RAY EFFECTS AND FALSE SCATTERING
Following from: Discrete ordinates and finite volume methods
The discrete ordinates method (DOM) and the finite volume method (FVM) have been widely applied in the last two decades to the solution of radiative heat transfer problems, owing to their good compromise between accuracy, flexibility, and moderate computational requirements. They can be applied to nonisothermal, nonhomogeneous, anisotropically scattering, and nongray media in complex geometries. However, several limitations of these methods have also been identified. Among them, ray effects and false scattering are perhaps the two most important ones. These two problems have been discussed, e.g., in Chai et al. (1993), Raithby (1999), and Coelho (2002a), and several proposals have been made to overcome them.
False Scattering
False scattering, also referred to in the literature as false diffusion, numerical scattering, numerical diffusion, or numerical smearing, is related to the spatial discretization scheme. It is independent of angular discretization. False scattering is the counterpart of false diffusion in computational fluid dynamics (CFD). It occurs in multidimensional problems when radiation propagates along a direction oblique to the gridlines and there is a gradient or a discontinuity of radiation intensity in the direction normal to the direction of propagation. There is no false scattering in steady 1D problems. The cause of false scattering can be attributed to calculating the radiation intensity across a cell face from the radiation intensity at points that lie along the normal to that cell face. Spatial discretization schemes that account for the multidimensional nature of radiation, i.e., schemes that calculate the radiation intensity at a cell face based on the radiation intensity at points that lie along the direction of propagation of radiation, are potentially more accurate and mitigate false scattering.
A simple example taken from Coelho (2002b) is given here to illustrate false scattering. Let us consider a 2D square enclosure of unity side (L = 1) containing a transparent medium and bounded by black walls. The blackbody radiation intensity is equal to one at the left side wall and equal to zero at the other walls. A single direction of propagation of radiation intensity, making an angle of 45 deg with the x and y coordinates axes, and a uniform grid with 11 × 11 control volumes are considered. The exact solution is equal to one for x < y, equal to zero for x > y, and exhibits a discontinuity for x = y. The numerical solution calculated using the DOM and the step scheme shows that the predicted radiation intensity field for the direction under consideration is strongly smoothed, as if a diffusion phenomenon were present [see Fig. 1(a)].
Figure 1. Radiation intensity field in a square enclosure with a transparent medium — the direction of propagation makes an angle of 45 deg with the coordinate axis: (a) step scheme; (b) SMART scheme.
It is worth pointing out that false scattering does not occur when the radiation intensity propagates along a direction aligned with a coordinate axis in a Cartesian mesh, and is more severe when the direction of propagation makes an angle of 45 deg with the axis, as in the example above. Although the previous example considered a transparent medium, false scattering is also present in emitting-absorbing and/or scattering media. In the case of an optically thick medium, the local radiation intensity is strongly dependent on the local blackbody radiation intensity, and false scattering effects are attenuated (Coelho, 2002b).
False scattering may be reduced by refining the grid or by using more accurate spatial discretization schemes. The diamond scheme is not a remedy. This scheme is prone to nonphysical oscillations and unrealistic negative radiation intensities, even though it may reduce the magnitude of false scattering. However, there are many high-order resolution schemes that significantly reduce false scattering, such as the second- or third-order accurate schemes that satisfy the boundedness criteria of the normalized variable diagram, the total variation diminishing schemes, the essentially nonoscillatory schemes, and the genuinely multidimensional schemes (see the article “Spatial discretization schemes”).
Figure 1(b) shows the significant reduction of false scattering, for the example given above, when the high-order resolution SMART scheme is used (see Coelho, 2002b, for details). The performance of several other high-order schemes is illustrated in Fig. 2, which refers to the same example, but for a finer mesh with 25 × 25 control volumes. This figure shows the radiation intensity profile at x/L = 0.5 for two different values of the direction of propagation, namely, for angles α = 45 deg and 30 deg between that direction and the x-axis (see Coelho, 2008, for details). The solution error is much larger for the STEP scheme than for all other schemes. The numerical smearing is particularly large when the direction of propagation makes an angle of 45 deg with the grid lines, but remains quite large for α = 30 deg.
Figure 2. Radiation intensity at x/L = 0.5: (a) α = 45 deg; (b) α = 30 deg.
