Breakdown of the velocity and turbulence in the wake of a wind turbine - Part 2: Analytical modelling

This work aims to develop an analytical model for the streamwise velocity and turbulence in the wake of a wind turbine where the expansion and the meandering are taken into account independently. The velocity and turbulence breakdown equations presented in the companion paper are simplified and resolved analytically, using shape functions chosen in the moving frame of reference. This methodology allows us to propose a physically based model for the added turbulence and thus to have a better interpretation of the physical phenomena at stake, in particular when it comes to wakes in a non-neutral atmosphere. Five input parameters are used: the widths (in vertical and horizontal directions) of the non-meandering wake, the standard deviation of wake meandering (in both directions) and a modified mixing length. Two calibrations for these parameters are proposed: one if the users have access to velocity time series and the other if they do not. The results are tested on a neutral and an unstable large-eddy simulation (LES) that were both computed with Meso-NH. The model shows good results for the streamwise velocity in both directions and can accurately predict modifications due to atmospheric instability. For the axial turbulence, the model misses the maximum turbulence at the top tip in the neutral case, and the proposed calibrations lead to an overestimation in the unstable case. However, the model shows encouraging behaviour as it can predict a modification of the shape function (from bimodal to unimodal) as instability and thus meandering increases.