Inertial Range Scaling in Rotations of Long Rods in Turbulence

We derive a scaling relationship for the mean square rotation rate of rods with lengths in the inertial range in turbulence: $⟨{\stackrel{.}{p}}_i{\stackrel{.}{p}}_i⟩\propto l^{-4/3}$. We present experimental measurements of the rotational statistics of neutrally buoyant rods with lengths $2.8<l/\eta <72.9$, and find that the measurements approach the predicted scaling. The approach to inertial range scaling is shown to be more complex than anticipated with an overshoot and approach to the scaling from above. For all rod lengths, the correlation time of the Lagrangian autocorrelation of the rotation rate scales as the turnover time of the eddies of the size of the rod. Measuring rotational dynamics of single long rods provides a new way to access the spatial structure of the flow at different length scales.

Figure 1

The PDF of the rotation rate squared for different rod lengths. The lengths of rods are $l/\eta =2.8$ (green crosses), 4.9 (black filled circles), 8.5 (right triangles), 14.5 (open squares), 19.1 (red open circles), 32.8 (blue diamond), 42.2 (brown left triangles), and 72.9 (purple asterisks), and tracers from simulation (solid gray line). The results are reported from two experiments at $R_{\lambda }=150$ and 210. The simulations [18] are for tracer rods at $R_{\lambda }=180$.

Figure 2

Mean square rotation rate as a function of rod length. (a) Comparison of the experimental data at $R_{\lambda }=150$ (red open circles) and 210 (red diamonds) with the model of randomly oriented rods at $R_{\lambda }=150$ (purple up triangles) and $R_{\lambda }=210$ (black right triangles). (b) Comparison of the mean square rotation rate of rods (red open circles and diamonds) and the model for randomly oriented rods (up and right triangles) with the $l^{-4/3}$ inertial range scaling law (solid green line) and the simulation by Shin and Koch [17] (grey squares). (c) Comparison of the inertial range scaling law (green solid line) with the model for randomly oriented rods using Batchelor’s parametrization for very large Reynolds number (purple triangles).

Figure 3

Lagrangian autocorrelation of the rotation rate for different rod length. The lengths of rods are $l/\eta =2.8$ (green crosses), 4.9 (black filled circles), 8.5 (right triangles), 14.5 (open squares), 19.1 (red open circles), 32.8 (blue diamond), 42.2 (brown left triangles), and 72.9 (purple asterisk). (a) Time is normalized by the Kolmogorov time. (b) Time is normalized by the turnover time of eddies with sizes equal to the length of the rods, ${\tau }_{\mathit{l}}$. The symbols are displayed at every other data point.