Discrete ordinates solution of the radiative transfer equation in media with strong forward and backward scattering subjected to collimated irradiation

Abstract

Highly anisotropic phase functions are often approximated by simpler ones given by the sum of a Dirac delta function and a smooth function. However, if both forward and backward scattering are important, two Dirac delta functions are needed. In problems without collimated radiation, those phase functions can be easily handled using the discrete ordinates method (DOM). However, when collimated irradiation is present, the DOM cannot be applied using the decomposition of the radiation intensity into a diffuse and a collimated component. A new formulation of the DOM to solve radiative transfer problems with collimated irradiation in anisotropically scattering media with such approximate phase functions is described in this work. The proposed method is based on a decomposition of the radiation intensity into three components, namely a collimated, a backscattered collimated and a diffuse component. The method is applied to problems without and with collimated radiation and its accuracy is assessed through comparison with results obtained from Monte Carlo simulations. When a linear approximate phase function is used, the new formulation of the DOM yields accurate results without collimated irradiation, but does not perform so well with collimated irradiation, particularly for media with an optical thickness of the order of unity or lower. However, if the linear approximation is replaced by a quadratic one, the results are significantly improved.