Comparison of Phase Function Normalization Techniques for Radiative Transfer Analysis Using Dom

Five phase-function (PF) normalization techniques are compared using the discrete-ordinate method (DOM) for modeling diffuse radiation heat transfer in participating media. Both the mathematical formulation and the impact on the conservation of both scattered energy and PF asymmetry factor for both Henyey-Greenstein (HG) and Legendre PF distributions are presented for each technique. DOM radiation transfer predictions generated using the five normalization techniques are compared to high-order finite-volume method, to gauge their accuracy. The commonly implemented scattered energy averaging technique cannot correct asymmetry factor distortion after angular discretization, and thus large errors due to angular false scattering are prevalent. Another three simple techniques via correction of one or two terms in the PF are shown to reduce normalization complexity whilst retaining diffuse radiation computation accuracy for HG PFs. However, for Legendre PFs, such simple normalization is found to result in unphysical negative PF values at one or few correction directions. The relatively complex Hunter and Guo 2012 technique, in which normalization is realized through a correction matrix covering all discrete directions, is shown to be highly applicable for both PF types. INTRODUCTION Originally proposed as a method of determining astrophysical radiation [1], and later adopted as a method of solving the neutron-transport equation [2], the DiscreteOrdinates Method (DOM) has become a popular numerical method for evaluating radiation heat transfer via solution of the Equation of Radiation Transfer (ERT). Use of the DOM for determining steady-state radiation heat transfer was pioneered in the 1980’s [3,4]. In the following two decades, an important extension of the DOM to solve the transient hyperbolic ERT was proposed [5,6], in order to accurately determine ultrafast radiation transfer in participating media. While popular and easy to implement, the DOM suffers from two major numerical shortcomings: numerical smearing error due to spatial discretization, and ray effect error due to angular discretization [7,8]. High-order numerical schemes were considered to mitigate numerical smearing error [8]. Reduction of ray effect error has been achieved via use of different quadrature schemes [9-11]. Additionally, DOM with unstructured grids has also been developed for use with irregular geometries [12]. It is well known that the conservation of anisotropically scattered energy is broken after discretization of the continuous angular variation of radiation scattering into a finite set of discrete directions using the DOM [13, 14]. However, it has recently become clear that angular discretization additionally distorts the phase-function (PF) asymmetry factor [15-17] for processes where scattering is highly anisotropic, which can include many practical scattering media such as packed beds and biological tissues. PF normalization techniques become a popular method to correct these non-conservations. Multiple normalization methods have been proposed in literature, with the majority of them able to conserve either scattered energy [13, 14, 19] or asymmetry factor [20], but not both. Hunter and Guo developed a technique that can simultaneously conserve both quantities for both Henyey-Greenstein (HG) [16] and Legendre PFs [18] through use of a normalization matrix, allowing for more accurate radiation transfer predictions in 3-D cubic enclosures [21]. Other recent normalization techniques developed by Mishchenko et al. [19] and Kamdem Tagne [20] wisely tackled the issue of conserving either scattered energy or asymmetry factor in a simple manner. Rather than normalizing every value of the discrete PF, as had been done in many previous techniques, they proposed normalization of solely the forwardscattering phase-function term for HG PFs. Such PF treatment retains the PF value for most discrete direction combinations