A Paradox concerning the Refined Similarity Hypothesis of Kolmogorov for Isotropic Turbulence

It is proved that the famous third-order structure function of the velocity in homogeneous isotropic turbulence derived by Kolmogorov implies the statistical interdependence of the difference and sum of the velocities at two points separated by a distance r. However, this is inconsistent with Kolmogorov's hypothesis regarding the difference between such velocities, made in 1962, in the sense that it implies that the statistics of the non-dimensional prefactor of the velocity difference are not universal but, instead, depend on the velocity sum, at least. The only resolution of this apparent paradox that can maintain the spirit of Kolmogorov's original hypothesis, which has been so important in the study of intermittency in turbulence, is given.