Instantaneous turbulent kinetic energy modelling based on Lagrangian stochastic approach in CFD and application to wind energy

We present the construction of an original stochastic model for the instantaneous turbulent kinetic energy at a given point of a flow, and we validate estimator methods on this model with observational data examples. Motivated by the need for wind energy industry of acquiring relevant statistical information of air motion at a local place, we adopt the Lagrangian description of fluid flows to derive, from the $3$D+time equations of the physics, a $0$D+time-stochastic model for the time series of the instantaneous turbulent kinetic energy at a given position. Specifically, based on the Lagrangian stochastic description of a generic fluid-particles, we derive a family of mean-field dynamics featuring the square norm of the turbulent velocity. By approximating at equilibrium the characteristic nonlinear terms of the dynamics, we recover the so called Cox-Ingersoll-Ross process, which was previously suggested in the literature for modelling wind speed. We then propose a calibration procedure for the parameters employing both direct methods and Bayesian inference. In particular, we show the consistency of the estimators and validate the model through the quantification of uncertainty, with respect to the range of values given in the literature for some physical constants of turbulence modelling.