The spherical harmonics (PN) method, especially its lowest order, i.e., the P1 or differential approximation, enjoys great popularity because of its relative simplicity and compatibility with standard models for the solution of the (overall) energy equation. Low-order PN approximations perform poorly in the presence of strongly nonisotropic intensity distributions, especially in optically thin situations within nonisothermal enclosures (due to variation in surface radiosities across the enclosure surface, causing rapid change of irradiation over incoming directions). A previous modification of the PN approximation, i.e., the modified differential approximation (MDA), separates wall emission from medium emission to reduce the nonisotropy of intensity. Although successful, the major drawback of this method is that the intensity at the walls is set to zero into outward directions, while incoming intensity is nonzero, resulting in a discontinuity at grazing angles. To alleviate this problem, a new approach, termed here the advanced differential approximation (ADA)is developed, in which the directional gradient of the intensity at the wall is minimized. This makes the intensity distribution continuous for the P1 method and mostly continuous for higher-order PN methods. The new method is tested for a 1D slab and concentric spheres and for a 2D medium. Results are compared with the exact analytical solutions for the 1D slab as well as the Monte Carlo-based simulations for 2D media.

The accuracy and computational expense of various radiation models in the simulation of turbulent jet flames are compared. Both nonluminous and luminous methane-air nonpremixed turbulent jet flames are simulated using a comprehensive combustion solver. The combustion solver consists of a finite-volume/probability density function-based flow-chemistry solver interfaced with a high-accuracy spectral radiation solver. Flame simulations were performed using various k-distribution-based spectral models and radiative transfer equation (RTE) solvers, such as P-1, P-3, finite volume/discrete ordinates method (FVM/DOM), and line-by-line (LBL) accurate Photon Monte Carlo (PMC) methods, with and without consideration of turbulence-radiation interaction (TRI). Various spectral models and RTE solvers are observed to have strong effects on peak flame temperature, total radiant heat source and NO emission. The P-1 method is found to be the computationally least expensive RTE solver and the FVM the most expensive for any spectral model. For optically thinner flames all radiation models yield excellent accuracy. For optically thicker flames the P-3 and the FVM with advanced k-distribution methods predict radiation more accurately than the P-1 method with any spectral model when compared to the benchmark LBL PMC. The LBL PMC yields exact results with sufficient number of samples and is found to be less expensive than the FVM (for all spectral models) and the P-3 (for some spectral models) in statistically stationary combustion simulations. TRI is found to drop the peak temperature by close to 150. K for a luminous flame (optically thicker) and 25-100. K for a nonluminous flame (optically thinner).