maximum of 10o . By doing so, results from numerical models (ORFEUS or generic wake models) could be compared to SCADA data, since the inlet sector is small enough. Since it still represents the industry standard, the Jensen model is included in the comparison as well. To mitigate the effect of different data processing, an ensemble of several publications and data analyses
(12)
The length scale follows from the expectation that the structure size increases with height Λy ¼
κz
σv
:
(13)
As mentioned before, it is possible that the behaviour of the eddies differs from the horizontal to the vertical direction, as suggested by ~ a et al. (2012): The meandering observed during their experiment Espan was more prominent in the horizontal direction: The influence of the ground limits the fluctuating motion in the vertical direction. Therefore, the length scale in the vertical direction was set to half of that of Λy (Kaimal and Finnigan, 1994) Λz ¼
κz with σ w ¼ σ v : 2σ w
(14)
The detailed calibration of the atmospheric parameters (12)–(14) has not been performed here, so that the present expressions are only educated guesses based on observations of the atmospheric boundary layer (Kaimal and Finnigan, 1994; Stull, 2012). A more detailed investigation from mast data is needed to properly quantify these values in situ.
Fig. 4. Streamwise velocity at hub height in m/s without meandering (a) and including the meandering description (b). The steady simulation had the same setup adopted in Fig. 2, while the meandering parameters Λy , Λz , σ v , σ w were set according to equations (12)-(14). 5
R. Braunbehrens, A. Segalini
Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
power production significantly when wake effects are important. However, ORFEUS is able to simulate global upstream blockage effects of the farm (Ebenhoch et al., 2017; Segalini, 2017; Bleeg et al., 2018), that decrease the power output of the turbines placed in the first row. The results of the LUT method without wake-meandering correction are similar to ORFEUS, i.e. significantly underestimated. It is interesting that, despite of the fact that the LUT method should give similar results to ORFEUS, a slight overestimation is present, probably due to the superposition scheme. The LUT method with wake-meandering correction (from now on referred to as ORFEUS wake model) predicted power output agrees very well with the SCADA data for most rows, with a small tendency to underestimate, but nevertheless providing a clear improvement compared to the ORFEUS simulation and the LUT method without wake-meandering correction. In rows 4 and 5 the ORFEUS wake model is able to account for the recovery of the velocity field after the gap. Furthermore, the measurement data typically show a slight power increase from the second (which is usually the lowest) to the third turbine. The ORFEUS wake model is also able to account for this behaviour. For the longer rows 2, 3 and 4 the measured power decreases towards the end, whereas the ORFEUS wake model predicts a constant power. This effect is a known shortcoming of wake models: Frandsen et al. (2006) argued that the natural boundary layer changes in large wind farms and the amount of available kinetic energy is reduced. The ORFEUS wake model appears to be unable to catch this effect, probably also due to the static wake-combination method. The Jensen model performs quite well in all rows. Typically the predicted power is a bit lower than from the ORFEUS wake model, even though the wake decay constant was set to the higher end of the recommended range (Gaumond et al., 2012) for offshore conditions with k ¼ 0:05. The Jensen model seems to have the same tendency to over-predict the power towards the end of longer rows. Fig. 7 shows the normalised power output for the 120o wind direction. For this inflow angle, two empty turbine spaces are located in row E, where the wind has more space to recover. The ORFEUS wake model seems again to agree well with the data and the model is able to correctly predict the velocity after the gap in row E. The Jensen results lie close to the data as well with a slightly stronger tendency to under-predict. Towards the end of the long rows, the same deviation observed for the other wind direction occurs as the wake models erroneously predict the velocity to be constant. However, the data show that there seems to be an increase in wind speed (e.g. Row B).
has been used. The wake-meandering correction will be included in the LUT method for most of the available validation cases, highlighting its role. The effect of the LUT method alone (namely without accounting for the wake meandering) will be shown in the Lillgrund case only to provide a quantification of its role. 4.1. Lillgrund The Lillgrund wind farm is located in the Øresund strait , it consists of 48 Siemens SWT-2.3-93 turbines and has a rated capacity of 110 MW. The hub height of the turbines is zhub ¼ 68:5 m, with a diameter of D ¼ 93 m. This farm is particularly interesting to analyse because of the gaps in the layout and the relatively tight spacing between the turbines. This leads to strong wake effects and to a rather low array efficiency given by AE ¼ Ptot =Nt P0 ¼ 77%, with Ptot indicating the instantaneous total power production, Nt the number of turbines and P0 the power produced by the first row facing the wind. The two wind directions that have been analysed here and in the literature are 222o and 120o , as depicted in Fig. 5, where the rows are numbered and lettered. The spacing is 4:3D and 3:3D respectively (Nilsson et al., 2015). Two gaps exist in the cluster, where no turbine has been built. In these region, the wind velocity has space to recover inside the farm. Therefore, rows 4 and 5 as well as row E are of particular interest. The dataset used for comparison was published by Nilsson et al. (2015). During the measurement period, the average atmospheric stratification was considered to be neutral. The considered mean-wind velocity is U∞ ¼ 8 m=s (bin size: 7:5 m =s - 8:5 m =s) and the direction bin sectors are each 5o wide. The averaging time is in 10 min intervals, so that the mean wind travels twice through the farm in one interval. The measured turbulence intensity was 5%. Fig. 6 shows the power output of all rows from the 222o wind direction (rows 1 to 8). The power has been normalised with the median power of the turbines working in undisturbed wind. In rows 4 and 5, the missing turbine in the gap can be seen in the interruption of the data points. The results from the tested models have been plotted to show how well the prediction of the proposed wake model (with and without wakemeandering correction), Jensen model and ORFEUS match with the measurements. It can be seen that the steady ORFEUS simulation underestimates the
4.2. Horns Rev The second farm that was here analysed is the Horns Rev 1 offshore farm, located on the West coast of Denmark. It consists of 80 Vestas V80 turbines with a hub height of zhub ¼ 70 m and a rotor diameter of D ¼ 80 m. The total farm capacity is 160 MW and the turbines are arranged in a regular grid of 8 10. The layout is depicted in Fig. 8. The minimum distance between the turbines is 7D in the western direction or 9:4D and 10:4D, respectively in the diagonals (Wu and Porte-Agel, 2015). This makes the spacing larger than in the Lillgrund farm and leads to a higher array efficiency, AE. The 270o full-wake situation is a popular case in the literature (Barthelmie et al., 2009), where again the free-stream velocity U∞ ¼ 8 m=s has received most of the attention and will be our investigated case, as well as the diagonal directions 222o and 312o . The SCADA data from Barthelmie et al. (2009) was obtained using 10 min averaging periods and features all three inflow angles. The bin size was with 10o , higher than for the Lillgrund dataset. Only neutrally-stratified cases were considered, with a turbulence intensity of approximately 8%. For the aligned case (270o ) additional data from Walker et al. (2016) is also shown, again for 8 m =s and a bin size of 10o . The data was collected under the period of one year during 2005. Further data for the 270o direction from Gaumond et al. (2012) features a 5o bin size.
