^{1,a)}, Charles Meneveau

^{2,b)}and Johan Meyers

^{3,c)}

### Abstract

It is well known that when wind turbines are deployed in large arrays, their efficiency decreases due to complex interactions among themselves and with the atmospheric boundary layer (ABL). For wind farms whose length exceeds the height of the ABL by over an order of magnitude, a “fully developed” flow regime can be established. In this asymptotic regime, changes in the streamwise direction can be neglected and the relevant exchanges occur in the vertical direction. Such a fully developed wind-turbine array boundary layer (WTABL) has not been studied systematically before. A suite of large eddy simulations(LES), in which wind turbines are modeled using the classical “drag disk” concept, is performed for various wind-turbine arrangements, turbine loading factors, and surface roughness values. The results are used to quantify the vertical transport of momentum and kinetic energy across the boundary layer. It is shown that the vertical fluxes of kinetic energy are of the same order of magnitude as the power extracted by the forces modeling the wind turbines. In the fully developed WTABL, the kinetic energy extracted by the wind turbines is transported into the wind-turbine region by vertical fluxes associated with turbulence. The results are also used to develop improved models for effective roughness length scales experienced by the ABL. The effective roughness scale is often used to model wind-turbine arrays in simulations of atmospheric dynamics at larger (regional and global) scales. The results from the LES are compared to several existing models for effective roughness lengths. Based on the observed trends, a modified model is proposed, showing improvement in the predicted effective roughness length.

M.C. was supported by (Swiss) SNF 200021-107910/1 land-atmosphere interaction over complex terrain: large eddy simulation and field experiments. He also thanks the encouragement, advice, and support of his main advisor Professor Marc B. Parlange at EPFL. C.M. acknowledges partial funding from the National Science Foundation’s Energy for Sustainability Program (Project No. CBET 0730922). J.M. acknowledges funding from the Research Foundation-Flanders (FWO-Vlaanderen).

I. INTRODUCTION

II. HORIZONTALLY AVERAGED WTABL AND EFFECTIVE ROUGHNESS OF WIND FARMS

A. Horizontally averaged WTABL structure

B. Models for effective roughness

1. Lettau’s formula

2. Frandsen theory

III. LES METHODOLOGY AND CASES

A. Governing equations and LES codes

B. Wind-turbine model

C. Suite of LES cases

IV. VERTICAL PROFILES OF HORIZONTAL AVERAGES

V. MEASURED EFFECTIVE ROUGHNESS AND FRICTION VELOCITY

VI. DISCUSSION: COMPARISON WITH MODELS

VII. CONCLUSIONS

### Key Topics

- Wind turbines
- 64.0
- Large eddy simulations
- 55.0
- Wind energy
- 26.0
- Reynolds stress modeling
- 19.0
- Friction
- 16.0

## Figures

Instantaneous contours of streamwise velocity from LES of a fully developed WTABL (baseline case A2); (a) on a plane cutting through the middle of a column of wind turbines. (The location of the wind turbine disks are indicated with vertical black lines). (b) On a cross-stream plane at a distance downstream of a row of wind turbines. (c) On a plane at a height corresponding to hub height (the wind turbine centers).

Instantaneous contours of streamwise velocity from LES of a fully developed WTABL (baseline case A2); (a) on a plane cutting through the middle of a column of wind turbines. (The location of the wind turbine disks are indicated with vertical black lines). (b) On a cross-stream plane at a distance downstream of a row of wind turbines. (c) On a plane at a height corresponding to hub height (the wind turbine centers).

(a) Mean velocity profile of the baseline cases A1 (—○), A2 (—◻), and A3 (dash-dotted line), comparing the effects of slightly different numerical implementation of the LES, subgrid models, drag-disk force implementation, and domain and grid selections. The top dashed line is , the log-law behavior expected without wind turbine models; (●, ◼) cases A1 and A2 without turbine loads. (b) Mean velocity profile of cases A3 (dash-dotted line) and A4 (full line). The vertical dotted lines mark the bottom and top of the turbine-rotor planes.

(a) Mean velocity profile of the baseline cases A1 (—○), A2 (—◻), and A3 (dash-dotted line), comparing the effects of slightly different numerical implementation of the LES, subgrid models, drag-disk force implementation, and domain and grid selections. The top dashed line is , the log-law behavior expected without wind turbine models; (●, ◼) cases A1 and A2 without turbine loads. (b) Mean velocity profile of cases A3 (dash-dotted line) and A4 (full line). The vertical dotted lines mark the bottom and top of the turbine-rotor planes.

Vertical profiles of shear stresses of the baseline cases A1 and A2. The Reynolds shear stresses are indicated using dot-dashed line (A2) and asterisks (A1), while the dispersive stresses are denoted with dashed line (A2) and open diamonds (A1). Their respective sum is shown by a solid line (A2) and open circles (A1).

Vertical profiles of shear stresses of the baseline cases A1 and A2. The Reynolds shear stresses are indicated using dot-dashed line (A2) and asterisks (A1), while the dispersive stresses are denoted with dashed line (A2) and open diamonds (A1). Their respective sum is shown by a solid line (A2) and open circles (A1).

