2.1 Validation of wake turbulence on the basis of the DWM modelpredicted Reynolds stress
The method described by Equations (19)(23) for estimating the turbulence intensity in the wake of the DWM model on the basis of the turbulent stresses is validated by comparing the mean turbulence intensity in the wake as a function of downstream distance to the Ellpisys3D AL model calculations of Keck et al.[17] Because the DWM model deficit equation is designed to handle the wake deficit development without meandering, it is compared with AL data in the MFoR (i.e. following the centre position of the wake deficit).
In the first analysis, the turbulence intensity found by Equation (23) was investigated without including the correction to ensure that the local turbulence intensity is always equal to or larger than the ambient turbulence intensity [i.e. using only the left term under the max operator of Equation (23)]. This term represents the turbulent stresses that act on the wake deficit in the DWM model. The results in Figure 8 show the mean turbulence intensity over a cross section of the wake as a function of downstream distance.
Figure 8 shows the effect of including the ABL shear contribution in the eddy viscosity calculation of the DWM wake deficit for three different ambient turbulence intensities: 6% (A), 10% (B) and 14% (C). All cases conducted for a single wind turbine with a rotor diameter of 80 m, operating in neutral stratification with a mean wind speed of 8 m s^{−1}. The figure shows that the inclusion of the ABL shear contribution results in a higher mean turbulence level in the wake of the DWM model, relative to the unmodified DWM model. The effect is larger for higher turbulence intensity as the ABL shear gradient increases with turbulence intensity under neutral stratification. Compared with the unmodified DWM version, the turbulence predictions of the modified DWM model agree better with the AL reference data. The largest improvements are seen in the farwake region (where the wake deficit shear is small), and the effect increases with increasing ambient turbulence intensity (as the ABL shear increases). The modified DWM model slightly overpredicts the turbulence intensity between 4 and 6 D behind the rotor. The overprediction is on the order of 0.5–1.0 percentage points (pp, i.e. ΔTI). The farwake turbulence agrees well for the 6% and 10% cases (A and B) but is underpredicted for the 14% case (C). At 10–12 D, the turbulence intensity of the 14% case is underpredicted by ~2.5 pp.
As mentioned earlier, before coupling the turbulence intensity estimation to calculate the average turbulence intensity at the downstream turbine, the low turbulence regions are corrected by the ambient turbulence intensity [right term of Equation (23)]. These low turbulence intensity regions in the DWM model exist at a location where the velocity gradient of the deficit is close to zero. The turbulence in these regions will become low as there is no transport equation for turbulent stresses in the DWM model, and the ABL shear correction will not have full effect because of a Wiener filter (included for numerical stability). To avoid having these effects influence the intraturbine wake turbulence coupling, Equation (23) ensures that the minimum wake turbulence is equal to the ambient turbulence intensity level. Applying this correction yields the results presented in Figure 9 for the modified DWM model. The main difference occurs in the farwake region at high turbulence intensities, where the deviations are significantly smaller than in Figure 8 and allows for a more realistic coupling of turbulence intensity to the downstream turbines.
2.2 The effect of the proposed DWM model improvements
The influence of the proposed atmospheric shear contribution and turbulence buildup is examined by simulating flow through a row of eight wind turbines. The wind speed is 8 m s^{−1}, the turbulence intensity is 6% and the mean flow direction is along the row of turbines. We studied two different turbine spacings: 6 D and 10 D.
Figure 10 shows the effect on power output of the individual turbines as the various effects in the DWM model are enabled and disabled. By comparing the power output for the row of turbines when all DWM effects are enabled (black solid lines) to the output if turbulence buildup is not considered (dotdashed black lines), it is apparent that the effect of turbulence buildup is sensitive to turbine spacing. At 6 D spacing, turbulence buildup has an effect on the wake loss that is twice as large as the effect of wake meandering, but with 10 D spacing, the effect is only about 55% of that caused by wake meandering. The effect on power caused by the atmospheric shear contribution to turbulence is about 5% and 20% of the effect of wake meandering at 6 D and 10 D, respectively. Compared with the total power losses due to wake effects of the unmodified DWM model, the inclusion of wakeadded turbulence and turbulence buildup over the row of turbines reduced the predicted wake losses by 9% for the 6 D case and 6% for the 10 D case. The atmospheric shear contribution to turbulence in the DWM model further reduced the power loss by about 0.8% at 6 D spacing and 1.5% for the 10 D case.
