Completely Spectral Collocation Solution of Radiative Heat Transfer in an Anisotropic Scattering Slab With a Graded Index Medium

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Jing Ma ◽  
Ya-Song Sun ◽  
Ben-Wen Li
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Anisotropic Scattering ◽  
Discrete Ordinates ◽  
Spectral Collocation ◽  
Graded Index ◽  
Wide Range ◽  
Derivative Term ◽  
Scattering Phase Function

A completely spectral collocation method (CSCM) is developed to solve radiative transfer equation in anisotropic scattering medium with graded index. Different from the Chebyshev collocation spectral method based on the discrete ordinates method (SP-DOM), the CSCM is used to discretize both the angular domain and the spatial domain of radiative transfer equation. In this approach, the angular derivative term and the integral term are approximated by the high order spectral collocation scheme instead of the low order finite difference approximations. Compared with those available data in literature, the CSCM has a good accuracy for a wide range of the extinction coefficient, the scattering albedo, the scattering phase function, the gradient of refractive index and the boundary emissivity. The CSCM can provide exponential convergence for the present problem. Meanwhile, the CSCM is much more economical than the SP-DOM. Moreover, for nonlinear anisotropic scattering and graded index medium with space-dependent albedo, the CSCM can provide smoother results and mitigate the ray effect.

Solving Radiative Transfer Equation in Scattering Plane-Parallel Medium Using Wavelets Approximation

2002 ◽  
Author(s):  
Oguzhan Guven ◽  
Bryan Y. Wang ◽  
Yildiz Bayazitoglu
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Heat ◽  
Numerical Solutions ◽  
Radiative Transfer Equation ◽  
Anisotropic Scattering ◽  
Wavelet Approximation ◽  
Plane Parallel ◽  
Scattering Phase ◽  
Scattering Phase Function

Wavelet analysis is presented for solving the radiative transfer equation for a scattering medium in a one-dimensional plane-parallel geometry. Some properties of the wavelet transform for numerical approximation of radiative heat transfer are demonstrated. The governing equations are reduced to a system of first-order ordinary differential equations. Linear anisotropic scattering is assumed in order to compare the results with the previous researchers. The method of analysis is quite general since it only requires that the scattering phase function is square integrable. The numerical solutions indicate that wavelet approximation is promising.

An Efficient Sparse Finite Element Solver for the Radiative Transfer Equation

2009 ◽  
Author(s):  
Gisela Widmer
Keyword(s):  
Linear System ◽  
Radiative Transfer ◽  
Transfer Equation ◽  
Degrees Of Freedom ◽  
Radiative Transfer Equation ◽  
Three Dimensional ◽  
Discrete Ordinates ◽  
Five Dimensions ◽  
Computational Costs ◽  
Dimensional Domain

The stationary monochromatic radiative transfer equation (RTE) is posed in five dimensions, with the intensity depending on both a position in a three-dimensional domain as well as a direction. For non-scattering radiative transfer, sparse finite elements [1, 2] have been shown to be an efficient discretization strategy if the intensity function is sufficiently smooth. Compared to the discrete ordinates method, they make it possible to significantly reduce the number of degrees of freedom N in the discretization with almost no loss of accuracy. However, using a direct solver to solve the resulting linear system requires O(N3) operations. In this paper, an efficient solver based on the conjugate gradient method (CG) with a subspace correction preconditioner is presented. Numerical experiments show that the linear system can be solved at computational costs that are nearly proportional to the number of degrees of freedom N in the discretization.

Solution of the Radiative Transfer Equation in Three-Dimensional Participating Media Using a Hybrid Discrete Ordinates: Spherical Harmonics Method

2012 ◽  
Vol 134 (11) ◽  
Author(s):  
Maathangi Sankar ◽  
Sandip Mazumder
Keyword(s):  
Monte Carlo ◽  
Radiative Transfer ◽  
Spherical Harmonics ◽  
Hybrid Method ◽  
Transfer Equation ◽  
Large Range ◽  
Radiative Transfer Equation ◽  
Three Dimensional ◽  
Discrete Ordinates ◽  
Medium Component

In this article, a new hybrid solution to the radiative transfer equation (RTE) is proposed. Following the modified differential approximation (MDA), the radiation intensity is first split into two components: a “wall” component, and a “medium” component. Traditionally, the wall component is determined using a viewfactor-based surface-to-surface exchange formulation, while the medium component is determined by invoking the first-order spherical harmonics (P1) approximation. Recent studies have shown that although the MDA approach is accurate over a large range of optical thicknesses, it is prohibitive for complex three-dimensional geometry with obstructions, both from a computational efficiency as well as memory standpoint. The inefficiency stems from the use of the viewfactor-based approach for determination of the wall-emitted component. In this work, instead, the wall component is determined directly using the control angle discrete ordinates method (CADOM). The new hybrid method was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results for gray media in which the optical thickness was varied over a large range. In all cases, the accuracy of the hybrid method was found to be within a few percent of Monte Carlo results, and comparable to the solutions of the RTE obtained directly using CADOM. Finally, the new hybrid method was explored for 3D nongray media in the presence of reflecting walls and various scattering albedos. As a noteworthy advantage, irrespective of the conditions used, it was always found to be computationally more efficient than standalone CADOM and up to 15 times more efficient than standalone CADOM for optically thick media with strong scattering.

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