^{1}, Hyung Suk Kang

^{2}, Charles Meneveau

^{3}and Raúl Bayoán Cal

^{1}

### Abstract

For large wind farms, kinetic energy must be entrained from the flow above the wind turbines to replenish wakes and enable power extraction in the array. Various statistical features of turbulence causing vertical entrainment of mean-flow kinetic energy are studied using hot-wire velocimetry data taken in a model wind farm in a scaled wind tunnel experiment. Conditional statistics and spectral decompositions are employed to characterize the most relevant turbulent flow structures and determine their length-scales. Sweep and ejection events are shown to be the largest contributors to the vertical kinetic energy flux, although their relative contribution depends upon the location in the wake. Sweeps are shown to be dominant in the region above the wind turbine array. A spectral analysis of the data shows that large scales of the flow, about the size of the rotor diameter in length or larger, dominate the vertical entrainment. The flow is less incoherent below the array, causing decreased vertical fluxes there. The results show that improving the rate of vertical kinetic energy entrainment into wind turbine arrays is a standing challenge and would require modifying the large-scale structures of the flow. Such an optimization would in the future aid recovery of the wind turbine wake towards conditions corresponding to the undisturbed atmospheric boundary layer.

This experiment was funded in part by the NSF (CBET-0730922, CBET-1133800, CBET-0953053). The wind turbine array experiment, during which these data were acquired, was performed in close collaboration with Dr. J. Lebrón-Bosques and Professor L. Castillo. Additional thanks are extended to Charles Gibson for the aid in developing the data analysis code and compiling the initial results. Their input is gratefully acknowledged.

I. INTRODUCTION

II. ANALYSIS METHODOLOGY

A. Conditional averaging

B. Spectral analysis

III. EXPERIMENTAL SETUP

IV. RESULTS

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Turbulent flows
- 48.0
- Wind energy
- 33.0
- Wind turbines
- 32.0
- Reynolds stress modeling
- 20.0
- Coherence
- 17.0

##### F03D

## Figures

Diagram showing four quadrants and their nomenclature used in quadrant analysis of turbulent shear flows. is the streamwise velocity fluctuation (deviation from local mean), while is the vertical velocity fluctuation (positive is the normal direction away from the surface).

Diagram showing four quadrants and their nomenclature used in quadrant analysis of turbulent shear flows. is the streamwise velocity fluctuation (deviation from local mean), while is the vertical velocity fluctuation (positive is the normal direction away from the surface).

Schematic diagram of conditional averaging domains used in “quadrant-hole” analysis of turbulent shear flows. (a) The traditional approach in which the threshold is identified at a constant level. (b) The “kinetic energy flux” quadrant hole analysis used here, with axes and so that lines of constant correspond to thresholds on .

Schematic diagram of conditional averaging domains used in “quadrant-hole” analysis of turbulent shear flows. (a) The traditional approach in which the threshold is identified at a constant level. (b) The “kinetic energy flux” quadrant hole analysis used here, with axes and so that lines of constant correspond to thresholds on .

Schematic of streamwise and transverse measurement locations behind the middle turbine of the last row in model wind farm. Each filled circle indicates a location where 21 vertical measurements (along *y*-direction) were made. The circle indicates the measurement location at *x/D* = 5, the profile location at which most of the present analysis is performed.

Schematic of streamwise and transverse measurement locations behind the middle turbine of the last row in model wind farm. Each filled circle indicates a location where 21 vertical measurements (along *y*-direction) were made. The circle indicates the measurement location at *x/D* = 5, the profile location at which most of the present analysis is performed.

Profiles of the average streamwise velocities at eight streamwise locations in the turbine wake. Horizontal lines indicate the rotor wake area at *y/D* = 0.5 and *y/D* = 1.5 (*D* = 0.12 m).

Profiles of the average streamwise velocities at eight streamwise locations in the turbine wake. Horizontal lines indicate the rotor wake area at *y/D* = 0.5 and *y/D* = 1.5 (*D* = 0.12 m).

Profiles of stream-wise velocity variance at eight locations in the turbine wake. Horizontal lines indicate the rotor wake area.

Profiles of stream-wise velocity variance at eight locations in the turbine wake. Horizontal lines indicate the rotor wake area.

Profiles of the Reynolds shear stress at eight streamwise locations in the turbine wake. Horizontal lines indicate the rotor wake area.

Profiles of the Reynolds shear stress at eight streamwise locations in the turbine wake. Horizontal lines indicate the rotor wake area.

Profiles of the Reynolds stress (solid line) and correlation coefficient (dot-dashed line) at the selected *x/D* = 5 downstream location used for additional analysis. Horizontal lines indicate the rotor wake area.

Profiles of the Reynolds stress (solid line) and correlation coefficient (dot-dashed line) at the selected *x/D* = 5 downstream location used for additional analysis. Horizontal lines indicate the rotor wake area.

