Microscopic origin of self-similarity in granular blast waves

Abstract

The self-similar expansion of a blast wave, well-studied in air, has peculiar counterparts in dense and dissipative media such as granular gases. Recent results have shown that, while the traditional Taylor-von Neumann-Sedov (TvNS) derivation is not applicable to such granular blasts, they can nevertheless be well understood via a combination of microscopic and hydrodynamic insights. In this article, we provide a detailed analysis of these methods associating Molecular Dynamics simulations and continuum equations, which successfully predict hydrodynamic profiles, scaling properties and the instability of the self-similar solution. We also present new results for the energy conserving case, including the particle-level analysis of the classic TvNS solution and its breakdown at higher densities.

Blast in a One-Dimensional Cold Gas: From Newtonian Dynamics to Hydrodynamics

“…Thus we numerically establish that the Euler equations completely describing the hydrodynamic behavior of the AHP gas. Earlier attempts in two-and three-dimensional systems [18][19][20][21][22][23] have not been able to obtain the unambiguous agreement as reported here-this is because the 1D hard point gas is an ideal model for this study. It's equilibrium properties including the equation of state are known exactly which allows us to obtain the TvNS scaling functions exactly.…”

“…Secondly, this system can be simulated very efficiently using event driven simulations. However, large scale molecular dynamics (MD) simulations [16][17][18][19][20][21][22][23] have so far not found clear and complete agreement with the TvNS solution from hydrodynamics. It was suggested that possible reasons for the differences could be the lack of local equilibration or due to the contribution of viscosity and heat conduction in the hydrodynamic equations (not included in the TvNS analysis).…”

Blast waves in two and three dimensions: Euler versus Navier Stokes equations

“…If local thermal equilibrium is assumed, as is usually done, then the local pressure p is expressed in terms of local density ρ and local temperature T through an equation of state (EOS). While the ideal gas EOS was used in the original TvNS solution, for a hard sphere gas, steric effects become important and a more realistic EOS would be required in order to compare with results from hard sphere simulations [35,21,36]. We choose the EOS to be the virial EOS, which takes the form…”

“…TvNS theory has been used to study systems where energy is continuously input in a localized region of space [18,19]. TvNS theory has also been generalized for granular systems, where conserved quantities are fewer in number because energy is no longer conserved [20,21]. There are many examples in granular systems where a blast wave is generated when perturbed by either a single impact or by continuous energy injection.…”

“…Later work focused on verifying the theoretical predictions of the scaling functions for density, radial velocity, temperature and pressure. Among these, one study of the scaling functions in two dimensions concluded that the TvNS theory describes well the simulation data for low to medium number densities except for a small difference near the shock front and slight discrepancy near the shock center [21]. However, by performing more extensive and large scale simulations, we found that the numerically obtained radial distribution of density, radial velocity as well as the temperature do not match with the TvNS theory (modified to account for steric effects) neither in two dimensions [35] nor in three dimensions [36].…”