Mean kinetic energy replenishment mechanisms in vertical-axis wind turbine farms

Vertical-axis wind turbines (VAWTs) are the subject of renewed interest due to the potential for higher power generation per unit land used, as well as their lower center of mass (the generator is at the bottom of tower), which renders them favorable for offshore deployment. However, VAWT farms have hardly been studied. In this paper, using a previously tested actuator line model in a large eddy simulation code, we investigate the transport of the mean kinetic energy (MKE) that replenishes the power in the farm. The primary sources of MKE are (1) the initial advective streamwise influx through the frontal area and (2) the vertical planform influx through the top and bottom interfaces of the farm. The results show that, for realistic finite-size farms, the planform MKE transport is a loss term over the first six rows: in this initial zone the mean flow adjusts by slowing down, and an upward mean advection develops that results in an efflux loss of MKE from the farm volume. The power extracted from farms is thus mainly from the frontal advection over the first few rows. When the initial streamwise advective flux is exhausted, the planform regeneration of MKE from above the wind farm becomes the dominant source; it is primarily affected by turbulent-mean interaction. This regeneration continues to adjust until rows 8 to 10 in our setups, beyond which a fully developed flow (similar to an infinite wind farm) can be observed. In the fully developed region, actual mechanical power generation by the turbines is about one third of replenishment. A primary conclusion is that more irregular farms designs should be studied, while the current literature continues to focus on the very classic layouts.

Figure 1

Streamwise velocity slices (normalized by the friction velocity at the ground surface $u_{*}$) in finite-fetch wind farms with inflow for $10D$ and $20D$ horizontal spacings, mean wind from left to right: (a) $10D$ Aligned, (b) $10D$ Staggered, (c) $20D$ Aligned, (d) $20D$ Staggered.

Figure 2

Streamwise velocity (normalized by the friction velocity at the ground surface $u_{*}$) in the longer wind farm with inflow and $20D$ cluster spacings, case 20-S-L, mean wind from left to right.

Figure 3

Schematic view of a finite length wind-farm control volume; fluxes are defined positive into the control volume. WTABL is the “wind turbine array boundary layer.” The control volume is overlaid on the plot shown in Fig. 4.

Figure 4

10-S case velocity streamlines overlaid on an MKE pseudocolor plot. (a) $x\text{−}y$ plane at midheight of turbines and (b) $x\text{−}z$ plane averaged in the $y$ direction. The location of the wind farm and the control volume are delineated by the black rectangles.

Figure 5

Vertical profiles of fluxes of MKE averaged horizontally over the whole wind-farm domains: positive fluxes are downwards, negative are upwards. The turbine layer is delineated by the dashed magenta lines. (a) Turbulence interaction transport, (b) mean and dispersive transport, (c) total transport, (d) vertical gradient of total transport that shows the net total replenishment at a given height. As noted before, $x$-averaging is only over the farm length.

Figure 6

Fluxes of MKE, averaged over the cross-stream $y$ direction at top of wind farm, as a function of the distance from the leading edge of the farm $x_f$.

Figure 7

Power replenishment from the top and bottom of wind farms for all staggered cases at a given streamwise point $x_f$. The streamwise distance is normalized by the distance between rows and as such is equal to the order of that row from the leading edge.

Figure 8

Row-averaged replenishment of available (a) and extracted (b) power for all staggered cases. Markers are at location of each row of clusters. The streamwise distance is normalized by the distance between rows and as such is equal to the order of that row from the leading edge. The frontal flux is not included in (a).

Figure 9

Cumulative (a) replenished available and (b) extracted power for all staggered cases. Markers are at the location of each row of clusters.

Figure 10

Power coefficient $C_P$ of each rows of clustered VAWT versus distance from leading edge, markers are at the location of each row of clusters.