Turbulence: Large-scale sweeping and the emergence of small-scale Kolmogorov spectra

The dynamics of fully developed hydrodynamic turbulence still is a basically unsolved theoretical problem, due to the strong-coupling long-range nonlinearities in the Navier-Stokes equations. The present analysis focuses on the small-scale fluctuations in a turbulent boundary layer with one external length scale yo. After taking a (2+1)D spatiotemporal spectral transform of the fluctuating vorticity fields, care is taken of large-scale sweeping which arises as a collective zero mode from the nonlinear flow terms. The "unswept" small-scale nonlinearities are then shown to be asymptotically locally isotropic (i.e., for wave numbers k→∞) by internal consistency, which allows to close the nonlinear hierarchy. The Navier-Stokes equations (without external forcing) are integrated to give the spectral response of the fluctuating small-scale velocity fields on the presence of a locally isotropic blob of turbulence while it is being swept around over an arbitrary steady state mean velocity profile, using viscous boundary conditions at y=0. Averaging the response spectrum over all possible orientational configurations and sweep velocities results in a novel self-consistency integral for the 4D energy spectrum function. The distribution of turbulence sweep velocities is modeled by means of Lévy-type densities, having an algebraic tail with power p>1. The generic case (which includes Von Kármán’s logarithmic mean velocity profile) is found to correspond to 1<p<3. Asymptotic analysis of the self-consistency integral leads to a differential equation which fixes the scaling exponent λ of the unswept frequency Δ and admits a nonempty, integrable and positive definite Airy-type frequency spectrum Eı(k,Δ/kλ)∼kμ with so-called "normal" Kolmogorov scaling, that is, μ=−7/3 and λ=2/3. Anomalous scaling is possible for one special mean profile.