The collocation spectral method with domain decomposition for radiative heat transfer in two-dimensional enclosures

In this paper, the collocation spectral method (CSM) combined with domain decomposition method is developed to solve the radiative transfer equation in two-dimensional irregular domains. Three benchmark problems consist of the square enclosure, the L-shaped enclosure and the square enclosure with a centered obstruction are solved and compared with the published data to validate the ability of present developed method. The comparison shows good agreements and indicates that the method has a good accuracy for all problems. Then, the performances of influence matrix technique and iterative substructuring technique to exchange the radiative information between subdomains are compared. It is found that, in the serial computations, the computational cost of influence matrix technique is hundreds of times more expensive than that of iterative substructuring technique. The high cost of influence matrix technique is due to that the radiative intensity is a high dimensional variable with angular dependence, and tremendous subproblems have to be solved to construct the influence matrix. Finally, the modified CSM with domain decomposition, in which the radiative intensity is decomposed into three components, the real wall-related one, the virtual shared interface-related one, and the medium-related one, is proposed. The first two components are solved analytically, and the last one is still solved by the CSM. With such treatments, the computational cost is slightly increased, but the ray effect originated from step-change temperature of medium or step-change optical parameters can be effectively mitigated. Besides, the modified CSM with domain decomposition can also avoid the ray effect due to shadowing singularities. But it should be noted that, in such case, the ray effect due to inhomogeneous spatial distribution of source term may emerge. In conclusion, the CSM combined with domain decomposition is a good alternative method for thermal radiation calculation in complex geometry which can be decomposed into regular subdomains.