The spatial organization of wind turbine wakes

TL;DR

This work investigates the spatial organization of wind turbine wakes using a localized multifractal framework applied to nacelle-mounted LiDAR scans. By extracting local roughness $c_1(g,L)$ and intermittency $c_2(g,L)$ across scales, the authors reveal four wake zones, a highly coherent mid-wake region, intermittency bubbles intruding into the wake interior, and an inverse-cascade-like boundary between wake and ambient flow. The approach provides a robust, data-efficient diagnostic that complements classical CWES metrics and demonstrates consistency with cup-anemometer observations under undisturbed inflow. The findings have practical implications for wake mitigation, turbine spacing, and wake-model validation in wind-farm design and control, and suggest real-time applicability through efficient LiDAR-based analysis.

Abstract

Wind turbine wakes play a central role in determining wind farm performance, yet their spatial organization remains only partially understood. Here, we apply a spatially localized multifractal analysis to quantify the strength of dependencies (local roughness) and extreme velocity fluctuations (local intermittency) in turbine wakes, and relate these properties to established metrics in wind energy research. Using two-dimensional nacelle-mounted LiDAR plan-position-indicator scans, we extract scale-invariant features that enable systematic comparisons across the wake without requiring time-resolved data. Designed to robustly handle irregular sampling, our analysis yields four main findings: i.) Four distinct wake zones are identified, each exhibiting unique patterns of roughness and intermittency. ii.) Coherent, strongly correlated patches emerge 2 to 5 rotor diameters D downstream, with intermittency strengthening periodically at multiple D positions and along the wake-free-flow interface. iii.) The classical "intermittency ring" is consequently redefined as a set of localized "intermittency bubbles", iv.) which interact dynamically with the ambient atmosphere through an inverse energy cascade, transferring energy from small to large scales. These findings, supported by concurrent cup anemometer observations under free-inflow conditions, demonstrate that local multifractal analysis provides a robust and cost-effective diagnostic framework for wake characterization and wake-model validation, with direct relevance for wind-farm design and control.

Figures (14)

• Figure 1: Top: Instantaneous results of the CWES metrics . Three important remarks apply here: i.) The velocity deficit $VD(g,L)$ shown in (a) was used to extract the wake centerline path using the computation method detailed in the Methods. ii.) Velocity deficit (a) and turbulence intensity $TI(g,L)$ (b) convey distinct physical information in theory, yet in practice the two fields are highly anti-correlated ($\rho = -0.78$) at resolution $L$. iii.) The wake zones derived from the LMF parameters (g,h), when projected onto the CWES maps (a,b), exhibit clear visual consistency with the underlying flow structure. Panels (c,d) show the short-term centerline statistics for the CWES metrics. Bottom: Instantaneous results of the LMF parameters . The estimates at each site $g$ for $c_1(g,L)$ (e) and $c_2(g,L)$ (f) exhibit distinct patterns from each other ($\rho = -0.01$) and from those of $VD(g,L)$ and $TI(g,L)$ in (a,b) - for all pairwise cross-correlations see ED Table 1. The corresponding centerline profiles (g,h) also differ markedly in their tendencies. Their landmarks are used jointly to interpret the wake structure and to delineate the distinct wake zones. The inflow wind speed at 00:07 is $6.19~\mathrm{m\,s^{-1}}$ (ED Fig. \ref{['fig:mfcup_48']}.a).
• Figure 1: Meteorological conditions on site . Wind direction (a) measured by the wind vane at 88 m height. The wind shear exponent (b) was computed using wind velocities at four different heights using the cup-anemometer signals (ED Fig.\ref{['fig:mfcup_48']}a). The dashed line indicates the timestamp of the scan analyzed in the LiDAR-results section in Fig. \ref{['fig:main_c1c2']}.
• Figure 2: Short-term results . The LMF wake centerline statistics (a,b) display key trends and landmarks that are used here to delineate the wake zones: $c_1(g, L) \nearrow .5$ (the first $0.5$ crossing), $c_1(g, L) \searrow .5$ the first $0.5$ crossing after $c_1(g, L)$ reaches its maximum $c_1(g, L)\,\max$, and the recurring negative peaks $c_2(g, L)\,\min$. Note that the landmarks shown here are computed as averages of the individual scan-wise $x/D$ landmark locations (see Fig.\ref{['fig:main_c1c2']}g and Fig.\ref{['fig:main_c1c2']}h), rather than being derived from the averaged centerline. For illustration, we also include $c_1(g,L){\max}^*$, defined as the median of these extracted locations, in contrast to the mean value used in all other analyses: $c_1(g,L){\max}$, which is also shown. Differences between the wake and periphery histograms of the LMF indicators (c,e), together with the lateral slices at five downstream positions (d,f), help to identify the dominant structures within the wake zones. The corresponding LMF fields are displayed in ED Fig. \ref{['fig:ext_c1c2_shortterm']}. The average inflow wind speed for the short-term window at hub height is $6.21~\mathrm{m\,s^{-1}}$ (ED Fig. \ref{['fig:mfcup_48']}a).
• Figure 2: Schematic overview of the local multifractal analysis methodology . In panel (a), colors denote the key components of the LMF methodology: grey - original, non-homogeneously sampled LiDAR data $\kappa_e$ at points $e(x_e, y_e)$; blue - length scales $r$ used for the wavelet decomposition centred on $e$; red - estimation sites $g(x_g, y_g)$; green - extent of the local neighbourhood $L$ within which computations are performed. The globally averaged scaling, $\overline{C}_q(g,r,L) = \sum_g C_q(g,r,L)$ for order $q = 1,2$, of $C_1(g,r,L)$ (b) and $C_2(g,r,L)$ (c) is used to determine the suitable value for the local environment $L$.
• Figure 3: Long-term results : Averaged results for $c_{1}(g,L)$ (a,c) and $c_{2}(g,L)$ (b,d) across 222 scans, corresponding to approximately two hours of data with an average inflow wind speed of $6.94~\mathrm{m,s^{-1}}$ (ED Fig. \ref{['fig:mfcup_48']}a). The global Pearson correlation between the two fields (a,b) is $\rho = -0.03$, indicating that they capture largely distinct structural properties. For comparison, the correlation between the long-term averaged CWES metrics in ED Fig. \ref{['fig:ext_TI_VD_long']} is $\rho = -0.77$. The centerlines in panels (a,b) are obtained by taking the median $y$-coordinate across all individually extracted centerlines. Panels (g,h) show the distribution of landmark points derived from the streamwise profiles in (c,d), all of which jointly contribute to characterizing the wake zones.