Size Scaling of Velocity Field in Granular Flows through Apertures

For vertical velocity field $v_{\mathrm{z}}(r,z;R)$ of granular flow through an aperture of radius $R$, we propose a size scaling form $v_{\rm z}(r,z;R)=v_{\rm z} (0,0;R)f (r/R_{\rm r}, z/R_{\rm z})$ in the region above the aperture. The length scales $R_{\mathrm{r}}=R-0.5d$ and $R_{\mathrm{z}}=R+k_{2}d$, where $k_2$ is a parameter to be determined and $d$ is the diameter of granule. The effective acceleration, which is derived from $v_{\mathrm{z}}$, follows also a size scaling form $a_{\rm eff} = v_{\rm z}^2(0,0;R)R_{\rm z}^{-1} \theta (r/R_{\rm r}, z/R_{\rm z})$\mathrm{.}Forgranularflowundergravity$g$, there is a boundary condition $a_{\mathrm{e}\mathrm{f}\mathrm{f}}(0,0;R)=-g$ which gives rise to $v_{\rm z} (0,0;R)= \sqrt{ \lambda g R_{\rm z}}$with$\lambda=-1/\theta (0,0)$. Using the size scaling form of vertical velocity field and its boundary condition, we can obtain the flow rate $W=C_2\rho \sqrt{g}{R}_{\mathrm{r}}^{D-1}{R}_{\mathrm{z}}^{1\mathrm{/}2}$, which agrees with the Beverloo law when $R\gg d$. The vertical velocity fields $v_z(r,z;R)$ in three-dimensional (3D) and two-dimensional (2D) hoppers have been simulated using the discrete element method (DEM) and GPU program. Simulation data confirm the size scaling form of $v_{\mathrm{z}}(r,z;R)$ and the $R$-dependence of $v_{\mathrm{z}}(0,0;R)$.