Fig. 5. Layout of the Lillgrund wind farm. The rows and the columns are marked as well as the two wind directions considered here. 6
R. Braunbehrens, A. Segalini
Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
Fig. 6. Normalised power of the rows (1)–(8) of the Lillgrund wind-farm with U∞ ¼ 8 m=s and wind direction 222o . Data from Nilsson et al. (2015) (solid line) compared with ORFEUS (þ), Jensen (x) and LUT with wake-meandering correction (∘) and without wake-meandering correction (gray ). The cT and cP -curve was taken from van der Laan et al. (2015), zhub ¼ 0:736D and sector averaging the wake models was 5o . The shear exponent of the simulation was set to α ¼ 0:1. The wake-decay constant for the Jensen model was k ¼ 0:05.
Fig. 7. Normalised power of the rows (A)-(H) of the Lillgrund wind-farm with U∞ ¼ 8 m=s and wind from 120o . The marker styles as well as model configuration are the same as in Fig. 6.
In the 270o direction, the data from the three different sources differ to some extend. This underlines the difficulties and differences that occur during the data processing. The ORFEUS wake model is able to match the measurements best, lying in the middle of the data for turbines 3–7.
Since the farm has a regular layout, only one row inside the farm can to be considered in the analysis. Both wake models were averaged over a 10o sector this time, as the majority of the data. Fig. 9 shows the normalised power production for the three respective wind directions. 7
R. Braunbehrens, A. Segalini
Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
phenomenon in a fast solving wake-model code was presented. First it was decided to assume the wake itself to be a passive tracer and the meandering motions a result of large atmospheric turbulence. Then, a model on the basis of Lagrangian particle dispersion was implemented, derived from the findings of Taylor (1921). Starting from a simplified example for the observation of a single particle under discrete random fluctuations, Taylor (1921) developed a theory for diffusion under continuous correlated movements. This theory for particle dispersion was applied for the meandering cross-stream motion of the wake, where it was assumed that the wake cross sections are displaced rigidly, without any distortion (a hypothesis proposed by Larsen et al., 2008). The probability density function of the wake centreline could then be superimposed to a static wake-model velocity-deficit distribution. Through this, a time-averaged description of the wake including the meandering was obtained. Like suggested by Espa~ na et al. (2012), the strength of meandering differs in the horizontal and vertical direction, so both directions were considered independently. Larsen et al. (2008) used the rigid-wake displacement coupled with synthetic turbulence to generate time series for fatigue-load estimation. This process is quite expensive if the goal is just to estimate the wake deficit and the present model, being probabilistic, bypasses the need to create artificial time series (i.e. Monte Carlo simulations) providing directly a statistical description based on simple parameters that can be measured. The model for the wake velocity itself is based on ORFEUS, a linearised code that simulates the turbines as actuator-disk body forces. To obtain a wake model, results of several ORFEUS single-turbine simulations were analysed and the velocity deficit decomposed into four parameters: The wake-centre velocity deficit, CðxÞ, the wake-width parameters, σ y ðxÞ and σ z ðxÞ, and the height of the maximum deficit over ground, zoc ðxÞ. This analysis was limited to the far wake, which was assumed to have some kind of self-similar characteristics. All parameters were consequently stored and could then be accessed through a look-up table to rapidly restore individual turbine wakes in a wind farm. Before superimposing the meandering model, the parameters that govern the rate of diffusion had to be chosen. The cross-stream fluctuations and the time-scale of the governing atmospheric eddies (σ v and Λy in the horizontal direction) were estimated tofrom general atmospheric turbulence features. The superposition of the estimated wake with the wake-meandering correction (given by a convolution of the deficit with
Fig. 8. Layout of the Horns Rev wind farm.
However, the curves from Gaumond et al. (2012) and Walker et al. (2016) show a decreasing output towards the last turbines, which is again not captured by the wake model. The Jensen model (with k ¼ 0:05) under-predicts the power, which was already noticed for the Lillgrund results. The ORFEUS solver underestimates the power output again. However, it has to be noted that the measurements were binned in a 10o sector which causes an increase of power in the data, even more than for the 5o sectors for Lillgrund. ORFEUS as a linearised code is better for comparing smaller sectors. In the 222o and 312o cases, the spacing between the turbines becomes larger, so that the power output increases as well. Data in both cases was only available until turbine 5. At 222o both wake models under-predict the power. Furthermore, there seems to be a trend of decreasing power in the data along the row, which only ORFEUS is able to predict. In the 312o direction, the ORFEUS wake model matches again the data best. 5. Conclusions In this paper a novel approach to include the wake-meandering
Figure 9. Normalized power production of the Horns Rev wind farm for inflow angles of ðaÞ270o , ðbÞ222o and ðcÞ 312o . Data from Barthelmie et al. (2009) (solid line), Walker et al. (2016) (dashed line), Gaumond et al. (2012) (dotted line) compared with ORFEUS (þ), Jensen (x) and ORFEUS wake model (o). Free-stream velocity U∞ ¼ 8 m=s, averaging sector for both wake models 10o , cT and cP -curve from Hansen et al. (2010). The shear exponent for ORFEUS and the ORFEUS wake model was α ¼ 0:09 with zhub ¼ 0:875D. The wake-decay constant for the Jensen model was k ¼ 0:05.
8
R. Braunbehrens, A. Segalini
Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
the wake-centre probability density function) resulted in a wider wake and a smaller wake deficit, providing a faster wake recovery and highlighting the key role of the meandering phenomenon. To validate the approach, the model was compared to energy production data sets from two offshore-wind farms. The LUT model with wake-meandering correction showed good agreement with the Lillgrund wind farm, where it was able to predict the wind-speed increase after the gaps in the array. An expected shortcoming was the increasing discrepancy towards the end of long rows, where the model started to overpredict the power, probably because of the wake-combination method. When compared to the Jensen model, the agreement was similar for the Lillgrund data set and the proposed model seemed to be more accurate for the Horns Rev data. The absence of the meandering effect in the full ORFEUS simulations, or in the LUT generated from ORFEUS (without the meandering), provided a significant underestimation of the power production due to a reduced wake recovery. The significant advantage of the present formulation is its ability to physically account for the wake-diffusion mechanism, splitting it into two phenomena: One where the wake diffuses due to turbulent transport of momentum, and the other where the wake displaces due to dispersion mechanisms (the “wake-meandering” phenomenon). Physical constants and parameters are introduced and they can be quantified through mast measurements or educated guesses as done here, reducing the level of empiricisms in power estimations of wind farms. This is a big advantage with respect to the Jensen model (or similar ones), where the wakespreading constant, k, is obtained empirically and implicitly accounting for all transport and dispersive phenomena in the wake. Here the model provides a clear separation between the two phenomena and suggests even the possibility to develop new wake models that account only for the wake meandering, being quite a dominant wake-recovery mechanism.
