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We use a lattice-gas automaton modeling the formation of density waves of granular flow through a vertical pipe. It is found that both the dissipation and the roughness of the walls of the pipe are essential to the emergence of density waves. The density waves can only be observed when the average density of the system is in a certain range. The power spectra of density fluctuations in one region in the pipe follow, apart from a sharp peak corresponding to the density wave, a power-law spectrum 1/f^ with close to 4/3.
Peng et al. (Tue,) studied this question.