Shock Front Width and Structure in Supersonic Granular Flows

Abstract

The full structure of a shock front around a blunt body in a quasi-two-dimensional granular flow is studied. Two features, a large density gradient and a very small thickness of the front, characterize this shock and make it different from shocks in molecular gases. Both of these features can be understood using a modified version of the granular kinetic theory. Our model separates the particles into two subpopulations: fast particles having experienced no collisions and randomly moving particles. This separation is motivated by direct measurements of the particle velocities which show a bimodal distribution. Our results not only shed new light on the use of the granular kinetic theory under extreme conditions (shock formation) but bring new insight into the physics of shocks in general.

Microscopic origin of self-similarity in granular blast waves

“…While the shock front may in principle be described in a kinetic theoretical approach 45 , it is singular from the point of view of hydrodynamics: its width ǫ is microscopic, of order a few mean free paths in the gas at rest, and the velocity distribution of particles in the front is multimodal as some are excited and other immobile (and yet other fall in-between).…”

“…Therefore, the aforementioned experiments 17,18 belong to a corpus of mostly empirical and numerical studies on shocks in other types of granular flow: nonlinear acoustic waves and solitons 37,38 with analytical unidimensional models [39][40][41] , and shocks caused by an obstacle in the flow [42][43][44][45] .…”

Unstable blast shocks in dilute granular flows

“…The front is advancing at a velocity U (t). In the high Mach number limit (which is the case here as M ∼ 10, since the velocity of sound is near 10 cm/s while typical velocities of the front are of order 100 cm/s) [5,8], and in the simplest possible form, we can write the shock conditions as follows:…”

“…As mentioned above, recent work has shown that dilute granular flows may be used to understand the structure of shock waves and the dynamics of blasts [4][5][6][7][8][9]20]. Besides the advantage that blast propagation can be captured with relative ease in granular flows, this system has the additional feature of possessing inelastic collisions so the role of dissipation can be gauged precisely.…”

“…Granular media can be subjected to strong-enough solicitations to produce shocks in dense [1][2][3] as well as dilute [4][5][6][7] materials and blastlike perturbations in the dilute gaseous limit [8,9]. In gases, a blast shock ensues when a large amount of energy is deposited during a short time in a small spatial extent.…”