Wenjun Ge

Implementation of High-Order Spherical Harmonics Methods for Radiative Heat Transfer on openfoam

Wenjun Ge School of Engineering, University of California, Merced, CA 95343 e-mail: wge@ucmerced.edu Ricardo Marquez School of Engineering, University of California, Merced, CA 95343 e-mail: rmarquez3@ucmerced.edu Michael F. Modest 1 Professor Fellow ASME School of Engineering, University of California, Merced, CA 95343 e-mail: mmodest@ucmerced.edu Somesh P. Roy School of Engineering, University of California, Merced, CA 95343 e-mail: sroy3@ucmerced.edu Implementation of High-Order Spherical Harmonics Methods for Radiative Heat Transfer on OPENFOAM A general formulation of the spherical harmonics (P N ) methods was developed recently to expand the method to high orders of P N . The set of N(N þ 1)/2 three-dimensional second-order elliptic PDEs formulation and their Marshak boundary conditions for arbi- trary geometries are implemented in the OPENFOAM finite volume based CFD software. The results are verified for four cases, including a 1D slab, a 2D square enclosure, a 3D cylindrical enclosure, and an axisymmetric flame. All cases have strongly varying radia- tive properties, and the results are compared with exact solutions and solutions from the photon Monte Carlo method (PMC). [DOI: 10.1115/1.4029546] Keywords: radiative heat transfer, RTE solvers, spherical harmonics, computer implementation Introduction The study of radiative heat transfer in a multidimensional ge- ometry with a strongly varying participating medium has become increasingly important in various practical applications like com- bustion, manufacturing, and environmental systems. The radiative transfer equation (RTE) is an integro-differential equation in five independent variables (three in space and two in direction), which is exceedingly difficult to solve. Many approximate methods have been developed over time. The most widely used approximate methods today are the discrete ordinates method (DOM) [1,2] or its finite volume version (FVM) [3], the PMC [4], and the spheri- cal harmonics method (SHM) [5]. The DOM/FVM method discre- tizes the entire solid angle by finite ordinate directions and is relatively simple to implement within modern CFD software. But an iterative solution is required for scattering media or reflecting surfaces, and computational cost is high for optically thick media. The method may also suffer from ray effects and false scattering due to the angular discretization [6]. The PMC method statisti- cally provides the exact solution with sufficient photon bundles, but accurate solutions are computationally expensive. The spheri- cal harmonics P N approximation is potentially more accurate than DOM/FVM at comparable computational cost, but higher order P N are mathematically very complex and difficult to implement. The P N method decouples spatial and directional dependencies by expanding the radiative intensity into a series of spherical har- monics. The lowest order of the P N family, the P 1 approximation, has been extensively applied to radiative transfer problems. Corresponding author. Contributed by the Heat Transfer Division of ASME for publication in the J OURNAL OF H EAT T RANSFER . Manuscript received August 19, 2014; final manuscript received December 29, 2014; published online February 10, 2015. Assoc. Editor: Zhuomin Zhang. Journal of Heat Transfer However, it loses accuracy when intensity is directionally very ani- sotropic [7], as is often the case in optically thin media. Applications of higher order SHM methods were limited to one-dimensional cases for a long time, because of the cumbersome mathematics. Recently, Modest and Yang [8,9] and Modest [10] have derived a general three-dimensional P N formulation consisting of N(N þ 1)/ 2 second-order elliptic partial differential equations (PDEs) and their Marshak boundary conditions for arbitrary geometries. The main purpose of this paper is to present the procedure of implementing high-order P N formulations within the OPENFOAM [11] open source libraries, and a preliminary version was pre- sented in Ref. [12]. The numerical methods used are summarized along with four example cases. The results of high-order P N meth- ods are found to be very close to the exact solution of the RTE and accurately predict the incident radiation and radiative heat source. Also, the time cost of the P N methods is summarized for different examples and orders. Formulation 2.1 Governing Equations. In the spherical harmonics approximation, the radiative intensity is expanded into a sum of spherical harmonics Iðs; ^ s Þ ¼ N X n X I n m ðsÞY n m ð^ s Þ n¼0 m¼n where s ¼ b r dr is an optical coordinate, and b r is the extinction coefficient. Y n m ð^ s Þ are the spherical harmonics and the upper limit N is the order of the approximation. The set of N(N þ 1)/2 elliptic PDEs [10] of the P N method for isotropic scattering in 3D Carte- sian coordinates are as follows: C 2015 by ASME Copyright V MAY 2015, Vol. 137 / 052701-1 Downloaded From: http://heattransfer.asmedigitalcollection.asme.org/ on 02/25/2015 Terms of Use: http://asme.org/terms