Energy spectra at low wavenumbers in homogeneous incompressible turbulence

The form E(k,t)≅Cmkm in the limit as k→0 and where Cm is independent of k is examined under the assumption that the turbulence is homogeneous and the three-dimensional energy spectrum function is continuous. By using fractional derivatives together with the integrals that relate E(k,t) to moments of the two-point correlation functions, it is possible to show that m has to be an even integer. Thus fractional and odd powers are not possible in an infinite domain. -- Highlights: → The energy function spectrum is examined using fractional derivatives. → The energy function spectrum was found to be a power law with an even power close to zero. → Fractional powers are not possible in an infinite domain.