Universal Intermittent Properties of Particle Trajectories in Highly Turbulent Flows

We present a collection of eight data sets from state-of-the-art experiments and numerical simulations on turbulent velocity statistics along particle trajectories obtained in different flows with Reynolds numbers in the range $R_{\lambda }\in [120:740]$. Lagrangian structure functions from all data sets are found to collapse onto each other on a wide range of time lags, pointing towards the existence of a universal behavior, within present statistical convergence, and calling for a unified theoretical description. Parisi-Frisch multifractal theory, suitably extended to the dissipative scales and to the Lagrangian domain, is found to capture the intermittency of velocity statistics over the whole three decades of temporal scales investigated here.

Figure 1

Log-lin plot of the fourth order local exponent $\zeta (4,\tau )$ averaged over velocity components, as a function of the normalized time lag $\tau /{\tau }_{\eta }$. Data sets come from three experiments (EXP) (see Table 1) and five direct numerical simulations (DNS) (see Table 2). Error bars are estimated from the spread between the three components, except in EXP3 where only two components were measured. Each data set is plotted only in the time range where recognized experimental or numerical limitations are not affecting the results. In particular, for each data set, the largest time lag always satisfies $\tau <T_L$. The minimal time lag is set by the highest fully resolved frequency. The shaded area displays the prediction obtained by the MF model by using $D_{\mathrm{lo}}(h)$ or $D_{\mathrm{tr}}(h)$, with $\beta =4$, for a range of $R_{\lambda }\in [150:800]$, comparable with the range of $R_{\lambda }$ in the data. Notice that the MF predictions have been obtained by fixing equal to 7 the multiplicative constant in the definition of ${\tau }_{\eta }$. The straight dashed line corresponds to the dimensional nonintermittent value $\zeta (4,\tau )=2$, achieved at small time lags where structure functions do become differentiable. Notice that two among the DNS are sufficiently resolved to get the mentioned dimensional scaling in the high frequency limit.