Size scaling relation of velocity field in granular flows and the Beverloo law

摘要

In a hopper with cylindrical symmetry and an aperture of radius R, the vertical velocity of granular flow v z depends on the distance from the hopper's center r and the height above the aperture z and v z = v z (r, z; R). We propose that the scaled vertical velocity v z (r, z; R)/v z (0, 0; R) is a function of scaled variables r/R r and z/R z, where R r = R - 0.5d and R z = R - k 2 d with the granule diameter d and a parameter k 2 to be determined. After scaled by v 2 z (0, 0; R)/R z, the effective acceleration a eff (r, z; R) derived from v z is a function of r/R r and z/R z also. The boundary condition a eff (0, 0; R) = - g of granular flows under earth gravity g gives rise to v z (0, 0; R). v g R - k 2 d 1/2. Our simulations using the discrete element method and GPU program in the three- dimensional and the two- dimensional hoppers confirm the size scaling relations of v z (r, z; R) and v z (0, 0; R). From the size scaling relations, we obtain the mass flow rate of D- dimensional hopper W. v g(R - 0.5d) D- 1 (R - k 2 d) 1/2, which agrees with the Beverloo law at R d. It is the size scaling of vertical velocity field that results in the dimensional R- dependence of W in the Beverloo law.