A modification of the differential approximation for radiative transfer.

Abstract

In radiation-gas dynamics problems, frequent use has been made of the differential approximation, which involves replacing the integral equation of transfer by a differential equation for the flux. This differential approximation may be inaccurate at small optical depths if external radiation sources are present. A procedure that correctly accounts for the external radiation is developed in this paper. The resulting modified differential approximation is first applied to the one-dimensional problem of a medium in radiative equilibrium between two parallel walls held at different temperatures. The calculated flux is superior to that obtained using the unmodified differential approximation, especially in the optically thin limit where the modified differential approximation gives the correct (though small) absorption of the medium. The modified differential approximation is also used to solve the two-dimensional problem of linearized flow of a radiating gas over a wall with a sinusoidal temperature distribution. With a characteristic optical depth defined with respect to the wavelength of the wall temperature variation, it is found that the modified differential approximation gives accurate values for the flow variables over the entire opacity range, yielding the correct values in the optically thin limit. On the other hand, the unmodified differential approximation yields values that become increasingly inaccurate as the characteristic optical depth becomes small.