^{1,a)}, Marc B. Parlange

^{1,b)}and Charles Meneveau

^{2,c)}

### Abstract

Wind harvesting is fast becoming an important alternative source of energy. As wind farms become larger, they begin to attain scales at which two-way interactions with the atmospheric boundary layer (ABL) must be taken into account. Several studies have shown that there is a quantifiable effect of wind farms on the local meteorology, mainly through changes in the land-atmosphere fluxes of heat and moisture. In particular, the observed trends suggest that wind farms increase fluxes at the surface and this could be due to increased turbulence in the wakes. Conversely, simulations and laboratory experiments show that underneath wind farms, the friction velocity is decreased due to extraction of momentum by the wind turbines, a factor that could decrease scalar fluxes at the surface. In order to study this issue in more detail, a suite of large eddy simulations of an infinite (fully developed) wind turbine array boundary layer, including scalar transport from the ground surface without stratification, is performed. Results show an overall increase in the scalar fluxes of about 10%–15% when wind turbines are present in the ABL, and that the increase does not strongly depend upon wind farm loading as described by the turbines’ thrust coefficient and the wind turbines spacings. A single-column analysis including scalar transport shows that the presence of wind farms can be expected to increase slightly the scalar transport from the bottom surface and that this slight increase is due to a delicate balance between two strong opposing trends.

M.C. and M.P. are supported by (Swiss) SNF 200021-107910/1 Land-atmosphere interaction over complex terrain: large eddy simulation and field experiments. We gratefully acknowledge the help of Chad Higgins (EPFL) with whom many discussions about the subject were held. M.P. and M.C. also thank the Swiss National Supercomputing Center (CSCS) for the computing time allocation and the ETH domain Competence Center for Environmental Sustainability. C.M. acknowledges funding from the National Science Foundation (Project AGS 1045189).

I. INTRODUCTION

II. LARGE EDDY SIMULATION METHODOLOGY AND CASES

A. Governing equations and LES code

B. Wind turbinemodel

C. Suite of LES cases

III. LARGE EDDY SIMULATION RESULTS

IV. SINGLE COLUMN MODELING OF THE WTABL

A. Double log-layer mean velocity distribution including wake layer

B. Scalar distribution in single column model of the WTABL

V. DISCUSSION: COMPARING LES WITH 1D WTABL MODEL

VI. CONCLUSIONS

### Key Topics

- Wind turbines
- 90.0
- Large eddy simulations
- 53.0
- Wind farms
- 40.0
- Friction
- 32.0
- Surface measurements
- 10.0

## Figures

(Color) Instantaneous contours of stream-wise velocity and scalar difference from LES of a fully developed wind turbine array boundary layer (case *F*); (a) stream-wise velocity on a *x–y* plane at a height *z* = *z _{h} * corresponding to hub-height (the wind turbine centers). (b) Normalized scalar difference distribution , at the same

*x–y*plane and the same time.

(Color) Instantaneous contours of stream-wise velocity and scalar difference from LES of a fully developed wind turbine array boundary layer (case *F*); (a) stream-wise velocity on a *x–y* plane at a height *z* = *z _{h} * corresponding to hub-height (the wind turbine centers). (b) Normalized scalar difference distribution , at the same

*x–y*plane and the same time.

Mean velocity profiles for wind farms with different loading parameters. The dotted straight line represents the theoretical logarithmic profile: , and the solid line with hollow circles shows the LES result for the case with no wind turbines. The two parallel dotted lines represent the lower and upper limit of the wind turbine rotor disk. These will be used for the remainder of the vertical profile plots.

Mean velocity profiles for wind farms with different loading parameters. The dotted straight line represents the theoretical logarithmic profile: , and the solid line with hollow circles shows the LES result for the case with no wind turbines. The two parallel dotted lines represent the lower and upper limit of the wind turbine rotor disk. These will be used for the remainder of the vertical profile plots.

Mean velocity profiles for wind farms with different loading parameters. The dotted straight line represents the theoretical logarithmic profile: , and the solid line with hollow circles shows the LES result for the case with no wind turbines.

