Interaction between low-level jets and wind farms in a stable atmospheric boundary layer

Low-level jets (LLJs) are the wind maxima in the lower regions of the atmosphere with a high wind energy potential. Here we use large-eddy simulations to study the effect of LLJ height on the flow dynamics in a wind farm with $10\times 4$ turbines. We change the LLJ height and atmospheric thermal stratification by varying the surface cooling rate. We find that the first row power production is higher in the presence of a LLJ compared to a neutral reference case without LLJ. Besides, we show that the first row power production increases with decreasing LLJ height. Due to the higher turbulence intensity, the wind turbine wakes recover faster in a neutral boundary layer than in a stably stratified one. However, for strong thermal stratification with a low-height LLJ, the wake recovery can be faster than for the neutral reference case as energy can be entrained from the LLJ. Flow visualizations reveal that under stable stratification the growth of wind farm's internal boundary layer is restricted and the wind flows around the wind farm. Wind farms extract energy from LLJs through wake meandering and turbulent entrainment depending on the LLJ height. Both effects are advantageous for wake recovery, which is beneficial for the performance of downwind turbines. This finding is confirmed by an energy budget analysis, which reveals a significant increase in the kinetic energy flux in the presence of a LLJ. The jet strength reduces as it passes through consecutive turbine rows. For strong stratification, the combined effect of buoyancy destruction and turbulence dissipation is larger than the turbulent entrainment. Therefore, the power production of turbines in the back of the wind farm is relatively low for strong atmospheric stratifications. We find that the pronounced wind veer in stably stratified boundary layers creates asymmetry in the available wind resource, which can only be studied in finite-size wind farm simulations. We emphasize that spanwise-infinite wind farm simulations may underpredict wind farm performance as the additional beneficial effect of LLJ cannot be observed.

Figure 1

Sketch of the essential flow phenomena in wind farms in a SBL, including wakes and their superposition, the entrainment of energy from above, and the development of the IBL. On the left, the typical temperature and velocity profiles, which reveal the LLJ and the top of the surface inversion, are sketched.

Figure 2

Schematic of the computational domain, showing the wind farm layout, the extent of Rayleigh damping layer, and the fringe layers. Black circles indicate the positions of the wind turbines. The statistics are sampled from the shaded region of size $70D\times 20D\times D$, which is centered around the wind farm.

Figure 3

(a) Horizontal velocity magnitude, (b) potential temperature, (c) wind angle, and (d) vertical momentum flux as a function of height for the different cases, see Table 1 for details. The inset in panel (d) shows the variation of the gradient Richardson number with height.

Figure 4

(a) Instantaneous velocity contours $u_{\text{mag}}/G$ for the SBL–3 case. Top panel: Side view of the wind farm in an $x-z$ plane through the middle of the third turbine column from the bottom. Middle panel: Velocity contours at hub height ($x-y$ plane). Bottom panel: Front view of the wind farm in an $y-z$ plane passing through the fifth row. (b) Streamwise variation of the jet strength. Dashed vertical lines represent the turbine positions.

Figure 5

(a) The development of the IBL height with streamwise distance. (b) The lines indicate the top of the surface inversion, and the lines with markers the IBL height as in panel (a). Note that for SBL–4 and SBL–5 the IBL grows above the surface inversion.

Figure 6

Streamlines at the hub height for the (a) SBL–1 and (b) SBL–5 cases. Note that for SBL–5 in which the IBL grows above the surface inversion, the streamlines indicate that there is very significant flow around the farm. (c) Pressure perturbation at the surface inversion top $z_c$ as a function of streamwise distance. The flow experiences maximum adverse pressure gradient for SBL–5. (d) The variation of pressure perturbation at hub height with the streamwise distance. In (c) and (d) $p_{\text{inlet}}^{*}$ is the pressure perturbation at the inlet.

Figure 7

Shaded area represents the control volume used in the budget analysis. The control volume for each column has a dimension of $7D\times 20D\times D$ and starts at a height of $z_h-D/2$.

Figure 8

Energy budget for cases (a) SBL–1, (b) SBL–2, (c) SBL–3, and (d) SBL–5. All the terms are normalized by the power production of the first turbine row. The symbols in the legend are defined in Eq. (13) and information about the cases can be found in Table 1.

Figure 9

Streamwise velocity contour for different cases. Note that the strength of the LLJ is negligible toward the rear for the wind farm for SBL–5.

Figure 10

(a) Turbulence destruction due to buoyancy and dissipation, $\mathbb{B}+\mathbb{D}$. (b) Flow work $\mathbb{F}$ represents the work done due to pressure drop.

Figure 11

(a) Power production normalized with the first-row average power of SBL–1. (b) Power production normalized with the power production of the first row.

Figure 12

(a) The development of the turbulent transport term ${\mathbb{T}}_t$, see Eq. (13), as function of the downwind position. (b) Visualization of the streamwise velocity development at the hub height normalized by the inlet velocity.

Figure 13

Power map for the case SBL–3. All the entries have been normalized by the power of the first column. Due to the wind veer, the first column produces more power compared to the other columns.

Figure 14

(a) Normalized horizontal velocity magnitude $u_{\mathrm{mag}}/G$ and (b) the energy flux $\overline{u}\overline{u^{\prime }{w}^{\prime }}$ in the $y-z$ plane, passing through the sixth turbine row. The black line represents the surface with zero spanwise velocity ($v=0$). Above the line, the flow goes to the right, and below the line, the flow is going to the left.