Log-stable law of energy dissipation as a framework of turbulence intermittency

To describe the small-scale intermittency of turbulence, a self-similarity is assumed for the probability density function of a logarithm of the rate of energy dissipation smoothed over a length scale among those in the inertial range. The result is an extension of Kolmogorov's classical theory [A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 301 (1941)], i.e., a one-parameter framework where the logarithm obeys some stable distribution. Scaling laws are obtained for the dissipation rate and for the two-point velocity difference. They are consistent with theoretical constraints and with the observed scaling laws. Also discussed is the physics that determines the value of the parameter.

Figure 1

Schematic illustration of smoothing regions for $D=1$ in Eq. (3a) and for $D=3$ in Eq. (4a) on a two-dimensional cut of the space. The gray and dark gray areas denote regions of intense dissipation and of very intense dissipation.

Figure 2

Comparison of our framework with numerical simulations of forced steady states of homogeneous and isotropic turbulence at the Reynolds number for the Taylor microscale of 216 [18], 460 [19], and 600 [20]: (a) dissipation exponent ${\tau }_m$ for $D=3$ in Eq. (10b) and (b) velocity exponent ${\zeta }_m$ for $D=1$ in Eq. (13). The dotted lines denote the 1941 theory of Kolmogorov [1].