Coudou, N., Buckingham, S., van Beeck, J., 2017. Experimental study on the wind-turbine wake meandering inside a scale model wind farm placed in an atmosphericboundary-layer wind tunnel. In: Journal of Physics: Conference Series, vol. 854. IOP Publishing, 012008. Crespo, A., Hernandez, J., Frandsen, S., 1999. Survey of modelling methods for wind turbine wakes and wind farms. Wind Energy 2 (1), 1–24. Dahlberg, J.-Å., 2009. Assessment of the Lillgrund wind farm: power performance wake effects. Vattenfall Vindkraft AB, 6_1 LG Pilot Report, http://www.vattenfall.se/sv/f ile/15_Assessment_of_the_Lillgrund_W.pdf_16596737. pdf (cited March 30, 2012), 2009. Ebenhoch, R., Muro, B., Dahlberg, J.-Å., Berkesten H€agglund, P., Segalini, A., 2017. A linearized numerical model of wind-farm flows. Wind Energy 20 (5), 859–875. Espa~ na, G., Aubrun, S., Loyer, S., Devinant, P., 2012. Wind tunnel study of the wake meandering downstream of a modelled wind turbine as an effect of large scale turbulent eddies. J. Wind Eng. Ind. Aerodyn. 101, 24–33. Frandsen, S., Barthelmie, R., Pryor, S., Rathmann, O., Larsen, S., Højstrup, J., Thøgersen, M., 2006. Analytical modelling of wind speed deficit in large offshore wind farms. Wind Energy 9 (1–2), 39–53. Gaumond, M., Rethore, P.-E., Bechmann, A., Ott, S., Larsen, G.C., Pena Diaz, A., Kurt, K., 2012. Benchmarking of wind turbine wake models in large offshore windfarms. In: Proceedings of the Science of Making Torque from Wind. Hansen, K.S., Barthelmie, R., Jensen, L.E., Sommer, A., Vattenfall, A., Sommer, A., 2010. Power deficits due to wind turbine wakes at Horns Rev wind farm. In: Proceedings of the Science of Making Torque from Wind Conference, pp. 40–46. Kaimal, J.C., Finnigan, J.J., 1994. Atmospheric Boundary Layer Flows: Their Structure and Measurement. Oxford University Press. Katic, I., Højstrup, J., Jensen, N.O., 1986. A simple model for cluster efficiency. In: European Wind Energy Association Conference and Exhibition, pp. 407–410. Larsen, G.C., Madsen, H.A., Thomsen, K., Larsen, T.J., 2008. Wake meandering: a pragmatic approach. Wind Energy 11 (4), 377–395. Medici, D., Alfredsson, P., 2006. Measurements on a wind turbine wake: 3d effects and bluff body vortex shedding. Wind Energy 9 (3), 219–236. Medici, D., Alfredsson, P.H., 2008. Measurements behind model wind turbines: further evidence of wake meandering. Wind Energy 11 (2), 211–217. Muller, Y.-A., Aubrun, S., Masson, C., 2015. Determination of real-time predictors of the wind turbine wake meandering. Exp. Fluid 56 (3), 53. Nilsson, K., Ivanell, S., Hansen, K.S., Mikkelsen, R., Sørensen, J.N., Breton, S.-P., Henningson, D., 2015. Large-eddy simulations of the Lillgrund wind farm. Wind Energy 18 (3), 449–467. Pope, S., Jan 2000. Turbulent Flows. Cambridge University Press, Cambridge, UK. Segalini, A., 2017. Linearized simulation of flow over wind farms and complex terrains. Phil. Trans. R. Soc. A 375 (2091), 20160099. Stull, R.B., 2012. An Introduction to Boundary Layer Meteorology, vol. 13. Springer Science & Business Media. Taylor, G., 1990. Wake Measurements on the Nibe Wind-Turbines in Denmark: Data Collection and Analysis. National Power, Technology and Environment Centre. Taylor, G.I., 1921. Diffusion by continuous movements. Proc. Lond. Math. Soc. 20, 196–211. Thøgersen, E., Tranberg, B., Herp, J., Greiner, M., 2017. Statistical meandering wake model and its application to yaw-angle optimisation of wind farms. J. Phys. Conf. Ser. 854. Trujillo, J.-J., Bing€ ol, F., Larsen, G.C., Mann, J., Kühn, M., 2011. Light detection and ranging measurements of wake dynamics. part ii: two-dimensional scanning. Wind Energy 14 (1), 61–75. van der Laan, P.M., Sørensen, N.N., Rethore, P.-E., Mann, J., Kelly, M.C., Troldborg, N., Hansen, K.S., Leon, J.P.M., 2015. The k-ε-f p model applied to wind farms. Wind Energy 18 (12). Vermeer, L.J., Sørensen, J.N., Crespo, A., 2003. Wind turbine wake aerodynamics. Prog. Aerosp. Sci. 39 (6), 467–510. Walker, K., Adams, N., Gribben, B., Gellatly, B., Nygaard, N.G., Henderson, A., Marchante Jimemez, M., Schmidt, S.R., Rodriguez Ruiz, J., Paredes, D., et al., 2016. An evaluation of the predictive accuracy of wake effects models for offshore wind farms. Wind Energy 19 (5), 979–996. Wu, Y.-T., Porte-Agel, F., 2015. Modeling turbine wakes and power losses within a wind farm using LES: an application to the Horns Rev offshore wind farm. Renew. Energy 75, 945–955.
Acknowledgement AS is funded by The Swedish Research Council (VR) under a framework grant for strategic energy research and STandUP for Wind. References Ainslie, J.F., 1988. Calculating the flowfield in the wake of wind turbines. J. Wind Eng. Ind. Aerodyn. 27 (1–3), 213–224. Baker, R.W., Walker, S.N., 1984. Wake measurements behind a large horizontal axis wind turbine generator. Sol. Energy 33 (1), 5–12. Barthelmie, R.J., Hansen, K., Frandsen, S.T., Rathmann, O., Schepers, J., Schlez, W., Phillips, J., Rados, K., Zervos, A., Politis, E., et al., 2009. Modelling and measuring flow and wind turbine wakes in large wind farms offshore. Wind Energy 12 (5), 431–444. Bastankhah, M., Port e-Agel, F., 2014. A new analytical model for wind-turbine wakes. Renew. Energy 70, 116–123. Bing€ ol, F., Mann, J., Larsen, G.C., 2010. Light detection and ranging measurements of wake dynamics part i: one-dimensional scanning. Wind Energy 13 (1), 51–61. Bleeg, J., Purcell, M., Ruisi, R., Traiger, E., 2018. Wind farm blockage and the consequences of neglecting its impact on energy production. Energies 11 (1609).
9
Contents lists available at ScienceDirect
Journal of Wind Engineering & Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia
A statistical model for wake meandering behind wind turbines Robert Braunbehrens a, Antonio Segalini b, * a b
Innogy SE, Wind Onshore, Kapstadtring 7, 22297, Hamburg, Germany Linne Flow Centre, STandUP for Wind, KTH Mechanics, Royal Institute of Technology, S-100 44, Stockholm, Sweden
A R T I C L E I N F O
A B S T R A C T
Keywords: Numerical wake model Linearised RANS Wake meandering Atmospheric dispersion
A new wake model is proposed to account for wake meandering in simulations of wind-turbine wakes performed on steady solvers, through a wake-meandering description based on the dispersion theory of Taylor (1921, P. Lond. Math Soc., vol. 20, pp. 196–211). Single-turbine simulations were performed by means of the linearised solver ORFEUS. By analysing the steady wake behind a turbine, a set of parameters describing the wake was first obtained and synthesised into a look-up table. The proposed meandering model extended the simulation results by superimposing the lateral and vertical meandering motions to the steady wake. As a result, the time-averaged velocity distribution of the wake was increased in width and reduced in intensity. Through this combination, the model provides rationale for the wake-deficit decrease and for the power underestimation effects of several wake models. The new wake model is validated against the Lillgrund and Horns Rev data sets.