Li et al. (2002) proposed the double rays method to reduce false scattering in the framework of the discrete ordinates interpolation method, which is described in the article “Alternative formulations.” The basic idea of this method is to ensure that the radiation intensity at the upstream location [see Eq. (13) of that article] is always interpolated in a region where the radiation intensity is continuous. This implies that two rays start from discontinuous boundary points along every discrete direction. Although the underlying principle of this method is quite simple, the interpolation procedure is cumbersome, and the computer code becomes complicated, even in 2D problems, the only ones considered in the article. A hybrid scheme, which is much simpler to implement, but only able to reduce false scattering in arbitrarily specified directions, while retaining false scattering in the other discrete directions, is reported in Li (2004). In this version, the radiation intensity along specified directions, where false scattering is to be reduced, is calculated from integration of the RTE along that direction. Two rays are employed for each one of these specified directions. However, in contrast to the previous method, only the radiation intensity leaving the boundary point, where a discontinuity occurs, is interpolated from a region where the leaving radiation intensity is continuous. The other discrete directions are handled using the standard DOM.
A good discussion of false scattering is reported in Raithby (1999). Estimation of false scattering in the DOM and FVM is presented in Tan et al. (2004) and Zhang and Tan (2009). Other relevant works concerning this issue are referenced in the article “Spatial discretization schemes,” which presents numerous spatial discretization schemes that effectively mitigate false scattering, as pointed out above.
Ray Effects
Ray effects (Lathrop, 1968; 1971, Briggs et al., 1975; Morel et al., 2003) are related to the discretization of the angular distribution of the radiation intensity. They are independent of the spatial discretization scheme. Ray effects arise from the approximation of a continuously varying function (the radiation intensity) by a stepwise one, i.e., the radiation intensity field at a given point in space is represented by the radiation intensities at a finite number of directions. This occurs both for the DOM, where the radiation intensity field is represented by the radiation intensity at a discrete set of directions, and for the FVM, where the radiation intensity is assumed to remain constant within a control angle, even though its direction is allowed to change.
An example based on the work of Ramankutty and Crosbie (1997) is given here to illustrate ray effects. Let us consider a narrow rectangular enclosure with height H equal to 10 times the width L, and containing a transparent medium. A hot strip with length equal to half the width of the enclosure is placed at the center of the top wall. The remaining part of the boundary is cold. The walls are black. The incident heat flux along the bottom wall should be continuous, and decrease from the center to the corners of that wall. However, calculations performed using the DOM along with the step scheme, the S_{8} quadrature, and a uniform grid with 125 × 125 control volumes yield an incident heat flux at the bottom wall equal to zero. The reason for this behavior is that the radiation intensity field is represented by the radiation intensity along a finite number of directions. Since the enclosure is very narrow, the bottom wall may not receive any energy from the top wall, leading to unphysical results.
Although the medium has been assumed to be transparent in this example, ray effects are also present in the case of a participating medium, even though they are attenuated when the optical thickness of the medium increases. In fact, calculations performed for an emitting-absorbing medium in radiative equilibrium with an optical thickness of unity, based on the height of the enclosure, still predict zero heat flux at the bottom boundary. Interestingly, calculations performed using a coarse mesh with 5 × 5 control volumes predict a heat flux at the bottom boundary in fair agreement with the exact solution. The explanation for this behavior lies in the interaction between ray effects and false scattering, and is discussed in the following section.
Lathrop (1971) proposed several remedies to mitigate or eliminate ray effects in the solution of neutron transport problems. He concluded that the most promising remedy to mitigate ray effects is to use more directions, i.e., to refine the angular discretization. This is also valid for the RTE, but the ray effects are reduced slowly with angular refinement, particularly in the case of discontinuous boundary conditions. The choice of the selected directions may help to keep the total number of directions relatively small, but that choice is problem dependent. Several quadratures have been proposed aiming at the improvement of the solution accuracy, as discussed in the article “Angular discretization methods,” but in general this choice has a marginal impact on the mitigation of ray effects. Another method proposed to reduce ray effects is the introduction of coupling terms into the representation of the transport divergence operator. However, the discrete directions become coupled, and there is no a priori estimate of how much the ray effect is reduced.
A successful approach to mitigate ray effects is the modified DOM (Ramankutty and Crosbie, 1997), which is discussed in the article “Modified discrete ordinates and finite volume methods.” This approach has also been applied to the finite volume method. Lathrop (1971) also proposed a method to eliminate ray effects, based on the transformation of the discrete ordinates equations so that they are equivalent to spherical harmonic equations. This method has not been applied to radiative heat transfer problems.
Ray effects are discussed in the paper by Raithby (1999) mentioned above, which also examines false scattering. An investigation of the reduction of ray effects with the increase of the number of discrete directions is presented in Li et al. (2003). Estimation of ray effects in the DOM and FVM is presented in Tan et al. (2004).
Interaction between Ray Effects and False Scattering
Ray effects and false scattering are both present in the solution of radiative transfer problems using the DOM or the FVM, as shown in Chai et al. (1993). It turns out that these two error sources have opposite effects and tend to compensate each other, as discussed in Raithby (1999). False scattering tends to smooth the radiation intensity field, while ray effects tend to enhance discontinuities or gradients of the radiation intensity field.
