Velocity probability density functions of high Reynolds number turbulence

This paper deals with the probability density function (PDF) of velocity differences between two points separated by distance r. Measurements of PDFs were made, for r lying in the inertial range, for two different flows: in a jet with R_λ = 852 and in a wind tunnel with R_λ = 2720. These PDFs have a characteristic, non-Gaussian, shape with “exponential” tails. Following Kolmogorov's general ideas of log-normality, a new model for the PDF is developed which contains two parameters determined by experiments. This empirical model agrees with the experimental results that the tails of the PDF deviate from a truly exponential behaviour, in particular for small r. In addition, the model leads to the general scaling law <( Δlnɛ) ²> ∼ ( r/ r₀) ^{- β} differenr from Kolmogorov's third hypothesis <( Δlnɛ) ²> ∼ -μ ln( r/ r₀) restricted to the inertial range only ( Δ( x) isx - < x>). We develop also a formalism, based on an extremum principle, which is consistent with both the log-normality of ɛ and the above mentioned power law. In this formalism, β can be interpreted as the codimension of dissipative structures and asymptotically varies as β = β₁/ lnRλ.