Free fall of homogeneous and heterogeneous cones

The late stage of the free fall of homogeneous and heterogeneous cones in a quiescent water medium and the induced flow are experimentally inspected using three-dimensional particle tracking velocimetry and volumetric particle image velocimetry. The structures shared the same geometry but had different combined specific gravity $\mathrm{SG}\in [1.2,8]$. Results showed four distinct free-fall patterns, which were modulated by the degree of heterogeneity and $\mathrm{SG}$. They included (i) straight fall, (ii) regular sinusoidal-like trajectories with a maximum angle of oscillation with respect to the vertical, ${\theta }_{\max }<\pi /2$, (iii) motions characterized by inclined translations and large rotations with ${\theta }_{\max }>\pi /2$, and (iv) tumbling followed by an irregular motion. The straight fall occurred in cones with $\mathrm{SG}\gtrsim 6$ regardless of the mass distribution. The sinusoidal motions arose in the homogeneous cones with $\mathrm{SG}<3$. The large rotations occurred with heterogeneous cones of $\mathrm{SG}<3$, whereas the tumbling with irregular motions occurred in the heterogeneous cone with the highest ratio between the center of mass and geometrical centroid. Inspection of the linear and angular components of the velocities and accelerations reveal the distinct patterns and the linkage with the surrounding flow.

Figure 1

(a) Characteristics of the nine cones; $①, ⑤$, and $⑨$ indicate the homogeneous structures. (b) Schematic of the experimental setup for the tracking of the cones using 3D particle tracking velocimetry (3D-PTV). (c) setup for the characterization of the induced 3D flow field via 3D particle image velocimetry (3D-PIV).

Figure 2

Illustration of the 3D posture reconstruction. (a) Boundary recognition, (b) detection of tip and edges based on the boundary and centroid (black circle), and (c) distances between the centroid and the boundary, $r_c$; global and local peaks indicate the edges (black cross) and tip (red cross). (d) Sequence of a falling cone 5, (e) evolution of the horizontal ($\overline{x}$) and vertical $\overline{z}$ locations of the cone centroid, and (f) inclination angle from the vertical.

Figure 3

Distinct patterns exhibited by the cones. Pattern 1: straight fall; Pattern 2: regular sinusoidal-like trajectories with a maximum angle of oscillation ${\theta }_{\max }<\pi /2$; Pattern 3: with large rotations ${\theta }_{\max }>\pi /2$ with inclined translations; and Pattern 4: tumbling followed by irregular trajectory. Superimposed images every $\mathrm{\Delta }t=0.1$ s. See Supplemental Material at [18] for visualization of the motions.

Figure 4

(a) Stable free fall for $\rho /{\rho }_w\ge 6$ cones; the inset shows terminal velocity $u_s$ normalized by the terminal velocity of $\rho /{\rho }_w\ge 8, U$. (b) 3D flow field of a stable fall; $w$ is the vertical component of the flow velocity. (c) $Q$ criterion; red and blue isosurfaces denote shed structures (with $Q=0.02{\mathrm{s}}^{-1}$ and $Q=-0.04{\mathrm{s}}^{-1}$).

Figure 5

Drag coefficient $C_d$ of the cones with straight fall as a function of $\rho /{\rho }_w$. Also included is the bulk estimation over one cycle for those cases with periodic oscillations.

Figure 6

3D trajectories of the (a) homogeneous cone 5 and (b) heterogeneous cone 6. (c) Summary table including frequency of oscillations $f_o$, maximum inclination angle ${\theta }_{\max }$, wavelength $\lambda$, and amplitude $A_o$ of the trajectories. (d) Basic relation between ${\theta }_{\max }$ and degree of heterogeneity $h_c/h$. $h_c$ is the center of mass, the distance from the apex along the cone axis; note that the geometric center and center of mass coincide when $h_c/h=0.75$ (homogeneous cones).

Figure 7

Kinematics of the cones that exhibited regular oscillations, and large rotations with inclined translations. Temporal evolution of the (a) angle $\theta$ with respect to the vertical, (b) lateral displacement $\overline{x}$, (c) vertical displacement $\overline{z}$. (d)–(f) Corresponding velocities and (g)–(i) corresponding accelerations.

Figure 8

Superimposed pair of raw images of the heterogeneous cone 6 at various fall instants: (a) no inclination, (b) large rotation, (c) inclined translation (gliding), and (d) backward rotation; $G, B, M_f, M_g, M$, and $L$ denote the gravity force acting on the center of mass, buoyancy force acting on the centroid, base torque, additional torque (misalignment between $G$ and $B$), total torque, and lift. (e) Location of the centroid, $\overline{x}$, and center of mass, $x_c$ with respect to the mean fall axis. (f) Difference between $\overline{x}$ and $x_c$ and oscillation angle. $\theta$.

Figure 9

Horizontal velocity induced by the homogeneous cone 5 with superimposed velocity vectors at (a) no and (b) maximum inclinations. Isocontours of the $Q$ criterion ($=0.01$, and $-0.04{\mathrm{s}}^{-1}$) are included for reference.

Figure 10

3D flow fields of a heterogeneous light cone (case 6) at (a) gliding, (b) rerolling, (c) $0^{\circ }$ inclination angle, and (d) over-rolling with in-plane contours indicating horizontal velocity. Isocontours of the $Q$ criterion ($=0.01$ and $-0.04{\mathrm{s}}^{-1}$) are added for reference.