Mean velocity profiles for wind farms with different loading parameters. The dotted straight line represents the theoretical logarithmic profile: , and the solid line with hollow circles shows the LES result for the case with no wind turbines.

Vertical profiles of the total shear stress for wind farms with different loading coefficients. (a) shows the profiles for cases A to G, where the thrust coefficient is changed. On the top, right corner, the lowest 15% of the bottom region of the shear stress profiles is magnified. (b) shows the same profiles but for cases E, E1-E3; where the spacing *s _{x} *,

*s*is changed.

_{y}Vertical profiles of the total shear stress for wind farms with different loading coefficients. (a) shows the profiles for cases A to G, where the thrust coefficient is changed. On the top, right corner, the lowest 15% of the bottom region of the shear stress profiles is magnified. (b) shows the same profiles but for cases E, E1-E3; where the spacing *s _{x} *,

*s*is changed.

_{y}Symbols: friction velocity ratios as function of loading coefficient. Lines: predictions of simple 1D model.^{5} (a) Ratio *u* _{*} _{ hi }/*u* _{*} _{ lo } (see Eq. (23)); (b) *u* _{*} _{ hi }/*U _{G} * (see Eq. ((7))) and (c)

*u*

_{*}

_{ lo }/

*U*(from (a) and (b)).

_{G}Symbols: friction velocity ratios as function of loading coefficient. Lines: predictions of simple 1D model.^{5} (a) Ratio *u* _{*} _{ hi }/*u* _{*} _{ lo } (see Eq. (23)); (b) *u* _{*} _{ hi }/*U _{G} * (see Eq. ((7))) and (c)

*u*

_{*}

_{ lo }/

*U*(from (a) and (b)).

_{G}Vertical profiles of the total passive scalar flux for wind farms with different loading coefficients. (a) shows the profiles for cases A to G, where only the thrust coefficient is changed. (b) shows the same profiles but for cases E and E1-E3, where the spacing *s _{x} *,

*s*is changed. On the top, right corner insert in both plots, the lower 20% of the domain height is shown to examine the scalar flux profiles in more detail.

_{y}Vertical profiles of the total passive scalar flux for wind farms with different loading coefficients. (a) shows the profiles for cases A to G, where only the thrust coefficient is changed. (b) shows the same profiles but for cases E and E1-E3, where the spacing *s _{x} *,

*s*is changed. On the top, right corner insert in both plots, the lower 20% of the domain height is shown to examine the scalar flux profiles in more detail.

_{y}Vertical profiles of the scalar difference between the surface and a given height , normalized by the scalar difference between the surface and the top of the domain, (*θ _{s} * −

*θ*) for the different study scenarios. (a) shows the profiles for cases A to G, where only the thrust coefficient is changed. (b) shows the same profiles for cases E and E1-E3; where the spacing

_{∞}*s*,

_{x}*s*is changed. The centered small inserts in both plots show the lower 15% of the domain height so it is possible to examine the profiles in more detail close to the ground.

_{y}Vertical profiles of the scalar difference between the surface and a given height , normalized by the scalar difference between the surface and the top of the domain, (*θ _{s} * −

*θ*) for the different study scenarios. (a) shows the profiles for cases A to G, where only the thrust coefficient is changed. (b) shows the same profiles for cases E and E1-E3; where the spacing

_{∞}*s*,

_{x}*s*is changed. The centered small inserts in both plots show the lower 15% of the domain height so it is possible to examine the profiles in more detail close to the ground.

_{y}Symbols: Ratio of scalar flux with and without wind turbines (evaluated at a height 1.5*z _{h} *), as function of wind farm loading parameter

*c*

_{ft}.

Symbols: Ratio of scalar flux with and without wind turbines (evaluated at a height 1.5*z _{h} *), as function of wind farm loading parameter

*c*

_{ft}.

Prandtl number ratio obtained from LES and empirical fits. (a) shows a sample vertical profile of the ratio of Prandtl numbers from LES case E, with *c* _{ft} = 0.0143. (b) shows the ratio as a function of *c* _{ft} for three different heights above the wind turbine region. The hollow squares are the LES values at *z*/*z _{h} * = 1.5. The dotted-dashed line represents an exponential fit to these values. Similarly, the hollow circles and the triangles represent the LES values at

*z*/

*z*= 2 and

_{h}*z*/

*z*= 3, respectively. The thick solid line and the dashed line are their corresponding exponential fits used in the model.

