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RESEARCH PAPERS: Solution Methods

# Finite-Volume Formulation and Solution of the $P3$ Equations of Radiative Transfer on Unstructured Meshes

[+] Author and Article Information
Mahesh Ravishankar, Ankan Kumar

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210

Sandip Mazumder1

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210mazumder.2@osu.edu

1

Corresponding author.

J. Heat Transfer 132(2), 023402 (Dec 01, 2009) (14 pages) doi:10.1115/1.4000184 History: Received October 23, 2008; Revised March 03, 2009; Published December 01, 2009; Online December 01, 2009

## Abstract

The method of spherical harmonics (or $PN$) is a popular method for approximate solution of the radiative transfer equation (RTE) in participating media. A rigorous conservative finite-volume (FV) procedure is presented for discretization of the $P3$ equations of radiative transfer in two-dimensional geometry—a set of four coupled, second-order partial differential equations. The FV procedure presented here is applicable to any arbitrary unstructured mesh topology. The resulting coupled set of discrete algebraic equations are solved implicitly using a coupled solver that involves decomposition of the computational domain into groups of geometrically contiguous cells using the binary spatial partitioning algorithm, followed by fully implicit coupled solution within each cell group using a preconditioned generalized minimum residual solver. The RTE solver is first verified by comparing predicted results with published Monte Carlo (MC) results for two benchmark problems. For completeness, results using the $P1$ approximation are also presented. As expected, results agree well with MC results for large/intermediate optical thicknesses, and the discrepancy between MC and $P3$ results increase as the optical thickness is decreased. The $P3$ approximation is found to be more accurate than the $P1$ approximation for optically thick cases. Finally, the new RTE solver is coupled to a reacting flow code and demonstrated for a laminar flame calculation using an unstructured mesh. It is found that the solution of the four $P3$ equations requires 14.5% additional CPU time, while the solution of the single $P1$ equation requires 9.3% additional CPU time over the case without radiation.

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## Figures

Figure 1

Unstructured stencil showing the various geometrical entities and vectors used in the finite-volume formulation

Figure 2

Block-diagram representation of the coupled implicit solver (IDD+GMRES) for the P3 equations and its relationship to the overall solution procedure

Figure 3

(a) Geometry and boundary conditions for the first benchmark problem, and (b) computational mesh used for the verification study

Figure 4

Comparison of the nondimensional heat flux computed at the bottom (hot wall) using various mesh sizes and the P3-based RTE solver: (a) various structured grids (quadrilateral cells), and (b) various unstructured grids (triangular cells)

Figure 5

Comparison of nondimensional heat fluxes predicted using the P1 and P3 methods with benchmark MC results (15) for τh=0.1: (a) bottom, (b) right (side), and (c) top walls. The heat fluxes are normalized by σTh4.

Figure 6

Comparison of nondimensional heat fluxes predicted using the P1 and P3 methods with benchmark MC results (15) for τh=1: (a) bottom, (b) right (side), and (c) top walls. The heat fluxes are normalized by σTh4.

Figure 7

Comparison of nondimensional heat fluxes predicted using the P1 and P3 methods with benchmark MC results (15) for τh=5: (a) bottom, (b) right (side), and (c) top walls. The heat fluxes are normalized by σTh4.

Figure 8

Geometry and boundary conditions for the second benchmark problem

Figure 9

Comparison of nondimensional heat fluxes predicted using the P1 and P3 methods with benchmark MC results (15) for τh=0.1: (a) partly heated bottom, (b) right (side), and (c) top walls. The heat fluxes are normalized by σTh4.

Figure 10

Comparison of nondimensional heat fluxes predicted using the P1 and P3 methods with benchmark MC results (15) for τh=1: (a) partly heated bottom, (b) right (side), and (c) top walls. The heat fluxes are normalized by σTh4.

Figure 11

Comparison of nondimensional heat fluxes predicted using the P1 and P3 methods with benchmark MC results (15) for τh=5: (a) partly heated bottom, (b) right (side), and (c) top walls. The heat fluxes are normalized by σTh4.

Figure 12

(a) Geometry and boundary conditions for the reaction flow test case (2D laminar methane-air flame), and (b) computational mesh: only one-third of the channel is shown for clarity

Figure 13

Temperature distributions for a laminar methane-air diffusion flame: (a) without including radiation, (b) including radiation calculated using P1, and (c) including radiation calculated using P3

Figure 14

Convergence of the overall flow solver using (a) P1 and (b) P3 approximations

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