Angular False Scattering in Radiative Heat Transfer Analysis Using the Discrete-ordinates Method with Higher-order Quadrature Sets

The SN quadrature set for the discrete-ordinates method is limited in overall discrete direction number in order to avoid physically unrealistic negative directional weight factors. Such a limitation can adversely impact radiative transfer predictions. Directional discretization results in errors due to ray effect, as well as angular false scattering error due to distortion of the scattering phase function. The use higher-order quadrature schemes in the discrete-ordinates method allows for improvement in discretization errors without an overall directional limitation. In this analysis, four higher-order quadrature sets (Legendre-Equal Weight, Legendre-Chebyshev, Triangle Tessellation, and Spherical Ring Approximation) are implemented for determination of radiative transfer in a 3-D cubic enclosure containing participating media. Radiative heat fluxes, calculated at low direction number, are compared to the SN quadrature and Monte Carlo predictions to gauge quadrature accuracy. Additionally, investigation into the reduction of angular false scattering with sufficient increase in direction number using higher-order quadrature, including heat flux accuracy with respect to Monte Carlo and computational efficiency, is presented. While higher-order quadrature sets are found to effectively minimize angular false scattering error, it is found to be much more computationally efficient to implement proper phase function normalization for accurate radiative transfer predictions. INTRODUCTION One of the most commonly implemented approximate methods for determining radiative transfer through numerical solution of the Equation of Radiative Transfer (ERT) is the Discrete-Ordinates Method (DOM) [1-3], first proposed by Chandrasekhar [1] for atmospheric and astrophysical radiation, and later adapted to solve both the neutron-transport equation [4] and for use in radiative heat transfer. In many processes, such as high-temperature combustion and material processing [5-8] or biomedical therapeutic applications involving the interaction of ultrafast laser light with living tissue [9-13], radiation is the dominant mode of heat transfer, and thus complete and accurate ERT solutions are required for accurate physical modeling. In practical applications where light scattering exists, analytic solution of the ERT is extremely difficult, mandating the necessity for accurate yet efficient numerical solutions of the ERT. Fiveland [14,15] and Truelove [16] were among the first to apply the DOM in the field of radiation, determining steadystate radiative transfer in both 2-D and 3-D enclosures containing participating media. Later, Guo and Kumar extended use of the DOM for accurate determination of ultrafast radiative transfer in participating media [17,18] through solution of the transient ERT. Further works by Guo and co-authors implemented the transient DOM (TDOM) to accurately and efficiently model short-pulsed irradiation of turbid media [12,13], laser-tissue welding and soldering [11], and pulse train irradiation using Duhamel’s superposition theorem [6,19]. The DOM uses a finite set of discrete radiation directions, with corresponding directional weight factors, to approximate the continuous angular variation. Although the choice of discrete quadrature set is generally arbitrary, the directions must satisfy certain moment conditions [20]. The levelsymmetric S! quadrature [20,21] is one of the most commonly implemented quadrature sets, where the total number of directions is M = N(N + 2). This quadrature set, however, is limited in total discrete direction number, as the directional weighting factors become unrealistically negative for S!" and greater in order to satisfy the specific moment-matching criteria [21]. The DOM is known to suffer from ray effect [22] due to the inexact approximation of the continuous angular variation,