Coordinate systems for the radiative transfer equation in curvilinear media

Abstract

Spatial/directional coordinate systems are presented on which general directional boundary conditions may be satisfied for the solution of the radiative transfer equation in curvilinear, participating media. The general directional boundary conditions described herein may be applied to curvilinear media without regard to spatial symmetry, although their application depends upon use of a local coordinate system, defining the intensity direction, which has a polar axis directed normal to any nearby surfaces. In the most commonly used spatial/directional coordinate systems for cylindrical and spherical media, the orientation of the directional polar axis indicates that the non-symmetric boundary conditions may be applied only to, respectively, parallel disk and spherical annulus boundary geometries. Application of the general directional boundary conditions to non-symmetric curvilinear media of these geometries, referred to only partially or obliquely in previous works, is demonstrated completely and consisely in the paper. Also, new spatial/directional coordinate systems are derived, based on reorientation of the local directional coordinate system in spatially cylindrical and spherical geometries. These new coordinate systems allow application of the general directional boundary conditions to other curvilinear geometries, such as a cylindrical annulus, cylindrical wedge, conical annulus, and spherical wedge. For each coordinate system, expressions for the pathlength derivative of radiation intensity are given in simple, conservative, and discrete ordinates form. The techniques illustrated in the derivation of these spatial/directional coordinate systems for non-symmetric curvilinear media may be applied to other, more complicated boundary geometries as well.