Vertical profiles of dissipation of mean kinetic energy or the baseline case A2 due to turbulent Reynolds stresses (dot-dashed line; also production of turbulent kinetic energy) and due to dispersive stresses (dashed line).

Vertical profiles of dissipation of mean kinetic energy or the baseline case A2 due to turbulent Reynolds stresses (dot-dashed line; also production of turbulent kinetic energy) and due to dispersive stresses (dashed line).

Vertical profiles of fluxes of kinetic energy for the baseline case A2 due to turbulent Reynolds stresses (dot-dashed line) and due to dispersive stresses (dashed line).

Vertical profiles of fluxes of kinetic energy for the baseline case A2 due to turbulent Reynolds stresses (dot-dashed line) and due to dispersive stresses (dashed line).

Extracted power density by four different turbines in simulation A2. In (a) four different WTs corresponding to the same row, but different columns: solid line, dashed line, dotted line and dot-dashed line. In (b) four different WTs all aligned in the same column, but from different rows: solid line, dashed line, dot-dashed line and dotted line.

Extracted power density by four different turbines in simulation A2. In (a) four different WTs corresponding to the same row, but different columns: solid line, dashed line, dotted line and dot-dashed line. In (b) four different WTs all aligned in the same column, but from different rows: solid line, dashed line, dot-dashed line and dotted line.

Mean velocity profiles for wind farms with different parameters. (a) Results for varying geometrical loading, with (case G), (case A3), (case H), and (case I). (b) Results with varying aspect ratios (case J), (case A3), and (case K). (c) Simulations results for different surface roughnesses (case D), (case A2), (case E), and (case F). In gray: log profiles; below the turbines (near the bottom): ; above the turbines: (values for and are listed in Table II). A detailed overview of the different cases is summarized in Table I.

Mean velocity profiles for wind farms with different parameters. (a) Results for varying geometrical loading, with (case G), (case A3), (case H), and (case I). (b) Results with varying aspect ratios (case J), (case A3), and (case K). (c) Simulations results for different surface roughnesses (case D), (case A2), (case E), and (case F). In gray: log profiles; below the turbines (near the bottom): ; above the turbines: (values for and are listed in Table II). A detailed overview of the different cases is summarized in Table I.

Profiles of the total shear stress (case F in solid line and case D in open circles), dispersive stress (dot-dashed line for case F and asterisks for D), and Reynolds stress (dashed line for case F and open diamonds for case D).

Profiles of the total shear stress (case F in solid line and case D in open circles), dispersive stress (dot-dashed line for case F and asterisks for D), and Reynolds stress (dashed line for case F and open diamonds for case D).

Comparison between roughness heights obtained from models and those arising from LES . The comparison includes the Lettau formula (open triangles), the original Frandsen formula (stars), and the proposed modified formula (open circles) [see Eq. (39)].

Comparison between roughness heights obtained from models and those arising from LES . The comparison includes the Lettau formula (open triangles), the original Frandsen formula (stars), and the proposed modified formula (open circles) [see Eq. (39)].

Wake eddy viscosity obtained from LES (○) using the measured mean velocity gradient at hub height, compared to the model of Eq. (40) (—), for different values—(○): variation in by varying the geometric loading (cases G, A3, H, and I); (●): variation in by varying (cases C, A2, and B).

Wake eddy viscosity obtained from LES (○) using the measured mean velocity gradient at hub height, compared to the model of Eq. (40) (—), for different values—(○): variation in by varying the geometric loading (cases G, A3, H, and I); (●): variation in by varying (cases C, A2, and B).

Evaluation of effective roughness height as a function of (with ) (a) and as a function of (with ) (b). Lines correspond to the modified formula for . Symbols correspond to LES evaluations. In (a)—(○): variation in by varying the geometric loading (cases G, A3, H, and I); (●): variation in by varying (cases C, A2, and B).

Evaluation of effective roughness height as a function of (with ) (a) and as a function of (with ) (b). Lines correspond to the modified formula for . Symbols correspond to LES evaluations. In (a)—(○): variation in by varying the geometric loading (cases G, A3, H, and I); (●): variation in by varying (cases C, A2, and B).

Comparison between bottom friction velocities obtained from simple models and those arising from LES . Stars: the original Frandsen formula. Open circles: the proposed modified Frandsen formula.

Comparison between bottom friction velocities obtained from simple models and those arising from LES . Stars: the original Frandsen formula. Open circles: the proposed modified Frandsen formula.

## Tables

Summarizing parameters of the various LES cases. Between brackets is indicated which code is used: “L” refers to the KULeuven code and “J” refers to the JHU-LES code.

Summarizing parameters of the various LES cases. Between brackets is indicated which code is used: “L” refers to the KULeuven code and “J” refers to the JHU-LES code.

Effective roughness heights and friction velocities as determined from LES for the different cases.

Effective roughness heights and friction velocities as determined from LES for the different cases.

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