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 and recently an adaptive quadrature has been proposed by Cumber (2000). Arguably, in absence of a comprehensive error analysis, the efficacy of all these quadratures has only been proved for limited collections of radiation problems. In recent work we have established sharp error bounds for the heat flux integral in the DTM for irradiation distributions of three different continuity classes: smooth fields, fields with discontinuous angular derivatives and piecewise constant fields (Versteeg et al, 1999a,b). The resulting error formulae have paved the way for a new adaptive quadrature strategy. We show results of its application to an idealised jet flame and to radiative exchanges inside a cube-shaped enclosure. We also briefly comment on the viability of this approach in general-purpose CFD/radiation computations. Our work demonstrates that the new adaptive angular quadrature has the following capabilities: • Evaluation of DTM heat flux integrals to a pre-specified tolerance for sufficiently smooth intensity distributions. • Excellent accuracy with very low ray numbers for irradiation due small view factor sources. • Good heat flux estimates for piecewise constant sources, provided that the truncation criterion is slightly adjusted and care is taken in specifying the starting mesh