Bottleneck effects in turbulence: Scaling phenomena in r- versus p-space

Abstract

We (analytically) calculate the energy spectrum corresponding to various experimental and numerical turbulence data analyzed by Benzi et al.. We find two bottleneck phenomena: While the local scaling exponent $ζ_r(r)$ of the structure function decreases monotonically, the local scaling exponent $ζ_p(p)$ of the corresponding spectrum has a minimum of $ζ_p(p_{min})\approx 0.45$ at $p_{min}\approx (10 η)^{-1}$ and a maximum of $ζ_p(p_{max})\approx 0.77$ at $p_{max}\approx 8 L^{-1}$. A physical argument starting from the constant energy flux in p--space reveals the general mechanism underlying the energy pileups at both ends of the p--space scaling range. In the case studied here, they are induced by viscous dissipation and the reduced spectral strength on the scale of the system size, respectively.