MODIFIED DISCRETE ORDINATES AND FINITE VOLUME METHODS
Following from: Discrete ordinates and finite volume methods
The discrete ordinates method (DOM) and the finite volume method (FVM) suffer from ray effects, as discussed in the article “Ray Effects and False Scattering”. Ray effects may be successfully reduced by increasing the number of discrete directions in the DOM or the number of control angles in the FVM. However, ray effects decrease slowly with the refinement of the angular quadrature. The modified discrete ordinates method, MDOM (Ramankutty and Crosbie, 1997), based on the modified differential approximation (Olfe, 1970, Modest, 1989), proved to significantly mitigate ray effects originated from discontinuities or abrupt changes of the wall temperature. The model was originally developed for two-dimensional enclosures with black walls and homogeneous, isotropically scattering, gray media, and extended in Ramankutty and Crosbie (1998) to three-dimensional enclosures. The formulation for two-dimensional enclosures, as developed by Ramankutty and Crosbie (1997), is presented below.
Modified Discrete Ordinates Method – Basic Formulation
In the MDOM the radiation intensity is expressed as a sum of two components, the direct intensity component, I^{1}, that accounts for extinction of the radiation intensity leaving the boundaries, and the diffuse intensity component, I^{2}:
(1) |
The method was originally developed for a rectangular enclosure (0 ≤ x ≤ L_{x} and 0 ≤ y ≤ L_{y}) that contains an emitting, absorbing and isotropically scattering medium. In such a case, the two radiation intensity components in Eq. (1) satisfy the following equations:
(2a) |
(2b) |
The boundary conditions for black walls are written as
(3a) |
(3b) |
where the radiation intensity leaving the wall, I(r_{w}), is prescribed. Equation (2a) may be solved analytically for a homogeneous medium, yielding
(4) |
where s = |r − r_{w}| is the distance from the point r under consideration to the boundary point r_{w}. Using this result, the integral in the in-scattering term associated with the I^{1} component in Eq. (2b) is computed as follows:
(5) |
This integral over a spherical surface may be transformed into one-dimensional integrals as described in detail in Ramankutty and Crosbie (1997). After the numerical calculation of these integrals, Eq. (2b) is solved using the standard DOM.
The incident radiation is calculated as
(6) |
The first term is calculated from Eq. (5) and the second one is obtained from the solution of Eq. (2b) using the standard DOM. The incident heat flux on the boundary, i.e., the hemispherical irradiation, is given by
(7) |
The contribution of I^{1} to the incident heat flux on the boundary is computed by inserting Eq. (4) in the first term on the right of Eq. (7). Again, the integral over a spherical surface may be transformed into one-dimensional integrals following the algebraic treatment fully described in Ramankutty and Crosbie (1997). The resulting integrals are calculated numerically. The contribution of the second term of Eq. (7) to the incident heat flux on the boundary is calculated using the standard DOM.
The basic formulation of the MDOM described above is identical to that of the modified finite volume method (MFVM), except that Eq. 2(b) is solved using the standard DOM in the case of the MDOM, and the standard FVM in the case of the MFVM.
The MDOM eliminates ray effects that would arise due to discontinuities or abrupt changes of the wall temperature, because Eq. (2a) is solved analytically and the boundary conditions for Eq. (2b) do not depend on the wall temperature, which implies that the solution of Eq. 2(b) does not exhibit ray effects. This is illustrated in Fig. 1, which shows the incident heat flux on the bottom boundary normalized by the emissive power of the top boundary of the two-dimensional square enclosure considered in the example given in section “Interaction between Ray Effects and False Scattering” of the article “Ray Effects and False Scattering”. In this example, the emissive power of the top wall is equal to one, and the emissive power of the other walls is zero. The optical thickness of the medium for absorption and extinction are zero and one, respectively, and the scattering is isotropic. Hence, the radiation intensity field for component I^{2} is smooth for all directions. False scattering is small because there are no sharp gradients of the radiation intensity field component I^{2}, regardless of the direction under consideration. Consequently, an accurate solution is obtained with the MDOM using a coarse grid (15 × 15 control volumes), the step scheme and the S_{8} quadrature. Further refinement of the grid or quadrature has no significant influence on the predicted results.
Figure 1. MDOM predictions of the normalized incident heat flux on the bottom boundary of a square enclosure containing a purely isotropically scattering medium in radiative equilibrium.
