Evolution of a shock wave in a granular gas

Abstract

Shock wave evolution arising during a steady one-dimensional motion of a piston in a granular gas, composed of inelastically colliding particles is treated by a computational fluid dynamics (CFD) method. It is shown that the flow reaches an asymptotic stationary (final) stage after large evolution time. At this stage particle kinetic energy dissipation leads to formation of two regions within the upstream flow: a fluidized region adjacent to the shock front and a “solid” region adjacent to the piston. In the latter the density is close to the maximum packing density and the kinetic energy of chaotic granular motion is almost zero. The shock wave velocity, mass, and the granular kinetic energy in the fluidized region are found to be constant values, whereas the mass of the solid region grows linearly with time. All properties calculated for the final evolution stage are in excellent agreement with the predictions of the asymptotic solution of the problem obtained earlier. We extend the existing hydrodynamic model of granular gas consisting of hard spheres to provide a plausible description of the static pressure and the speed of sound in the solid block. This is achieved by introducing a cut density ρcut such that dissipation is allowed only if the hydrodynamic density ρ is smaller than ρcut. The computational model reveals that all the above quantities tend to their limiting values in a nonmonotonic manner. In particular, the fluidized mass reaches its maximum when the density at the piston reaches its maximal value. The kinetic energy reaches its maximum earlier than does the fluidized mass. The maximal values of the fluidized mass and the kinetic energy are shown to exceed the comparable long-time limiting values with the difference amounting to about 1.5-fold factor. The calculated results are discussed in relation to several granular flows, characterized by formation of fluidized and compacted regions.