Dynamics of projectile impact in a granular material, and the dynamics of a single sedimenting sphere in fluid

We have studied the dynamics of impact and sedimentation in two experimental systems. Tracking the incident object with high precision we examined the details of resulting trajectories through the ambient material in both cases. For each system direct numerical simulations have enabled us to gain insight into the forces leading to the resulting trajectories. A static bed of granular material in the absence of external forcing will remain in its steady state configuration. During impact, a projectile of sufficient energy causes a static bed of particles to become locally fluidized. The properties of the fluidized granular material allow for the projectile to penetrate into the bed. The granular drag force on the projectile as it penetrates is caused by dissipation and collisions between interacting particles. As the projectile velocity decreases the energy will become lower than the threshold for fluidization causing the granular bed to return to a static pile of material. The projectile rapidly slows to a stop. We have created a quasi-two-dimensional experiment to measure the granular drag force on a projectile as it penetrates a granular bed of particles. Our results show that the average drag force on the projectile is constant during penetration. Moreover, the magnitude of the drag force is proportional to the velocity of impact. We also report the surprising result that for a range of projectile impact velocities (50 < v < 400 cm/s) the projectile decelerates to v = 0 in t ≈ 0.15 s for all cases. Simulation results also show that the probability distribution of forces between individual particles in the bed remains unchanged during penetration. Based upon these simulation results we conclude that the granular drag force on the projectile is caused by the localized forces within the bed of particles that are within one diameter below the projectile. After a single sphere is released under the force of gravity within a Newtonian fluid medium, the sphere will reach terminal velocity. For large enough terminal velocities the sphere will begin to shed vortices. We have studied the case of a single sphere sedimenting between two closely spaced glass plates in a narrow aspect ratio cell (a Hele-Shaw cell). Using a very accurate tracking method we were able to track the position of a sphere during its fall for a range of Reynolds numbers (based on the terminal velocity and the diameter of a sphere), 20 < Re < 330. Simultaneous results of digital particle image velocimetry images provided a picture of the fluid wake behind the sphere. We found that the trajectory of the sphere is highly sensitive to the size of the gap, Γ = dgap/dsphere. For Γ ≤ 1.05 we observed nearly two dimensional vortices behind the sphere. These vortices caused very small oscillations in the motion of the sphere in the transverse direction . For larger gap sizes (1.10 ≤ Γ ≤ 1.40), the trajectory of the sphere was very different due to a change in the wake structure from nearly two-dimensional to fully three-dimensional. For these cases we observed oscillatory behavior in both the transverse and streamwise directions. Simulations confirmed spiral trajectories and indicated that vortices are formed in parallel tubes which are shed for large Re. Our detailed transition map shows that for small gap cases there are similarities to results for fixed cylinders while the transitions for the large gap cases are similar to those for flow past a fixed sphere without the presence of confining walls.