Extensions of the Modified Discrete Ordinates and Finite Volume Methods
The MDOM may be applied to problems with anisotropic scattering. In such a case, the contribution of the direct intensity component I^{1} to the in-scattering term of Eq. (2b) yields a more complicated integral, because the phase function is no longer equal to one. Therefore, the integral is a function of spatial location and direction of propagation, while it was only a function of spatial location for an isotropic scattering medium. Ramankutty and Crosbie (1997) suggested the application of the isotropic scaling technique (Kim and Lee, 1990a) or δ-M scaling technique (Kim and Lee, 1990b) to MDOM. In the former case, the integral becomes only a function of the spatial location, and the complexity and computational requirements are similar for anisotropic and isotropic scattering media. However, the isotropic scaling technique does not give accurate results for non-isothermal media. If the δ-M scaling technique is employed, as suggested, the integral remains a function of spatial location and direction, and therefore does not significantly reduce the computational requirements.
If the medium is not homogeneous, the optical thickness is given by ∫^{s}_{0}β(s′)ds′ instead of βs. Accordingly, Eqs. (4) and (5) are modified as follows
(8) |
(9) |
The numerical integration of Eq. (9) is more involved than the integration of Eq. (5).
The basic formulation described in the previous section is also applicable to gray boundaries. However, in this case I^{1}(r_{w}, s) in Eq. (4) is not prescribed. This means that the integrals in Eqs. (5) and (7) need to be recomputed whenever I^{1}(r_{w}, s) is updated according to the boundary conditions.
The MDOM was applied by Sakami and Charette (2000) to two-dimensional enclosures of irregular geometry containing an emitting-absorbing-scattering medium. Both isotropic and anisotropic scattering were considered in this work. The analytical solution of the equation for the direct radiation intensity component is still straight forward. However, the calculation of the in-scattering term for that radiation intensity component, as well as the calculation of the heat flux incident on the walls, is more difficult, due to the irregular geometry. These calculations were performed as in the zonal method, dividing the boundary of the enclosure into sub-surfaces of uniform leaving intensity, and evaluating numerically the integrals by means of Gaussian quadrature. An application to complex two-dimensional enclosures with obstacles is reported in Sakami et al. (2001). Two-dimensional irregular geometries were studied by Baek et al. (2000) using similar ideas, but applying the MFVM. They employed the Monte-Carlo method instead of the zonal method to calculate the in-scattering term and the incident heat flux, and the standard FVM instead of the standard DOM to solve the governing equation for the diffuse radiation intensity component. Amiri et al. (2011) also used the MDOM to solve radiation problems in complex enclosures, but relied on the blocked-off concept to approximate inclined or curved boundaries.
The MDOM or the MFVM, as used in the works mentioned above, are unable to mitigate ray effects originated from sharp gradients of the emissive power of the medium. A new modified version (NMDOM) that successfully mitigates ray effects originated from discontinuities or abrupt changes of both the wall and medium emissive powers was proposed in Coelho (2002). The method was developed for two-dimensional rectangular enclosures with black walls, containing a homogeneous, isotropically scattering, gray medium. The extension to non-homogeneous media, anisotropic scattering and gray boundaries is described in Coelho (2004). In this method, the two components of the radiation intensity satisfy the following equations
(10a) |
(10b) |
with the boundary conditions given by
(11a) |
(11b) |
The NMDOM coincides with the MDOM if the absorption coefficient of the medium is zero. In non-scattering media, the NMDOM is identical to the method of Crosbie and Schrenker (1984).The formal solution to Eq. (10a) is given by
(12) |
where τ = βs is the optical thickness of the medium. The contribution of I^{1} to the in-scattering term of Eq. (10b) includes two terms, one due to radiative energy leaving the wall and the other due to emission from the medium. The former term is evaluated as described above for the MDOM, while the latter is calculated according to the method of Crosbie and Schrenker (1984). In the calculation of the incident heat flux on the boundary of the enclosure, the contribution of the radiative energy emitted by the medium is also calculated following Crosbie and Schrenker (1984). Further details on the NMDOM may be found in Coelho (2004).
