Plane shock waves and Haff’s law in a granular gas

Abstract

The Riemann problem of planar shock waves is analysed for a dilute granular gas by solving Euler-and Navier-Stokes-order equations numerically. The density and temperature profiles are found to be asymmetric, with the maxima of both density and temperature occurring within the shock layer. The density peak increases with increasing Mach number and inelasticity, and is found to propagate at a steady speed at late times. The granular temperature at the upstream end of the shock decays according to Haff's law (θ(t) ∼ t −2 ), but the downstream temperature decays faster than its upstream counterpart. Haff's law seems to hold inside the shock up to a certain time for weak shocks, but deviations occur for strong shocks. The time at which the maximum temperature deviates from Haff's law follows a power-law scaling with the upstream Mach number and the restitution coefficient. The origin of the continual build-up of density with time is discussed, and it is shown that the granular energy equation must be 'regularized' to arrest the maximum density.

Oblique shock waves in granular flows over bluff bodies

“…The upstream Mach number can thus be a misleading metric to characterize the "granular" shock since the obstacle would see a different (and in-fact larger) Mach number upstream from the point of collision with granular particles. A similar conclusion has recently been noted in the case of plane granular shock waves [5]. The non-dimensional detachment length, δ/D, (see Fig.…”

“…Granular flows around an object have been the focus of numerous analytical [1][2][3], experimental [1,2] and simulation studies [1,4].The formation of plane shock waves [5], when a granular medium is perturbed faster than the velocity of sound, has been reported by previous studies. Experimental studies [6,7] have shown that a stagnant zone formation, similar to the classical "bow-shock" detachment (see Fig.…”

“…1), is possible in granular flows. Analytical relations [5] for granular shocks have been developed and Discrete Element Modelling (DEM) simulation studies [4,8] of single component granular flow past obstacles have been reported in literature.…”

Recasting Navier–Stokes equations

“…Namely, mass, momentum and energy conservation laws that govern the motion of a fluid. The classical Navier-Stokes equations in an Eulerian reference frame are given as [16,19]: mass balance/continuity equation where ρ is the mass-density of the fluid, U is the flow mass velocity, p is the hydrostatic pressure and e in is the specific internal energy of the fluid. Further, NS P ( ) and q NS ( ) represent the shear stress tensor and the heat flux vector, respectively, and I is the identity matrix.…”

“…A late suggestion to substitute the Navier-Stokes is given by Svärd [18].Generally, there are a number of experimental data that standard Navier-Stokes fail to predict. Shock wave structure predictions are a ubiquitous example of Navier-Stokes failure [4,6,19,20]. Experimental data of water flows in carbon nanotubes or confined pores is another research topic showing large deviations from the classical Hagen-Poiseuille equation [21,22].…”