Assessing Spacing Impact on the Wind Turbine Array Boundary 1 Layer via Proper Orthogonal Decomposition

A 4 x 3 array of wind turbines was assembled in a wind tunnel with four cases to study the influence based on streamwise and spanwise spacings. Data are extracted using stereo particleimage velocimetry and analyzed statistically. The maximum mean velocity is displayed at the upstream of the turbine with the spacing of 6D and 3D, in streamwise and spanwise direction, respectively. The region of interest downstream to the turbine confirms a notable influence of the streamwise spacing is shown when the spanwise spacing equals to 3D. Thus the significant impact of the spanwise spacing is observed when the streamwise spacing equals to 3D. Streamwise averaging is performed after shifting the upstream windows toward the downstream flow. The largest and smallest averaged Reynolds stress, and flux locates at cases 3D x 3D and 6D x 1.5D, respectively. Snapshot proper orthogonal decomposition is employed to identify the flow coherence depending on the turbulent kinetic energy content. The case of spacing 6D x 1.5D possesses highest energy content in the first mode compared with other cases. The impact of the streamwise and spanwise spacings in power produce is quantified, where the maximum power is found in the spacing of 6D x 3D. PACS numbers: 6 1 Wind Energ. Sci. Discuss., doi:10.5194/wes-2016-23, 2016 Manuscript under review for journal Wind Energ. Sci. Published: 11 July 2016 c © Author(s) 2016. CC-BY 3.0 License.

