Finite Reynolds Number Effect on Small-Scale Statistics in Decaying Grid Turbulence

Since about 1997, the realisation that the finite Reynolds number (FRN) effect needs to be carefully taken into account when assessing the behaviour of small-scale statistics came to the fore. The FRN effect can be analysed either in the real domain or in the spectral domain via the scale-by-scale energy budget equation or the transport equation for the energy spectrum. This analysis indicates that the inertial range (IR) is established only when the Taylor microscale Reynolds number $R{e}_{\lambda }$ is infinitely large, thus raising doubts about published power-law exponents at finite values of $R{e}_{\lambda }$, for either the second-order velocity structure function $\overline{{\left(\delta u\right)}^2}$ or the energy spectrum. Here, we focus on the transport equation of $\overline{{\left(\delta u\right)}^2}$ in decaying grid turbulence, which represents a close approximation to homogeneous isotropic turbulence. The effect on the small-scales of the large-scale forcing term associated with the streamwise advection decreases as $R{e}_{\lambda }$ increases and finally disappears when $R{e}_{\lambda }$ is sufficiently large. An approach based on the dual scaling of $\overline{{\left(\delta u\right)}^2}$, i.e., a scaling based on the Kolmogorov scales (when the separation r is small) and another based on the integral scales (when r is large), yields $\overline{{\left(\delta u\right)}^2}\sim r^{2/3}$ when $R{e}_{\lambda }$ is infinitely large. This approach also yields $\overline{{\left(\delta u\right)}^n}\sim r^{n/3}$ when $R{e}_{\lambda }$ is infinitely large. These results seem to be supported by the trend, as $R{e}_{\lambda }$ increases, of available experimental data. Overall, the results for decaying grid turbulence strongly suggest that a tendency towards the predictions of K41 cannot be dismissed at least at Reynolds numbers which are currently beyond the reach of experiments and direct numerical simulations.