_{h}Prandtl number ratio obtained from LES and empirical fits. (a) shows a sample vertical profile of the ratio of Prandtl numbers from LES case E, with *c* _{ft} = 0.0143. (b) shows the ratio as a function of *c* _{ft} for three different heights above the wind turbine region. The hollow squares are the LES values at *z*/*z _{h} * = 1.5. The dotted-dashed line represents an exponential fit to these values. Similarly, the hollow circles and the triangles represent the LES values at

*z*/

*z*= 2 and

_{h}*z*/

*z*= 3, respectively. The thick solid line and the dashed line are their corresponding exponential fits used in the model.

_{h}Solid line: ratio of scalar fluxes obtained using Eq. (31), as a function of *c* _{ft}. Hollow triangles: ratio of scalar fluxes obtained from LES data. The ratio *u* _{*} _{ hi }/*u* _{*} (first factor in right-hand-side of Eq. (31)) is plotted in the same figure with dotted-dashed line, while the open circles show the LES values. Finally, the remaining factors on the right-hand-side of Eq. (31) are also plotted (dashed line for the model, and open squares for the LES results).

Solid line: ratio of scalar fluxes obtained using Eq. (31), as a function of *c* _{ft}. Hollow triangles: ratio of scalar fluxes obtained from LES data. The ratio *u* _{*} _{ hi }/*u* _{*} (first factor in right-hand-side of Eq. (31)) is plotted in the same figure with dotted-dashed line, while the open circles show the LES values. Finally, the remaining factors on the right-hand-side of Eq. (31) are also plotted (dashed line for the model, and open squares for the LES results).

Further detailed analysis of scalar flux ratios with and without wind turbines and comparison between single column model and LES results. The analytical results are presented as solid lines while LES results are shown as triangles. (a) shows the ratio of scalar fluxes; (b) shows the ratio of scalar fluxes divided by the ratio of Prandtl numbers; and (c) shows the ratio of scalar fluxes divided by the ratio of Prandtl numbers and also divided by the ratio *u* _{*} _{ hi }/*u* _{*} (corresponding to the last term in the right hand side of Eq. (31)).

Further detailed analysis of scalar flux ratios with and without wind turbines and comparison between single column model and LES results. The analytical results are presented as solid lines while LES results are shown as triangles. (a) shows the ratio of scalar fluxes; (b) shows the ratio of scalar fluxes divided by the ratio of Prandtl numbers; and (c) shows the ratio of scalar fluxes divided by the ratio of Prandtl numbers and also divided by the ratio *u* _{*} _{ hi }/*u* _{*} (corresponding to the last term in the right hand side of Eq. (31)).

Vertical profiles of the *φ _{θ} * function (a) and the scalar sub-grid coefficient (b).

Vertical profiles of the *φ _{θ} * function (a) and the scalar sub-grid coefficient (b).

Vertical profile of the subgrid turbulent Prandtl number. The solid line shows the case with no wind turbines. The dashed line shows the less loaded wind farm scenario (A; ), with increasing thrust coefficient and the further the lines are shifted towards the right.

Vertical profile of the subgrid turbulent Prandtl number. The solid line shows the case with no wind turbines. The dashed line shows the less loaded wind farm scenario (A; ), with increasing thrust coefficient and the further the lines are shifted towards the right.

## Tables

Table summarizing parameters of the various LES cases.

Table summarizing parameters of the various LES cases.

Table summarizing the measured effective surface roughness for the different study cases and the corresponding re-normalization factors *u* _{*} _{ hi }/*U _{G} *, computed using Eq. (7) for the fixed

*Ro*= 1000.

_{h}Table summarizing the measured effective surface roughness for the different study cases and the corresponding re-normalization factors *u* _{*} _{ hi }/*U _{G} *, computed using Eq. (7) for the fixed

*Ro*= 1000.

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