1. Introduction As offshore wind farms are built with increasing size and investments, an accurate estimation of the wake losses is important in the design phase. However, for a computational simulation of a turbine wake, a wide range of turbulent length scales needs to be accounted for. This is why full RANS simulations of wind farms are still too computationally costly in an iterative design process. The industry relies therefore heavily on wake models, which do not solve for the flow field with the turbines, but rather in a post-processing manner: They are usually semi-empirical formulations that require the introduction of model parameters, that resemble physical conditions like the ambient turbulence or ground roughness. Remarkably, most wake models used in industry do not explicitly account for the phenomenon of wake meandering, and this could be a reason for the tendency to under-predict the power outputs of wind farms (Vermeer et al., 2003). Wake meandering was observed in full-scale measurements by Baker and Walker (1984) and Taylor (1990). It refers to a fluctuating movement of the whole wake region in the lateral and vertical directions. As a consequence of the wake being displaced, the power output of downstream turbines increases. Since these first studies, several attempts to model and verify experimentally the proposed dynamics have been made. Ainslie (1988) created a description in his wake model based on the thin-shear layer equations. He suggested that the meandering motion was due to the variation in the wind direction,
excluding however short timescale fluctuations. Further research was not pursued until the first wind-tunnel measurements were carried out by Medici and Alfredsson (2006, 2008) with a small-scale wind-turbine model. They observed a periodically-fluctuating helical movement of the wake, which made them suggest that an instability of the wake itself was causing the meandering, much like the vortex shedding behind a bluff body. Although the experimental setup had no free-stream large-scale turbulence, it became a reference case in the wind community. In contrast to Medici and Alfredsson, Larsen et al. (2008) introduced the Dynamic Wake Meandering model (DWM) with the fundamental assumption that the wake behaves like a passive tracer, so that no wake instability is accounted for. The large-scale atmospheric turbulence is responsible for the displacement of the wake as a whole and thus for the wake meandering. Larsen et al. defined that the threshold in characteristic length scales influencing the wake is two wake diameters, which means that only eddies with a length scale larger than that will induce wake meandering. This is based on the consideration that, in order for a wake-displacing eddy to move all points of a cross section, it has to be twice the size, since half of the displacement is going to be in the positive direction and the other half in the negative direction. In the final DWM model, the velocity-deficit shape is based on the wake model from Ainslie (1988) that assumes a steady eddy viscosity. The meandering motion is obtained through the low-pass filtering of a synthetic turbulent field to obtain the effect of the large-scale turbulence. The DWM purpose is mainly to provide a rapid option to calculate blade loadings.
* Corresponding author. E-mail address: [email protected] (A. Segalini). https://doi.org/10.1016/j.jweia.2019.103954 Received 4 January 2019; Received in revised form 9 June 2019; Accepted 10 July 2019 Available online 29 August 2019 0167-6105/© 2019 Elsevier Ltd. All rights reserved.
R. Braunbehrens, A. Segalini
Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
To verify the model, Bing€ ol et al. (2010) and Trujillo et al. (2011) measured the wake velocity with a LIDAR system mounted on the back of a full-scale turbine. They were able to detect a meandering motion of the wake due to large-scale turbulence and no periodic vortex shedding was observed. Wind-tunnel investigations were continued by Espa~ na et al. (2012) who used a porous disk submerged by an atmospheric boundary layer with length scale 10 times larger than the disk model. They were able to detect a meandering motion, more prominent in the horizontal than in the vertical direction. The assumption of the wake as a passive tracer was further supported by Muller et al. (2015), although they found that the characteristic length scale of the governing eddies was larger than assumed by Larsen et al. (2008). Finally, Coudou et al. (2017) used a setup of small wind-turbine models to investigate whether the wake meandering persisted inside a wind farm. With an incoming flow characterised by a turbulence length scale larger than three turbine diameters, they were able to assess that the meandering occurred throughout the whole farm model. Furthermore, in contrast to Espa~ na et al. (2012) and Muller et al. (2015), they observed a periodic movement of the wake similar to Medici and Alfredsson (2006) and concluded that intrinsic instability is amplified by large-scale turbulence. Several works assume that the success of the Jensen (also known as PARK) model (Katic et al., 1986) is associated to the fact that the empirical wake-expansion parameter already accounts for the wake meandering (Thøgersen et al., 2017). To avoid this implicit consideration of the wake-meandering mechanism, a simple model for wake meandering model based on Lagrangian particle dispersion (Taylor, 1921) is presented in this paper. It is implemented into a wake model for power prediction that was derived from the linearised code ORFEUS, which has been introduced by Segalini (2017) and Ebenhoch et al. (2017). The paper is structured as follows. Section 2 describes how the original theory of Taylor can be used to characterise wake meandering. It is followed by section 3 describing how the simulations of the linearised code ORFEUS were combined to create look-up tables to construct the wake. The proposed wake model is a combination of sections 2 and 3. Thereafter, section 4 assesses the performance of the new wake model for the two reference cases given by the Lillgrund and Horn Rev wind farms, while some final conclusions are stated in section 5.
Fig. 1. Schematic representation of the approach proposed in this paper. The “instantaneous” field is obtained from steady ORFEUS simulations (or LUT method) without wake meandering. The “time-averaged” field results from the statistical ensemble of the steady ORFEUS fields averaged according to the assumption that the wake centre behaves as a passive tracer.
