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

In order to better integrate the underlying meteorological processes with the developing technologies within wind energy industry, acquiring relevant statistical information of air motion at a local place, and quantifying the subsequent uncertainty of involved parameters in the models, are fundamental tasks. Special emphasis should be made on the growing interest in energy production forecasting and modelling for wind energy developments that rises the issue of accounting for the uncertain nature of the local forecast. Taking this into consideration, 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 from annual historic of a 10 Hz anemometer wind measurements.
More precisely, starting from the viewpoint of Lagrangian modelling of the wind in the boundary layer, we establish a mathematical link between 3D+time computational fluid dynamics (CDF) models for turbulent near-wall flows and stochastic time series models by deriving a family of mean-field dynamics featuring the square norm of the turbulent velocity. Then, by approximating at equilibrium the characteristic nonlinear terms of the dynamics, we recover the so called Cox-Ingersoll-Ross stochastic model, which was previously suggested in the literature for modelling wind speed. Remarkably, our stochastic model for the instantaneous turbulent kinetic energy is parametrised by physical constants in CFD, which provides a more direct link between the stochastic nature of the underlying processes and the classical physics behind these phenomena. Nevertheless, these physical parameters may vary with the flow characteristics and situations, so we consider it relevant to adjust their values while constructing the forecasts. Such tuning of the physical parameters was previously proposed in the literature from a deterministic modelling context with RANS equations. We then propose a two-step procedure for the calibration of the parameters: a training stage where we construct a priori distribution for the parameter vector using direct methods and wind measurements, and a stage of refinement of the uncertainty distribution using Bayesian inference combined with Markov Chain Monte Carlo sample techniques. In particular, we show the accuracy of the calibration method and the performance of the calibrated model in predicting the wind distribution through the quantification of uncertainty.