A Maximum Principle for Determining the Intermittency Exponent of Fully Developed Steady Turbulence

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Abstract
The energy spectrum in the inertial range takes the form E(k) ∝ε2/3k-5/3(kl0)-B, where ε is the mean rate of energy transfer and l0 is the length scale of the production range. The β-model of intermittency presented by Frisch, Sulem and Nelkin (1978), leads to the Mandelbrot relation B=µ/3(3-D)/3, where D is a fractional dimension. A new statistical hypothesis on the energy cascade is introduced determine the exponent D. The vortex-stretching picture implies that vortices change into thinner, more extended ribbon-like structures, eventually producing a random spatial distribution of eddies of different sizes. With the aid of this picture, an information entropy of intermittency H(D) is defined and is assumed to take a maximum value in the steady state. This determines D. In fact, an explicit expression for H(D) is derived for the β-model and the maximum principle is shown to give µ=0.34, D=2.66 in agreement with the experiments µ=0.3∼0.5. The mean number of offspring for one cascade step turns out to be 6.32.

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