Rotations of large inertial cubes, cuboids, cones, and cylinders in turbulence

We conduct experiments to investigate the rotations of freely moving particles in a homogeneous isotropic turbulent flow. The particles are nearly neutrally buoyant and the particle size exceeds the Kolmogorov scale so that they are too large to be considered passive tracers. Particles of several different shapes are considered including those that break axisymmetry and fore-aft symmetry. We find that regardless of shape the mean-square particle angular velocity scales as ${d_{eq}}^{-4/3}$, where $d_{eq}$ is the equivalent diameter of a volume-matched sphere. This scaling behavior is consistent with the notion that velocity differences across a length $d_{eq}$ in the flow are responsible for particle rotation. We also find that the probability density functions (PDFs) of particle angular velocity collapse for particles of different shapes and similar $d_{eq}$. The significance of these results is that the rotations of an inertial, nonspherical particle are only functions of its volume and not its shape. The magnitude of particle angular velocity appears log-normally distributed and individual Cartesian components show long tails. With increasing $d_{eq}$, the tails of the PDF become less pronounced, meaning that extreme events of angular velocity become less common for larger particles.

Figure 1

Top view schematic of the turbulence tank and measurement system. All dimensions are in meters (m).

Figure 2

Particle angular velocity for a subset of cube data calculated from two different methods described in the text. The solid line shows the 1:1 correspondence.

Figure 3

Dimensionless mean-square particle angular velocity plotted against the dimensionless particle size measured by the diameter of a volume-matched sphere. The mean-square uncertainty, $\langle {\delta {\omega }_p}^2\rangle =\langle {\delta {\omega }_p}_x^2\rangle +\langle {\delta {\omega }_p}_y^2\rangle +\langle {\delta {\omega }_p}_z^2\rangle$, is subtracted from the mean-square particle angular velocity, $\langle {{\omega }_p}^2\rangle =\langle {{\omega }_p}_x^2\rangle +\langle {{\omega }_p}_y^2\rangle +\langle {{\omega }_p}_z^2\rangle$, to remove the effects of measurement noise as explained in the text. Data of cubes (red square), cuboids (red diamond), and cones (red triangle) are plotted along with small, medium, and large cylinders (black squares) and long cylinders (black circle). Cylinder data (black symbols) have been made dimensionless by $\eta =0.3\times {10}^{-3}$ m and ${\tau }_{\eta }=0.11$ s, as reported in Ref. [26]. The 95% confidence interval uncertainty bars are calculated from a bootstrap analysis. The dashed lines are given by ${(d_{eq}/\eta )}^{-4/3}$ and $2{(d_{eq}/\eta )}^{-4/3}$.

Figure 4

Probability density functions (PDFs) of particle angular velocity components normalized by their rms values: blue crosses, ${{\omega }_p}_x$; magenta squares, ${{\omega }_p}_y$; red circles, ${{\omega }_p}_z$. (a) Cubes, (b) cuboids, (c) cones, (d) small cylinders, (e) medium cylinders, (f) large cylinders, and (g) long cylinders. The PDFs are fitted to Eq. (3) and the fitting parameter, $s$, is listed in Table 4.

Figure 5

Probability density functions (PDFs) of one component of particle angular velocity normalized by its rms value. Data of cubes (red squares), cuboids (red diamonds), and cones (red triangles) are plotted along with Eq. (3) with $s=0.63$, which is the average value across the three shapes (see Table 4).

Figure 6

Kurtosis values of particle angular velocity components and transverse velocity increments in the flow. A Gaussian distribution would have a value of $K=3$. The symbols are the same as Fig. 3 and their error bars are calculated from the 95% confidence intervals on the fitting parameter, $s$, and listed in Table 4. Cylinder data (black symbols) have been made dimensionless by $\eta =0.3\times {10}^{-3}$ m, as reported in Ref. [26]. The solid line is $\langle {\mathrm{\Delta }}_x{u}_y^4\rangle /{\langle {\mathrm{\Delta }}_x{u}_y^2\rangle }^2$ and the dashed line shows the power law ${d_{eq}}^{-0.11}$.