Contribution of large scale coherence to wind turbine power: A large eddy simulation study in periodic wind farms

Length scales of eddies involved in the power generation of infinite wind farms are studied by analyzing the spectra of the turbulent flux of mean kinetic energy (MKE) from large eddy simulations (LES). Large-scale structures with an order of magnitude bigger than the turbine rotor diameter $\left(D\right)$ are shown to have substantial contribution to wind power. Varying dynamics in the intermediate scales $\left(D–10D\right)$ are also observed from a parametric study involving interturbine distances and hub height of the turbines. Further insight about the eddies responsible for the power generation have been provided from the scaling analysis of two-dimensional premultiplied spectra of MKE flux. The LES code is developed in a high Reynolds number near-wall modeling framework, using an open-source spectral element code Nek5000, and the wind turbines have been modelled using a state-of-the-art actuator line model. The LES of infinite wind farms have been validated against the statistical results from the previous literature. The study is expected to improve our understanding of the complex multiscale dynamics in the domain of large wind farms and identify the length scales that contribute to the power. This information can be useful for design of wind farm layout and turbine placement that take advantage of the large-scale structures contributing to wind turbine power.

Figure 1

Actuator line forces obtained at the nodal points of the turbine blades. $V_{\text{rel}}$ is obtained from the velocity triangle. $F_t$ and $F_{\theta }$ are the axial thrust and rotational forces on the turbine blades due to aerodynamic lift $\left(L\right)$ and drag $\left(D\right)$ forces.

Figure 2

Computational domain showing the $8\times 6$ periodic arrangement of wind turbines for the baseline Case I. Hub height $z_h$, and rotor diameter $D$ set to $0.1H$. The dashed white arrows indicate the direction of mean wind flow.

Figure 3

(a) Mean streamwise velocity $\langle \overline{u}\rangle$ normalized by $u_{*,hi}$ vs. $z/H$. (b) Total (Reynolds + dispersive) stresses ${\tau }_{xz}$ normalized by $u_{*,hi}$ vs. $z/H$. Solid black—Case I, dashed black—Case IIa, dashed gray—Case IIb, solid gray—Case III. Solid gray $\square$ in (a)—neutral ABL; open $\circ$ in (a), (b)—Calaf et al. (2010) [9] (same wind farm setup parameters as in Case I). Black, gray dashed-dotted vertical (a) and horizontal (b) lines: rotor swept area of Cases I–III.

Figure 4

(a) Vertical variation of the MKE flux for different cases. Solid black—Case I, dashed black—Case IIa, dashed gray—Case IIb, solid gray—Case III. Gray square—neutral ABL. Open $\circ$—Calaf et al. (2010) [9] (same wind farm setup parameters as in Case I). Dashed dotted lines: rotor swept area of Cases I–III. (b) Correlation between the difference in MKE flux $\mathrm{\Delta }{\mathrm{\Phi }}_p$ and mean power density ${\rho }_{\text{mean}}$ for different cases.

Figure 5

Temporal variation of the array-averaged power $P\left(t\right)=\frac{1}{N_t}{\sum }_{i=1}^{N_t}{\overline{P}}_i$ of the turbines in different layouts. Solid black—Case I, dashed black—Case IIa, dashed gray—Case IIb and solid gray—Case III. Time scale is normalized with $H/U_{\infty }$. Mean power gain: IIa/I—$5\%$, IIb/I—$2\%$, III/I—$40\%$.

Figure 6

Premultiplied 1D spectra $k_x\xi$ vs. normalized streamwise wavelengths ${\lambda }_x/D$ for (a) $z=0.25D$, (b) $z=8.75D$. Black lines, Case I; gray lines, neutral ABL. $k_x{E}_{uu}$ (–), $k_x{E}_{ww}\left(--\right)$, $k_x{\varphi }_{uw}\left(-.-\right)$. All spectra normalized by $U_{\infty }^2$. Gray patch corresponds to the resolved part of the spectra directly influenced by SGS viscosity.

Figure 7

2D premultiplied energy spectra vs. normalized streamwise and spanwise wavelengths, ${\lambda }_x/D,{\lambda }_y/D$. Black lines, Case I; gray lines, neutral ABL. $k_x{k}_y{E}_{uu}$ (–), $k_x{k}_y{E}_{ww}\left(--\right)$, $k_x{k}_y{\varphi }_{uw}\left(-.-\right)$. (a) $z=0.25D$ (b) $z=D$ (c) $z=8.75D$. Contours are 0.25 of maximum at that level. All spectra normalized by $U_{\infty }^2$.

Figure 8

One-dimensional normalized premultiplied difference in the MKE flux spectra $k_{\eta }\mathrm{\Delta }{\widehat{\mathrm{\Phi }}}_p\left({\lambda }_{\eta }\right)$ [Eq. (12)]between the top and bottom wake region, $z=z_h\pm D/2$: (a) versus streamwise wavelength, $\eta =x$; (b) versus spanwise wavelength, $\eta =y$. Solid black—Case I; dashed black—Case IIa, dashed gray—Case IIb; solid gray—Case III. $A=s_x{s}_y{D}^2$. Gray patch—resolved part of the spectra directly influenced by SGS viscosity.