Rev farm is 7D, optimal spacing along the bulk flow direction and 9.4D or 10.4D along the diagonal.Barthelmie and Jensen [9] showed that the spacing in the Nysted farm is responsible for 68-76% of the farm efficiency variation and for wind speed below 15 ms −1 , the efficiency will increase 1.3% for every one diameter increasing in spacing.Hansen et al. [10] pointed out that the variations in the power deficit for different spacing were almost negligible at approximately 10D into Horns Rev farm in spite of a large power deficit resulting from smaller turbine spacing.In addition, the mean power deficit is similar along single wind turbine rows when inflow direction is unified and the wind speed interval from 6 to 10 ms −1 .
Furthermore, the maximum deficit happens between the first and the second row of turbines and minimum deficit in the remaining downstream.González-Longatt [11] found that when the downstream and spanwise spacing increased, the wake coefficient representing the total power output with wake effect over total power without wake effect increased, and the effect of the incoming flow direction on the wake coefficient increased when the spacing of the array is reduced.Meyers and Meneveau [12] studied the optimal spacing in fully developed wind farm with considerable limitations including neutral stratification and flat terrain with no topography.The results highlight that depending on the ratio of land cost and turbine cost, the optimal spacing might be 15D instead of 7D.Stevens [13] used the effective roughness length performed by LES to predict the wind velocity at hub height depending on the streamwise and spanwise spacing, and the turbine loading factors.Also showing that optimal spacing depends on the wind farm length in addition to the factors suggested in [12].Stevens et al. [14] used LES model to investigate the power output and wake effects in aligned and staggered wind farms with different streamwise and spanwise turbine spacing.
In the staggered configuration, power output in fully developed flow depends mainly on the spanwise and streamwise spacings, whereas in the aligned configuration, power strongly depends on the streamwise spacing.
In this article, the proper orthogonal decomposition (POD) analysis will be employed to identify the structure of the turbulent wake associated with variation in spacing and understand the effect of the streamwise and spanwise on the characteristic flow of the wind turbine array, including Reynolds shear stress, turbulent flux and energy production.
Balancing between the gain and loss in energy can be quantified through the mean kinetic energy equation [15].One of the main gain sources can be obtained by the spatial transport of energy by Reynolds shear stress, named the energy flux.The Reynolds shear stress is the center of the energy flux, therefore this study will focus on the energy flux to quantify the impact of the streamwise and spanwise spacing through the statistical analysis and using Proper orthogonal decomposition.POD is a mathematical tool that depends on a set of snapshots to obtain the optimal basis functions and decompose the flow into modes that express the most dominant features.This technique, which is presented in the frame of turbulence by Lumely [16], categorizes structures within the turbulent flow depending on their energy content and allows for filtering the structures associated with the low energy level.Sirovich [17] presented the snapshot POD that relaxes the difficulties of the classical orthogonal decomposition.
The flow field, taken as the fluctuating velocity, can be represented as u i = u( x, t n ), where x and t n refer to the spatial coordinates and time at sample n, respectively.A set of the orthonormal basis functions, φ, can be presented as The optimal functions have minimum averaged error and maximum averaged projection in mean square sense.The largest projection can be determined using the two point correlation tensor and Fredholm integral equation where R( x, x ) is a spatial correlation between two points x and x , N is the number of snapshots, T is the transpose of the matrix, and λ are the eigenvalues.The optimal deterministic problem is solved numerically as the eigenvalue problem.The eigenfunctions are orthogonal and have a corresponding positive and real eigenvalues organized by descending arrangement.The POD eigenvectors illustrate the spatial structure of the turbulent flow and the eigenvalues measure the energy associated with corresponding eigenvectors.The summation of the eigenvalues presents the total turbulent kinetic energy (E) in the flow domain.The fraction of the cumulative energy, η and the normalized energy content of each mode, ξ, can be represented as, POD tool is particularly useful in rebuilding the Reynolds shear stress using a set of eigenfunctions as follows, POD used to describe coherent structures of different types of flow such that axisymmetric mixing layer [18], channel flow [19], atmospheric boundary layer [20], wake behind disk [21], and subsonic jet [22].In the frame of a wind turbine wake flow, Anderson et al. [23] applied POD to the flow in a wind farm simulated using LES.They showed the large scale motion and dynamic wake meandering are strongly governed by turbine spacing.The number of modes required to reconstruct the flow is related to the flow homogeneity.grid, which consists of 7 horizontal and 6 vertical rods, to introduce large-scale turbulence.
Nine vertical Plexiglas strakes located at 0.25 m downstream of the passive grid and 2.15 m upstream the first row of the wind turbine were used to modify the inflow.The thickness of the strakes is 0.0125 m with a spanwise spacing of 0.136 m.Surface roughness elements were placed on the wall as a series of chains with diameter of 0.0075 m and spaced 0.11 m apart.Figure 1 shows the schematic of experimental setup.
A 0.0005 m thick steel was used to construct 3 bladed wind turbine rotor.The diameter of the rotor was 0.12 m, equal to the height of the turbine tower.Each rotor blade was pitched at 15 • out of plane at the root and 5 • at the tip.These angles were chosen to provide angular velocity that correlates with required ranges of tip-speed ratio.A DC electrical motor of 0.0013 m diameter and 0.0312 m long formed the nacelle of the turbine and was aligned with flow direction.A torque sensing system was connected to the DC motor shaft following the design outlined in [27].Torque sensor consists of a strain gauge, Wheatstone bridge and the Data Acquisition with measuring software to collect the data.For more information on the experiment conditions and data processing, see [7].
In this study, the flow field was sampled in four configurations of a model-scale wind turbine array, classified as Π n , where n varies from 1 through 4 and considered in Table I.
Permutations of streamwise spacing of 6D and 3D, and spanwise spacing of 3D and 1.5D are IV. POWER MEASUREMENTS.that the maximum power are extracted approximately at angular velocity of 1500 ± 100 rpm.The optimal power of 0.078 W is harvested at the largest spacing, i.e., case Π 1 .
Reducing streamwise spacing shows a significant decreasing in extracted power especially at 1000 < ω < 1800 rpm.The maximum power of case Π 2 is 33% less than case Π 1 .The reduction ratio between cases Π 3 and Π 4 is 22%.Reducing spanwise spacing displays a majority at x/D = 6 where the reduction ratio of 20% is noticed.Small reduction ratio of 6% is identified between cases Π 2 and Π 3 .

