Abstract
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.
Highlights
The maximum entropy principle is very useful, in determining blackbody radiation spectra [1], energy distribution in particles [2], and in specifying drop size distributions [3], as some examples.This principle states that the energy distribution of particles will tend toward the state of maximum entropy under the given constraints of the physical system
The total energy and the range of length scales that exist in the turbulent flow primarily depend on the Reynolds number
Kolmogorov’s k−5/3 scaling is plotted for comparison, and we can see that for large Reynolds numbers this scaling is tangent to the lognormal distribution in the so-called inertial subrange
Summary
The maximum entropy principle is very useful, in determining blackbody radiation spectra [1], energy distribution in particles [2], and in specifying drop size distributions [3], as some examples. The total energy and the range of length scales that exist in the turbulent flow primarily depend on the Reynolds number. The total turbulence kinetic energy contained in the energy spectrum will be specified by the initial mean velocity and length scale of the flow, in other words by the Reynolds number. The range of its applicability tends to be limited as tangent lines do not fully express the curved energy spectrum
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