  Granular flow dynamics on a vertically vibrated pile is studied by means of
both laboratory experiments and numerical simulations. As already revealed, the
depth-averaged velocity of a fully-fluidized granular pile under strong
vibration, which is measured by a high-speed laser profiler in the experiment,
can be explained by the nonlinear diffusion transport model proposed by our
previous paper (Tsuji et al, Phys. Rev. Lett. 120, 128001 (2018)). In this
paper, we report that a similar transport model can be applied to the relation
between the surface velocity and slope in the experiment. These facts are also
reproduced by particle-scale numerical simulations based on the discrete
element method. In addition, using these numerical results, the velocity
profile inside the fluidized pile is measured. As a result, we show that the
flow velocity decreases exponentially with depth from the surface of the pile,
which means that a clearly fluidized region, also known as shear band
structure, is localized around the surface. However, its thickness grows
proportionally with the local height of the pile, i.e., the shear band does not
consist of a fluidized layer with a constant thickness. From these features, we
finally demonstrate that the integration of this exponentially-decreasing
velocity profile is consistent with the depth-averaged velocity predicted by
the nonlinear diffusion transport model.