density function of the position yp ðtN Þ tends to be Gaussian as N → ∞ pffiffiffiffi N Δt σ v . If one expffiffiffiffi presses the total time as T ¼ NΔt, it is clear that σ yp ∝ T , so that the standard deviation does not grow linearly with time, but at a slower rate. This is true regardless of the velocity probability density function. At this point, one should remove the assumption of independence of the velocity displacements and consider a continuous velocity field correlated in time. The position of the particle is now given by with mean yp ¼ NΔtv and standard deviation σ yp ¼
Z
t
xp ðtÞ ¼ 2. Statistical description of wake meandering
N X
vðti Þ :
(2)
0
The mean position is simply obtained as xp ðtÞ ¼ vt, while the variance of the generic component of the position is given by
The fundamental assumptions of the present model were stated by Larsen et al. (2008) with the hypothesis that the wake behaves as a passive tracer and that the entire wake preserves its shape when laterally displaced. Under these two hypotheses, the instantaneous wake characteristics are defined by the wake centre position and by the undisturbed geometry of the wake. This splits the characterisation of wake meandering into two independent problems as schematically represented in Fig. 1: One unsteady, where the wake centre must be determined with a Lagrangian approach in the turbulent field and a time-averaged one, where the wake properties are determined in statistically-steady (but still turbulent) conditions. The first problem is tackled by following the same formulation of Taylor (1921) for the diffusion of a passive scalar. Since only a statistical model is desired, the statistical properties of the wake centre are the target here. Let us consider the idea of a simplified problem where a particle, p, with no inertia in a one-dimensional space is subjected to a random velocity field vi ¼ vðti Þ during the time interval ½ti ; ti þΔt with position yp ðt0 ¼ 0Þ ¼ 0. The velocity field has mean v and variance σ 2v . Here and in the following the over-bar will indicate the average operator assuming ergodicity. The position of the particle after N time intervals is given by yp ðtN Þ ¼ v1 Δt þ v2 Δt þ … ¼ Δt
v xp ðτÞ; τ dτ :
σ 2xp ðtÞ ¼
Z
t
Z
0
t
v’ xp ðτÞ; τ v’ xp ðζÞ; ζ dτdζ ;
(3)
0
with v’ ¼ v v as the velocity fluctuation in the same direction pointed by x. Here one introduces the assumption that the two-point two-time correlation v’ðxp ðτÞ; τÞv’ðxp ðζÞ; ζÞ depends exclusively on the separation time ξ ¼ ζ τ since the turbulent velocity field is homogeneous in space and time and their respective positions also depend on time. Indeed, one can introduce the correlation function as Rðξ ¼ ζ τÞ ¼
1
σ 2v
v’ xp ðτÞ; τ v’ xp ðζÞ; ζ ;
(4)
simplifying equation (3) to
σ 2xp ðtÞ ¼ 2σ 2v
Z
t
ðt ξÞRðξÞdξ :
(5)
0
Equation (5) is of utmost importance as it gives the variance of the particle position for given characteristics of the incoming velocity field, like the velocity variance and the correlation function. The latter can be modelled by an exponential function of the form RðξÞ ¼ expð ξ=ΛÞ, allowing for the analytical integration of (5) to
(1)
i¼1
By using the central limit theorem, one obtains that the probability 2
R. Braunbehrens, A. Segalini
σ 2xp ðtÞ ¼ 2σ 2v Λ2
ht Λ
Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
i þ et=Λ 1 :
"
(6)
Equation (6) is one of the key results of the present analysis as it relates the variance of the position to the convection time of the wake, t ¼ Δx=Uc (where Δx is the streamwise distance from the turbine and Uc is some unknown convection velocity). The integral time scale Λ is also unknown a priori but it depends on the characteristics of the turbulent field where the wake evolves. This is dependent on the atmospheric turbulence and the operating conditions of the turbine: However, one can neglect the contribution from wake-induced turbulence since the atmospheric turbulence is dominating being characterised by large energetic eddies. Equation (6) indicates also that, at small distances from the rotor, the standard deviation of the wake centre goes as σ xp ∝ Δx, while instead pffiffiffiffiffiffiffi at large distances it goes as σ xp ∝ Δx, similarly to the smoke of a chimney (Taylor, 1921). If we assume that the wake deficit is a passive scalar, it makes sense that only the characteristics of the atmospheric turbulence, minimally synthesised by the parameters σ v and Λ, play a key role in the wake-centre transport. This could be true even when considering the convective instability of the wake that acts as a noise amplifier. Now, according to the central limit theorem, since the wake is given by sequential correlated displacements, it is expected that the wakecentre position should be a Gaussian random variable with mean given by xp ¼ vΔx=Uc and variance given by (6). Since the transport in the streamwise direction should be identical, xp ¼ Δx, from which follows that Uc ¼ v and the wake velocity can be used as convection velocity. In the lateral directions, the mean transport velocity should be zero, leaving dispersion as the only mechanism to provide meandering. The streamwise dispersion is however complex since the wake must be present continuously downstream of the rotor: Therefore, the streamwise dispersion of the wake is here neglected when compared to the mean advection. However, this is not the case for the lateral dispersion that can be statistically described with a Gaussian distribution with known mean and variance. It is possible at this point to move the focus to the wake geometry to solve the second problem posed at the beginning of this section. Similar to the model proposed by Bastankhah and Porte-Agel (2014), the wake deficit is assumed to be Gaussian and given by the formula " ΔU ¼ C exp
# ðy yc Þ2 ; 2σ 2y
σ yc σy
"
2 #1=2
exp
# ðy yoc Þ2 ; 2σ 2y þ 2σ 2yc
(10)
indicating that the wake meandering has the effect to increase the wake 1=2
size (from σ y to ðσ 2y þ σ 2yc Þ ) and to decrease the wake deficit of the same relative amount. To summarise, the complete formula for the mean velocity deficit, including the vertical and lateral wake displacements, is " ΔUðy; zÞ ¼ C 1 þ
σ yc σy
2 #1=2 "
1þ
# ðy yoc Þ2 ðz zoc Þ2 ; exp 2 2σ y þ 2σ 2yc 2σ 2z þ 2σ 2zc "
σ zc σz
2 #1=2
(11)
where the parameters C, σ y and σ z are the wake-deficit maximum and the wake-widths parameter in the lateral and vertical directions (approximately 95% of the velocity deficit is included in a radius from the wake centre of 2σ ) while yoc and zoc represent the wake centre (yoc can be assumed to be zero due to symmetry, although zoc might not be zero allowing for a downward motion of the wake). These four parameters are obtained from steady simulations and are dependent on the turbine operating conditions (like thrust coefficient, hub height, etc.) and on the streamwise distance from the rotor Δx. The other two parameters σ yc and σ zc are obtained from equation (6), necessitating of the measure (or the estimate) of the velocity standard deviations, σ v and σ w , and integral time scales, Λv and Λw . The determination of these parameters with the current setup is described in the next section. 3. Model parameters and setup The basic wake parameters have been here obtained by means of the linearised simulation code ORFEUS developed at KTH (Segalini, 2017). The code solves the linearised Navier-Stokes equations with a spectral approach by using the Fourier transform in the horizontal directions and Chebyshev polynomials in the vertical direction. The turbine is introduced as an actuator disk in the simulation. By analysing the wake region of a single-turbine simulation with ORFEUS, the defining parameters of the velocity deficit distribution can be synthesised into look-up tables (LUTs). These LUTs allow for the rapid wake reconstruction for all individual turbines in a wind farm by means of the proposed wake model. The defining wake parameters C, σ y , σ z and zoc , (since it is assumed that yoc ¼ 0), were obtained from ORFEUS simulations with different free-stream velocities, shear exponents, hub heights and thrust coefficients. However, for a farm with turbines operating with the same thrust coefficient curve and hub height, only few pre-run simulations over all the relevant free-stream velocities are required. To illustrate this process, the velocity deficits in the wake of a single
(7)
where C indicates the wake-deficit intensity, yc the wake centre and σ y the size of the wake. The inclusion of a displacement in the other lateral direction is trivial since it is assumed that the displacements in the two directions are uncorrelated, as expected from symmetry arguments (this might not be the case if the wind turbine is yawed). The only stochastic entry in (7) is given by the wake centre, yc , while the wake-deficit intensity and the wake size can be determined from steady non-meandering simulations. The probability density function of the wake centre is assumed to be given by " # 1 ðyc yoc Þ2 f ðyc Þ ¼ pffiffiffiffiffi exp ; 2σ 2yc 2π σ yc
ΔUðyÞ ¼ C 1 þ
(8)
where the offset centre of the wake, yoc , is obtained from the steady simulation and the wake centre standard deviation from (6). The average wake deficit profile can then be obtained from the integral Z ΔUðyÞ ¼ C ¼ pffiffiffiffiffi 2π σ yc
Z
∞
∞
"
∞
∞
ΔUðy; yc Þf ðyc Þdyc ¼
# " # ðy yc Þ2 ðyc yoc Þ2 exp exp dyc : 2σ 2y 2σ 2yc
(9) Fig. 2. Evolution of the velocity deficit at hub height in the horizontal plane z ¼ zhub (a) and in the vertical plane y ¼ 0 (b) for x =D ¼ 1; 2; 4; 7 (sorted according to the arrows). The dashed line in (b) indicates the hub height zhub ¼ 0:875D. The present ORFEUS simulation was performed with: U∞ ¼ 8 m=s, cT ¼ 0:82, and a shear exponent of α ¼ 0:09.
