An adaptive angular quadrature for the discrete transfer method based on error estimation

The ray effect is a serious problem in radiative heat transfer computations. Continuously varying radiation fields are approximated numerically by sampling a limited number of angular directions. The discrete transfer method (DTM) is a conceptually simple technique suitable for general-purpose calculations of thermal radiation in complex geometries. Over the years a large variety of quadratures based on fixed ray firing patterns has been suggested for use in conjunction with the DTM. Arguably, in absence of a comprehensive error analysis, the efficacy of all these quadratures has only been proved for limited collections of radiation problems. Recently, sharp error bounds for the heat flux integral in the DTM have been established for irradiation distributions of three different continuity classes: smooth fields, fields with discontinuous angular derivatives and piecewise constant fields. The resulting error formulas have paved the way for a new adaptive quadrature strategy. Results are presented of its application to an idealized jet flame and to radiative exchanges inside a cube-shaped enclosure, along with brief comments on the viability of this approach in general-purpose CFD/radiation computations. In this paper, the following capabilities of the new adaptive angular quadrature are demonstrated: Evaluation of DTM heat flux integrals to a pre-specified tolerance for any intensity distribution; Excellent accuracy with very low ray numbers for irradiation due to small view factor sources; and Good heat flux estimates for piecewise constant sources, provided that the starting mesh is selected carefully.