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.

Figure 1

Schematic representation of the DP and IB methods. In the DP method, the motion of the small granular particles is solved, taking into account detailed contact forces, as well as drag forces caused by motion relative to the interstitial gas. The particles are smaller than the grid on which the gas phase equations are solved (left), requiring empirical drag relations. The intruder is much larger than the gas grid and coupled to the gas phase through the IB method, which enforces no-slip boundary conditions by a distribution of force points across the surface of the intruder (right). A schematic representation of half the bed geometry is also shown (center); particles are color coded according to their initial position in the $z$ direction. Visualization was carried out using OVITO [9].

Figure 2

Granular bed structure under different impact velocities of a 10 mm spherical intruder at the time when the intruder has moved one diameter since initial contact $(z=D)$. Only half of the bed is shown and granular particles are color coded according to their velocities.

Figure 3

Plunging force $f$ of a 10 mm intruder with different intruding velocities (velocity increasing from bottom to top) as a function of penetration depth $z$. The inset shows the measured force on the intruder, nondimensionalized by its weight, on a linear scale. The main plot shows the measured force on a double-logarithmic scale, emphasizing its power-law dependence. Squares indicate the location of the concave-to-convex transition.

Figure 4

Plunging force $f$ of intruders with different sizes (size increasing from top to bottom) under an impact velocity of 0.01 m/s as a function of penetration depth $z$. The inset shows the measured force on the intruder, nondimensionalized by its weight, on a linear scale. The main plot shows the measured force nondimensionalized by the intruder volume $D^3$ versus penetration depth nondimensionalized by the intruder diameter $D$, both on a logarithmic scale.

Figure 5

Contact force on the upper (top) and lower (bottom) hemispheres as function of penetration depth $z$ (intruding velocities increasing from bottom to top). Note that the contribution of the upper hemisphere is much smaller.

Figure 6

Direct gas-induced drag force on the upper (top) and lower (bottom) hemispheres as function of penetration depth $z$ (intruding velocities increasing from bottom to top). Oscillations are caused by stick-slip behavior of the surrounding granular fluid, which is coupled to the gas phase. Color legend is the same as in Fig. 5.

Figure 7

Granular drag force $f_v$ as function of penetration depth $z$. Different colors correspond to different intruder velocities (velocities increasing from bottom to top). Symbols are measurements from the simulations. Straight dashed lines are theoretical predictions based on Froude and Reynolds numbers (see main text).

Figure 8

The transition between two regimes of the drag force (black line) coincides with the point at which the intruder is engulfed by granular particles (gray line shows the contact force on the upper hemisphere). In this example $v=0.40$ m/s and $D=10$ mm.

Figure 9

Normalized transition depth $z^{*}/D$ versus the Froude number $\mathrm{Fr}=v^2/\left(gD\right)$ for different intruder velocities and different intruder diameters (symbols). The straight line is a fit to the prediction, Eq. (14).

Figure 10

Effective granular drag force $f_v^{*}$ from the plateau region $z<z^{*}$ for different intruder velocities and different intruder diameters (symbols). The line corresponds to the Schiller and Naumann correlation for a sphere in a Newtonian fluid, Eq. (16). The inset shows the same data as a function of estimated effective Reynolds number.

Figure 11

Average overlap with neighboring particles, representing the pressure field in the granular fluid, for the impact of a $D=10$ mm intruder moving at a constant velocity of $v=0.1$ m/s. Results are shown as a function of axial and radial positions, where the intruder is moving along $r=0$.