Curvature of Lagrangian Trajectories in Turbulence

We report measurements of the curvature of Lagrangian trajectories in an intensely turbulent laboratory water flow measured with a high-speed particle-tracking system. The probability density function (PDF) of the instantaneous curvature is shown to have robust power-law tails. We propose a model for the instantaneous curvature PDF, assuming that the acceleration and velocity are uncorrelated Gaussian random variables, and show that our model reproduces the tails of our measured PDFs. We also predict the scaling of the most probable vorticity magnitude in turbulence, assuming Heisenberg-Yaglom scaling. Finally, we average the curvature along trajectories and show that, by removing the effects of large-scale flow reversals, the filtered curvature reveals the turbulent features.

Figure 1

PDFs of the curvature $\kappa$ nondimensionalized by $\eta$ for eight different Reynolds numbers, ranging from $R_{\lambda }=200$ (top curve) to $R_{\lambda }=815$ (bottom curve). The PDFs have been shifted vertically for clarity. The solid lines are reference ${\kappa }^1$ and ${\kappa }^{-5/2}$ power laws.

Figure 2

Joint PDFs of the curvature, nondimensionalized by $\eta$, and (a) the velocity magnitude and (c) normal acceleration magnitude, normalized by their standard deviations, at $R_{\lambda }=815$. (b) and (d) show the PDF quotients [Eq. (1)] for the same quantities. The behavior of these PDFs at other Reynolds numbers is similar.

Figure 3

The curvature PDFs now plotted as a function of $\kappa \eta R_{\lambda }$, including corrections (of less than 30%) to the acceleration variance, and compared with Eq. (2) (solid line). The inset shows the data plotted on logarithmic axes; our theory clearly captures the power-law tails of the PDF. In the main panel, the data are plotted on linear axes. The data collapse for the different Reynolds numbers, but the peak of the model PDF is different from that of the experimental PDFs.

Figure 4

PDFs of the logarithmically filtered curvature at $R_{\lambda }=815$ for $\Delta =0$ (no filtering), ${\tau }_{\eta }$, $5{\tau }_{\eta }$, $10{\tau }_{\eta }$, and $15{\tau }_{\eta }$ from top to bottom, offset vertically for clarity. The shape of the PDF changes dramatically when the curvature is filtered.

Figure 5

The same quantities as in Fig. 2 at $R_{\lambda }=815$ but now with the curvature filtered over $\Delta =15{\tau }_{\eta }$.