Rotation Rate of Rods in Turbulent Fluid Flow

The rotational dynamics of anisotropic particles advected in a turbulent fluid flow are important in many industrial and natural settings. Particle rotations are controlled by small scale properties of turbulence that are nearly universal, and so provide a rich system where experiments can be directly compared with theory and simulations. Here we report the first three-dimensional experimental measurements of the orientation dynamics of rodlike particles as they are advected in a turbulent fluid flow. We also present numerical simulations that show good agreement with the experiments and allow extension to a wide range of particle shapes. Anisotropic tracer particles preferentially sample the flow since their orientations become correlated with the velocity gradient tensor. The rotation rate is heavily influenced by this preferential alignment, and the alignment depends strongly on particle shape.

Figure 1

 Three-dimensional view of a rod trajectory with a large rotation rate from the experiment at $R_{\lambda }=214$. The color of the rod represents the rotation rate. This rod is tracked for 284 ms. The green lines show the projection of the center of the rod onto the $y\mathrm{\text{−}}z$ and $x\mathrm{\text{−}}y$ planes. The rod is a circular cylinder with length 1 mm and diameter 0.2 mm.

Figure 2

The PDF of rotation rate squared of rods. (a) Comparison of simulation (blue line, $R_{\lambda }=180$) with experiment (black circle, $R_{\lambda }=160$; red plus, $R_{\lambda }=214$) for aspect ratio $\alpha =5$. Error bars represent both random uncertainty and the systematic uncertainty produced by measuring rotation rates from experimental orientation data over a time interval. (b) Comparison of the PDF of rotation rate squared for rods at $\alpha =5$ (blue plus) and spheres at $\alpha =1$ (black circle) with PDFs of enstrophy (green square) and energy dissipation rate (red triangle) for the simulation at $R_{\lambda }=180$. One symbol is displayed for every twenty bins.

Figure 3

Mean square rotation rate as a function of aspect ratio. (a) Blue plus is the DNS at $R_{\lambda }=180$. Black open square is the experiment at $R_{\lambda }=214$. Red open circle is the experiment at $R_{\lambda }=160$. The error bars on the experimental points represent systematic error due to extrapolation from the finite fit time required to measure the rotation rate [29]. The purple open circle shows the result for infinite aspect ratio rods from simulations by Shin and Koch [16] at $R_{\lambda }=53.3$. The green line is the analytic prediction for randomly oriented rods from Eq. (2). The dashed line indicates the rotation due to vorticity. (b) Simple cases of particles integrated through different velocity gradient fields: Blue plus uses velocity gradients along Lagrangian trajectories at $R_{\lambda }=180$. Black open square uses velocity gradients sampled from the DNS and delta correlated in time. These data match the analytic prediction for randomly oriented rods shown as the green line. Red closed circle uses velocity gradients sampled from the DNS and held fixed in time. The purple right handed triangle uses the velocity gradient of a plane shear flow. (c) Comparison of simulations at different Reynolds numbers.

Figure 4

Fourth moment of the rotation rate of ellipsoids. Blue plus is the DNS at $R_{\lambda }=180$. Black closed square is the experiment at $R_{\lambda }=214$. Red open circle is the experiment at $R_{\lambda }=160$. Green dashed line is the prediction for spheres uncorrelated with the direction of ${\Omega }_{ij}$. The purple dash-dotted line is the fourth moment of $S_{ij}$ and the black dashed line represents the case of a Gaussian distribution.