Numerical investigation of the spatial integration effect on the velocity spectrum: Consequences in the wind or tidal turbine power spectrum

We numerically investigate the effects of spatially integrating turbulent velocity signals and their impact on the power spectrum. Possible applications concern turbine power spectra. To this end, a 2D synthetic database built with the aim to mimic a realistic Atmospheric turbulent Boundary Layer (ABL) is used to infer previous observations from laboratory experiments. The size of the computational domain is supposed to be smaller than the integral length scale of the flow. In this case, it is demonstrated that, when spatially averaged along N direction(s), an anisotropic turbulent flow initially satisfying the Kolmogorov’s “−5/3 law” exhibits a −5/3−2N/3 slope in the inertial range of its velocity spectrum. The synthetic database is further used to assess an equivalent turbine power spectrum computed from a 2D-spatial average (over the blade area) coupled with an average over the number of blades of the incoming velocity field. It is then confirmed that such an equivalent turbine power spectrum exhibits a power law decay exponent of −11/3 in the inertial range. Finally, several 1D databases are generated to study the spatial averaging procedure as a function of the number of points per integral length scale and the number of statistically independent velocity samples. We show that the additional −2/3 in the power law decay of the spatially averaged velocity spectrum is favorably observed in the case of a sufficient number of points (superior to 90) per integral length scale, regardless of the number of statistically independent velocity samples, even when the length of the domain is smaller than the integral length scale.