Turbulence models and probability distributions of dissipation and relevant quantities in isotropic turbulence

A generalized class of Cantor-set models are introduced in a deductive way to explain the intermittent and multifractal structure in the inertial range of isotropic turbulence. All such Cantor-set models are produced by taking the probability density of the stochastic ratio of averaged disspations in the basic breakdown process with scale ratio ${\mathit{A}}^{\mathrm{-}1}$, ${\mathrm{\varepsilon }}_{\mathit{r}}$/${\mathrm{\varepsilon }}_{\mathit{A}\mathit{r}}$, as a polynomial δ function, and it is proved that only some can guarantee the f-α spectra with a non-negative f. The whole statistics of the dissipation density, its square root, and the scaling index α which these models should imply are revealed, and typical models are compared with a recent direct numerical simulation to indicate promising applications.