A logarithmic scaling for structure functions, in the formS_p∼ [ln(r/ η)]^ζp, where η is the Kolmogorov dissipation scale and ζ_p are the scaling exponents, is suggested for the statistical description of the near-dissipation range for which classical power-law scaling does not apply. From experimental data at moderate Reynolds numbers, it is shown that the logarithmic scaling, deduced from general considerations for the near-dissipation range, covers almost the entire range of scales (about two decades) of structure functions, for both velocity and passive scalar fields. This new scaling requires two empirical constants, just as the classical scaling does, and can be considered the basis for extended self-similarity.

A tentative suggestion is made that the flatness of the velocity derivative could reach an infinite value at finite (though very large) Reynolds number, with possible implications for the singularities of the Navier—Stokes equations. A direct test of this suggestion requires measurements at Reynolds numbers presently outside the experimental capacity, so an alternative suggestion that can be tested at accessible Reynolds numbers is also made.

I summarize here the remarks made at the closing of the Conference and Research Workshop: Perspectives on Nonlinear Dynamics, held in Trieste in July 2007.