Discrete Ordinate Interpolation Method for Solution of Radiative Transfer Equation in Arbitraty 2-D Geometry and Unstructured Grid

Abstract

The discrete ordinates interpolation method developed for numerical of radiative transfer equation is applied to unstructured grid system in arbitrary two dimensional geometry. Basic solution method is briefly explained and five sample geobetries with absorbing-emitting and nonscattering media with a uniform temperature are taken to demonstrate the applicability and accuracy. The typical optical depths in the problem are from highly transparent cases(0.1) to highly thick cases(about 10). In the first problem, a triangle is treated with triangular grid lines. The wall heat flux is calculated and compared with the exact solution. Th e error is less than a percent with about 60 grid points. In the second problem, a quadrilateral enclosure with either unstructured or structured grids are taken. Both grids show good agreement with the exact solution with a maximum of about 1 percent error, when the number of grids is roughly 100 for both of them. A hexagonal geometry is taken to employ a mixture of different kinds of grid, and sililar accuracy in the wall heat flux as the preceding geometries with comparable density of the grids is obtained too. The fourth problem with a J-shaped enclosure shows th most complicated distribution of heat flux, however, it is successfully calculated within the resolution square geometry with structured of imbedded grids. The computational accuracy is higher for finer grids, however, the imbedded grids can significantly reduce the computation time with similar accuracy compared with the structured dense grid system. In all of the tested cases, the effect of the optical depth is relatively small and no general tendency with the optical depth is observed. The results successfully reveal the applicability of the discrete ordinates interpolation method for any geometry and optical depth. Any grid system employed in FDM, FEM or FVM may be thus adopted and any desired level of numerical accuracy can be obtained with finer of imbedded grids.