ABSTRACT
In the granular system, although pointed out already by several authors, there are very few studies related to the turbulence, since the dissipation mechanism of viscosity in NS equation is viewed as completely different from that of the inelastic collisions. Here, we summarize previous studies on the energy spectra in the granular systems: First Taguchi obtained numerically that the energy spectra from the particle displacements in the mid-depth of 2D vibrated granular bed, which was composed of a few hundred particles, have a −5/3 exponent, similar to Kolmogorov scaling in 3D turbulence (Taguchi 1993). This was a pioneer work on this topic, however, there seem to be many indefinite factors to determine the energy spectra and the granular dynamics (e.g. a periodical cycle of energy injection, a strong dependence of the time resolution of particle displacements, the influence of the anisotropic gravity.) Then, except for the influence of gravity, a 2D granular gas with a Langevin type thermostat was studied in a system with about 1000 particles (Peng & Ohta 1998). However, these authors also introduced a periodical cycle of a driven thermostat like in a vibrated bed, and they cannot show any exponent related to the NS turbulence. Recently, (Radjai & Roux 2002) have reported the spatial power-law spectrum of the particle displacements like Taguchi’s work in the quasi-static motions of the dense granular flow with a boundary of Parrinello-Rahman type in a 2D MD model. They showed that the scaling behavior of the energy spectrum was different from that of 2D turbulence. All these studies were done in 2D and showed the
power-law scaling of the energy spectrum and pointed out some relationships to NS turbulence. However, the special features of the 2D NS turbulence have not been shown. The reasons might be the following: (1) Statistical errors are quite large because of the small particle number, (2) the system size is too small and is comparable to the minimal spatial correlation length (such as the Kolmogorov dissipative length), (3) no systematic parameters survey exists, and (4) the unique features of 2D turbulence have not been well-known in the granular physics community, and the enstrophy cascades in 2D turbulence seem to be not motivated.
