The decay of perturbations in a radiating gas

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Abstract
The decay of perturbations in a radiating gas is analyzed by first carrying out complete solutions for the decay of initial sinusoidal perturbations in the temperature, gas velocity, and pressure. These sinusoidal perturbations are superposed to yield solutions for the decay of initial ‘step’ temperature profiles consisting of constant initial temperature perturbations inside finite planar, cylindrical and spherical regions, with zero initial temperature perturbations outside. In contrast to the sinusoidal case, which may be described by a single radiation parameter, the decay of the step profile is determined by both the optical depth of the initial profile and the Boltzmann number, which is inversely proportional to the blackbody radiative flux. As the limits of zero and infinite Boltzmann numbers are approached, constant-density and constant-pressure cooling expressions are recovered. For a broad range of intermediate and small Boltzmann numbers the cooling proceeds in time from a constant-density process to a constant-pressure process. This transition is produced by gasdynamic waves generated near the profile edges by the radiative cooling. The temperature near the profile centre may increase during the transition period.

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