VARIABLE REFRACTIVE INDEX MEDIA
Following from: Discrete ordinates and finite volume methods
In many applications, the refractive index of the medium is uniform, and photons propagate along straight lines. In a few other applications, however, the refractive index is a function of the spatial coordinate. In these media, referred to as graded index media, the photons propagate along curved trajectories that depend on the local refractive index according to the principle of Fermat. This principle states that the optical path between two points is the path that minimizes the travel time. Graded index media result from changes in the chemical and physical properties of the medium, e.g., the concentration of salt in the ocean, and the variation of density in planetary atmospheres and in biological tissues. This article describes the application of the discrete ordinates method (DOM) and finite volume method (FVM) to graded index media.
The radiative transfer equation (RTE) in a medium with variable refractive index is written as (Liu, 2006a)
(1) |
where n is the refractive index of the medium, which is a function of spatial position, r (see article “Mathematical formulation” for the definition of the other symbols). According to Fermat’s principle, the path of a ray between two points is the path that minimizes the travel time. This implies that rays travel along curved paths in graded index media, as given by the following equation (Born and Wolf, 1999):
(2) |
Equation (1) may be written as follows in Cartesian coordinates (Liu, 2006a):
(3) |
The corresponding equations in cylindrical and spherical coordinates are given in Liu et al. (2006b). In Eq. (3), i, j, and k denote the unit vectors along the x, y, and z directions, respectively, and θ is the polar angle and φ the azimuthal angle of the direction of propagation of radiation, s. The direction cosines of the direction of propagation along the x, y, and z directions are ξ, η, and μ, respectively. The RTE for graded index media has two additional terms in comparison with the RTE for uniform refractive index media, namely, the second and third terms on the left side of this equation. They are referred to as angular redistribution terms, and account for the curvature of the optical path.
The boundary condition for a gray surface that emits and reflects diffusely is given by
(4) |
Lemonnier and Le Dez (2002) were the first to apply the DOM to a graded index medium. They considered a 1D semitransparent slab with a transverse continuous and monotonic variation of the refractive index. Chang and Wu (2008) studied azimuthally dependent radiative transfer in an anisotropically scattering slab with variable refractive index and oblique irradiation using a similar formulation. A more general approach is described in Liu (2006a), who derived Eq. (3) and solved it for multidimensional problems using the FVM. Asllanaj and Fumeron (2010) applied the FVM to 2D complex geometries using a slightly different procedure to discretize the angular redistribution terms. The method developed by Liu (2006a), which relies on the step scheme for spatial discretization and piecewise constant angular discretization, is summarized below.
The total solid angle is subdivided into N_{θ} × N_{φ} control angles with polar steps and azimuthal steps, given by
(5a) |
(5b) |
The discretization of the terms on the right side of Eq. (3) is carried out as described in the article “Mathematical formulation.” The integration of the first term on the left side of that equation over a control volume and a control angle, and application of the Gauss divergence theorem, leads to
(6) |
Note that in this article two superscripts are used to identify the direction of propagation of radiation, i.e., l for the polar angle and n for the azimuthal angle. The subscript f identifies the cell face, with F being the total number of cell faces of the control volume under consideration. The definition of D^{l,n}_{cf} is given in Eq. (24a) of the article “Mathematical formulation.”
If the radiation intensity at a cell face is evaluated using the step scheme, the following approximation is obtained for the terms in Eq. (6):
(7) |
where subscript U identifies the upstream grid node when radiative energy propagates across cell face f from the control volume centered at grid node U to the control volume centered at grid node P. The factors n^{2}_{f} / n^{2}_{P} and n^{2}_{f} / n^{2}_{U,f} account for the change of solid angle when a radiative energy bundle passes into a medium with different refractive index (Liu, 2006a).
