The collocation spectral method with domain decomposition for radiative heat transfer in two-dimensional enclosures. (English) Zbl 1524.80028
Summary: 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.
MSC:
| 80M22 | Spectral, collocation and related (meshless) methods applied to problems in thermodynamics and heat transfer |
| 76M22 | Spectral methods applied to problems in fluid mechanics |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 45K05 | Integro-partial differential equations |
| 78A40 | Waves and radiation in optics and electromagnetic theory |
| 80A21 | Radiative heat transfer |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
References:
| [1] | Chandrasekhar, S., Radiative Transfer (1950), Clarendon Press: Clarendon Press Oxford · Zbl 0037.43201 |
| [2] | Raithby, G. D.; Chui, E. H., A finite-volume method for predicting a radiant heat transfer in enclosures with participating media, J. Heat Transf., 112, 415-423 (1990) |
| [3] | Kisselev, V. B.; Roberti, L.; Perona, G., An application of the finite element method to the solution of the radiative transfer equation, J. Quant. Spectrosc. Radiat. Transf., 51, 603-614 (1994) |
| [4] | Lockwood, F. C.; Shah, N. G., A new radiation solution method for incorporation in general combustion prediction procedures, Symp. Combust., 18, 1405-1414 (1981) |
| [5] | Howell, J. R., Application of Monte Carlo to heat transfer problems, Adv. Heat Transf., 5, 1-54 (1969) |
| [6] | Zhou, H. C.; Chen, D. L.; Cheng, Q., A new way to calculate radiative intensity and solve radiative transfer equation through using the Monte Carlo method, J. Quant. Spectrosc. Radiat. Transf., 83, 459-481 (2004) |
| [7] | Li, B. W.; Sun, Y. S.; Yu, Y., Iterative and direct Chebyshev collocation spectral methods for one-dimensional radiative heat transfer, Int. J. Heat Mass Transf., 51, 5887-5894 (2008) · Zbl 1153.80323 |
| [8] | Zhao, J. M.; Liu, L. H., Discontinuous spectral element method for solving radiative heat transfer in multidimensional semitransparent media, J. Quant. Spectrosc. Radiat. Transf., 107, 1-16 (2007) |
| [9] | Ehrenstein, U.; Peyret, R., A Chebyshev collocation method for the Navier-Stokes equations with application to double-diffusive convection, Int. J. Numer. Methods Fluids, 9, 427-452 (1989) · Zbl 0665.76107 |
| [10] | Peyret, R., Spectral Methods for Incompressible Viscous Flow (2002), Springer-Verlag: Springer-Verlag New York · Zbl 1005.76001 |
| [11] | Leriche, E., Direct numerical simulation in a lid-driven cavity at high Reynolds number by a Chebyshev spectral method, J. Sci. Comput., 27, 335-345 (2006) · Zbl 1115.76379 |
| [12] | Zhou, R. R.; Li, B. W.; Sun, Y. S., Predicting radiative heat transfer in axisymmetric cylindrical enclosures using the collocation spectral method, Int. Commun. Heat Mass Transf., 117, Article 104789 pp. (2020) |
| [13] | Piotrowska, J.; Miller, J. M.; Schnetter, E., Spectral methods in the presence of discontinuities, J. Comput. Phys., 390, 527-547 (2019) · Zbl 1452.65282 |
| [14] | Li, B. W.; Tian, S.; Sun, Y. S.; Hu, Z. M., Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method, J. Comput. Phys., 229, 1198-1212 (2010) · Zbl 1183.65152 |
| [15] | Ma, J.; Sun, Y. S.; Li, B. W., Completely spectral collocation solution of radiative heat transfer in an anisotropic scattering slab with a graded index medium, J. Heat Transf., 136, Article 012701 pp. (2014) |
