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.
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 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.
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).
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).
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.
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.
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