Profiles of contributions, , from each quadrant to the total Reynolds stress profile. The hole-size threshold, *H*, here is zero and all realizations of the stress are included. The sum of all quadrants (solid line, ) is also shown. Summing the contributions of the Reynolds stress profiles from all 4 quadrants recovers the profile depicted in Figure 7 .

Profiles of contributions, , from each quadrant to the total Reynolds stress profile. The hole-size threshold, *H*, here is zero and all realizations of the stress are included. The sum of all quadrants (solid line, ) is also shown. Summing the contributions of the Reynolds stress profiles from all 4 quadrants recovers the profile depicted in Figure 7 .

Profiles of contributions, , from each quadrant to the flux of turbulent kinetic energy, again with a hole-size threshold, *H*, of zero. The sum of all quadrants (solid line, ) is also shown. Each profile above is the product of the mean streamwise velocity, , and the respective profile of shown in Figure 8 . The net power entrained into the wake at the measurement location is the difference between *F* at the top and bottom of the rotor wake area (at and shown as horizontal lines).

Profiles of contributions, , from each quadrant to the flux of turbulent kinetic energy, again with a hole-size threshold, *H*, of zero. The sum of all quadrants (solid line, ) is also shown. Each profile above is the product of the mean streamwise velocity, , and the respective profile of shown in Figure 8 . The net power entrained into the wake at the measurement location is the difference between *F* at the top and bottom of the rotor wake area (at and shown as horizontal lines).

Quadrant stress fractions, , at four vertical locations: proximate to the wall (*y/D* = 0.042, solid line and circles), bottom of the wake (*y/D* = 0.458, dashed line and circles), top of the wake (*y/D* = 1.458, dot-dashed line and circles), and the above wake (*y/D* = 1.71, dotted line and circles). Hole size *H* is given in .

Quadrant stress fractions, , at four vertical locations: proximate to the wall (*y/D* = 0.042, solid line and circles), bottom of the wake (*y/D* = 0.458, dashed line and circles), top of the wake (*y/D* = 1.458, dot-dashed line and circles), and the above wake (*y/D* = 1.71, dotted line and circles). Hole size *H* is given in .

Quadrant kinetic energy flux fractions, , at four vertical locations: proximate to the wall (*y/D* = 0.042), bottom of the wake (*y/D* = 0.458), top of the wake (*y/D* = 1.458), and above wake (*y/D* = 1.71). In this figure, the flux hole size is given in .

Quadrant kinetic energy flux fractions, , at four vertical locations: proximate to the wall (*y/D* = 0.042), bottom of the wake (*y/D* = 0.458), top of the wake (*y/D* = 1.458), and above wake (*y/D* = 1.71). In this figure, the flux hole size is given in .

Quadrant duration fractions, , at four vertical locations (legend is the same as in Figs. 10 and 11 ).

Quadrant duration fractions, , at four vertical locations (legend is the same as in Figs. 10 and 11 ).

Profiles of the fractional quadrant contributions to (surrogate) viscous dissipation. Taylor's hypothesis is used to approximate the spatial derivative using measured time derivatives of streamwise velocity.

Profiles of the fractional quadrant contributions to (surrogate) viscous dissipation. Taylor's hypothesis is used to approximate the spatial derivative using measured time derivatives of streamwise velocity.

Cospectra of and fluctuations at vertical locations corresponding to the bottom-most hotwire location (*y/D* = 0.042), the bottom of the turbine wake (*y/D* = 0.458), the top of the turbine wake (*y/D* = 1.458), and the topmost hotwire position (*y/D* = 1.708). The figure omits the highest wavenumber range for clarity. The classic power-law slope line is shown for reference.

Cospectra of and fluctuations at vertical locations corresponding to the bottom-most hotwire location (*y/D* = 0.042), the bottom of the turbine wake (*y/D* = 0.458), the top of the turbine wake (*y/D* = 1.458), and the topmost hotwire position (*y/D* = 1.708). The figure omits the highest wavenumber range for clarity. The classic power-law slope line is shown for reference.

Profile of the premultiplied cospectral density contributions to the flux of kinetic energy. The wavenumbers shown denote the center wavenumber of the corresponding wavenumber bins, over which the spectral density has been averaged. The bins have a width of around the indicated wavenumber location.

Profile of the premultiplied cospectral density contributions to the flux of kinetic energy. The wavenumbers shown denote the center wavenumber of the corresponding wavenumber bins, over which the spectral density has been averaged. The bins have a width of around the indicated wavenumber location.

Coherence spectra at the top and bottom wake locations. The line (Kolmogorov theory) is shown for reference.

Coherence spectra at the top and bottom wake locations. The line (Kolmogorov theory) is shown for reference.

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