An example of the interaction between ray effects and false scattering taken from Coelho (2002a) is presented here. A 2D square enclosure of side L and black walls is considered. The emissive power of the top boundary is equal to one, and the emissive power of the bottom and side boundaries is zero. The medium neither emits nor absorbs, but scatters isotropically, and is in radiative equilibrium. The optical thickness of the medium is equal to one. A quasi-exact solution of this problem is available (Crosbie and Schrenker, 1984), which is labeled as “exact” in Fig. 3. This figure shows the incident heat flux on the bottom boundary normalized by the emissive power of the top boundary. The calculations were carried out using the DOM. Only the results in the range 0.5 ≤ x/L ≤ 1.0 are shown for symmetry reasons.
Figure 3. Normalized incident heat flux on the bottom boundary of a square enclosure containing a purely isotropically scattering medium in radiative equilibrium: (a) S_{8} quadrature; (b) S_{16} quadrature; (c) N_{θ} = N_{φ} = 100 angles per octant
A relatively coarse grid with 15 × 15 control volumes, the step scheme, and the S_{8} quadrature yield reasonably accurate results, as shown in Fig. 3(a), but the maximum relative error exceeds 5%. If a more accurate spatial discretization is employed, either refining the grid or using the CLAM scheme, which is a high-resolution scheme (see the article “Spatial discretization schemes”), the solution error does not decrease, and an oscillatory solution is obtained. It is worth stressing that the CLAM scheme is bounded, as is the step scheme, and does not originate wiggles characteristic of unbounded spatial differencing schemes, such as the diamond scheme. The explanation for the observed deterioration of the solution accuracy is the following. The smooth and reasonable solution obtained using the coarse grid and the step scheme is due to compensation of the errors arising from false scattering and ray effects, which have an opposite effect and tend to compensate each other. The diffusive nature of the step scheme tends to smooth the radiation intensity field, while the ray effects tend to enhance discontinuities or gradients of the radiation intensity field. The grid refinement, as well as the use of a high-order accurate scheme, reduce the spatial discretization error, but have no influence on the error due to ray effects. The oscillatory solutions observed in Fig. 3(a) are due to ray effects, which are no longer compensated by spatial discretization errors. A finer angular discretization, such as the S_{16} quadrature, tends to reduce the ray effects. However, the results displayed in Fig. 3(b) reveal that ray effects are still significant.
On the other hand, if the step scheme and a 15 × 15 uniform grid are employed, refinement of the angular discretization does not improve the solution accuracy, as revealed by Figs. 3(a)–3(c). In fact, the solutions obtained using the S_{16} quadrature and a polar/azimuthal discretization with N_{θ} = N_{φ} = 100 angles per octant are less accurate than that computed using the S_{8} quadrature. Although the ray effects are largely suppressed by finer angular discretization, the spatial discretization error remains, and is no longer compensated by the error due to ray effects. The reduction of false scattering or ray effects may decrease solution accuracy if only one of these error sources is eliminated. The improvement of the solution accuracy requires the simultaneous reduction of false scattering and ray effects. This implies simultaneous spatial and angular refinement, or a more accurate spatial differencing scheme and angular refinement, as shown in Fig. 3(c). If only one of these two refinements is made, the solution accuracy may decrease.
REFERENCES
Briggs, L. L., Miller, Jr., W. F., and Lewis, E. E., Ray-Effect Mitigation in Discrete Ordinate-Like Angular Finite Element Approximations in Neutron Transport, Nucl. Sci. Eng., vol. 57, pp. 205−217, 1975.
Chai, J. C., Lee, H. S., and Patankar, S. V., Ray Effect and False Scattering in the Discrete Ordinates Method, Numer. Heat Transfer, Part B, vol. 24, pp. 373−389, 1993.
Coelho, P. J., The Role of Ray Effects and False Scattering on the Accuracy of the Standard and Modified Discrete Ordinates Methods, J. Quant. Spectrosc. Radiat. Transfer, vol. 73, pp. 231−238, 2002a.
Coelho, P. J., Bounded Skew High-Order Resolution Schemes for the Discrete Ordinates Method, J. Comput. Phys., vol. 175, pp. 412−437, 2002b.
Coelho, P. J., A Comparison of Spatial Discretization Schemes for Differential Solution Methods of the Radiative Transfer Equation, J. Quant. Spectrosc. Radiat. Transfer, vol. 109, no. 2, pp. 189−200, 2008.
Crosbie, A. L. and Schrenker, R. G., Radiative Transfer in a Two-Dimensional Rectangular Medium Exposed to Diffuse Radiation, J. Quant. Spectrosc. Radiat. Transfer, vol. 31, pp. 339−372, 1984.