| [16] | Ma, J.; Li, B. W.; Howell, J. R., Thermal radiation heat transfer in one- and two-dimensional enclosures using the spectral collocation method with full spectrum k-distribution model, Int. J. Heat Mass Transf., 71, 35-43 (2014) |
| [17] | Zhou, R. R.; Li, B. W., Chebyshev collocation spectral method to solve radiative transfer equation in one-dimensional cylindrical medium, Int. J. Heat Mass Transf., 111, 1206-1217 (2017) |
| [18] | Zhou, R. R.; Li, B. W.; Qian, Z. D., Performance comparisons of the DOM, CCSM, and hybrid CCS-DOM for thermal radiation analysis in one-dimensional cylindrical coordinates, Int. J. Therm. Sci., 129, 434-450 (2018) |
| [19] | Chen, S. S.; Li, B. W., Application of collocation spectral domain decomposition method to solve radiative heat transfer in 2D partitioned domains, J. Quant. Spectrosc. Radiat. Transf., 149, 275-284 (2014) |
| [20] | Coelho, P. J.; Gonçalves, J., Parallelization of the finite volume method for radiation heat transfer, Int. J. Numer. Methods Heat Fluid Flow, 9, 388-406 (1999) · Zbl 1157.80315 |
| [21] | Kim, G.; Kim, S.; Kim, Y., Parallelized unstructured-grid finite volume method for modeling radiative heat transfer, J. Mech. Sci. Technol., 19, 1006-1017 (2005) |
| [22] | Heinemann, T.; Dobler, W.; Nordlund, Å.; Brandenburg, A., Radiative transfer in decomposed domains, Astron. Astrophys., 448, 731-737 (2006) · Zbl 1104.85008 |
| [23] | Lygidakis, G. N.; Nikolos, I. K., Using the finite-volume method and hybrid unstructured meshes to compute radiative heat transfer in 3-D geometries, Numer. Heat Transf., Part B, Fundam., 62, 289-314 (2012) |
| [24] | Poitou, D.; Amaya, J.; Duchaine, F., Parallel computation for radiative heat transfer using the DOM in combustion applications: direction, frequency, subdomain decompositions, and hybrid methods, Numer. Heat Transf., Part B, Fundam., 62, 28-49 (2012) |
| [25] | Colomer, G.; Borrell, R.; Trias, F. X.; Rodríguez, I., Parallel algorithms for Sn transport sweeps on unstructured meshes, J. Comput. Phys., 232, 118-135 (2013) |
| [26] | Plimpton, S. J.; Hendrickson, B.; Burns, S. P.; McLendon, W.; Rauchwerger, L., Parallel Sn sweeps on unstructured grids: algorithms for prioritization, grid partitioning, and cycle detection, Nucl. Sci. Eng., 150, 267-283 (2015) |
| [27] | Xu, L.; Cao, L.; Zheng, Y.; Wu, H., Development of a new parallel SN code for neutron-photon transport calculation in 3-D cylindrical geometry, Prog. Nucl. Energy, 94, 1-21 (2017) |
| [28] | Badri, M. A.; Jolivet, P.; Rousseau, B.; Favennec, Y., High performance computation of radiative transfer equation using the finite element method, J. Comput. Phys., 360, 74-92 (2018) · Zbl 1395.65136 |
| [29] | N’Kaoua, T., Solution of the nonlinear radiative transfer equations by a fully implicit matrix Monte Carlo method coupled with the Rosseland diffusion equation via domain decomposition, SIAM J. Sci. Stat. Comput., 12, 505-520 (1991) · Zbl 0726.65161 |
| [30] | Klar, A.; Siedow, N., Boundary layers and domain decomposition for radiative heat transfer and diffusion equations: applications to glass manufacturing process, Eur. J. Appl. Math., 9, 351-372 (1998) · Zbl 0927.45004 |
| [31] | Roger, M.; Crouseilles, N., A dynamic multi-scale model for transient radiative transfer calculations, J. Quant. Spectrosc. Radiat. Transf., 116, 110-121 (2013) |
| [32] | Chai, J. C.; Moder, J. P., Spatial-multiblock procedure for radiation heat transfer, Numer. Heat Transf., Part B, Fundam., 31, 277-293 (1997) |