These findings show that the proposed DWM model improvements have significant effects on the DWM deficit development. They not only influence the wake turbulence (which is the direct effect) but they also affect the mean wind speed and thereby power predictions. Table 1 shows the development of turbulence intensity and wind speed along the rows for the two presented cases. The results suggest that an equilibrium wind speed is reached already at the second or third turbine inside the park. The turbulence requires much longer to become fully developed, and an equilibrium value is not reached before the fifth or sixth turbine. This suggests that an approach in which only the nearest upstream wake deficit affects a given turbine might be an acceptable simplification. However, for an accurate turbulence representation, the influence of more upstream wakes should be considered.
Table 1. The development of turbulence intensity and wind speed over the row of wind turbines with the proposed atmospheric shear effect and wake turbulence buildup.Mean wind speed (m s^{−1})       

 WTG 1  WTG 2  WTG 3  WTG 4  WTG 5  WTG 6  WTG 7  WTG 8 


6D  8.00  5.90  6.00  6.00  5.97  5.95  5.95  5.95 
10D  8.00  6.65  6.62  6.57  6.55  6.54  6.54  6.54 
Turbulence intensity (%)       
 WTG 1  WTG 2  WTG 3  WTG 4  WTG 5  WTG 6  WTG 7  WTG 8 
6D  6.0  12.1  13.4  13.9  14.3  14.5  14.6  14.7 
10D  6.0  9.7  10.1  10.3  10.4  10.5  10.6  10.6 
2.3 Lillgrund wind farm
The performance of the suggested DWM improvements are tested against field data and OpenFOAM AL results for the offshore Lillgrund wind farm. The power production of the Lillgrund wind farm is estimated by both the unmodified and the modified DWM models and compared with the field data presented by Dahlberg[26] and the OpenFOAM simulations of Churchfield et al.[27] The wind speed and turbulence fields of the DWM models were compared with the OpenFOAM data, and the agreement between the two models was evaluated by using Equation (16).
To simulate the Lillgrund wind farm with the DWM model, the rotor induction for the turbines is required. Since this information is not publically available, an approximate rotor description was developed on the basis of the publically available power curve and blade profile family. This rotor description is used in the OpenFOAM AL simulation of the wind farm. The induction vector used in the DWM model for the Lillgrund turbines is based on the timeaveraged velocity field at the rotors in the OpenFOAM simulation. First, the velocity field at the rotor is timeaveraged and transformed from the Cartesian grid to a vector of axisymmetric values. Then, the azimuthally averaged induction vectors required by the DWM model are found by dividing the axisymmetric velocity vector at the rotor by the oncoming wind speed. For the first rotor, the ambient wind speed is used. For the wakeaffected turbines, the oncoming wind speed is found by running a DWM simulation for a single rotor and calculating the mean wind speed at the location of the downstream turbine using Equations (21) and (22).
The layout of the Lillgrund wind farm is shown in Figure 11(a). The analysis presented in this paper is performed for rows B (turbines 15–8) and D (turbines 30–24). Row B contains eight wind turbines that are equidistantly spaced 4.4 D apart. Row D contains seven wind turbines separated by 4.4 D; however, the turbine that would have been the fourth in this row was omitted from the actual wind farm because the water depth is too shallow to allow access by construction boats, so there is an 8.8 D gap between turbines 28 and 27.
The simulations are run in a neutral atmosphere (although there is a stable capping inversion at roughly 800 m above the surface), with a mean wind speed of 9 m s^{−1} and a turbulence intensity of 6.2%. The mean flow direction is perpendicular to the rows. The OpenFOAM data is based on a fullscale simulation of the entire wind farm. The duration of the AL simulation is a tradeoff between computational cost and uncertainty in the resulting flow field. As a ‘standard’ AL simulation over the wind farm requires on the order of 500,000 CPUhours to simulate the flow field for a 10 min period, the simulations times were required to be kept around 10 min to maintain acceptable time consumption with the computational resources available. As shown in Figure 11(b), a consequence of the relatively short simulation time is that the average wind speed approaching the turbines was not entirely uniform in the whole domain. In the figure, streaks of lower (higher) mean wind speed are seen as lighter (darker) areas. An example of a lowvelocity streak that could have some influence on the results is hitting turbine 15 (i.e. the first turbine of row B). This inhomogeneity of the incoming flow field should be seen as a source of uncertainty on the order of 0.165 m s^{−1} in terms of wind speed and 0.53 pp in terms of turbulence intensity. This corresponds to the spatial standard deviation of the average wind speed and turbulence intensity upstream of the wind farm.