The second term on the left side of Eq. (3) is integrated over a control volume and a control angle, and discretized as follows:
(8) |
where the following definitions were used:
(9a) |
(9b) |
In the cases of l = 1 and l = N_{θ}, we have
(10) |
The radiation intensities at grid node P and polar angles l ± 1/2, which appear in Eq. (8), are not available, and must be related to the radiation intensity at that grid node for polar angles with an integer index (l − 1, l, and l + 1). This is accomplished using a step-like scheme for the polar angle, according to the following approximation:
(11) |
The third term on the left side of Eq. (3) is integrated over a control volume and a control angle, and discretized as follows:
(12) |
where
(13a) |
(13b) |
In the cases of n = 1 and n = N_{θ}, we have
(14) |
The radiation intensities at grid node P and azimuthal angles n ± 1/2, which appear in Eq. (12), are related to the radiation intensity at that grid node for azimuthal angles with an integer index (n = 0 and n = ±1). The step scheme is applied to the azimuthal angle, yielding
(15) |
The final discretized equation for the FVM is obtained by considering the discretization of the terms on the left side of Eq. (3), namely, Eqs. (6) and (7) for the first term, Eqs. (8) and (11) for the second term, and Eqs. (12) and (15) for the third one, along with the discretization of the terms on the right side of Eq. (3). This yields the following equation:
(16) |
where the definitions of ΔΩ^{l,n} and Φ^{l′,n′; l,n} are given in Eqs. (24b) and (24c), respectively, of the article “Mathematical formulation,” except that two superscripts, i.e., l and n, are used here to define the direction of propagation, while only one superscript was used in that article. This equation may be rewritten as follows, by grouping on the left side all the terms that depend on the radiation intensity at grid node P and on the right side the remaining terms:
(17) |
The solution algorithm is identical to that described in the article “Solution algorithm.”
If the DOM is used instead of the FVM, Eq. (3) is written as follows for every discrete direction s^{l,n} (Liu et al., 2006a):
(18) |
where the polar and azimuthal angles of the discrete directions are defined as
(19a) |
(19b) |
and the quadrature weights are determined as
(20a) |
(20b) |
Integration of the first term on the left of Eq. (18) over a control volume and application of the Gauss divergence theorem yields
(21) |
If the step scheme is employed to evaluate the cell face faces intensities, the following approximation is obtained:
(22) |
The discretization of the second term on the left side of Eq. (18) is carried out using a finite difference approximation as follows (e.g., Liu et al., 2006a):
(23) |
The following recursion formula for χ_{θ}^{l + 1/2, n} may be derived (see Liu, 2006a, for details) as
(24a) |
(24b) |
Equation (11) still holds. Proceeding similarly for the third term on the left of Eq. (18), we have
(25) |
The recursion formula for χ_{θ}^{l + 1/2, n} is written as (Liu, 2006a)
(26a) |
(26b) |
Equation (15) remains valid. The discretization of the terms on the right side of Eq. (18) follows standard practices (see article “Mathematical formulation”). The final discretized form of the DOM equations may now be written as
(27) |
This equation may be rewritten as follows, by grouping on the left side all the terms that depend on the radiation intensity at grid node P and on the right side the remaining terms:
(28) |
A few papers have been published in the last few years for graded index media where the DOM is used for angular discretization, and various alternative formulations are employed for spatial discretization. These include the finite element method (FEM) for Cartesian (Liu et al., 2006a) and cylindrical (Zhang et al., 2009) geometries, the discontinuous FEM (Liu and Liu, 2007), the least squares FEM (Liu, 2007), the least-squares spectral element method (Zhao and Liu, 2007), the Chebyshev collocation spectral method (Sun and Li, 2009), and the meshless method (Liu, 2006b).
REFERENCES
Asllanaj, F. and Fumeron, S., Modified Finite Volume Method Applied to Radiative Transfer in 2D Complex Geometries and Graded Index Media, J. Quant. Spectrosc. Radiat. Transfer, vol. 11, pp. 274−279, 2010.
Born, M. and Wolf, E., Principles of Optics, 7th ed., Cambridge: Cambridge University Press, 1999.
Chang, C. C. and Wu, C. Y., Azimuthally Dependent Radiative Transfer in a Slab with Variable Refractive Index, Int. J. Heat Mass Transfer, vol. 51, pp. 2701−2710, 2008.
Lemonnier, D. and Dez, V. Le, Discrete Ordinates Solution of Radiative Transfer Across a Slab with Variable Refractive Index, J. Quant. Spectrosc. Radiat. Transfer, vol. 73, pp. 195−204, 2002.
Liu, L. H., Finite Volume Method for Radiation Heat Transfer in Graded Index Medium, J. Thermophys. Heat Transfer, vol. 20, no. 1, pp. 59−66, 2006a.
Liu, L. H., Meshless Method for Radiation Heat Transfer in Graded Index Medium, Int. J. Heat Mass Transfer, vol. 49, pp. 219−229, 2006b.
Liu, L. H., Least–Squares Finite Element Method for Radiation Heat Transfer in Graded Index Medium, J. Quant. Spectrosc. Radiat. Transfer, vol. 103, pp. 536−544, 2007.