After some algebra, equation (2) becomes
3
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Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
height above ground. The two parameters that remain to be determined are the two width parameters in the lateral as well as in the vertical directions, σ y and σ z . If they are equal, the wake will be axisymmetric, otherwise it will be oval. The parameters CðxÞ, zoc , σ y and σ z are determined by least-square fitting a Gaussian curve to the simulation data at every streamwise location. Fig. 3 shows the parameters obtained for the same turbine simulated with ORFEUS and used in Fig. 2. The curve for the maximum velocity deficit, CðxÞ, in Fig. 3a has the expected behaviour and tends towards zero from the beginning of the far wake at x 2D. The comparison of the present result with the wake model proposed by Bastankhah and Porte-Agel (2014) indicates good agreement in the far field region, although the latter has a faster wake decay. The height of the maximum velocity deficit, zoc , on the other hand does not remain constant in 3b and decreases in the downstream direction of 0:15D after 10D. The wake-width parameters, σ y and σ z , develop differently from each other: They both start from 0:5D, but the increase for σ z is stronger making the wake more oval shaped than round after 10D. This could be due to the larger shear in the top of the wake, where there is more turbulent mixing, compared to the lateral direction where only the wake deficit provides shear. The agreement with the model of Bastankhah and Porte-Agel (2014) is reasonably good everywhere in the far wake. Since the thrust coefficient of a wind turbine changes depending on the free-stream velocity, analyses are carried out from simulations ranging from 4 m =s to 20 m =s with steps of 1 m =s and the obtained set of parameters form a turbine-specific LUT. In the wake-model code, the incoming wind speed for a turbine is obtained through averaging the velocity field in front of the rotor. This accounts for partial-wake situations (namely when the wind turbine is partially experiencing the wake of upstream turbines) as the rotor can be considered an integrator over the whole area. For the wake-combination scheme, it was found that the method of adding the square of the velocity deficits, as proposed by Katic et al. (1986), agreed best with the data. The LUT model has now lost some of the features of ORFEUS. Turbines will no longer influence other upstream turbines and the wake-combination scheme usually starts deviating from measurements for long turbine rows. However, the static
turbine in the hub-height plane and in the vertical middle plane are shown in Fig. 2 at several downstream stations. For both planes, the decay of the wake deficit is visible as one moves further downstream. In the literature, the decay process is often separated into two regions: In the near-wake region the flow is still directly influenced by the presence of the rotor and the ring-shaped shear layer has not yet reached the centreline. In the far-wake region the wake should have forgotten about its initial shape, achieving some kind of self-similarity and can therefore be considered to have an axisymmetric Gaussian velocity distribution (Pope, 2000). The starting point for the far wake is suggested by Crespo et al. (1999) to be in the range of 2D-5D downstream of the rotor. In Fig. 2a the profile seems to have reached a Gaussian shape already at the first station x =D ¼ 1, which should still be located in the near-wake region. However, when moving to the profile at x =D ¼ 2, it can be seen that no substantial velocity recovery has occurred. This means that the wake has not yet started to mix with the surrounding flow and the region 0-2D should be still considered as near wake. In the vertical plane, the wake deficit has no symmetric Gaussian shape. It is smoothing out in downstream direction but the shape below hub height differs from the shape above the turbine. This is probably due to the combined effect of wind shear and the confinement of the ground: The incoming boundary layer causes a downwards displacement of the wake and prevents a real axisymmetric wake profile. Nevertheless, many authors have reported Gaussian velocity deficits also in the vertical plane, and we will assume as well that the velocity deficit is Gaussian as in the horizontal plane. This is consistent with the observations (not shown here) that the ORFEUS wake becomes increasingly axisymmetric as the turbine hub height increases and as the velocity shear decreases. For the sake of simplicity and usefulness, it was decided to limit the analysis to the far-wake region motivated by the fact that the turbine spacing in wind farms is typically on the order of several diameters. The far wake starting point was decided to be at x =D ¼ 2 and the analysis and fitting process was consequently started from there. The wake can be characterised by the maximum velocity deficit, CðxÞ, defining also the centreline locus. The downwards movement of the maximum velocity deficit is captured by the zoc parameter that describes the centreline
Fig. 3. Distribution of (a) the wake intensity, C, (b) wake-centre height, zoc , and (c) wake widths, σ y (dashed line) and σ z (solid line), for the different downstream positions. The same simulation setup used in Fig. 2 was adopted here. (Dash-dotted line) wake model from Bastankhah and Porte-Agel (2014). 4
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Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
Fig. 4 shows a comparison of the streamwise velocity field at hub height. The wake in 4a was composed with equation (7). Fig. 4b was composed with the meandering-model superposition given by equation (11). From x =D ¼ 2 the increase in width is visible and it accounted for faster recovery of the centerline deficit when meandering is present.