A second term of uncertainty due to the short averaging time is the misalignment of the wake deficits, relative to the mean wind direction, because of the stochastic wake meandering (Figure 12). This uncertainty component, σ_{misalignment}, is estimated using Equation (31),
 (31)
where n is the number of independent samples of wake centre position, σ_{M} is the wake meandering given as the standard deviation of the wake centre position and is the radial gradient of the studied parameter (for this application axial velocity or turbulence intensity). An estimation of the uncertainty is obtained by assuming the following: (i) 100 independent samples of wake centre position over the 10 min simulations, (ii) a representative standard deviation of the wake centre position equal to 11.2 m (based on simulations by Keck et al.[17]), (iii) a wake velocity gradient of 0.06 m s^{−1} m^{−1} and turbulence intensity gradient of 0.25 pp m^{−1}. Applying these numbers in Equation (31) yields an uncertainty of 0.067 m s^{−1} in terms of wind speed and 0.28 pp in terms of turbulence intensity. However, the average wake deficit and turbulence profiles due to the turbines are still on an order of magnitude larger than the combined uncertainty caused by the low averaging time, which allowed for a highquality comparative analysis.
The uncertainty in terms of power production is based on the combined rootsquare sum of the aforementioned sources of uncertainties in wind speed (which is equal to 0.178 m s^{−1}). The uncertainty in the power production is estimated as the ratio of power production based on the mean wind plus the combined uncertainty of the wind speed, to power production based on the mean wind speed alone. This yielded an uncertainty in power production of 6%.
By comparing the power production estimates presented in Figure 13, we see that the unmodified DWM model (red lines) predicts lower power production under multiple wake conditions compared with the modified DWM model (black lines). This prediction is expected, as the turbulence buildup in the modified DWM model will lead to a faster wake recovery. Compared with the OpenFOAM AL model (green lines) and the field data (blue lines), the unmodified DWM model underpredicts power production of the row of turbines.
In general, the power prediction made by the modified DWM model agrees with the OpenFOAM AL model and the field data. The main deviation seen is the power estimation for the fourth turbine in row D, i.e. the turbine with a larger separation to the upstream turbine, where the DWM model increases slightly more than the OpenFOAM AL model and the field data. We also see that both the modified DWM model and the OpenFOAM AL model overpredict power production for the last three turbines in the rows, compared with the field data. A likely explanation for these deviations is that the field data contain some effect that is not included in the numerical simulations. The DWM model code was run with the same input as the OpenFOAM simulation, which was a neutrally stratified atmosphere with a wind speed of 9.0 m s^{−1} and a turbulence intensity of 6.2%. Although the simulations matched the observed average inflow wind speed and turbulence intensity for this wind direction, the difference in predictions for the rear of the wind farm may be caused by the fact that the field data was collected over many months. Over that period, there was undoubtedly a range of atmospheric stability and wind speed, unlike in the fixed condition simulations. This range may affect the wake propagation and power production. The field data is only binned by wind direction at the first turbine, and wind speeds outside of the turbines' region 2 (in which wake effects are maximum) are not included in the average.
Table 2 gives a quantitative analysis of the differences in mean wind speed and turbulence intensity distribution over the two rows of turbines. The table shows the STE between the OpenFOAM AL model and the DWM models calculated on the basis of the mean flow field according to Equation (16). The values in the table are based on a single case of ambient conditions: wind speed of 9 m s^{−1} and turbulence intensity of 6.2%. For all turbines, data were collected at the cross sections located 3 and 4 D downstream of the rotor, apart from turbine 3 in row D (i.e. turbine 28 in Figure 11(a)), where data were collected at 3, 4, 5, 6, 7 and 8 D behind the rotor. This is due to the longer undisturbed wake evolution of turbine 28. The data at used in the flow field analysis were collected from along a lateral line at hub height at each cross section. The line was 3.2 D of length, oriented perpendicular to the mean flow direction and centred on the axis of the turbine (thus extending 1.6 D in lateral direction from the mean wake centre).
