Brian Hunter

Comparison of the Discrete-Ordinates Method and the Finite-Volume Method for Steady-State and Ultrafast Radiative Transfer Analysis in Cylindrical Coordinates

The time-dependent equation of radiative transfer is solved for an axisymmetric cylindrical medium using both the discrete-ordinates method and the finite-volume method. Steady and transient flux profiles are determined for absorbing and scattering media. Results for each solution method are compared and shown for various grid numbers, scattering albedos, and optical thicknesses. A comparison of computational time and memory usage between the methods is presented. It is found that the finite-volume method uses more memory and has a longer convergence time than the discrete-ordinates method for all cases, due to the difference in angular treatment.

Phase-function normalization for accurate analysis of ultrafast collimated radiative transfer

The scattering of radiation from collimated irradiation is accurately treated via normalization of phase function. This approach is applicable to any numerical method with directional discretization. In this study it is applied to the transient discrete-ordinates method for ultrafast collimated radiative transfer analysis in turbid media. A technique recently developed by the authors, which conserves a phase-function asymmetry factor as well as scattered energy for the Henyey-Greenstein phase function in steady-state diffuse radiative transfer analysis, is applied to the general Legendre scattering phase function in ultrafast collimated radiative transfer. Heat flux profiles in a model tissue cylinder are generated for various phase functions and compared to those generated when normalization of the collimated phase function is neglected. Energy deposition in the medium is also investigated. Lack of conservation of scattered energy and the asymmetry factor for the collimated scattering phase function causes overpredictions in both heat flux and energy deposition for highly anisotropic scattering media. In addition, a discussion is presented to clarify the time-dependent formulation of divergence of radiative heat flux.

Phase-Function Normalization in the 3-D Discrete-Ordinates Solution of Radiative Transfer—PART I: Conservation of Scattered Energy and Asymmetry Factor

The conditions for which conversation of scattered energy and phase-function asymmetry factor after discrete-ordinates methods (DOM) directional discretization for 3-D radiative transfer in anisotropic scattering media breaks down are examined. Directional discretization in anisotropic scattering media is found to alter the scattering asymmetry factor—a second-type of “false scattering.” Phase-function normalization which conserves scattered energy alone cannot correct this problem, and conservation of the asymmetry factor is simultaneously required. A normalization technique developed by the authors, which was successfully tested in 2-D asymmetric cylindrical-coordinate radiative transfer analysis, is intensively examined and validated with benchmark problems in 3-D Cartesian coordinates. In Part I of this study, the degree of anisotropy for which normalization is necessary to conserve these inherent quantities is presented for various phase-function approximations and discrete quadrature sets.

Phase-Function Normalization in the 3-D Discrete-Ordinates Solution of Radiative Transfer—PART II: Benchmark Comparisons

Radiative transfer in a cubic enclosure, subject to varying conditions, is determined using the discrete-ordinates method (DOM) with the two normalization techniques introduced in Part I of this study. Their predictions are compared with Monte Carlo simulations. For all cases, false scattering due to directional discretization cannot be corrected when the old technique, which solely conserves scattered energy, is implemented; and thus, significant discrepancies exist when compared to Monte Carlo results. The new technique, which conserves both scattered energy and the asymmetry factor, is able to retain original scattering properties after directional discretization, leading to improved accuracy when compared to Monte Carlo. In addition, a parametric study is presented to gauge the impact of asymmetry-factor conservation on media with various optical properties. Finally, the impact of normalization is investigated for both ultrafast radiative transfer and ballistic incidence with varying incident angle.

Comparison of Quadrature Schemes in DOM for Anisotropic Scattering Radiative Transfer Analysis

The commonly implemented level-symmetric S N quadrature set for the discrete-ordinates method suffers from a limitation in discrete direction number to avoid physically unrealistic weighting factors. This limitation can have an adverse impact for determining radiative transfer, as directional discretization results in angular false scattering errors due to distortion of the scattering phase function in addition to the ray effect. To combat this limitation, several higher-order quadrature schemes with no directional limitation have been developed. Here, 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. Heat fluxes obtained at low direction number are compared to the S N quadrature and Monte Carlo predictions to gauge and compare quadrature accuracy. Investigation into the reduction/elimination of angular false scattering with increase in direction number, including heat flux accuracy with respect to Monte Carlo and computational efficiency, is presented. It is found that while the higher-order quadrature sets are able to effectively minimize angular false scattering, the number of directions required is extremely large, and thus it is more computationally efficient to implement proper phase-function normalization to obtain accurate results.

