Recent Studies on Two-Dimensional Radiative Transfer Problems in Anisotropic Scattering Media

Abstract

In this work, an explicit formulation to solve two-dimensional radiative transfer problems in anisotropic scattering media is developed. A nodal technique along with the Analytical Discrete Ordinates (ADO) method are used to solve the discrete ordinates approximation of the radiative transfer equation, in Cartesian geometry. To make it possible, the discrete ordinates equations are transversally integrated over regions of the domain reducing the complexity of the model, yielding two one-dimensional equations for average angular intensities in x and y directions. The one-dimensional equations, with approximations for the unknown intensities on the contours of the regions, are then explicitly solved by the ADO method, with respect to the spatial variables, whose solutions are written in terms of eigenvalues and eigenfunctions. The phase function is expanded in terms of Legendre polynomials up to arbitrary order L, to model higher order anisotropy. The eigenvalue problem is derived for this general case and it preserves a relevant feature of the ADO method, which is the reduced order equal to half of the number of discrete directions. Numerical results for the average radiation density and radiative heat flux are presented, for test cases in which the degree of anisotropy can be up to twelve and the albedo assumes different values. A comparative analysis with results available in the literature allows the verification of the formulation and indicates a good performance of the proposed method in coarser meshes.

Keywords:

• Radiative transfer
• anisotropic scattering
• two-dimensional geometry
• discrete ordinates
• ADO method

Additional information

Funding

This study was financed in part by: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001; Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brasil (CNPq); Fundação de Amparo à Pesquisa do Estado do RS - Brasil (FAPERGS) - Proc. 19/2551–0001766–9. The authors would like to thank also the support of the Centro Nacional de Supercomputação (CESUP), Universidade Federal do Rio Grande do Sul (UFRGS).