Added-mass force in dry granular matter

From two-dimensional (2D) numerical simulations of the motion of a circular intruder into a dry granular packing, we provide evidence for a specific force term in the case of unsteady motion in addition to the force corresponding to a steady motion. We show that this additional term is proportional to the acceleration of the intruder relative to the grains as the added-mass force known for simple fluids. This force term corresponds to a variation in the kinetic energy of the surrounding flow and is characterized by a coefficient $C_{AM}$ which is intrinsically linked to the nature of the granular media. An analytical calculation of the added-mass coefficient $C_{AM}$ is developed based on the specific velocity field known for 2D granular flow around a cylinder. The theoretical value is shown to depend on the grain-cylinder size ratio, in good agreement with the measurements from our unsteady simulations.

Figure 1

Snapshot of the flow configuration. A large disk of diameter $D$ is moved horizontally at an imposed velocity $U$ along the $x$ direction at the depth $h$ of a two-dimensional layer of small grains of diameter $d$ within the vertical gravity field $g$. The simulation parameters here are $U=0.5$ m/s, $D=15d$, and $h=15.5d$ $[\mathrm{Fr}=U/{\left(gh\right)}^{1/2}=1.3]$, with $d=1$ mm grains colored according to their velocity $u$, from blue at $u=0$ to dark red at $u/U=2.5$.

Figure 2

(a) Dimensionless drag force $F/\left(\rho ghDd\right)$ as a function of the dimensionless time $tU/D=x/D$ for a disk of diameter $D=15d$ moving with $U=0.5$ m/s $\simeq 1.3{\left(gh\right)}^{1/2}$ at the depth $h=15.5d$ of a packing of grains of density $\rho ={10}^3{\mathrm{kg}\mathrm{m}}^{-3}$. Solid line shows instantaneous force (in gray) and force averaged over ${10}^5$ successive points (time step) corresponding here to $\mathrm{\Delta }x/D=0.2$ (in black). Dashed line shows mean value $F_S/\left(\rho ghDd\right)=3.3$ over the entire time window ($\mathrm{\Delta }x/D=6$). (b) Dimensionless steady drag force $F_S/\left(\rho ghDd\right)$ as a function of the Froude number $\mathrm{Fr}=U/\sqrt{gh}$ in a log-log plot for the whole range of Fr and (inset) in a lin-lin plot for a narrow range of Fr around 4. Circles show data from numerical simulations for an intruder disk of diameter $D=15d$ moving at different constant velocities $U$ and thus different Froude numbers $\mathrm{Fr}=U/{\left(gh\right)}^{1/2}$, dashed line shows asymptotic behaviors with a constant plateau value $F_S/\left(\rho ghDd\right)=1$ at $\mathrm{Fr}\ll 1$ or a quadratic law $F_S/\left(\rho ghDd\right)=0.4{\mathrm{Fr}}^2$ at $\mathrm{Fr}\gg 1$, and solid line shows best linear fit $F_S/\left(\rho ghDd\right)=3.2\mathrm{Fr}-3.3$ in the range $2⩽\mathrm{Fr}⩽6$.

Figure 3

Time evolution of the force $F\left(t\right)/F_{S_i}$ for a disk of diameter $D=15d$ accelerating from $U_i=1$ m/s $\simeq 2.6{\left(gh\right)}^{1/2}$ to $U_f=2$ m/s $\simeq 5.1{\left(gh\right)}^{1/2}$, with $\mathrm{\Gamma }=0.9g$ (blue line), $12.5g$ (yellow line), and $25g$ (red line). The force signal is averaged over ${10}^5$ successive points corresponding to $\tau /10$. Dashed lines show expected initial and final steady forces $F_{S_i}$ and $F_{S_f}=2.6F_{S_i}$.

Figure 4

Added-mass coefficient $C_{AM}$ as a function of the grain-cylinder size ratio $d/D$. Diamonds show simulations results, and solid and dashed lines show the theoretical prediction from Eq. (1) with $\alpha =1/4+2d/D$. Inset shows dimensionless unsteady force $F_{AM}/F_{S_i}$ as a function of $\rho V\mathrm{\Gamma }/F_{S_i}$ for $4<\mathrm{\Gamma }⩽500$ m ${\mathrm{s}}^{-2}$, $D=15d$, and $\rho ={10}^3$ kg ${\mathrm{m}}^{-3}$ ($\square$), and for $\rho ={10}^2$ kg ${\mathrm{m}}^{-3}$ and ${10}^4$ kg ${\mathrm{m}}^{-3}$ ($\times$). Dashed line shows best linear fit $F_{AM}=0.8\rho V\mathrm{\Gamma }$.