An alternative form of the super-Gaussian wind turbine wake model

Abstract

Abstract. A new analytical wind turbine wake model, based on a super-Gaussian shape function, is presented. The super-Gaussian function evolves from a nearly top-hat shape in the near wake to a Gaussian shape in the far wake, which is consistent with observations and measurements of wind turbine wakes. Using such a shape function allows the recovery of the mass and momentum conservation that is violated when applying a near-wake regularization function to the expression of the maximum velocity deficit of the Gaussian wake model. After a brief introduction of the theoretical aspects, an easy-to-implement model with a limited number of parameters is derived. The super-Gaussian model predictions are compared to wind tunnel measurements, full-scale measurements, and a large-eddy simulation (LES), showing a good agreement and an improvement compared with predictions based on the Gaussian model.

Theoretical modelling of the three-dimensional wake of vertical axis turbines

“…Further downstream these eventually converge towards a value of 2.4, i.e. / = 1.0, similarly to other HAT super-Gaussian models (Shapiro et al, 2019;Blondel & Cathelain, 2020). For a squareshaped rotor, i.e.…”

“…In the derivation of a super-Gaussian model, the wake is assumed to feature a velocity deficit (Δ ) distribution that evolves in the streamwise direction according to the local scale of velocity ( ), determined as the maximum normalised velocity deficit (Δ / 0 ); and spatial length scales ( ) and ( ) that define an three-dimensional super-Gaussian distribution according to exponents and , respectively. In the case of HATs (Blondel & Cathelain, 2020) or VATs, the wake starts with a rectangular (or nearly top-hat) shape immediately and evolves with an elliptical shape until attaining a Gaussian shape in the far wake ( = = 2) assuming it is self-similar (Bastankhah & Porté-Agel, 2014). Thus, and are deemed to vary in the streamwise direction following an exponential decay as…”

“…Recent wake models adopting super-Gaussian functions have been introduced for HATs, providing good results, especially in the near wake compared to standard Gaussian models. The former have an initial shape close to a top-hat distribution behind the turbine and evolve towards a nearly Gaussian shape further downstream (Shapiro et al, 2019;Blondel & Cathelain, 2020).…”

“…Current theoretical VAT wake models do not account for the uneven wake expansion over the horizontal and vertical planes (Figure 1), and represent its shape with standard Gaussian functions (Abkar & Dabiri, 2017;Abkar, 2019). To overcome these limitations, a super-Gaussian shape enables a more physically realistic description of the VAT wake velocity field compared to top-hat models that assume a uniform value across the wake or Gaussian models that fail to reproduce the elliptical shape (Blondel & Cathelain, 2020).…”

Analytical solution for the cumulative wake of wind turbines in wind farms

“…Shapiro et al. (2019 b ), Blondel & Cathelain (2020), Schreiber, Balbaa & Bottasso (2020), among others) can be consulted.…”