Radiative heat transfer in strongly forward scattering media using the discrete ordinates method

Abstract

International audienceThe discrete ordinates method (DOM) is widely used to solve the radiative transfer equation, often yielding satisfactory results. However, in the presence of strongly forward scattering media, this method does not generally conserve the scattering energy and the phase function asymmetry factor. Because of this, the normalization of the phase function has been proposed to guarantee that the scattering energy and the asymmetry factor are conserved. Various authors have used different normalization techniques. Three of these are compared in the present work, along with two other methods, one based on the finite volume method (FVM) and another one based on the spherical harmonics discrete ordinates method (SHDOM). In addition, the approximation of the Henyey–Greenstein phase function by a different one is investigated as an alternative to the phase function normalization. The approximate phase function is given by the sum of a Dirac delta function, which accounts for the forward scattering peak, and a smoother scaled phase function. In this study, these techniques are applied to three scalar radiative transfer test cases, namely a three-dimensional cubic domain with a purely scattering medium, an axisymmetric cylindrical enclosure containing an emitting–absorbing–scattering medium, and a three-dimensional transient problem with collimated irradiation. The present results show that accurate predictions are achieved for strongly forward scattering media when the phase function is normalized in such a way that both the scattered energy and the phase function asymmetry factor are conserved. The normalization of the phase function may be avoided using the FVM or the SHDOM to evaluate the in-scattering term of the radiative transfer equation. Both methods yield results whose accuracy is similar to that obtained using the DOM along with normalization of the phase function. Very satisfactory predictions were also achieved using the delta-M phase function, while the delta-Eddington phase function and the transport approximation may perform poorly

The impact of radiative heat transfer in combustion processes and its modeling – with a focus on turbulent flames

“…A simple correction factor can be used to overcome this problem, but it may change the shape of the scattering phase function when the asymmetry factor of the phase function is large. Several methods have been proposed to simultaneously satisfy conservation of energy and maintain the shape of the phase function [58], [59]. Modifications of the angular discretization procedure have been used to reduce the ray effects, namely by averaging the computed solution for several reference frame orientations [60], by employing a goal-based angular adaptivity method [61] or a goal-oriented regional angular adaptive algorithm [62].…”

Radiative heat transfer in strongly forward scattering media of circulating fluidized bed combustors

“…Proposed normalization technique is also tested in a recent study for an axisymmetric cylindrical test problem with optically thin and thick media [39]. It is shown that CPU times required for the normalization of the scattering phase function are negligible and accurate results are obtained with DOM for various optical thicknesses and asymmetry factors.…”

“…In addition to the above described method to solve the conservation problem of scattered energy and asymmetry factor for strongly forward scattering particles, HenyeyGreenstein phase function can be approximated by simpler phase functions which require no normalization. One of these methods is transport approximation [24,39] where the forward scattering peak is represented by a Dirac-delta function while the rest of the scattering is taken as isotropic. This is achieved by solving RTE as if the phase function is isotropic with the modified scattering coefficient:…”