RECENT BENCHAMRKINGS OF RADIATIVE HEAT TRANSFER WITHIN NONHOMOGENEOUS PARTICIPATING MEDIA AND THE IMPROVED YIX METHOD

Abstract

A review of the recent benchmark efforts since the First Symposium on Solution Methods for Radiative Heat Transfer in Participating Media is presented. The Symposium was first held at 1992 28th National Heat Transfer Conference and then at 1994 6th AIAA/ASME Joint Thermophysics and Heat Transfer Conference. Also presented is the continuing effort to improve the solution accuracy of the YIX method for benchmarking. The latest work is focused on multi-dimensional, gray, and nonhomogeneous participating media. Three higher order interpolation schemes, i.e., piecewise linear, tri-linear, and tri-quadratic, are presented to improve solution accuracy in generating benchmarks with YIX method. Detail and systematic error analyses indicate that superconvergence exists for these interpolations. Significant saving in computational time and memory can be achieved with high order interpolation. This has important implications when coupling the RTE calculations with flow codes. Alongside the interpolation error, the integration errors of the YIX method, which include distance and angular quadratures, are also examined. The use of discrete ordinates sets in the angular quadrature is studied rigorously and compared with the use of Simpson rule. The use of high order distance quadrature is discussed. The main intent of these results is to provide a verified set of solutions which can be useful as benchmarks when developing other methods. NOMENCLATURE Cm function space where a function and its partial derivatives of order less than or equal to m are continuous within the definition domain e blackbody emissive power of the medium [W/m2] h grid spacing K kernel function of the radiation transport integral equations L characteristic length of the geometry [m] N order of the discrete ordinates (S-N) set Nw number of angular quadrature points n unit surface normal vector q radiative heat flux vector [W/m2] q (surface) radiative heat flux [W/m2] r position vector, (x, y, z) β coefficient of the linear anisotropic scattering phase function κ extinction coefficient [m-1] τ optical thickness, κL Φ scattering phase function Ω computational domain ω direction vector Subscripts g volume quantity s surface quantity x, y, z coordinate axes Superscript ' in-scattering direction, dummy argument