wake description allows for the combination with the wake-meandering model. The LUT obtained so far did not include wake meandering since that is an unsteady phenomenon associated to the local increase of momentum diffusivity. The proposed approach combines the wake parametrisation discussed above with the wake-meandering theory developed in section 2, as represented in Fig. 1. Assuming some characteristics of atmospheric turbulence like the velocity variances and the integral time scales, it is possible to quickly account for wake meandering statistically, i.e. increasing the wake width and decreasing the wake deficit. Here, it was simply assumed that the fluctuation intensity σ v is related to the momentum transfer to the ground in the atmospheric boundary layer (namely to the friction velocity, u ). This is usually stated as (Stull, 2012)
σ v ¼ 2:5u :
4. Results For validation purposes, the proposed wake model was tested against production data from two different offshore wind farms. Since during the measurement period the wind direction changed continuously, the data had to be binned in sectors of arbitrary width. The sector width can vary from 3o to 30o and include all measurements from the investigated bin. To account for this spatial averaging, multiple flow fields with 1o increment of the model are averaged within the sector. With ORFEUS, or a generic Computational Fluid Dynamics (CFD) code, this procedure would be computationally costly and this is one reason for the use of wake models in industry. In different studies from Gaumond et al. (2012), Wu and Porte-Agel (2015) and Walker et al. (2016) the authors found that wake-model results agree better with data from wider wind sectors (say larger than 10o ). In order to test the model with strong wake effects, inline layout configurations were analysed, namely when the wind direction is aligned with the wind-farm rows. The wakes will directly affect the next turbines and therefore cause significant power losses. However, a large sector width includes most likely situations where turbines are not or just partly covered by wakes, decreasing the wake effect and the error of any wake model. For the Lillgrund wind farm, Dahlberg (2009) claims that a sector width smaller or equal to 5o is necessary to clearly see the wake effects on a turbine from the preceding one. As the goal is to get an insight about the models performance under full-wake conditions, it was decided to compare data from sector widths between 5o to a maximum of 10o . By doing so, results from numerical models (ORFEUS or generic wake models) could be compared to SCADA data, since the inlet sector is small enough. Since it still represents the industry standard, the Jensen model is included in the comparison as well. To mitigate the effect of different data processing, an ensemble of several publications and data analyses
(12)
The length scale follows from the expectation that the structure size increases with height Λy ¼
κz
σv
:
(13)
As mentioned before, it is possible that the behaviour of the eddies differs from the horizontal to the vertical direction, as suggested by ~ a et al. (2012): The meandering observed during their experiment Espan was more prominent in the horizontal direction: The influence of the ground limits the fluctuating motion in the vertical direction. Therefore, the length scale in the vertical direction was set to half of that of Λy (Kaimal and Finnigan, 1994) Λz ¼
κz with σ w ¼ σ v : 2σ w
(14)
The detailed calibration of the atmospheric parameters (12)–(14) has not been performed here, so that the present expressions are only educated guesses based on observations of the atmospheric boundary layer (Kaimal and Finnigan, 1994; Stull, 2012). A more detailed investigation from mast data is needed to properly quantify these values in situ.
Fig. 4. Streamwise velocity at hub height in m/s without meandering (a) and including the meandering description (b). The steady simulation had the same setup adopted in Fig. 2, while the meandering parameters Λy , Λz , σ v , σ w were set according to equations (12)-(14). 5
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Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
power production significantly when wake effects are important. However, ORFEUS is able to simulate global upstream blockage effects of the farm (Ebenhoch et al., 2017; Segalini, 2017; Bleeg et al., 2018), that decrease the power output of the turbines placed in the first row. The results of the LUT method without wake-meandering correction are similar to ORFEUS, i.e. significantly underestimated. It is interesting that, despite of the fact that the LUT method should give similar results to ORFEUS, a slight overestimation is present, probably due to the superposition scheme. The LUT method with wake-meandering correction (from now on referred to as ORFEUS wake model) predicted power output agrees very well with the SCADA data for most rows, with a small tendency to underestimate, but nevertheless providing a clear improvement compared to the ORFEUS simulation and the LUT method without wake-meandering correction. In rows 4 and 5 the ORFEUS wake model is able to account for the recovery of the velocity field after the gap. Furthermore, the measurement data typically show a slight power increase from the second (which is usually the lowest) to the third turbine. The ORFEUS wake model is also able to account for this behaviour. For the longer rows 2, 3 and 4 the measured power decreases towards the end, whereas the ORFEUS wake model predicts a constant power. This effect is a known shortcoming of wake models: Frandsen et al. (2006) argued that the natural boundary layer changes in large wind farms and the amount of available kinetic energy is reduced. The ORFEUS wake model appears to be unable to catch this effect, probably also due to the static wake-combination method. The Jensen model performs quite well in all rows. Typically the predicted power is a bit lower than from the ORFEUS wake model, even though the wake decay constant was set to the higher end of the recommended range (Gaumond et al., 2012) for offshore conditions with k ¼ 0:05. The Jensen model seems to have the same tendency to over-predict the power towards the end of longer rows. Fig. 7 shows the normalised power output for the 120o wind direction. For this inflow angle, two empty turbine spaces are located in row E, where the wind has more space to recover. The ORFEUS wake model seems again to agree well with the data and the model is able to correctly predict the velocity after the gap in row E. The Jensen results lie close to the data as well with a slightly stronger tendency to under-predict. Towards the end of the long rows, the same deviation observed for the other wind direction occurs as the wake models erroneously predict the velocity to be constant. However, the data show that there seems to be an increase in wind speed (e.g. Row B).
has been used. The wake-meandering correction will be included in the LUT method for most of the available validation cases, highlighting its role. The effect of the LUT method alone (namely without accounting for the wake meandering) will be shown in the Lillgrund case only to provide a quantification of its role. 4.1. Lillgrund The Lillgrund wind farm is located in the Øresund strait , it consists of 48 Siemens SWT-2.3-93 turbines and has a rated capacity of 110 MW. The hub height of the turbines is zhub ¼ 68:5 m, with a diameter of D ¼ 93 m. This farm is particularly interesting to analyse because of the gaps in the layout and the relatively tight spacing between the turbines. This leads to strong wake effects and to a rather low array efficiency given by AE ¼ Ptot =Nt P0 ¼ 77%, with Ptot indicating the instantaneous total power production, Nt the number of turbines and P0 the power produced by the first row facing the wind. The two wind directions that have been analysed here and in the literature are 222o and 120o , as depicted in Fig. 5, where the rows are numbered and lettered. The spacing is 4:3D and 3:3D respectively (Nilsson et al., 2015). Two gaps exist in the cluster, where no turbine has been built. In these region, the wind velocity has space to recover inside the farm. Therefore, rows 4 and 5 as well as row E are of particular interest. The dataset used for comparison was published by Nilsson et al. (2015). During the measurement period, the average atmospheric stratification was considered to be neutral. The considered mean-wind velocity is U∞ ¼ 8 m=s (bin size: 7:5 m =s - 8:5 m =s) and the direction bin sectors are each 5o wide. The averaging time is in 10 min intervals, so that the mean wind travels twice through the farm in one interval. The measured turbulence intensity was 5%. Fig. 6 shows the power output of all rows from the 222o wind direction (rows 1 to 8). The power has been normalised with the median power of the turbines working in undisturbed wind. In rows 4 and 5, the missing turbine in the gap can be seen in the interruption of the data points. The results from the tested models have been plotted to show how well the prediction of the proposed wake model (with and without wakemeandering correction), Jensen model and ORFEUS match with the measurements. It can be seen that the steady ORFEUS simulation underestimates the
4.2. Horns Rev The second farm that was here analysed is the Horns Rev 1 offshore farm, located on the West coast of Denmark. It consists of 80 Vestas V80 turbines with a hub height of zhub ¼ 70 m and a rotor diameter of D ¼ 80 m. The total farm capacity is 160 MW and the turbines are arranged in a regular grid of 8 10. The layout is depicted in Fig. 8. The minimum distance between the turbines is 7D in the western direction or 9:4D and 10:4D, respectively in the diagonals (Wu and Porte-Agel, 2015). This makes the spacing larger than in the Lillgrund farm and leads to a higher array efficiency, AE. The 270o full-wake situation is a popular case in the literature (Barthelmie et al., 2009), where again the free-stream velocity U∞ ¼ 8 m=s has received most of the attention and will be our investigated case, as well as the diagonal directions 222o and 312o . The SCADA data from Barthelmie et al. (2009) was obtained using 10 min averaging periods and features all three inflow angles. The bin size was with 10o , higher than for the Lillgrund dataset. Only neutrally-stratified cases were considered, with a turbulence intensity of approximately 8%. For the aligned case (270o ) additional data from Walker et al. (2016) is also shown, again for 8 m =s and a bin size of 10o . The data was collected under the period of one year during 2005. Further data for the 270o direction from Gaumond et al. (2012) features a 5o bin size.
