Scaling of the maximum-entropy turbulence energy spectra

The log-normal type of turbulence energy spectral function, derived from the maximum entropy principle, is shown to be parameterizable in terms of root turbulence variables including the Reynolds number. The spectral function is first compared with a number of experimental data sets, showing a very close agreement across the entire energy and length (wavenumber) scales. The peak wavenumber (m) and the width parameter (C₂) prescribe the spectral location and broadening, respectively, when the Reynolds number increases, where C₂ has ∼1/Re^m dependence. The energy scale is adjusted with a multiplicative factor. In this perspective, the inertial scaling from k⁻³ to k^−5/3 when the flow transitions from two- to three-dimensions is explained as the increase in spectral width since the range of scales varies as Re^1∕2 and Re^3/4 for two and three-dimensional turbulence, respectively. Energy spectra at various locations in channel flows are also reproduced using the same function, indicating applicability wherever local equilibrium is achieved. Therefore, based on a small number of scaling parameters the full energy spectra can be prescribed using the maximum-entropy formalism.