V. RESULTS
A. Statistical Analysis.
Herein, characterization of the wind turbine wake flow via mean velocity, Reynolds shear stress and kinetic energy, with the aim to understand the effect of turbine-to-turbine spacing.compared with the other cases due to greater recovery of the flow upstream of the turbine.
Although the streamwise spacing of case Π 4 is similar to case Π 1 , the former shows reduced hub height velocity.The mean velocity is about 2.88 ms −1 compared with 3.3 ms −1 in case Π 1 , confirming the influence of the spanwise spacing on wake evolution and flow recovery.
Small variations are observed between case Π 2 and Π 3 above the top tip (y/D = 1.5) and below the bottom tip (y/D = 0.5), where case Π 2 demonstrates higher velocities.
Downstream of the turbine, the four cases show clear differences especially above the top tip and below the bottom tip, where case Π 1 , once again, shows the largest velocities.
Case Π 2 also shows higher velocities below the bottom tip compared with cases Π 3 and Π 4 .
The comparison between case Π 3 and case Π 4 shows resemblance in velocity contour with exception at region x/D < 0.8, where case Π 4 displays the most significant velocity deficit.shows increasing of 16% in Reynolds shear stress of case Π 2 .A similar effect is observed in case Π 3 where it exhibits higher stress than case Π 4 with increasing average of 2% is noticed.
The spanwise spacing effect is more pronounced when the streamwise spacing is 3D as can be shown when comparing between case Π 3 and case Π 2 that shows increasing 20% in over domain average .However, decreasing spanwise spacing increases − uv slightly as shown when comparing between case Π 1 and Π 4 .The difference of 6% is shown and the variation is observed only in a small region at ( y/D ≈ 1.3 and x/D > 1.2), where higher Reynolds shear stress is found in case Π 1 .demonstrates higher energy flux.The average over downstream domain shows decreasing of 14% in − uv U of case Π 4 .The same tendency is observed when comparing between cases Π 2 and Π 3 that shows decreasing about 24.5% in the the vertical flux.This result confirms that when the spanwise spacing decreases, the energy flux decreases also.Decreasing the streamwise spacing, case Π 2 exhibits higher − uv U than case Π 1 mainly when x/D > 1 and the increasing average is 15%.The similar behavior is observed when comparing between case Π 3 and Π 4 .Case Π 3 displays higher − uv U of 5% than case Π 4 and the mainly differences are seen at x/D > 1 and y/D ≈ 1.5.In general, the impact of streamwise spacing on energy flux is more pronounced when spacing z = 3D than 1.5D.The impact of spanwise spacing on energy flux is more pronounced when the spacing x = 3D than 6D.Also, case Π 2 shows higher − uv U comparing with other cases.
Spatial averaging of the variables is determined via shifting the upstream domain of each case beyond its respective downstream flow and performing streamwise averaging according to the procedure used in Cal et al. the different cases while removing the streamwise dependence.Here, streamwise averaging is denoted by • x .Figure 7(a) shows profiles of streamwise averaged mean velocity for all four cases.Case Π 1 and case Π 3 show the largest and smallest velocity deficits, respectively.
At hub height, the velocity of case Π 1 is approximately 2.25 ms −1 whereas case Π 3 shows approximately velocity of 1.6 ms −1 .The difference between case Π 1 with case Π 4 is less than the difference between case Π 1 with case Π 2 confirming that the impact of reducing streamwise spacing is greater than changing the spanwise spacing.The influence of streamwise spacing is also observed when comparing cases Π 3 and Π 4 .Interestingly, a reduction in streamwise spacing show less effect when the spanwise spacing z/D = 1.5.For example, the utmost disparity in streamwise velocity between the cases Π 1 and Π 2 is 0.57 ms −1 as opposed to the dissimilarity of 0.42 ms −1 between cases Π 3 and Π 4 .Negligible variations are shown between the profile of cases Π 2 and Π 3 .The cases Π 2 , Π 3 and Π 4 converge at y/D > 1.4 while the case Π 2 and case Π 3 coalesce at the regions above the hub height.
The trend of the averaged profiles of the streamwise velocity follows the same trend that is observed in the power curves, see figure 3, and that verify the relation between the power on the turbine with the deficit velocity.between case Π 2 and Π 3 are observed below the hub height due to the significant difference between the Reynolds shear stress of these two cases as can be shown in figure 7(b).Above the hub height, the difference between these cases is diminished.In general, when spanwise spacing decreases, the energy flux also decreases as shown when comparing between case Π 4 with case Π 2 and case Π 2 with case Π 3 .In contrast, when streamwise spacing decreases, the energy flux increases as observed in comparing between case Π 1 with case Π 2 and case Π 3 with Π 4 .The maximum and minimum flux are observed at case Π 2 and case Π 4 , respectively.