Fig. 5. Layout of the Lillgrund wind farm. The rows and the columns are marked as well as the two wind directions considered here. 6
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Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
Fig. 6. Normalised power of the rows (1)–(8) of the Lillgrund wind-farm with U∞ ¼ 8 m=s and wind direction 222o . Data from Nilsson et al. (2015) (solid line) compared with ORFEUS (þ), Jensen (x) and LUT with wake-meandering correction (∘) and without wake-meandering correction (gray ). The cT and cP -curve was taken from van der Laan et al. (2015), zhub ¼ 0:736D and sector averaging the wake models was 5o . The shear exponent of the simulation was set to α ¼ 0:1. The wake-decay constant for the Jensen model was k ¼ 0:05.
Fig. 7. Normalised power of the rows (A)-(H) of the Lillgrund wind-farm with U∞ ¼ 8 m=s and wind from 120o . The marker styles as well as model configuration are the same as in Fig. 6.
In the 270o direction, the data from the three different sources differ to some extend. This underlines the difficulties and differences that occur during the data processing. The ORFEUS wake model is able to match the measurements best, lying in the middle of the data for turbines 3–7.
Since the farm has a regular layout, only one row inside the farm can to be considered in the analysis. Both wake models were averaged over a 10o sector this time, as the majority of the data. Fig. 9 shows the normalised power production for the three respective wind directions. 7
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Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
phenomenon in a fast solving wake-model code was presented. First it was decided to assume the wake itself to be a passive tracer and the meandering motions a result of large atmospheric turbulence. Then, a model on the basis of Lagrangian particle dispersion was implemented, derived from the findings of Taylor (1921). Starting from a simplified example for the observation of a single particle under discrete random fluctuations, Taylor (1921) developed a theory for diffusion under continuous correlated movements. This theory for particle dispersion was applied for the meandering cross-stream motion of the wake, where it was assumed that the wake cross sections are displaced rigidly, without any distortion (a hypothesis proposed by Larsen et al., 2008). The probability density function of the wake centreline could then be superimposed to a static wake-model velocity-deficit distribution. Through this, a time-averaged description of the wake including the meandering was obtained. Like suggested by Espa~ na et al. (2012), the strength of meandering differs in the horizontal and vertical direction, so both directions were considered independently. Larsen et al. (2008) used the rigid-wake displacement coupled with synthetic turbulence to generate time series for fatigue-load estimation. This process is quite expensive if the goal is just to estimate the wake deficit and the present model, being probabilistic, bypasses the need to create artificial time series (i.e. Monte Carlo simulations) providing directly a statistical description based on simple parameters that can be measured. The model for the wake velocity itself is based on ORFEUS, a linearised code that simulates the turbines as actuator-disk body forces. To obtain a wake model, results of several ORFEUS single-turbine simulations were analysed and the velocity deficit decomposed into four parameters: The wake-centre velocity deficit, CðxÞ, the wake-width parameters, σ y ðxÞ and σ z ðxÞ, and the height of the maximum deficit over ground, zoc ðxÞ. This analysis was limited to the far wake, which was assumed to have some kind of self-similar characteristics. All parameters were consequently stored and could then be accessed through a look-up table to rapidly restore individual turbine wakes in a wind farm. Before superimposing the meandering model, the parameters that govern the rate of diffusion had to be chosen. The cross-stream fluctuations and the time-scale of the governing atmospheric eddies (σ v and Λy in the horizontal direction) were estimated tofrom general atmospheric turbulence features. The superposition of the estimated wake with the wake-meandering correction (given by a convolution of the deficit with
Fig. 8. Layout of the Horns Rev wind farm.
However, the curves from Gaumond et al. (2012) and Walker et al. (2016) show a decreasing output towards the last turbines, which is again not captured by the wake model. The Jensen model (with k ¼ 0:05) under-predicts the power, which was already noticed for the Lillgrund results. The ORFEUS solver underestimates the power output again. However, it has to be noted that the measurements were binned in a 10o sector which causes an increase of power in the data, even more than for the 5o sectors for Lillgrund. ORFEUS as a linearised code is better for comparing smaller sectors. In the 222o and 312o cases, the spacing between the turbines becomes larger, so that the power output increases as well. Data in both cases was only available until turbine 5. At 222o both wake models under-predict the power. Furthermore, there seems to be a trend of decreasing power in the data along the row, which only ORFEUS is able to predict. In the 312o direction, the ORFEUS wake model matches again the data best. 5. Conclusions In this paper a novel approach to include the wake-meandering
Figure 9. Normalized power production of the Horns Rev wind farm for inflow angles of ðaÞ270o , ðbÞ222o and ðcÞ 312o . Data from Barthelmie et al. (2009) (solid line), Walker et al. (2016) (dashed line), Gaumond et al. (2012) (dotted line) compared with ORFEUS (þ), Jensen (x) and ORFEUS wake model (o). Free-stream velocity U∞ ¼ 8 m=s, averaging sector for both wake models 10o , cT and cP -curve from Hansen et al. (2010). The shear exponent for ORFEUS and the ORFEUS wake model was α ¼ 0:09 with zhub ¼ 0:875D. The wake-decay constant for the Jensen model was k ¼ 0:05.
8
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Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103954
the wake-centre probability density function) resulted in a wider wake and a smaller wake deficit, providing a faster wake recovery and highlighting the key role of the meandering phenomenon. To validate the approach, the model was compared to energy production data sets from two offshore-wind farms. The LUT model with wake-meandering correction showed good agreement with the Lillgrund wind farm, where it was able to predict the wind-speed increase after the gaps in the array. An expected shortcoming was the increasing discrepancy towards the end of long rows, where the model started to overpredict the power, probably because of the wake-combination method. When compared to the Jensen model, the agreement was similar for the Lillgrund data set and the proposed model seemed to be more accurate for the Horns Rev data. The absence of the meandering effect in the full ORFEUS simulations, or in the LUT generated from ORFEUS (without the meandering), provided a significant underestimation of the power production due to a reduced wake recovery. The significant advantage of the present formulation is its ability to physically account for the wake-diffusion mechanism, splitting it into two phenomena: One where the wake diffuses due to turbulent transport of momentum, and the other where the wake displaces due to dispersion mechanisms (the “wake-meandering” phenomenon). Physical constants and parameters are introduced and they can be quantified through mast measurements or educated guesses as done here, reducing the level of empiricisms in power estimations of wind farms. This is a big advantage with respect to the Jensen model (or similar ones), where the wakespreading constant, k, is obtained empirically and implicitly accounting for all transport and dispersive phenomena in the wake. Here the model provides a clear separation between the two phenomena and suggests even the possibility to develop new wake models that account only for the wake meandering, being quite a dominant wake-recovery mechanism.
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