The sudden expansion of a gas cloud into vacuum revisited

The sudden expansion into vacuum of a gas cloud, with initial centrally symmetric density and temperature profiles, is studied theoretically for different values of the specific heat ratio γ. Models treating the expansion are discussed, in particular, a model for isentropic expansion and a model for spatially isothermal expansion. For γ→1, the density of the gas obtained from the former model for late stages of the expansion, approaches a Gaussian spatial profile which is the exact solution to the latter model. A description by a Gaussian profile can be, for some important cases, approximately correct even for large deviations of γ from one. For a spherically symmetric flow, the maximum difference (for any given time and distance from the center of symmetry) between the densities obtained from the above two models is 11% for γ=7/5. For γ=1.28, which corresponds to the expansion of lead azide detonation products previously studied in the authors' laboratory, the difference is 9%. It is also shown that in practice it is more convenient to use the model for isothermal expansion to describe the density profile since it does not depend on γ, which is very often not exactly known. Finally, for γ→1, a relation between the density and the temperature is obtained which is not dependent on their initial distributions.