Magnus effect in granular media

A spinning ball moving through air experiences a net lift due to the “Magnus effect” resulting from the pressure difference across its two sides. In this work, with the help of numerical simulations, we present a similar phenomenon on a circular intruder undergoing both translational and rotational motion in a two-dimensional granular medium. The direction of this “Magnus” lift in a granular medium, irrespective of its area fraction, is opposite to the general direction in viscous fluids in the range of velocities studied. We relate this effect to the switching in the direction of tangential forces, uneven shearing of the force chains, and uneven distribution in the number of contacts across the rotating intruder. Moreover, this is observed for area fractions $\mathrm{\Phi }$ as low as 0.35 to as high as 0.82, which lies just below the jamming point. Distribution of the contact force around the intruder surface is also studied with respect to the nondimensionalized rotation speed of the intruder. A saturation in the lift to drag ratio is achieved at very high rotation speed, and the ratio is seen to be weakly dependent on the area fraction of the granular medium. The change in local flow fields of velocity, area fraction, and granular temperature around the intruder for several spin ratios is also discussed. The downstream wake of the intruder also deflects in the opposite direction when compared to the case in viscous fluids.

Figure 1

The initial configuration for $\mathrm{\Phi }=0.352$. The red disk at the center shows the intruder of diameter $d_{\mathrm{intr}}$ located at the origin surrounded by granular particles (light blue particles) of mean diameter $d$. The two confining walls are located at $-99.5d$ to $99.5d$ (black particles).

Figure 2

(a) Variation of $D$ with $V_0$ for various $\mathrm{\Phi }$ on a log-log plot. A linear fit gives a slope of roughly 2 suggesting quadratic dependence of the form $D\propto V_0^2$. (b) Variation of $N$ with $V_0$ for various $\mathrm{\Phi }$. The symbols mean the same as in (a).

Figure 3

The lift to drag ratio $L/D$ against spin ratio $(\mathrm{\Omega }={\omega }_{0}R/V_0)$ for various $V_0$ for $\mathrm{\Phi }=$ (a) 0.352, (b) 0.585, and (c) 0.818. The symbols mean the same as in (b).

Figure 4

The figure shows tangential force and its direction at a point on the surface of an intruder. Here the intruder is moving in the $x$ direction and rotating in the counterclockwise direction. The front side of the intruder, which faces the majority of contacts, is depicted in this figure with vectors drawn along the surface of the intruder. The vectors are color-coded to suggest whether the contacts are undergoing slipping (yellow vectors) or not (violet vectors). The figure shows three cases of $\mathrm{\Omega }=$ (a) 0, (b) 1, and (c) 20. When the spin ratio is zero, the sum of tangential forces cancels each other out in the $y$ direction. However, as $\mathrm{\Omega }$ is increased, a resultant vector is produced from the sum of these tangential forces. This resultant vector is the major cause for the Magnus effect in granular media.

Figure 5

Plot of $D_{\mathrm{\Omega }}$ normalized by $D_{\mathrm{\Omega }=0}$, which is the drag at a given spin ratio normalized by the case with zero spin ratio, against $\mathrm{\Omega }$ (top row). Plot of $N_{\mathrm{\Omega }}/N_{\mathrm{\Omega }=0}$ vs $\mathrm{\Omega }$, where the former is the number of contacts at $\mathrm{\Omega }$ normalized by $N$ at $\mathrm{\Omega }=0$ (bottom row). The plots correspond to $\mathrm{\Phi }=0.352$, 0.585, and 0.818 from left to right. The symbols mean the same as in Fig. 3.

Figure 6

(a) $F_{N,\theta }/D_{\mathrm{\Omega }=0}$ vs $\theta$, (b) $F_{T,\theta }/\mu F_{N,\theta }$ vs $\theta$, and (c) $N_{\theta }/N_{\mathrm{\Omega }}$ vs $\theta$ for $\mathrm{\Phi }=0.818$. Please refer to (c) for the legends.

Figure 7

The variation of normalized velocity field [(a), (b), and (c)], local area fraction [(d), (e), and (f)] and normalized local granular temperature [(g), (h), and (i)] in the vicinity of the intruder for $\mathrm{\Omega }=0,$ 1, and 10, respectively. The black disk in the velocity and granular temperature fields represents the intruder while in the local area fraction plot it is represented by the yellow disk.

Figure 8

$L_{\infty }/D_{\infty }$ vs $\mu$ for various $\mathrm{\Phi }$. The more dilute the system is, the closer the value of $L_{\infty }/D_{\infty }$ to the proposed model.