Liu, L. H. and Liu, L. J., Discontinuous Finite Element Method for Radiative Heat Transfer in Semitransparent Graded Index Medium, J. Quant. Spectrosc. Radiat. Transfer, vol. 105, pp. 377−387, 2007.
Liu, L. H., Zhang, L., and Tan, H. P., Finite Element Method for Radiation Heat Transfer in Multi-Dimensional Graded Index Medium, J. Quant. Spectrosc. Radiat. Transfer, vol. 97, pp. 436−445, 2006a.
Liu, L. H., Zhang, L., and Tan, H. P., Radiative Transfer Equation for Graded Index Medium in Cylindrical and Spherical Coordinate Systems, J. Quant. Spectrosc. Radiat. Transfer, vol. 97, pp. 446−456, 2006b.
Sun, Y.-S. and Li, B.-W., Chebyshev Collocation Spectral Method for One-Dimensional Radiative Heat Transfer in Graded Index Media, Int. J. Thermal Sci., vol. 48, no. 4, pp. 691−698, 2009
Zhang, L., Zhao, J. M., and Liu, L.H., Finite Element Method for Modeling Radiative Transfer in Semitransparent Graded Index Cylindrical Medium, J. Quant. Spectrosc. Radiat. Transfer, vol. 110, pp. 1085−1096, 2009.
Zhao, J. M. and Liu, L. H., Solution of Radiative Heat Transfer in Graded Index Media by Least Square Spectral Element Method, Int. J. Heat Mass Transfer, vol. 50, pp. 2634−2642, 2007.
Referencias
- Asllanaj, F. and Fumeron, S., Modified Finite Volume Method Applied to Radiative Transfer in 2D Complex Geometries and Graded Index Media, J. Quant. Spectrosc. Radiat. Transfer, vol. 11, pp. 274−279, 2010.
- Born, M. and Wolf, E., Principles of Optics, 7th ed., Cambridge: Cambridge University Press, 1999.
- Chang, C. C. and Wu, C. Y., Azimuthally Dependent Radiative Transfer in a Slab with Variable Refractive Index, Int. J. Heat Mass Transfer, vol. 51, pp. 2701−2710, 2008.
- Lemonnier, D. and Dez, V. Le, Discrete Ordinates Solution of Radiative Transfer Across a Slab with Variable Refractive Index, J. Quant. Spectrosc. Radiat. Transfer, vol. 73, pp. 195−204, 2002.
- Liu, L. H., Finite Volume Method for Radiation Heat Transfer in Graded Index Medium, J. Thermophys. Heat Transfer, vol. 20, no. 1, pp. 59−66, 2006a.
- Liu, L. H., Meshless Method for Radiation Heat Transfer in Graded Index Medium, Int. J. Heat Mass Transfer, vol. 49, pp. 219−229, 2006b.
- Liu, L. H., Leastâ€“Squares Finite Element Method for Radiation Heat Transfer in Graded Index Medium, J. Quant. Spectrosc. Radiat. Transfer, vol. 103, pp. 536−544, 2007.
- Liu, L. H. and Liu, L. J., Discontinuous Finite Element Method for Radiative Heat Transfer in Semitransparent Graded Index Medium, J. Quant. Spectrosc. Radiat. Transfer, vol. 105, pp. 377−387, 2007.
- Liu, L. H., Zhang, L., and Tan, H. P., Finite Element Method for Radiation Heat Transfer in Multi-Dimensional Graded Index Medium, J. Quant. Spectrosc. Radiat. Transfer, vol. 97, pp. 436−445, 2006a.
- Liu, L. H., Zhang, L., and Tan, H. P., Radiative Transfer Equation for Graded Index Medium in Cylindrical and Spherical Coordinate Systems, J. Quant. Spectrosc. Radiat. Transfer, vol. 97, pp. 446−456, 2006b.
- Sun, Y.-S. and Li, B.-W., Chebyshev Collocation Spectral Method for One-Dimensional Radiative Heat Transfer in Graded Index Media, Int. J. Thermal Sci., vol. 48, no. 4, pp. 691−698, 2009
- Zhang, L., Zhao, J. M., and Liu, L.H., Finite Element Method for Modeling Radiative Transfer in Semitransparent Graded Index Cylindrical Medium, J. Quant. Spectrosc. Radiat. Transfer, vol. 110, pp. 1085−1096, 2009.
- Zhao, J. M. and Liu, L. H., Solution of Radiative Heat Transfer in Graded Index Media by Least Square Spectral Element Method, Int. J. Heat Mass Transfer, vol. 50, pp. 2634−2642, 2007.