Numerical investigation of the vertical plunging force of a spherical intruder into a prefluidized granular bed.

The plunging of a large intruder sphere into a prefluidized granular bed with various constant velocities and various sphere diameters is investigated using a state-of-the-art hybrid discrete particle and immersed boundary method, in which both the gas-induced drag force and the contact force exerted on the intruder can be investigated separately. We investigate low velocities, where velocity dependent effects first begin to appear. The results show a concave-to-convex dependence of the plunging force as a function of intruder depth. In the concave region the force fits to a power law with an exponent around 1.3, which is in good agreement with existing experimental observations. Our simulation results further show that the force exerted on the frontal hemisphere of the intruder is dominant. At larger intruder velocities, friction with the granular medium causes a velocity-dependent drag force. As long as the granular particles have not yet closed the gap behind the intruder, this drag force is independent of the actual intruder depth. In this regime, the drag force experienced by intruders of different diameter moving at different velocities all fall onto a single master curve if plotted against the Reynolds number, using a single value for the effective viscosity of the granular medium. This master curve corresponds well to the Schiller-Naumann correlation for the drag force between a sphere and a Newtonian fluid. After the gap behind the intruder has closed, the drag force increases not only with velocity but also with depth. We attribute this to the effect of increasing hydrostatic particle pressure in the granular medium, leading to an increase in effective viscosity.