Internal boundary layer growth over large wind farms under conventionally neutral conditions

In wind farms, where large numbers of wind turbines are placed relatively close to each other, the characteristics of the atmospheric boundary layer (ABL) are changed significantly. For instance, the flow in the turbine disk region slows down due to the turbine thrust force, causing an increased mean shear which in turn increases the turbulent kinetic energy production. In very large wind farms, the ABL adapts to the presence of the wind turbines, and a new equilibrium state is attained in which the turbulent dissipation and wind-farm power extraction are balanced by energy coming from large scale forces. This ABL state is called a wind-farm boundary layer, the dynamics of which are governed by complex interactions between individual wind turbine wakes. Numerical modelling of this wind-farm boundary layer therefore requires detailed spatial and temporal resolutions, which can be provided by large-eddy simulations (LES). In recent years, many studies have used LES to investigate both the fully developed wind-farm boundary layer as well as the transition from an undisturbed ABL.

In the current study, we use the LES technique to model wind-farm boundary layers developing in a conventionally neutral ABL. Under these conditions, the ABL consists of a well-mixed layer near the surface and a stably stratified free atmosphere. A thin inversion layer or capping inversion with very strong stability is found at the interface between the mixed layer and the free atmosphere, and the height of the inversion layer is correlated with the sharp temperature increase over this layer. Furthermore, Coriolis forces strongly affect the boundary layer flow and cause the wind direction to change with height. Given these atmospheric conditions, we investigate how wind farms are affected by the boundary layer height of the flow upstream. For this study, we performed a suite of LES cases with the in-house LES code SP-Wind, in which the ABL height is varied by imposing a capping inversion with varying inversion strength.

For fully developed wind-farm boundary layers, the average wind speed profile has been shown to contain two logarithmic layers under both truly neutral and conventionally neutral conditions [1,2]. This finding suggests that, from a larger point of view, the presence of a wind farm can be viewed as an increase in the surface roughness. Consequently, the development of a wind-farm boundary layer from an undisturbed ABL shows similarities with the flow structure after a surface roughness transition, where the sudden change of surface drag causes the growth of an internal boundary layer (IBL). The development of an IBL over large wind farms has been confirmed by a number of LES studies, and is also visible in our LES results. Moreover, we find that the IBL height closely follows a 0.8 slope, which is a well-known heuristic power law for IBLs after a surface roughness transition. However, this slope is only present when the capping inversion is situated high above the wind farm. For lower ABL heights, the capping inversion coincides with the IBL at some point above the farm, thereby reducing the IBL growth rate further downstream.

The LES results further indicate that the mean flow changes direction throughout the farm. This effect is related to the decrease in Coriolis forces due to the flow deceleration throughout the farm. For very low boundary layers, the relatively large reduction in wind velocity leads to considerable turbine wake deflection near the end of the farm. A detailed kinetic energy budget analysis shows that the turbulent stress adapts rapidly to the new equilibrium, whereas the mean flow evolves more slowly. In other words, the length scales related to the adaptation of the wind direction and the turbulent stress are very different, which was also found for surface roughness transition flows subject to Coriolis forces.

Similar to topographic flows, we find that wind farms generate internal gravity waves in the inversion layer and in the free atmosphere. The mechanism triggering the waves is the vertical displacement of the capping inversion due to the horizontal flow divergence in the farm. Although gravity waves cannot exist inside the neutral boundary layer, the pressure perturbations in the free atmosphere are imposed on the boundary layer, thereby allowing a feedback effect on the wind farm. Lowering the inflow height increases the capping inversion displacement, resulting in stronger wave perturbations and induced pressure gradients. Regarding wind-farm performance, decreasing the inversion-layer height from 1000 to 250 m yielded differences of up to 17 % in power deficit in downstream turbines.

The authors acknowledge support from the European Research Council (FP7--Ideas, grant no. 306471). The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government -- department EWI.

[1] Calaf M. et al., Phys. Fluids, 22, 015110 (2010) [2] Allaerts D. and Meyers J., Phys. Fluids, 27, 065108 (2015)