Probability distribution functions for Navier–Stokes turbulence

Abstract

The probability distribution function (PDF), P(Δur), of a velocity difference, Δur, over a distance r in incompressible fluid turbulence is derived from the Navier–Stokes equations using the underlying functional probability distribution. Two different types of approximation are used to evaluate the resulting functional integral for P(Δur). The first is based on the saddle-point technique. It is used to examine the non-Gaussian features of P(Δur) and to demonstrate, in particular, that its tail has the characteristic exponential form associated with intermittency. The second approximation is developed for the purpose of deriving the anomalous scaling exponents of the structure functions from P(Δur). It represents P(Δur) as the integral with respect to the spatially averaged dissipation rate, εr, of the product of the PDF of Δur conditioned to a particular εr, and the PDF of εr. These are coarse-grained PDFs, obtained using the renormalization group. The former is approximately Gaussian, whereas the latter is given by a constrained Gaussian functional integral, which is evaluated approximately using a transformation that enables it to be represented in terms of a Poisson process. This approach yields the scaling exponents in the form discussed by She and Lévêque [Phys. Rev. Lett. 72, 336 (1994)], and also gives the complete third-order structure function exactly.