The region very close to the hub height also shows zero energy flux and changes the sign of the energy flux.Based on the POD analysis, the spatially integrated turbulent kinetic energy is expressed by the eigenvalue of each mode.Normalized cumulative energy, η n , from Eq. ( 9) for upstream and downstream measurement windows are presented in the figure 8(a) and (b), respectively.Insets show the normalized energy content per mode, ξ n , given by Eq. ( 10).
These results can be attributed to the reduction the streamwise spacing.Convergence of case Π 1 oscillates around the curve of case Π 4 .The same trend is observed between case Π 2 and Π 3 but with fewer alternations.Modes 2 through 5 and modes 40 through 100 coincide in cases Π 1 and Π 4 .Thus, convergence of case Π 2 is approximately coincident with case Π 3 except at mode 1 and modes 3 through 20.The inset of figure 8(a) indicates that the first mode of case Π 4 and case Π 3 contain higher energy content than the first mode of case Π 1 and case Π 2 , respectively.The second mode of case Π 4 shows a greater decrease in energy content than case Π 1 .Accounting for the convergence profile of cases Π 1 and case Π 4 at mode 2. The energy content, ξ n , shows a trivial difference, O(10 −3 ), between the four cases after mode 10.For the downstream flow, case Π 4 converges faster than the other cases, thereafter it is ordered as Π 1 , Π 2 and Π 3 in succession.The oscillating behavior observed in the upstream flow, is noticed only between case Π 2 and Π 3 .Beyond the tenth mode, the difference in energy content between four cases is lessened.
The comparison between the upstream and downstream reveals that energy accumulates in fewer modes in the upstream of each case, e.g., case Π 1 requires 14 modes to obtain 50% of the total kinetic energy in upstream, whereas 26 modes are required to obtain the same percentage of energy downstream.A greater dissimilarity is observed between the convergence profile of case Π 1 and Π 4 at the downstream than the difference at the upstream.
The contrast between case Π 1 and Π 4 is larger than the discrepancy between case Π 2 and Π 3 especially at downstream.The disparity between the upstream and downstream windows can be identified in the most energetic mode that shows the maximum and minimum variations at case Π 4 and case Π 3 , respectively.This observation can be attributed to structure of the upstream flow of case Π 4 that is rather recovered, whereas the downstream show high deficit.However, the upstream and downstream of case Π 3 both show high velocity deficit, therefore the structure might be similar especially for large scale.For mode 2 through 10,  the biggest difference between the upstream and downstream is found in case Π 1 .
Figure 9 presents the first modes at the upstream and downstream of the four different cases.The four cases show that small gradients in the streamwise direction compared with high gradient in the wall-normal direction.Although the four cases show divergence between the eigenvalues of the first mode, the eigenfunctions display rather analogous structures.
The first POD mode shows variation of 1.25% when comparing between the upstream and downstream of case Π 1 .Less important variations of 0.68% and 0.32% are observed in cases Π 2 and Π 3 , respectively.Therefore, the structures of upstream and downstream of these cases are approximately equivalent.Upstream of case Π 3 looks like the opposite of its downstream.Similarity is observed between case Π 1 and Π 4 although the energy difference between them about 3%.Case Π 4 presents significant differences between the upstream and downstream mainly at y/D ≈ 1.5 and the region between the hub height and bottom tip.
Figure 10 presents the fifth mode at the upstream and downstream of the four cases that show a mixture of POD and Fourier (homogenous) modes in the streamwise direction.
Although the fifth mode of the four cases contain ≈ 74% less energy of than the first mode, large scales are still pronounced.Small scales also appeared in the upstream and the downstream windows of the four cases.Upstream windows of cases Π 1 , Π 2 , and Π 3 show the opposite structure of its own downstream windows.Interestingly, the upstream and downstream widows of case Π 3 look like the reduced scale of the it own first mode.The same trend is observed in the downstream window of case Π 4 .Power produced is measure directly using torque sensing system.The power curves exactly follow the trend of the velocity profiles.The maximum power extracted at angular velocity of 1500 ± 100 and it is harvested in case Π 1 .Small difference in harvested power is observed between cases Π 2 and Π 3 .The findings of this study have a number of the practical implications especially in the tight wind farm when the large areas are not available.A continue efforts are required to understand the impact of streamwise and spanwise spacing in infinity array flow with different stratification conditions.

