Orientation dynamics of nonspherical particles under surface gravity waves

Experiments were conducted with three-dimensional printed disks and rods to study the orientation dynamics of nonspherical particles under surface gravity waves. Trials were run with both neutrally buoyant and slightly negatively buoyant particles, which were large enough that inertia due to their finite size was important. Although the particles had a broad distribution of initial orientations, over time the waves were observed to focus these orientations toward a preferred angle that agreed with theory. Negatively buoyant particles additionally exhibited a tendency to adopt an orientation that maximized vertical drag. The overall orientation can be described as the result of a competition between the orientation favored by waves and the orientation favored by settling. The spread about the mean orientation was also observed to increase with wave strength. Finally, the stabilization of out-of-plane orientations of disk-shaped particles was observed due to their finite size.

Figure 1

Coordinate system and geometry of the 3D-printed particles. Elongated and flattened cylinders were used to approximate rods and disks, respectively. Their orientation is described by a unit vector $p$ directed along the symmetry axis. The azimuthal angle $\theta$ captures the orientation of the particles out of the plane of the wave motion, and the polar angle $\varphi$ is the angle measured from the vertical ($z$) axis.

Figure 2

Schematic (not to scale) of the experimental setup. Waves are generated on the left side of the tank by a plunging wave maker and are dissipated on the right side by the horse-hair beach. A laser is directed upward through the bottom of the tank to illuminate tracer particles. Bright LEDs provide illumination for the nonspherical particles through the side of the tank. The horse-hair beach is arranged to maximize the amount of horsehair at the top of the water column where the waves are strongest but minimize its lateral extent into the tank.

Figure 3

Photo of the particles used in the experiments after having been dyed with Rhodamine. Clockwise from the top left are small disks, large disks, long rods, and short rods. The ruler provides a scale in centimeters.

Figure 4

Mean flow characteristics. Measured average wave amplitudes (a) $U$ and (b) $W$ as a function of depth for the three wave cases (symbols) are plotted along with the predictions from linear wave theory (solid lines). (c) Measured Stokes drift $u_s$ velocities (symbols) as a function of depth along with the theoretical predictions (solid lines). (d) Measured mean Eulerian flow velocities $\overline{u}$ as a function of depth. For all panels, case WC2 is shown with circles, case WC3 is shown with triangles, and case WC4 is shown with squares.

Figure 5

Time series of the polar angle $\varphi$ for several neutrally buoyant ($\beta =1$) long rods in WC2 waves. Though their initial $\varphi$ values were uncontrolled, the particles all settle onto an oscillatory limit cycle at the wave frequency about a similar angle. The wave-preferred angle ${\varphi }_w^{*}$ expected from theory is marked with a dashed line.

Figure 6

PDFs of the polar angle $\varphi$ of long rods in waves with (a) $\beta =1.01$ and (b) $\beta =1$ at different times, as indicated by the shading. The dotted line shows the theoretical wave-preferred angle ${\varphi }_w^{*}$; the settling-preferred angle is ${\varphi }_s^{*}=\pi /2$. Data are combined from all three wave cases but only for particles in regions where $ka\left(z\right)\ge 0.05$.

Figure 7

Evolution of the (a) mean and (b) standard deviation of the polar angle $\varphi$ of neutrally buoyant ($\beta =1$) long rods in waves as a function time. Data from WC2, WC3, and WC4 are shown by $•$, $▴$, and $\blacksquare$, respectively. For all cases, data are included only for particles in regions with $ka\left(z\right)\ge 0.05$.

Figure 8

Evolution of the (a) mean and (b) standard deviation of the polar angle $\varphi$ of negatively buoyant ($\beta =1.005$) short rods in waves as a function time. Data from WC2, WC3, and WC4 are shown by $•$, $▴$, and $\blacksquare$, respectively. For all cases, data are included only for particles in regions with $ka\left(z\right)\ge 0.05$.

Figure 9

PDFs of the polar angle $\varphi$ for (a) large disks ($\beta =1.005$) and (b) long rods ($\beta =1.01$) for different values of the nondimensional wave shear $ka\left(z\right)$, as indicated by the shading. The dotted lines show the theoretical wave-preferred angle ${\varphi }_w^{*}$. The preferred settling angle is ${\varphi }_s^{*}=0$ for the disks and ${\varphi }_s^{*}=\pi /2$ for the rods. Data are combined from all three wave cases, but only for particles that had evolved in the waves for at least 10 wave periods.

Figure 10

Standard deviation of the wave-period-averaged polar angle $\varphi$ as a function of nondimensional wave shear $ka\left(z\right)$ for different particle shapes: long rods (closed triangles), short rods (open triangles), large disks (closed circles), and small disks (open circles). The data come from $\beta =1.005$ for all shapes except long rods which have $\beta =1.01$, and data are included only for particles that had evolved in the waves for at least 10 wave periods.

Figure 11

PDFs of the apparent aspect ratio ${\mathrm{\Gamma }}_{\mathrm{obs}}$ of large disks with (a) $\beta =1.005$ and (b) $\beta =1$ for different values of the nondimensional wave shear $ka\left(z\right)$, as indicated by the shading. Data are combined from all three waves cases over the entire experimental record. The dashed line indicates the actual aspect ratio of the particles.

Figure 12

PDFs of the apparent aspect ratio ${\mathrm{\Gamma }}_{\mathrm{obs}}$ of small disks with (a) $\beta =1.005$ and (b) $\beta =1$ for different values of the nondimensional wave shear $ka\left(z\right)$, as indicated by the shading. Data are combined from all three waves cases over the entire experimental record. The dashed line indicates the actual aspect ratio of the particles.