Normalization of Various Phase Functions for Radiative Heat Transfer Analysis in a Solar Absorber Tube

Normalization of various phase functions is considered for accurately predicting radiative heat transfer. A solar absorber tube filled with anisotropic scattering working medium is used as an example. Analysis of a previous normalization technique shows that while it does conserve scattered energy exactly after discrete-ordinates method (DOM) discretization, the overall asymmetry factor of the phase function is distorted, leading to substantial changes in overall scattering effect. A new normalization technique that conserves asymmetry factor and scattered energy simultaneously is investigated. The impact of lack of asymmetry factor conservation is analyzed for both the Legendre polynomial and the Henyey–Greenstein phase function approximations. Variations of medium optical thickness, scattering albedo, asymmetry factor, and side-wall emissivity are scrutinized to determine the effects of said parameters on wall heat flux and energy absorbing rate inside the absorber tube. Side-wall heat flux is found to increase with increases in asymmetry factor, optical thickness, and wall emissivity, and with decreases in scattering albedo. Energy absorbing rate profiles are found to depend greatly on optical thickness and scattering albedo.

Improved Treatment of Anisotropic Scattering in Radiation Transfer Analysis Using the Finite Volume Method

Discretization of the integral anisotropic-scattering term in the equation of radiative transfer will result in two kinds of numerical errors: alterations in scattered energy and asymmetry factor. Though quadrature flexibility with large angular directions and further solid-angle splitting in the finite volume method (FVM) allow for reduction/minimization of these errors, computational efficiency is adversely impacted. A phase-function normalization technique to get rid of these errors is simpler and is applied to the three-dimensional (3-D) FVM for the first time to improve anisotropic radiation transfer computation accuracy and efficiency. FVM results are compared to Monte Carlo and discrete-ordinates method predictions of radiative heat transfer in a cubic enclosure housing a highly anisotropic participating medium. It is found that the FVM results generated using the normalization technique conform accurately to the results of the other two methods with little impact on computational efficiency.

Applicability of Phase-Function Normalization Techniques for Radiation Transfer Computation

The applicability of recently developed four phase-function (PF) normalization techniques for modeling radiation transfer in strongly anisotropic scattering media is intensively examined using the discrete-ordinates method. The three simple techniques via normalization of only the forward- and/or backward-scattering directions have been shown to reduce normalization complexity while retaining diffuse radiation computation accuracy for Henyey-Greenstein (HG) PFs. For Legendre PFs, however, such simple techniques are found to result in unphysical negative PF values at one or a few correction directions in some cases. Additionally, negative PF values can occur with these simple techniques for ballistic radiation transfer for both HG and Legendre PF types. If negative-intensity correction is applied, however, radiative heat transfer calculation can still converge regardless of the appearance of negative PF values. 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 applicable for diffuse and ballistic radiation for both PF types.

Improved Treatment of Anisotropic Scattering for Ultrafast Radiative Transfer Analysis

The necessity of conserving both scattered energy and asymmetry factor for ballistic incidence after FVM or DOM discretization is shown. A phase-function normalization technique introduced previously by the present authors is applied to scattering of ballistic incidence in 3-D FVM/DOM to improve treatment of anisotropic scattering through reduction of angular false scattering errors. Ultrafast radiative transfer predictions generated using FVM and DOM are compared to benchmark Monte Carlo to illustrate the necessity of ballistic phase-function normalization. Proper ballistic phase-function treatment greatly improves predicted heat fluxes and energy deposition for anisotropic scattering and for situations where accurate numerical modeling is crucial. 1 Author to whom correspondence should be addressed Journal of Heat Transfer SEPTEMBER 2015,Vol.137, ar. no. 091004. DOI: 10.1115/1.4030211 2

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.Copyright