# Angular discretization methods

﻿

## ANGULAR DISCRETIZATION METHODS

Following from: Discrete ordinates and finite volume methods

In the article Mathematical formulation, we have seen that the angular discretization of the radiative transfer equation (RTE) requires the selection of a finite number of directions of propagation of radiation intensity and the associated quadrature weights in the discrete ordinates method (DOM), and the selection of discrete solid angles, also referred to as control angles, in the finite volume method (FVM). In general, any angular discretization method employed in the FVM may also be applied in the DOM, since the value of a solid angle defined in the FVM may be regarded as a weight in the DOM, and the center of that solid angle may be taken as the direction of propagation of radiation. The reverse is not true. In fact, although the weight of a quadrature in the DOM may be thought of as a solid an ...

#### References

1. Carlson, B. G., Tables of Equal Weight Quadrature Over the Unit Sphere, Los Alamos Scientific Laboratory, Report LA–4737, 1971.
2. Carlson, B. G. and Lathrop, K. D., Discrete Ordinates Angular Quadrature of the Neutron Transport Equation, Los Alamos Scientific Laboratory, Report LA–3186, 1965.
3. Cumber, P. S., Application of Adaptive Quadrature to Fire Modeling, J. Heat Transfer, vol. 121, pp. 203–205, 1999..
4. Cumber, P. S., Ray Effect Mitigation in Jet Fire Radiation Modelling, Int. J. Heat Mass Transfer, vol. 43, pp. 935–943, 2000.
5. El Wakil, N. and Sacadura, J. F., Some Improvements of the Discrete Ordinates Method for the Solution of the Radiative Transport Equation in Multidimensional Anisotropically Scattering Media, Developments in Radiative Heat Transfer, S. T. Thynell, M. F. Modest, L. C. Burmeister, M. L. Hunt, T. W. Tong, R. D. Skocypec, W. W. Yuen, and W. A. Fiveland, Eds., ASME HTD-vol. 103, pp. 119–127, 1992.
6. Fiveland, W. A., Discrete Ordinate Methods for Radiative Heat Transfer in Isotropically Scattering Media, J. Heat Transfer, vol. 109, pp. 809–812, 1987.
7. Fiveland, W. A., The Selection of Discrete Ordinate Quadrature Sets for Anisotropic Scattering, Fundamentals of Radiation Heat Transfer, W.A. Fiveland W. A. Fiveland, A. L. Crosbie, A. M. Smith and T. F. Smith, Eds., ASME HTD- vol. 160, pp. 89–96, 1991.
8. Kim, S. H. and Huh, K. Y., A New Angular Discretization Scheme of the Finite Volume Method for 3-D Radiative Heat Transfer in Absorbing, Emitting and Anisotropically Scattering Media, Int. J. Heat Mass Transfer, vol. 43, pp. 1233–1242, 2000.
9. Koch, R. and Becker, R., Evaluation of Quadrature Schemes for the Discrete Ordinates Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 84, pp. 423–435, 2004.
10. Koch, R., Krebs, W., Wittig, S., and Viskanta, R., Discrete Ordinates Quadrature Schemes for Multidimensional Radiative Transfer, J. Quant. Spectrosc. Radiat. Transfer, vol. 53(4), pp. 353–372, 1995.
11. Lebedev, V. Values of the Nodes and Weights of Ninth to Seventeenth Order Gauss-Markov Quadrature Formulae Invariant under the Octahedron Group with Inversion, USSR Comput. Math. Math. Phys., vol. 15, pp. 44–51, 1975.
12. Lebedev, V., Quadratures on a Sphere, USSR Comput. Math. Math. Phys., vol. 16, pp. 10–24, 1976.
13. Li, B. W., Yao, Q, Cao, X.-Y., and Cen K.-F., A New Discrete Ordinates Quadrature Scheme for Three-Dimensional Radiative Heat Transfer, J. Heat Transfer, vol. 120, pp. 514–518, 1998.
14. Li, B. W., Chen, H.-G., Zhou, J.-H., Cao, X.-Y., and Cen K.-F., The Spherical Surface Symmetrical Equal Dividing Angular Quadrature Scheme for Discrete Ordinates Method, J. Heat Transfer, vol. 124, pp. 482–490, 2002.
15. Rukolaine, S. A. and Yuferev, V. S., Discrete Ordinates Quadrature Schemes Based on the Angular Interpolation of Radiation Intensity, J. Quant. Spectrosc. Radiat. Transfer, vol. 69, pp. 257–275, 2001.
16. Thurgood, C. P., Pollard, A., and Becker, H. A., The TN Quadrature Set for the Discrete Ordinates Method, J. Heat Transfer, vol. 117, pp. 1068–1070, 1995.
17. Verteeg, H. K., Henson, J. C., and Malalasekera, W., An Adaptive Angular Quadrature for the Discrete Transfer Method Based on Error Estimation, J. Heat Transfer, vol. 125, pp. 301–311, 2003.