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 Rλ = 852 and in a wind tunnel with Rλ = 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 〈(Δ ln ε)2〉 ∼ (r/r0)-β differenr from Kolmogorov's third hypothesis 〈(Δ ln ε)2〉 ∼ -μ ln(r/r0) restricted to the inertial range only (Δ(x) is x - 〈x〉). We develop also a formalism, based on an extremum principle, which is consistent with both the log-normality of ε 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/ln Rλ. © 1990.