In a hopper with cylindrical symmetry and an aperture of radius R, the vertical velocity of granular flow $$v_z$$ vz depends on the distance from the hopper’s center r and the height above the aperture z and $$v_z = v_z (r,z;\,R)$$ vz=vz(r,z;R) . We propose that the scaled vertical velocity $$v_{z}(r,z;\,R)/v_{z} (0,0;\,R)$$ vz(r,z;R)/vz(0,0;R) is a function of scaled variables $$r/R_r$$ r/Rr and $$z/R_z$$ z/Rz , where $$R_{ r}=R- 0.5 d$$ Rr=R-0.5d and $$R_{ z}=R-k_2 d$$ Rz=R-k2d with the granule diameter d and a parameter $$k_2$$ k2 to be determined. After scaled by $$v_{ z}^2 (0,0;\,R)/R_z $$ vz2(0,0;R)/Rz , the effective acceleration $$a_{\mathrm{eff}} (r,z;\,R)$$ aeff(r,z;R) derived from $$v_z$$ vz is a function of $$r/R_r$$ r/Rr and $$z/R_z$$ z/Rz also. The boundary condition $$a_\mathrm{eff} (0,0;\,R)=-\,g$$ aeff(0,0;R)=-g of granular flows under earth gravity g gives rise to $$v_{ z} (0,0;\,R) \propto \sqrt{g}\left( R -k_2 d\right) ^{1/2}$$ vz(0,0;R)∝gR-k2d1/2 . 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 $$v_{ z} (r,z;\,R)$$ vz(r,z;R) and $$v_{ z} (0,0;\,R)$$ vz(0,0;R) . From the size scaling relations, we obtain the mass flow rate of D-dimensional hopper $$W \propto \sqrt{g } (R-0.5 d)^{D-1} (R-k_2 d)^{1/2}$$ W∝g(R-0.5d)D-1(R-k2d)1/2 , which agrees with the Beverloo law at $$R\gg d$$ R≫d . It is the size scaling of vertical velocity field that results in the dimensional R-dependence of W in the Beverloo law.