Unsteady expansion of an ideal gas into a vacuum

Abstract

The unsteady expansion of an ideal gas into a vacuum is studied in one-dimensional planar and spherical geometries. The free-molecular expansion of a Maxwell-distributed gas is compared to the continuum expansion of a perfect gas with $\gamma = \frac{5}{3}$. Time histories of density, temperature, and wall pressure (i.e. pressure on a wall surface oriented normal to the flow) are given at four near-field locations, and the approach to far-field behaviour is illustrated. In the free-molecular limit, closed-form expressions for the wall pressure, translational temperature, and fluxes of momentum, kinetic energy, and thermal energy have been obtained in addition to previously published results for density and velocity. The density and dynamic fluxes are observed to decay more rapidly in the tails of continuum pulses than in free-molecular pulses. The reverse is true for wall pressure, which decays less rapidly in continuum flow. Translational temperature, in the free-molecular case, rises discontinuously upon pulse arrival, and, at long times approaches $\frac{2}{3}$ for planar flow and tends to zero for spherical flow. Continuum thermodynamic temperature pulses, on the other hand, rise and fall in simple relation to continuum density. The far-field peak wall pressure in both Knudsen-number limits is found to decrease in inverse (or inverse cubic) proportion to the distance from the initial planar (or spherical) region. This result for the spherical case is at odds with the experiments of Ahrens, Allen & Kovach (1971) which indicate a more rapid (ξ^−3.5) fall-off of peak overpressure with distance from a point source in a vacuum.