The response of a compressible gas to extremely rapid transient, spatially resolved energy addition: an asymptotic formulation

Abstract

An asymptotics-based analysis of the Navier–Stokes equations is used to study the response of an inert gas volume (the near-field) to localized, spatially distributed, transient energy deposition per unit mass, significantly larger than the ambient specific internal energy in the volume. The specified heat-addition time scale is chosen to be much less than the initial characteristic acoustic time of the affected volume. The evolutionary process in the near-field depends fundamentally on the ratio of energy deposition relative to the initial internal energy in the heated volume. A wide range of ratio values is found to be consistent with near-inertial confinement in the heated gas. The pressure rises directly with the very large temperature, the density is nearly constant and the internal Mach number of expansion is very small. The near-field response provides a piston-like source of compression waves in the surrounding cold environment (the far-field). When the energy ratio reaches an explicit, extreme value, inertial confinement fails, the gas dynamics are described by fully compressible flow equations, the temperature and pressure increases are extreme, density variations are modest and the internal and expansion Mach numbers are sonic. These near-field gas physics are shown to be compatible with the singular properties of far-field similarity solutions for strong blast waves, considered originally by von Neumann (The point source solution, NDRC, Div. B Report AM-9, 1941; revised in 1944; reprinted in von Neumann J (1963) In: Taub AH (ed) Collected works, vol VI. Pergamon Press, New York, pp. 219–237), Taylor (Proc Roy Soc Lond A 186:273–292, 1946), and Sedov (Prikl Mat Mekh 10:241–250, 1946). An asymptotic reformulation of the strong-blast-wave problem, using concepts similar to those in the near-field study, is used to rationalize and quantify ad hoc approximations by the aforementioned researchers.