Multi-scale/fractal processes in the wake of a wind turbine array boundary layer

ABSTRACT

Multi-scale statistics are used to analyse the flow structure of wake flow in the boundary layer of a wind turbine array. Experimentally, a $3\times 3$ wind turbine array is tested with X-type hot-wire anemometry, providing a velocity signal at discrete locations downstream of the array along the centreline of the centre turbine. Based on the Markov property, the turbulent cascade can be taken as a stochastic process in scale, for which an underlying Fokker-Planck equation and its Kramers-Moyal coefficients are assigned. The first two terms of the Kramers-Moyal expansion (drift and diffusion coefficients) are estimated directly from the measured data by an optimisation procedure, which includes reconstruction of the joint probability density functions via short-time propagator. To quantify the accuracy of estimated the Fokker-Planck equation for describing the turbulent cascade process, the validity of a fundamental law of nonequilibrium thermodynamics named integral fluctuation theorem is verified. The results highlight that multi-scale analysis separates the stochastic cascade into universal and non-universal portions with respect to physical location downstream of the rotor. In addition, the Kramer-Moyal coefficients reveal the impact of a specific generation mechanism of turbulence and its large and small scale motions. Velocity-intermittency quadrant method is used to characterise the flow structure of the wake flow. Multifractal framework presents the intermittency as a pointwise Hölder exponent. The relationship between large and small scales in wake flow is considered by quantifying the impact of the small scales on the large scales in terms of the pointwise Hölder condition. A negative correlation between the velocity and the intermittency is shown at the hub height and bottom tip, whereas the top tip regions show a positive correlation. The second and fourth quadrants are dominant downstream from the rotor. The pointwise results reflect large-scale organisation of the flow and velocity-intermittency events corresponding to a foreshortened recirculation region near the hub height and the bottom tip. A linear regression approach based on the Gram-Charlier series expansion of the joint probability density function is used to model the contribution of the second and fourth quadrants arriving at an excellent agreement between the model and the experiment. The model shows the best fit with the correlation of 0.9864.

KEYWORDS:

• Atmospheric turbulence
• chaos and fractals
• small scale intermittency

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Naseem Ali http://orcid.org/0000-0002-8560-4406