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

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.

💡 Research Summary

This paper presents a rigorous validation and extension of the analytic momentum‑availability (M) model that underpins the two‑scale momentum theory (TSMT) for wind‑farm flows. The authors use high‑resolution large‑eddy simulation (LES) data to directly compute M, thereby testing the assumptions of existing analytic formulations and proposing a more general model that accounts for the influence of the atmospheric boundary‑layer (ABL) Rossby number.

Background and Theory
TSMT separates wind‑farm dynamics into an internal (turbine/array) sub‑problem and an external (farm/ABL) sub‑problem. The external sub‑problem is characterized by the momentum‑availability factor M, defined as the ratio of total momentum flux into the farm region with turbines present to that without turbines. By applying a control‑volume analysis to the streamwise‑averaged momentum equation, M can be expressed as a sum of contributions from five physical mechanisms: advection (X_adv), pressure‑gradient forcing (X_PGF), Coriolis (X_Cor), turbulence (X_turb), and unsteady effects (X_uns).

Kirby et al. (2023) derived an analytic expression for each contribution (Eqs. 2.11‑2.15) and combined them using the “adv‑pressure (AP) approximation” to obtain a compact form for the advection‑plus‑pressure term ΔM_AP = (H_F L C_f0)(1 − β²). The turbulence term involves the ratio of shear stress at the top of the control volume (τ_t) to the surface shear stress (τ_w0). The final model (M_KDN1, Eq. 2.18) requires five inputs: β (farm‑scale wind‑speed reduction), H_F (effective farm height), L (farm length), C_f0 (surface friction coefficient), and τ_t/τ_w0. Simpler versions (M_KDN2, M_KDN3) introduce additional linearizations, while the most basic models are M_constant = 1 and the linear response model M_lin = 1 + ζ(1 − β).

LES Database and Test Cases
Six LES cases from the Lanzilao & Meyers database are examined. Three cases share the same turbine layout but differ in ABL height (H = 300, 500, 1000 m); three others keep H = 500 m while varying layout (aligned, double‑spacing, half‑farm). All simulations are performed in a conventionally neutral boundary layer (CNBL) with geostrophic wind G = 10 m s⁻¹, Coriolis parameter f_c = 1.14 × 10⁻⁴ s⁻¹, and roughness length z₀ = 10⁻⁴ m. Turbines are modeled as actuator disks (D = 198 m, hub height z_h = 119 m). Time‑averaged three‑dimensional fields of velocity, pressure, and Reynolds stresses are available, enabling a direct evaluation of M_exact via Eq. 2.8.

Model Validation

• M_constant fails dramatically, confirming that wind‑farm‑ABL feedback cannot be ignored.
• M_lin performs acceptably for low ABL (H = 300 m) but increasingly over‑predicts M for taller ABLs; the fixed response factor ζ ≈ 10 does not capture the reduced sensitivity at higher ABL heights.
• M_KDN1 yields the smallest errors (3‑8 %) across all cases, yet its turbulence term relies on a linear shear‑stress profile that becomes inaccurate when the ABL is deep.
• M_KDN2 and M_KDN3 reduce input requirements but exhibit error growth (up to ~12 %) when the ABL Rossby number Ro = U_g/(f_c H) exceeds ~0.2.
• The authors discover that the over‑prediction of the linear model correlates strongly with Ro, indicating that Coriolis effects, previously assumed negligible (ΔM_Cor = 0), actually influence momentum availability in tall ABLs.

Proposed Extension (M_BNK)
To remedy this, a new non‑linear shear‑stress correction is introduced:

 τ_t/τ_w0 = 1 − (H_F/h₀)·(1 + α·Ro)

where α is calibrated from LES (α ≈ 0.6). This formulation embeds the Rossby number directly into the turbulence contribution, effectively capturing the weakening of the linear‑shear assumption at high Ro. The resulting model, denoted M_BNK, is tested against the six LES cases. It reduces the mean absolute error to < 4 % and, notably for the H = 1000 m aligned case, improves accuracy by ~12 % relative to M_KDN1.

Physical Interpretation
In deep ABLs the Coriolis term, though small, alters the vertical shear distribution, leading to a non‑zero ΔM_Cor that the original analytic model neglects. Moreover, the turbulent shear stress profile deviates from linearity, especially near the top of the farm where entrainment is enhanced. By scaling the shear‑stress ratio with Ro, the new model captures both effects without requiring full 3‑D LES data.

Conclusions and Outlook
The study demonstrates that LES‑based direct evaluation of momentum availability can expose hidden deficiencies in analytic models. Incorporating the ABL Rossby number yields a more robust, generally applicable momentum‑availability formulation suitable for a wide range of atmospheric conditions, including high‑latitude offshore sites where Coriolis forces are strong. Future work should extend the framework to include unsteady contributions (ΔM_uns) and to test the model under unstable stratification, temperature inversions, and complex terrain.

Overall, the paper provides a solid validation of the two‑scale momentum theory, clarifies the limits of existing linearized models, and offers a practical, physics‑based extension that enhances predictive capability for large‑scale wind‑farm performance assessments.