Comment on Energy spectra at low wavenumbers in homogeneous incompressible turbulence [Phys. Lett. A 375 (2011) 2850]

A full and explicit relationship is derived for homogeneous isotropic turbulence between the respective power expansions of the two-point second-order longitudinal correlation function of velocity f (r) at large separations r, and the kinetic energy spectrum E(kappa) at small wavenumbers kappa. The result contradicts a recent publication by Gustafsson and George [Phys. Lett. A 375 (2011) 2850] which claims to prove, on purely mathematical grounds, the oft-held conjecture that E(kappa) similar to kappa(mu) at small wavenumbers with only two allowed values mu = 2 or 4. Actually, such constraints could only come from supplementary physical properties or conditions. The relationship shows that any values below the threshold mu = 4 are acceptable (and could be produced in a relatively simple way) whereas values above the threshold further require the cancellation of the integral integral fr(4) dr. (C) 2013 Elsevier B.V. All rights reserved.