On von Kármán spectrum from a view of fractal
Von Kármán originally deduced his spectrum of wind speed fluctuation based on the Stockes-Navier equation. That derivation, however, is insufficient to exhibit the fractal information of time series, such as wind velocity fluctuation. This paper gives a novel derivation of the von Kármán spectrum based on fractional Langevin equation, aiming at establishing the relationship between the conventional von Kármán spectrum and fractal dimension. Thus, the present results imply that a time series that follows the von Kármán spectrum can be taken as a specifically fractional Ornstein-Uhlenbeck process with the fractal dimension 5/3, providing a new view of the famous spectrum of von Kármán’s from the point of view of fractals. More importantly, that also implies a novel relationship between two famous spectra in fluid mechanics, namely, the Kolmogorov’s spectrum and the von Kármán’s. Consequently, the paper may yet be useful in practice, such as ocean engineering and shipbuilding.
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