Energy spectra at low wavenumbers in homogeneous incompressible turbulence

The form E(k,t)≃CmkmE(k,t)≃Cmkm in the limit as k→0k→0 and where CmCm 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)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.

► 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.