Entrainment model for fully-developed wind farms: Effects of atmospheric stability and an ideal limit for wind farm performance

While a theoretical limit has long been established for the performance of a single turbine, no corresponding upper bound exists for the power output from a large wind farm, making it difficult to evaluate the available potential for further performance gains. Recent work involving vertical-axis turbines has achieved large increases in power density relative to traditional wind farms [J. O. Dabiri, J. Renewable Sustainable Energy 3, 043104 (2011)], thereby adding motivation to the search for an upper bound. Here, we build a model describing the essential features of a large array of turbines with arbitrary design and layout, by considering a fully-developed wind farm whose upper edge is bounded by a self-similar boundary layer. The exchanges between the wind farm, the overlaying boundary layer, and the outer flow are parametrized by means of the classical entrainment hypothesis. We obtain a concise expression for the wind farm's power density (corresponding to power output per unit planform area), as a function of three coefficients, which represent the array thrust and the turbulent exchanges at each of the two interfaces. Before seeking an upper bound on farm performance, we assess the performance of our simple model by comparing the predicted power density to field data, laboratory measurements, and large-eddy simulations for the fully-developed regions of wind farms, finding good agreement. Furthermore, we extend our model to include the effect of atmospheric stability on power output, by using a parametrization (which had been previously developed in the context of geophysical fluid dynamics) relating entrainment coefficients to local Froude numbers. Our predictions for power variation with atmospheric stability are in agreement with field measurements and large-eddy simulations. Finally, we consider an ideal limit for array operation, whereby turbines are designed to maximize momentum exchange with the overlying boundary layer. This enables us to obtain an upper bound for the performance of large wind farms, which we determine to be an order of magnitude larger than the output of contemporary turbine arrays.

Figure 1

(a) Aerial view of the Horns Rev wind farm; turbine trails are visualized by condensation (photograph by Christian Steiness, adapted from Ref. [9]). (b) Power produced by each turbine row, normalized by the power output from the first row, when the wind direction is aligned with the turbine grid [4].

Figure 2

Sketch of our entrainment model for a wind farm (the boundary layer is not to scale).

Figure 3

(a) Sketch of a spatially developing mixing layer, separating regions moving at velocities $U_b$ and $U_f$. The characteristic velocity of the mixing layer is $U_{\text{mix}}=\frac{1}{2}(U_b+U_f)$. Fluid from the top and bottom regions is entrained with velocities of magnitude $E(U_b-U_{\text{mix}})$ and $E(U_{\text{mix}}-U_f)$, respectively. (b) Schematic diagram of the flow past a sequence of porous obstacles. As each shear layer impinges on the following obstacle, its momentum is partitioned between the farm and boundary layer regions, and a new mixing layer starts.

Figure 4

Schematic illustration of blockage effects on the outer velocity $U_o$, in wind tunnel measurements and complete-farm simulations.

Figure 5

Displacement thickness growth rate $d{\delta }^{*}/dx$ as a function of the combined planform-averaged thrust and drag coefficients, that is, $c_{ft}^{\prime }+c_d^{\prime }$. The dashed line corresponds to our model, assuming $E=0.16$ and $C_M=0.04$. Symbols are laboratory experiments for model wind farms [47] as well as for model canopies [32].

Figure 6

Farm power coefficient vs planform-averaged thrust coefficient. Symbols correspond to sources reported in Table 1; sources of uncertainty for the field measurements are described in Table 2. The dashed line corresponds to our theory with $E=0.16$, $C_M=E/4=0.04$, and $c_d^{\prime }=0.008$, whereas the gray region corresponds to the effect of a change of $\pm 20\%$ in $E$ and $C_M$. (a) shows results for $0<c_{ft}^{\prime }<0.1$; available experimental, numerical, and field data are all within this range of $c_{ft}^{\prime }$. (b) illustrates the trend predicted for larger $0<c_{ft}^{\prime }<0.3$, which exhibits a maximum for $c_{ft}^{\prime }\simeq 0.179$.

Figure 7

Entrainment coefficient $E$ vs Froude number Fr, as Re is varied, adapted from Ref. [25], which compiles data for ocean overflows. A scale for the corresponding Richardson number Ri is also shown to ease comparison with atmospheric data. Symbols show experimental and field data. The colored lines show the parametrization proposed in Ref. [25], which yields $E\to 1$ at large Re and finite Fr. Since the available data do not seem to support $E$ of order 1, here we propose a slightly revised parametrization, which caps $E$ at 0.16. The difference between the original and revised parametrizations is shown by the red dashed and continuous lines. The two parametrizations are identical for Re less than about ${10}^4$.

Figure 8

Effect of atmospheric stability on power production, as expressed by farm power coefficient in a fully-developed wind farm, as a function of Obukhov length $L$ (normalized by wind farm height $h_f$). (a) Bars correspond to the Horns Rev wind farm [15], whereas the purple circle is from Ref. [17]. The values of $p$ show the reported probabilities, for westerly winds, of each stability class. The dashed line corresponds to our theory (with $c_{ft}^{\prime }=0.0291$ matching the estimated $c_{ft}^{\prime }$ for Horns Rev, and assuming $c_d^{\prime }\simeq 0.008$). Error bars as in Fig. 6. (b) Large-eddy simulation data. Since these span a wide range of $c_{ft}^{\prime }$ values, we ease comparison by normalizing each stable $c_{fp}$ using the values for unstable conditions. The dashed lines show our theoretical results, with $c_{ft}^{\prime }$ bracketing the values in the LES.

Figure 9

Farm power coefficient $c_{fp}$ in a fully-developed wind farm, maximized over $c_{ft}^{\prime }$, as a function of momentum transfer coefficient $C_M$. Symbols show available field data for existing wind farms (for which we assume $C_M=E/4\simeq 0.04$). The dashed lines corresponds to our theory (with $E=0.16$). The shaded region shows the effect of a $\pm 20\%$ change in $E$. Symbols as in Fig. 6 and Table 1; these symbols are displayed to show that LES, experiments, and field measurements are consistent with the upper bound proposed here.

Figure 10

Illustrations of turbines in dense layouts. (a) Lillgrund wind farm (comprising horizontal-axis wind turbines) [38]. The next sketches show hypothetical (b) aligned and (c) staggered configurations that would provide an order of magnitude increase in power density, provided that a nearly complete wake recovery were to be achieved between turbine pairs. These are only slightly more closely spaced than existing large wind farms. (d) shows a sketch of the recently developed FLOWE facility (consisting of pairs of vertical-axis turbines) [54].