Dynamical equations for high-order structure functions, and a comparison of a mean-field theory with experiments in three-dimensional turbulence.

Two recent papers [V. Yakhot, Phys. Rev. E 63, 026307, (2001) and R. J. Hill, J. Fluid Mech. 434, 379, (2001)] derive, through two different approaches that have the Navier-Stokes equations as the common starting point, a set of dynamic equations for structure functions of arbitrary order in turbulence. These equations are not closed. Yakhot proposed a "mean-field theory" to close the equations for locally isotropic turbulence, and obtained scaling exponents of structure functions and expressions for the peak in the probability density function of transverse velocity increments, and for its behavior for intermediate amplitudes. At high Reynolds numbers, some relevant experimental data on pressure gradient and dissipation terms are presented that are needed to provide closure, as well as on other aspects predicted by the theory. Comparison between the theory and the data shows varying levels of agreement, and reveals gaps inherent to the implementation of the theory.