| [33] | Byun, D. Y.; Baek, S. W.; Kim, M. Y., Investigation of radiative heat transfer in complex geometries using blocked-off, multiblock, and embedded boundary treatments, Numer. Heat Transf., Part A, Appl., 43, 807-825 (2003) |
| [34] | Talukdar, P.; Steven, M.; Issendorff, F. V.; Trimis, D., Finite volume method in 3-D curvilinear coordinates with multiblocking procedure for radiative transport problems, Int. J. Heat Mass Transf., 48, 4657-4666 (2005) · Zbl 1189.80048 |
| [35] | Yavuz, M.; Larsen, E. W., Spatial domain decomposition for neutron transport problems, Transp. Theory Stat. Phys., 18, 205-219 (1989) · Zbl 0742.65103 |
| [36] | Previti, A.; Furfaro, R.; Picca, P.; Ganapol, B. D.; Mostacci, D., Solving radiative transfer problems in highly heterogeneous media via domain decomposition and convergence acceleration techniques, Appl. Radiat. Isot., 69, 1146-1150 (2011) |
| [37] | Hu, Z.; Zhu, X.; Guo, Z.; Tian, H.; Li, B., The spatial and angular domain decomposition method for radiation heat transfer in 2D rectangular enclosures with discontinuous boundary conditions, Int. J. Therm. Sci., 146, Article 106091 pp. (2019) |
| [38] | Modest, M. F., Radiative Heat Transfer (2003), Academic Press: Academic Press San Diego |
| [39] | Zhou, R.-R.; Li, B.-W., Chebyshev collocation spectral method for one-dimensional radiative heat transfer in linearly anisotropic-scattering cylindrical medium, J. Quant. Spectrosc. Radiat. Transf., 189, 206-220 (2017) |
| [40] | Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods: Fundamentals in Single Domains (2006), Springer-Verlag: Springer-Verlag Berlin Heidelberg · Zbl 1093.76002 |
| [41] | Quarteroni, A.; Valli, A., Domain Decomposition Methods for Partial Differential Equations (1999), Clarendon Press: Clarendon Press Oxford · Zbl 0931.65118 |
| [42] | Sànchez, A.; Smith, T. F., Surface radiation exchange for two-dimensional rectangular enclosures using the discrete-ordinates method, J. Heat Transf., 114, 465-472 (1992) |
| [43] | Talukdar, P., Discrete transfer method with the concept of blocked-off region for irregular geometries, J. Quant. Spectrosc. Radiat. Transf., 98, 238-248 (2006) |
| [44] | Zhou, R.-R.; Li, B.-W.; Wang, W.-K.; Sun, Y.-S., Improved integration strategies for the singularity subtraction method to solve radiative integral transfer equations with specified temperature field, Int. J. Therm. Sci., 149, Article 106158 pp. (2020) |
| [45] | Zhou, R.-R.; Sun, Y.-S.; Li, B.-W.; Ma, J., The composite Chebyshev integration method for radiative integral transfer equations in rectangular enclosures, Int. J. Therm. Sci., 170, Article 107141 pp. (2021) |
| [46] | Ramankutty, M. A.; Crosbie, A. L., Modified discrete ordinates solution of radiative transfer in two-dimensional rectangular enclosures, J. Quant. Spectrosc. Radiat. Transf., 57, 107-140 (1997) |
| [47] | Zhou, R.-R.; Li, B.-W., The modified discrete ordinates method for radiative heat transfer in two-dimensional cylindrical medium, Int. J. Heat Mass Transf., 139, 1018-1030 (2019) |
| [48] | Chai, J. C.; Moder, J. P., Radiation heat transfer calculation using an angular-multiblock procedure, Numer. Heat Transf., Part B, Fundam., 38, 1-13 (2000) |
| [49] | Coelho, P. J., A modified version of the discrete ordinates method for radiative heat transfer modelling, Comput. Mech., 33, 375-388 (2004) · Zbl 1145.80301 |
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