FIG. 2 :
FIG. 2: Top view of 4 by 3 wind turbine array.The dash lines at the last row centerline turbine represent the measurement locations.

Figure 3 7 WindFIG. 3 :
Figure3demonstrates the power produced, F x , that is obtained directly via the torque sensing system, versus the angular velocity, ω, for all cases.It is apparent from this figure

Figure 4
Figure 4 presents the streamwise velocity in upstream and downstream of the cases Π 1 through Π 4 .The left and right contours of each case present the upstream and downstream flow, respectively.At upstream, case Π 1 attains the largest streamwise mean velocities

Figure 5 FIG. 5 :
Figure 5 contains the in-plane Reynolds shear stress − uv for the same cases as shown in figure 3.At upstream, cases Π 2 and Π 3 display higher stress compared with Π 1 and Π 4 .Although the spanwise spacing of case Π 3 is half of case Π 2 , no significant difference is apparent.The differences are quite revealing 0.5 ≤ y/D ≤ 1, where case Π 2 exhibits heightened magnitudes of − uv .At the downstream, comparison between the cases indicates that 9

Figure 6 1 10FIG. 6 :
Figure6displays the vertical flux of kinetic energy, − uv U .At upstream, small variations are shown between case Π 1 and Π 4 mainly above the top tip as a result to higher mean velocity of case Π 1 at this location.The maximum − uv U is found at case Π 2 and Π 3 .The variation between cases Π 2 and Π 3 shows that maximum negative flux is found at the regions between the hub height and bottom tip of case Π 2 ; higher positive flux is found above the top tip of the case Π 3 .At downstream, case Π 1 displays the same energy flux distribution of case Π 4 with significant differences at the regions x/D > 1.3, where case Π 1 [3].Spatial averaging makes it possible to compare 11 Wind Energ.Sci.Discuss., doi:10.5194/wes-2016-23,2016 Manuscript under review for journal Wind Energ.Sci.Published: 11 July 2016 c Author(s) 2016.CC-BY 3.0 License.

Figure 7 ( 1 through Π 4 .
Figure 7(b) contains the streamwise averaged Reynolds shear stress − uv for cases Π 1through Π 4 .Slight decreasing in − uv is attained in case Π 4 where the spanwise spacing is reduced.Reducing spanwise spacing shows an important influence when the streamwise spacing is x/D = 3.The noticeable discrepancies between case Π 2 and case Π 3 are found at the region below the hub height.Streamwise spacing differences play a more noteworthy role than variations in spanwise spacing.There are a significant variations between Reynolds shear stress of case Π 1 and case Π 2 .The same trend holds when comparing between case Π 3 and Π 4 .Interestingly, the largest difference between the Reynolds shear stress of cases is found between case Π 1 and case Π 2 , located at y/D ≈ 0.7 and y/D ≈ 1.4.Furthermore, all four cases have approximately zero Reynolds shear stress at the inflection point located at hub height.In addition, the most striking result to emerge from averaged profiles is that case Π 2 displays the maximum Reynolds stress and case Π 4 presents the minimum stress.

Figure 7 (FIG. 7 :
Figure 7(c) presents streamwise average profile of the vertical flux of kinetic energy.Below the hub height, the difference cases Π 1 and Π 4 is small.The variation begins above the hub height and increases with increasing wall-normal distance due to the variation of the streamwise velocity of these two case as shown in figure7(a).The significant variations

Figure 11 FIG. 11 :
Figure 11 presents the twentieth mode at the upstream and downstream of the four cases.Small structures become noticeable in both upstream and downstream windows.The upstream of cases Π 1 and Π 4 show large scale structure compared with the other two

WindFIG. 12 :
FIG. 12: Reconstruction Reynolds shear stress using: first mode (− − −), first 5 modes (− − −), first 10 modes (− − −), first 25 modes (− − −) and first 50 modes (− − −).Full data statistics (− − −).The insets show the reconstruction using modes 5-10, 5-25, and 5-50 (− − −).The first five modes display exactly the form of the full data profile in each case.Maximum difference between the successive reconstruction profiles displays between the first mode and first five profiles.Cases Π 1 and Π 2 show a moderate variation between the first five and first ten (green line) profiles.After the first ten profiles, the contribution in reconstruction is small as shown magenta and gray lines.Using more successive modes leads to more accurate reconstruction.Generally, the maximum difference between the full data profiles and the reconstructed profiles is located at y/D ≈ 0.75 and y/D ≈ 1.4 where the extrema in uv x are located.To quantify the contribution of the small scale structures, Reynolds shear stress is recon-

TABLE I :
Streamwise and spanwise spacing of the experimental tests.