Steady flow dynamics during granular impact

We study experimentally and computationally the dynamics of granular flow during impacts where intruders strike a collection of disks from above. In the regime where granular force dynamics are much more rapid than the intruder motion, we find that the particle flow near the intruder is proportional to the instantaneous intruder speed; it is essentially constant when normalized by that speed. The granular flow is nearly divergence free and remains in balance with the intruder, despite the latter's rapid deceleration. Simulations indicate that this observation is insensitive to grain properties, which can be explained by the separation of time scales between intergrain force dynamics and intruder dynamics. Assuming there is a comparable separation of time scales, we expect that our results are applicable to a broad class of dynamic or transient granular flows. Our results suggest that descriptions of static-in-time granular flows might be extended or modified to describe these dynamic flows. Additionally, we find that accurate grain-grain interactions are not necessary to correctly capture the granular flow in this regime.

Figure 1

(a) PIV flow field for a circular intruder with a radius of $R=6.35\mathrm{cm}$ at a particular frame. (b) Vertical and horizontal components of the spatially smoothed PIV flow field from experiment, normalized by the instantaneous intruder velocity ($v=3.05\mathrm{m}/\mathrm{s}$). The left panel shows the normalized vertical velocity $u_z/v$ with downward as positive, and the right panel shows the horizontal velocity $u_x/v$ with rightward as positive. The white color in the intruder in the left panel denotes its downward motion. The red box encloses the region used to obtain the steady-state velocity field, which will move along with the intruder. (c) The instantaneous flow fields $u_z/v$ and $u_x/v$ at a particular time from simulations with Hertzian frictional interactions between grains.

Figure 2

(a) The average flow field $\mathbf{A}$ from experiments for a circular intruder with a radius of $R=6.35\mathrm{cm}$. The left side shows the vertical velocity $A_z$ (down is positive), and the right side shows the horizontal velocity $A_x$ (right is positive). (b) The instantaneous fluctuations $A_z^{\prime }$ and $A_x^{\prime }$ in the flow field at a particular frame from experiments for a circular intruder with a radius of $R=6.35\mathrm{cm}$. (c) A spatial plot of $A_{\mathrm{rms}}^{\prime }$, the root-mean-squared value of ${\mathbf{A}}^{\prime }$. A time series of the spatial mean of $A_{\mathrm{rms}}^{\prime }$ within the outlined region (where the fluctuations are most prominent) is plotted (thick dashed line) in panel (d). This quantity is essentially constant in time. The red, blue, and green solid lines in (d) show $A_z^{\prime }$ at three points beneath the intruder. These signals fluctuate around zero with a correlation time of roughly 3 ms.

Figure 3

The local shear rate $\dot{\gamma }/v$ and the divergence of $\mathbf{A}$, computed numerically from the flow field shown in Fig. 2, showing that $\mathbf{A}$ is a divergence-free shear flow.

Figure 4

(a) Data (solid blue lines) and corresponding fit line (dashed black lines) of the form shown in Eq. (3) for one intruder (radius of $R=3.18$ cm) at $r=1.5R, 2R$, and $3R$, where $r=R$ corresponds to the intruder boundary. (b)–(e) A comparison of the fit parameters—$a_r\left(r\right),b_r\left(r\right),a_{\theta }\left(r\right),b_{\theta }\left(r\right)$—for circular intruders with $R=3.18$ cm (small red circles) $R=6.35$ cm (medium blue circles), $R=10.15$ cm (large green circles), as well as the circular-nosed intruder with $R=4.65$ cm and a rectangular tail (black squares). The inset of (b) shows semilogarithmic plots of $1-a_r\left(r\right)$ versus $r/R$; the thick black reference lines show exponential decay with decay lengths of $0.7R$ (upper) and $0.25R$ (lower). The inset of (c) shows semilogarithmic plots of $b_r\left(r\right)-1$ versus $r/R$; the thick black reference line shows exponential decay with a decay length of $1.85R$.

Figure 5

Comparison of the parameters $a_r\left(r\right),b_r\left(r\right),a_r\left(r\right)$, and $a_{\theta }\left(r\right)$ between experimental data (blue circles) from Fig. 4 and simulations using frictional linear (magenta crosses), frictional Hertzian (magenta triangles), and frictionless linear (light blue asterisks) interactions, all under the same conditions with $R=6.35\mathrm{cm}$. The insets of (a) and (b) show semilogarithmic plots of $1-a_r\left(r\right)$ and $b_r\left(r\right)-1$, respectively, versus $r/R$. The thick black reference lines in the insets show exponential decay with decay lengths of (a) of $1.3R$ (upper) and $0.45R$ (lower) and in (b) of $1.85R$.