Universality in quasi-two-dimensional granular shock fronts above an intruder

We experimentally study quasi-two-dimensional dilute granular flow around intruders whose shape, size, and relative impact speed are systematically varied. Direct measurement of the flow field reveals that three in-principle independent measurements of the nonuniformity of the flow field are in fact all linearly related: (1) granular temperature, (2) flow-field divergence, and (3) shear-strain rate. The shock front is defined as the local maxima in each of these measurements. The shape of the shock front is well described by an inverted catenary and is driven by the formation of a dynamic arch during steady flow. We find universality in the functional form of the shock front within the range of experimental values probed. Changing the intruder size, concavity, and impact speed only results in a scaling and shifting of the shock front. We independently measure the horizontal lift force on the intruder and find that it can be understood as a result of the interplay between the shock profile and the intruder shape.

Figure 1

(a) Front view of experimental setup. The intruder in the picture corresponds to a superdisk exponent $n=7.5$. (b) Intruder-sensor system schematic. (c) Examples of intruders with different superdisk exponents. As $n$ increases, the shapes become more convex, with $n=\infty$ being a square.

Figure 2

(a)–(c) Intensity plot of scaled granular temperature, absolute divergence of the flow field, and local shear-strain rate for $n=0.75$. (d)–(f) Scatter plots of scaled flow-field divergence vs temperature, flow-field divergence vs local shear rate, and shear rate vs temperature, respectively, measured from all points enclosed by the white dotted regions.

Figure 3

(a) Plots of shock fronts. Scaled $x$ and $y$ coordinates are in units of $R$, the vertical length of the intruder. The data are shown for $n=0$ ($+$), 0.75 ($\times$), 0.85 ($*$), 0.95 ($\circ$), 1 ($\square$), 1.5 ($\diamond$), 2 ($▵$), 2.5 ($▿$), 5 ($◃$), 7.5 ($▹$), 10 (☆), $\infty$ ($✡$). Solid black lines are fits to Eq. (7). (b) Plot of scaled shock boundaries for all probed values of $n$ with constant $R=4$ in., shown in circles. Squares represent all tested intruder locations $h\in \{23,34,46,57,69,84\}\mathrm{cm}$ and intruder radii $R\in \{2,3,4,7\}\mathrm{in}.$ for constant $n=2$. Diamond symbols represent data from an asymmetric intruder with random features on it, as shown in Fig. 4. The black curve is the inverted catenary given by Eq. (7). (c) Top to bottom: Plots of the fit parameters $w, c$, and $w+p$ as functions of $n$, respectively. Red circles represent data from varying $n$ with constant $R$ and blue squares represent data from fixed $n=2$ but different combinations of $R$ and impact speed $v_0$. The dashed line shows the mean width $|\overline{w}|\approx 0.8$. In the $c$ vs $n$ plot, the dark solid curve shows the center of mass of the intruders shifts with varying $n$.

Figure 4

Representative mean flow field around an intruder with no line of symmetry. The masked out region is the intruder and the arrows are velocity vectors calculated from PIV.

Figure 5

(a) Plot of horizontal lift forces from force sensor (green circle) and calculated lift forces (dark blue squares) as a function of $n$. Lift is maximum for $n\to 0$ and decreases to 0 as $n\to \infty$. The inset is an illustration showing the intruder, outlined in black, trapped grains in gray, and the shock profile in red. The solid green arrow represents the tangential net force $\vec{F}(s,\theta )$ on a small segment of the shock front. Dotted green arrows are components of this vector. Purple arrows labeled $v_{x,0/1}$ represent horizontal ejected grain velocities from the trapped region. (b) Plot of the left (red) and right (blue) exit heights $l$ as a function of $n$. (c) Plot of absolute horizontal left (red) and right (blue) exit velocities $|v_x|$ vs $n$.

Figure 6

Plots of total measured lift force $F_{\text{meas}}$, calculated lift force $F_L$ (blue squares), force due to mass ejection from trapped pile $F_{\text{flow}}$ (green diamond), quasistatic load contribution $F_S$ (downward pointing arrow), and force due to impacts $F_C$ (upward pointing arrow) as a function of $n$.