Table 2. The STE in terms of mean wind speed and turbulence intensity level as a function of turbine position for the modified DWM and unmodified DWM models compared with the OpenFOAM Al model.  STE WS unmodified  STE TI unmodified  STE WS modified  STE TI modified 

DWM (m s^{−1})  DWM (%)  DWM (m s^{−1})  DWM (%) 


WTG 1  0.38  2.11  0.38  2.11 
WTG 2  0.51  2.39  0.46  1.81 
WTG 3  0.63  2.88  0.40  1.93 
WTG 4  0.69  3.48  0.47  2.21 
WTG 5  0.80  3.41  0.48  1.78 
WTG 6  0.86  2.80  0.55  2.02 
WTG 7  0.98  3.21  0.69  1.83 
WTG 8  1.02  4.24  0.61  1.18 
Mean  0.73  3.07  0.51  1.86 
WTG 1  0.60  2.10  0.60  2.15 
WTG 2  0.46  1.70  0.43  1.28 
WTG 3  0.20  1.54  0.17  1.05 
WTG 4  0.55  2.67  0.40  1.50 
WTG 5  0.65  3.01  0.43  1.55 
WTG 6  0.71  3.62  0.44  1.70 
WTG 7  0.68  3.38  0.32  1.88 
Mean  0.55  2.58  0.40  1.59 
The two leftmost columns in the tables are the STE between the OpenFOAM AL model and the unmodified DWM model, and the two rightmost columns represent the OpenFOAM AL results compared with the modified DWM model. The table shows that the STE, in terms of mean wind speed, is reduced from 0.73 to 0.51 m s^{−1} for row B (31.1%), and 0.55 to 0.40 m s^{−1} for row D (27.4%). The reduction in STE for turbulence intensity for row B is 3.07 to 1.86 pp (39.4%), and 2.58 to 1.59 pp (38.4%) for row D. Excluding the first turbine, the reduction in STE for waked turbines is found to be 31.8% for wind speed and 43.5% for turbulence intensity, by applying the proposed correction to the DWM model. The STE number should be evaluated with the uncertainty of the AL data in mind. As discussed earlier, the uncertainty of the AL simulations, which is caused by the relatively short simulation times, will increase the STE linearly. The uncertainty, 0.178 m s^{−1} and 0.60 pp for turbulence intensity, could therefore be considered to be the cause of ~39% of the discrepancies in wind speed profiles and ~35% for the turbulence intensity profiles over the rows of turbines.
Note that the STE for the modified DWM model does not increase significantly as a function of turbine position in the row. In fact, the combined STE of the first turbines is actually higher than the mean for the entire row, considering both wind speed and turbulence intensity. The unmodified DWM model has a clear trend of increasing STE numbers with turbine position. Furthermore, by comparing the STE of wind turbine generator (WTG) 3 in row D (i.e. the turbine with the longer undisturbed wake) to the STE of the other turbines, the DWM model agrees better with the AL model at larger downstream distances. The main reason for the increased agreement at downstream is that the gradients of the deficit are smaller, so the comparison is less sensitive to misalignment of the wake centre or difference in the wake width.
The improved agreement between the modified DWM model and the OpenFOAM AL model is also illustrated by showing the turbulence intensity, Figure 14, and mean velocity, Figure 15, at cross sections located 2D, 3D and 4D downstream of turbine 1, 5 and 8 in row B of the Lillgrund wind farm. From the top row of panels (A, B and C), it can be seen that the suggested DWM modifications has little effect on the flow field in a single wake. This is due to the short distances and low turbulence intensity (6.2%), which means that the ABL shear correction has little effect, and since it is the first turbine of the row, the turbulence buildup does not affect the solution.
The middle row of panels (D, E and F) shows that the DWM modification improves the agreement both in terms of wind speed and turbulence intensity when studying the wake after the fifth turbine. In terms of turbulence intensity, Figure 14 shows that the unmodified DWM model (dashed red lines) underpredicts the turbulence in the wake centre and outside of the wake shear layer but overpredicts turbulence intensity of the wake shear layer compared with the AL model. The modified DWM model (solid black lines) overpredicts the turbulence levels in the wake shear layer slightly more than the unmodified model but overall captures the turbulence level outside of the ‘peaks’ well. The average turbulence intensity level predicted by the modified DWM model is therefore closer than the AL model results. The middle row of Figure 15 shows that the increased turbulence level of the modified DWM model results in a fast recovery, which is closer to that of the AL model. Both DWM implementations overpredict the depth of the deficit in the centre region and underpredict the width of the wake deficit relative to the AL model results.
The bottom row of panels (G, H and I) shows the same trends as the previously discussed panels (D, E and F). The main differences are that both DWM implementations captures turbulence intensity of the wake shear layer better and that both implementations overpredict the depth of the wake deficit slightly more after the eighth turbine.
It is worth noticing that the wind speed and turbulence fields of the DWM models, compared with the OpenFOAM AL flow fields, also agree fairly well at 2 D (panels A, D and G). This observation suggests that the pressure effects of the upstream rotor are small in that region and that DWM could be valid down to 2 D. The STE at 2 D, however, is roughly four times larger than the STE at 3 D.