Validation and extension of an analytic momentum availability model for the two-scale momentum theory of wind farm flows

Abstract

A key parameter in the two-scale momentum theory of wind farm flows is the momentum availability, which quantifies the supply of momentum to a wind farm from various different momentum transport mechanisms (advection, pressure gradient, Coriolis, turbulence and unsteadiness). In this study, the contribution of each of these mechanisms to the momentum availability is evaluated directly from large-eddy simulation (LES) data in order to validate an analytic momentum availability model (Kirby, Dunstan, & Nishino, J. Fluid Mech., vol. 976, 2023, A24). Application of the model to six wind farm cases, three with different atmospheric boundary-layer (ABL) heights and three with different turbine layouts, shows that the full model performs well across all cases, but that its linearized version increasingly overpredicts the momentum availability for increasing ABL heights. It is found that the overprediction is related to the ABL Rossby number, and based on this observation, we propose an extension of the original linear model, which improves its accuracy for the considered cases and makes it more generally applicable, in particular to cases with tall ABL heights or strong Coriolis forcing.

Figures (21)

• Figure 1: Hierarchy of momentum availability models. We propose a new model, $M_{\rm BNK}$, in this paper, see § \ref{['sec:rossby']}. The arrows indicate assumptions to move from one model to another.
• Figure 2: Contours at hub-height of the time-averaged (a-f) streamwise velocity and (g-l) pressure perturbation (subtracted by inlet value). All cases have $N_t = 160$ turbines, except the double-spacing and half-farm cases, which have $N_t = 40$ and $N_t = 80$, respectively. The inter-spacing is $s_x/D = s_y/D = 5$, except in the double-spacing case. A staggered arrangement is used for all cases, except the aligned case. The farm width ($W \equiv N_y s_y$) is the same for all cases, while the farm length ($L \equiv N_x s_x$) is reduced for the half-farm case. This results in an array density of $\lambda = N_t A_d/S_{\rm cv} = 0.0314$ for all cases, except the double-spacing case, which has $\lambda = 0.00785$.
• Figure 3: Test of existing momentum availability models with LES data.
• Figure 4: Momentum sinks in the NDFM equation, i.e. left-hand side of (\ref{['eq:NDFM']}) and (\ref{['eq:NDFM_LHS']}). For completeness, the momentum sinks in the aligned, double-spacing and half-farm cases are also included, although these cases will not be further investigated in this section.
• Figure 5: (a) Local wind-speed reduction factor and (b) normalized perturbation pressure (subtracted by inlet value) spanwise-averaged over the width $y=[y_{\rm center}-W/2,y_{\rm center}+W/2]$ and height $z=[0,H_F]$ of the CV. The CV-region enclosing the wind-farm is marked with gray.