Figure 9

One-dimensional premultiplied vertical velocity spectra $k_x{E}_{ww}$ at $z=z_h+D/2$ vs. normalized streamwise wavelength ${\lambda }_x/D$ with and without wind turbines. Solid black—Case I; dashed black—Case IIa, dashed gray—Case IIb; solid gray—Case III. Solid light gray—neutral ABL, rotor region of I. Dashed light gray—neutral ABL, rotor region of III. Gray patch—resolved part of the spectra directly influenced by SGS viscosity.

Figure 10

One-dimensional premultiplied MKE flux spectra $k_x{\widehat{\mathrm{\Phi }}}_p$ versus normalized streamwise wavelength: (a) at the top wake region $z=z_h+D/2$; (b) at the bottom wake region $z=z_h-D/2$. Solid black—Case I; dashed black—Case IIa, dashed gray—Case IIb; solid gray—Case III.

Figure 11

Modulation of large-scale structures. Isosurface of normalized velocity magnitude $\sqrt{u^2+v^2+w^2}/U_{\infty }$ for $z\le z_h+0.75D$. (a) Wind turbine array, Case I. (b) neutral ABL, without wind turbines. Light gray patch—$0.65U_{\infty }$, dark gray patch—$0.95U_{\infty }$.

Figure 12

Vertical variation of (a) integral length scales ${\mathfrak{L}}_{uu}$ based on streamwise fluctuations and (b) normalized mean-squared streamwise fluctuations $\overline{u^{\prime 2}}/U_{\infty }^2$ (averaged over $xy$ planes), with and without the presence of wind turbines. Solid black—Case I; dashed black—Case IIa; dashed gray—Case IIb; solid gray—Case III; light gray $\circ$—neutral ABL, without wind turbines. Dotted lines—hub heights, $z_h=0.1H$ (Cases I, IIa, and IIb), $z_h=0.33H$ (Case III). See Appendix pp3 for the definition of ${\mathfrak{L}}_{uu}$ used in the current study.

Figure 13

Cumulative spectral content of the difference in MKE flux, ${\gamma }_{\eta }\left({\lambda }_{\eta }\right)$ [Eq. (13)] at different (a) streamwise wavelengths, $\eta =x$, (b) spanwise wavelengths, $\eta =y$. Solid black—Case I; dashed black—Case IIa; dashed gray—Case IIb; solid gray—Case III.

Figure 14

2D premultiplied spectra of the difference in MKE flux, $k_x{k}_y\mathrm{\Delta }{\widehat{\mathrm{\Phi }}}_p({\lambda }_x,{\lambda }_y)/U_{\infty }^3$, versus normalized streamwise and spanwise wavelengths. (a) Case I, (b) Case IIa, (c) Case IIb, (d) Case III. Solid gray—positive contours, dashed gray—negative contours. Chain dotted gray—${\lambda }_y\sim {\lambda }_x^{1/3}$, chain dotted black—${\lambda }_y\sim {\lambda }_x^{1/2}$. Gray dotted—${\lambda }_y\sim {\lambda }_x^{1/5}$. Contour levels: positive—10 levels, from 10% to 100% of maximum; negative—10 levels from 10% to 100% of minimum.

Figure 15

2D premultiplied spectra of MKE flux for baseline Case I at locations $z=z_h\pm D/2$. Solid—$z=z_h-D/2$, dashed—$z=z_h+D/2$. Black—$25\%$ of the maximum contour. Gray—$12.5\%$ of the maximum contour. Chain dotted black line—${\lambda }_y\sim {\lambda }_x^{1/3}$

Figure 16

Snapshot of normalized velocity magnitude $\sqrt{u^2+v^2+w^2}/U_{\infty }$ contour for Case I, in $xy$ plane at hub-height location $z=z_h$, taken at 15 flow-through times $T_e$ after obtaining statistical stationarity.

Figure 17

Schematic of grid refinements of Case I compared to neutral ABL in the region near the WT rotors in (a) streamwise, (b) spanwise, and (c) vertical direction. Location of a hypothetical turbine (in gray) shown in the neutral ABL grid. $D=0.1H$ is the rotor diameter. Thick dashed black line—location of the bottom rough wall surface.

Figure 18

Streamwise 2D correlation coefficient ${\rho }_{u^{\prime }{u}^{\prime }}(\mathit{r};z=\xi )$ for flows with and without wind turbines. Panels (a)–(e) correspond to the Cases neutral ABL at $\xi =D$ and Cases I, IIa, IIb, and III at $\xi =z_h$ (hub-height), respectively.

Figure 19

Streamwise 2D correlation coefficient ${\rho }_{u^{\prime }{u}^{\prime }}(\mathit{r};z=8.75D)$ for flows with and without wind turbines. Panels (a)–(e) correspond to the Cases neutral ABL, I, IIa, IIb, and III, respectively.