An example is given here to illustrate the performance of the NMDOM. The enclosure is identical to that considered in the example of the previous section, as well as the boundary conditions. The blackbody radiation intensity of the medium is independent of x, but varies along y according to a Gaussian profile centered at y = 0.5:
(13) |
The constants a and σ are set to 1.0 and 0.1, respectively. The radiative properties of the medium are κL = 1 and σ_{s}L = 1. The results are shown in Fig. 2. The standard DOM (SDOM) and the MDOM behave in a similar way, yielding a smooth, but inaccurate, profile if a relatively coarse grid (15 × 15 control volumes), the step scheme and the S_{8} quadrature are used. The smooth profile is due to a compensation of errors due to false scattering and ray effects. Oscillatory profiles arise if the grid is fine or the CLAM scheme is employed, but the S_{8} quadrature is retained, because the false scattering errors are largely suppressed, but ray effects remain present, so that there is no compensation between these two error sources (see article “Ray Effects and False Scattering”). The MDOM is unable to suppress ray effects in this example, as illustrated in Fig. 2(b), because their origin is the significant temperature gradient in the medium, and not gradients or discontinuities of the temperature of the boundary.Good predictions without wiggles are only obtained if fine spatial and angular discretizations are employed. The NMDOM overcomes the problem and yields a smooth profile, which is almost insensitive to grid refinement, showing that a coarse grid is sufficient to obtain an accurate solution (see Fig. 2c).
The constants a and σ are set to 1.0 and 0.1, respectively. The radiative properties of the medium are κL = 1 and σ_{s}L = 1. The results are shown in Fig. 2. The standard DOM (SDOM) and the MDOM behave in a similar way, yieldinga smooth, but inaccurate, profile if a relatively coarse grid (15 × 15 control volumes), the step scheme and the S_{8} quadrature are used. The smooth profile is due to a compensation of errors due to false scattering and ray effects. Oscillatory profiles arise if the grid is fine or the CLAM scheme is employed, but the S_{8} quadrature is retained, because the false scattering errors are largely suppressed, but ray effects remain present, so that there is no compensation between these two error sources (see article “Ray Effects and False Scattering”). The MDOM is unable to suppress ray effects in this example, as illustrated in Fig. 2(b), because their origin is the significant temperature gradient in the medium, and not gradients or discontinuities of the temperature of the boundary.Good predictions without wiggles are only obtained if fine spatial and angular discretizations are employed. The NMDOM overcomes the problem and yields a smooth profile, which is almost insensitive to grid refinement, showing that a coarse grid is sufficient to obtain an accurate solution (see Fig. 2c). In conclusion, the MDOM mitigates ray effects caused by discontinuities or abrupt changes of the wall temperature. However, it is ineffective if ray effects are caused by sharp gradients of the emissive power of the medium. The NMDOM successfully mitigates ray effects caused by discontinuities or abrupt changes of both the wall and/or the medium temperatures, and yields accurate solutions using relatively coarse grids and quadratures. Although the MDOM and NMDOM are generally slower than the standard DOM if the same grid, quadrature and spatial differencing scheme are employed, they may become faster if ray effects are present and the same level of accuracy is sought.
Figure 2. Normalized incident heat flux on the bottom boundary of a square enclosure containing a non-isothermal medium. (I_{b} = I_{b}(y) – Gaussian profile, κL = 1, σ_{s}L = 1).
REFERENCES
Amiri, H., Mansouri, S. H., and Coelho, P. J., Application of the Modified Discrete OrdinatesMethod with the Concept of Blocked-off Region to Irregular Geometries, Int. J. Thermal Sciences, vol. 50, no. 4, pp. 515−524, 2011.
Baek, S. W., Byun, D. Y., and Kang, S. J., The Combined Monte-Carlo and Finite-Volume Method for Radiation in a Two-Dimensional Irregular Geometry, Int. J. Heat Mass Transfer, vol. 43, pp. 2337−2344, 2000.
Coelho, P. J., The Role of Ray Effects and False Scattering on the Accuracy of the Standard and Modified Discrete Ordinates Methods, J. Quantitative Spectroscopy Radiative Transfer, vol. 73, pp. 231−238, 2002.
Coelho, P. J., A Modified Version of the Discrete Ordinates Method for Radiative Heat Transfer Modelling, Computational Mech., vol. 33, no. 5, pp. 375−388, 2004.
Crosbie, A. L. and Schrenker, R. G., Radiative Transfer in a Two-Dimensional Rectangular Medium Exposed to Diffuse Radiation, J. Quantitative Spectroscopy and Radiative Transfer, vol. 31, pp. 339−372, 1984.
Kim, T. -K. and Lee, H. S., Scaled Isotropic Results for Two-Dimensional Anisotropic Scattering Media, J. Heat Transfer, vol. 112, pp. 721−727., 1990a.
Kim, T. -K. and Lee, H. S., Modified δ-M Scaling Results for Mie-Anisotropic Scattering Media, J. Heat Transfer, vol. 112, pp. 988−994, 1990b.
