Drag Coefficient for a Circular Obstacle in a Quasi-Two-Dimensional Dilute Supersonic Granular Flow

The measurement of the drag coefficient of a dilute granular flow around a cylinder is carried out over a wide range of Knudsen numbers. The variation of this coefficient shows a smooth transition from a freely falling grains regime to a continuous flow regime. This is reminiscent of the behavior of gases in the supersonic regime. This transition is accompanied by remarkable changes of the density and velocity profiles near the cylinder. A simple model is proposed for the transition regime which is in agreement with the experimental measurements.

Figure 1

Pictures of the flow of grains around a cylindrical obstacle sandwiched between two transparent vertical plates for three different flow volume fractions or Knudsen numbers (steel spheres of $d=1\mathrm{mm}$ and aluminum cylinder of $D=11\mathrm{cm}$). Note that as $k_D$ decreases, the density in the close proximity of the cylinder increases considerably going from the dilute limit at $k_D=0.5$ to the dense limit at $k_D=0.02$.

Figure 2

(a) Force versus ${\varphi }_a$ for the high $k_D$ range for which the mass of accumulated grains on top of the cylinder is negligible. The solid line is the expected result for a constant $C_d$. (Steel spheres of $d=1\mathrm{mm}$ and a cylinder of $D=2.7\mathrm{cm}$) (b) Force versus the weight of the dense zone on the obstacle for the low $k_D$ range. (Steel spheres of $d=1\mathrm{mm}$ for $D=11\mathrm{cm}$). Open symbols are measurements, closed symbols are obtained using the model in the text.

Figure 3

$C_d$ versus $k_D$ (solid circles: steel spheres $d=1\mathrm{mm}$, $D=11\mathrm{cm}$ and 2.7 cm, squares: glass beads $d=1.2\mathrm{mm}$ and $D=11\mathrm{cm}$, triangles: estimates from momentum flux difference, crosses: estimates from integrating the pressure around the cylinder). The horizontal dashed line is the expected result for high $k_D$. The solid line is the best fit using the proposed model.

Figure 4

Volume fraction (gray dots) and vertical velocity (black circles) profiles measured at mid level of the cylinder for different ${\varphi }_a$. Measurements are along the $z$ axis with origin at the center of the cylinder. The insets show representative illustrations of the flow. For small $k_D$, the flow is dense while for the higher $k_D$ (0.5) the flow is less dense with particles rebounding away from the obstacle.

Figure 5

${\varphi }_{\max }$ on top of the obstacle, shear rate $\Gamma$ and the standard deviation of the vertical velocity $\delta V_y$ near the mid level of the cylinder versus $k_D$ for steel spheres. Note the changes occurring near $k_D=0.15$. Upper graph: the closed symbols are for steel spheres while the open symbols are for glass beads (squares and diamonds: different $D$ but fixed ${\varphi }_a$, circles: varying ${\varphi }_a$ and fixed $D$).