Lognormality in turbulence energy spectra

The maximum entropy principle states that the energy distribution will tend toward a state of maximum entropy under the physical constraints, such as the zero energy at the boundaries and a fixed total energy content. For the turbulence energy spectra, a distribution function that maximizes entropy with these physical constraints is a lognormal function due to its asymmetrical descent to zero energy at the boundary lengths scales. This distribution function agrees quite well with the experimental data over a wide range of energy and length scales. For turbulent flows, this approach is effective since the energy and length scales are determined primarily by the Reynolds number. The total turbulence kinetic energy will set the height of the distribution, while the ratio of length scales will determine the width. This makes it possible to reconstruct the power spectra using the Reynolds number as a parameter.