Olfe, D. B., Radiative Equilibrium of a Gray Medium Bounded by Nonisothermal Walls, Progress in Astronautics and Aeronautics, vol. 23, pp. 295−317, 1970.
Modest, M. F., The Modified Differential Approximation for Radiative Transfer in Three-Dimensional Media, J. Thermophysics and Heat Transfer, vol. 3, pp. 283−288, 1989.
Ramankutty, M. A. and Crosbie, A. L., Modified Discrete Ordinates Solution of Radiative Transfer in Two-Dimensional Rectangular Enclosures, J. Quantitative Spectroscopy Radiative Transfer, vol. 57, pp. 107−140, 1997.
Ramankutty, M. A., and Crosbie, A. L., Modified Discrete Ordinates Solution of Radiative Transfer in Three-Dimensional Rectangular, J. Quantitative Spectroscopy Radiative Transfer, vol. 60, pp. 103−134, 1998.
Sakami, M. and Charette, A., Application of a Modified Discrete Ordinates Method to Two-dimensional Enclosures of Irregular Geometry, J. Quantitative Spectroscopy Radiative Transfer, vol. 64, pp. 275−298, 2000.
Sakami, M., El Kasmi, A., and Charette, A., Analysis of Radiative Heat Transfer in Complex Two-dimensional Enclosures with Obstacles Using the Modified Discrete Ordinates Method, J. Heat Transfer, vol. 123, pp. 892−900, 2001.
References
- Amiri, H., Mansouri, S. H., and Coelho, P. J., Application of the Modified Discrete OrdinatesMethod with the Concept of Blocked-off Region to Irregular Geometries, Int. J. Thermal Sciences, vol. 50, no. 4, pp. 515−524, 2011.
- Baek, S. W., Byun, D. Y., and Kang, S. J., The Combined Monte-Carlo and Finite-Volume Method for Radiation in a Two-Dimensional Irregular Geometry, Int. J. Heat Mass Transfer, vol. 43, pp. 2337−2344, 2000.
- Coelho, P. J., The Role of Ray Effects and False Scattering on the Accuracy of the Standard and Modified Discrete Ordinates Methods, J. Quantitative Spectroscopy Radiative Transfer, vol. 73, pp. 231−238, 2002.
- Coelho, P. J., A Modified Version of the Discrete Ordinates Method for Radiative Heat Transfer Modelling, Computational Mech., vol. 33, no. 5, pp. 375−388, 2004.
- Crosbie, A. L. and Schrenker, R. G., Radiative Transfer in a Two-Dimensional Rectangular Medium Exposed to Diffuse Radiation, J. Quantitative Spectroscopy and Radiative Transfer, vol. 31, pp. 339−372, 1984.
- Kim, T. -K. and Lee, H. S., Scaled Isotropic Results for Two-Dimensional Anisotropic Scattering Media, J. Heat Transfer, vol. 112, pp. 721−727., 1990a.
- Kim, T. -K. and Lee, H. S., Modified δ-M Scaling Results for Mie-Anisotropic Scattering Media, J. Heat Transfer, vol. 112, pp. 988−994, 1990b.
- Olfe, D. B., Radiative Equilibrium of a Gray Medium Bounded by Nonisothermal Walls, Progress in Astronautics and Aeronautics, vol. 23, pp. 295−317, 1970.
- Modest, M. F., The Modified Differential Approximation for Radiative Transfer in Three-Dimensional Media, J. Thermophysics and Heat Transfer, vol. 3, pp. 283−288, 1989.
- Ramankutty, M. A. and Crosbie, A. L., Modified Discrete Ordinates Solution of Radiative Transfer in Two-Dimensional Rectangular Enclosures, J. Quantitative Spectroscopy Radiative Transfer, vol. 57, pp. 107−140, 1997.
- Ramankutty, M. A., and Crosbie, A. L., Modified Discrete Ordinates Solution of Radiative Transfer in Three-Dimensional Rectangular, J. Quantitative Spectroscopy Radiative Transfer, vol. 60, pp. 103−134, 1998.
- Sakami, M. and Charette, A., Application of a Modified Discrete Ordinates Method to Two-dimensional Enclosures of Irregular Geometry, J. Quantitative Spectroscopy Radiative Transfer, vol. 64, pp. 275−298, 2000.
- Sakami, M., El Kasmi, A., and Charette, A., Analysis of Radiative Heat Transfer in Complex Two-dimensional Enclosures with Obstacles Using the Modified Discrete Ordinates Method, J. Heat Transfer, vol. 123, pp. 892−900, 2001.
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