Velocity gradient distributions in fully developed turbulence: An experimental study

We report measurements of the probability distribution function of the velocity derivatives, and the corresponding hyperflatness factors, up to order 6, as a function of the microscale Reynolds number ${\mathrm{R}}_{\mathrm{\lambda }}.$ The measurements are performed in a flow produced between counter-rotating disks, using low-temperature helium gas as the working fluid, in a range of microscale Reynolds numbers lying between 150 and 2300. Consistently with previous studies, a transitional behavior is found around ${\mathrm{R}}_{\mathrm{\lambda }}\text{≈700.}$ We determine a simple scaling law, in terms of ${\mathrm{R}}_{\mathrm{\lambda }},$ which allows the collapse of the tails of the pdf of the velocity derivatives onto a single curve, below the transition. We find well-defined relative power laws for the hyperflatness factors $H_p$ and $H_p^{*},$ throughout the entire range of variation of ${\mathrm{R}}_{\mathrm{\lambda }}:$ $H_4{\text{=F=(0.99±0.05)H}}_6^{\text{0.376±0.015}}$ and $H_5^{*}{\text{=(0.95±0.05)H}}_6^{\text{0.67±0.022}}.$ These results are compared to those of previous investigators and to various theoretical approaches both statistical (multifractal model) and structural (i.e., based on a model of fine scales).