Analysis of the momentum availability factor in the two-scale momentum theory of wind farm flows

The two-scale momentum theory was developed by Nishino & Dunstan (2020, JFM) and describes the efficiency of a wind farm for a given set of input parameters, e.g. internal thrust coefficient, effective array density, surface friction coefficient, and the momentum availability factor (denoted M). A simple analytical model for M was derived by Kirby, Dunstan & Nishino (2023, JFM) through considering each component of M and applying various physical principles. While some of the assumptions were tested and the final model appears to perform well, the validity and generality of some other underlying assumptions remain uncertain.

In this work, we analyze large-eddy simulations (LESs) of wind farm flows by Lanzilao & Meyers (2025, JFM) to quantify the individual components of the momentum availability factor, M. With this dataset we can also exactly pinpoint which assumptions are reasonable and which need further development. It is found that the overall model works well with M being within +/- 10% for five of the six considered cases, but that the good agreement to some degree is due to error cancellation of various contributions. It is especially clear that the contribution from the turbulent diffusion (i.e. stress divergence) is overpredicted due to how the farm boundary-layer height is modelled, which however is compensated for by the underprediction of the advection and pressure-gradient terms.