Lathrop, K. D., Ray Effects in Discrete Ordinates Equations, Nucl. Sci. Eng., vol. 32, pp. 357−369, 1968.
Lathrop, K. D., Remedies for Ray Effects, Nucl. Sci. Eng., vol. 45, pp. 255−268, 1971.
Li, H.-S., Flamant, G., and Lu, J.-D., Reduction of False Scattering of the Discrete Ordinates Method, J. Heat Transfer, vol. 124, pp. 837−844, 2002.
Li, H.-S., Flamant, G., and Lu, J.-D., Mitigation of Ray Effects in the Discrete Ordinates Method, Numer. Heat Transfer, Part B, vol. 43, pp. 445−466, 2003.
Li, H.-S., Reduction of False Scattering in Arbitrarily Specified Discrete Directions of the Discrete Ordinates Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 86, pp. 215−222, 2004.
Morel, J. E., Wareing, T. A., Lowrie, R. B., and Parsons, D. K., Analysis of Ray-Effect Mitigation Techniques, Nucl. Sci. Eng., vol. 144, pp. 1−22, 2003.
Raithby, G. D., Evaluation of Discretization Errors in Finite-Volume Radiant Heat Transfer Predictions, Numer. Heat Transfer, Part B, vol. 36, pp. 241−264, 1999.
Ramankutty, M. A. and Crosbie, A. L., Modified Discrete Ordinates Solution of Radiative Transfer in Two-Dimensional Rectangular Enclosures, J. Quant. Spectrosc. Radiat. Transfer, vol. 57, pp. 107−140, 1997.
Tan, H.-P., Zhang, H.-C., and Zhen, B., Estimation of Ray Effect and False Scattering in Approximate Solution Method for Thermal Radiative Transfer Equation, Numer. Heat Transfer, Part A, vol. 46, pp. 807−829, 2004.
Zhang, H.-C. and Tan, H.-P., Evaluation of Numerical Scattering in the Finite volume Method for Solving Radiative Transfer Equation by a Central Laser Incidence Model, J. Quant. Spectrosc. Radiat. Transfer, vol. 110, pp. 1965−1977, 2009.
参考文献列表
- Briggs, L. L., Miller, Jr., W. F., and Lewis, E. E., Ray-Effect Mitigation in Discrete Ordinate-Like Angular Finite Element Approximations in Neutron Transport, Nucl. Sci. Eng., vol. 57, pp. 205−217, 1975.
- Chai, J. C., Lee, H. S., and Patankar, S. V., Ray Effect and False Scattering in the Discrete Ordinates Method, Numer. Heat Transfer, Part B, vol. 24, pp. 373−389, 1993.
- Coelho, P. J., The Role of Ray Effects and False Scattering on the Accuracy of the Standard and Modified Discrete Ordinates Methods, J. Quant. Spectrosc. Radiat. Transfer, vol. 73, pp. 231−238, 2002a.
- Coelho, P. J., Bounded Skew High-Order Resolution Schemes for the Discrete Ordinates Method, J. Comput. Phys., vol. 175, pp. 412−437, 2002b.
- Coelho, P. J., A Comparison of Spatial Discretization Schemes for Differential Solution Methods of the Radiative Transfer Equation, J. Quant. Spectrosc. Radiat. Transfer, vol. 109, no. 2, pp. 189−200, 2008.
- Crosbie, A. L. and Schrenker, R. G., Radiative Transfer in a Two-Dimensional Rectangular Medium Exposed to Diffuse Radiation, J. Quant. Spectrosc. Radiat. Transfer, vol. 31, pp. 339−372, 1984.
- Lathrop, K. D., Ray Effects in Discrete Ordinates Equations, Nucl. Sci. Eng., vol. 32, pp. 357−369, 1968.
- Lathrop, K. D., Remedies for Ray Effects, Nucl. Sci. Eng., vol. 45, pp. 255−268, 1971.
- Li, H.-S., Flamant, G., and Lu, J.-D., Reduction of False Scattering of the Discrete Ordinates Method, J. Heat Transfer, vol. 124, pp. 837−844, 2002.
- Li, H.-S., Flamant, G., and Lu, J.-D., Mitigation of Ray Effects in the Discrete Ordinates Method, Numer. Heat Transfer, Part B, vol. 43, pp. 445−466, 2003.
- Li, H.-S., Reduction of False Scattering in Arbitrarily Specified Discrete Directions of the Discrete Ordinates Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 86, pp. 215−222, 2004.
- Morel, J. E., Wareing, T. A., Lowrie, R. B., and Parsons, D. K., Analysis of Ray-Effect Mitigation Techniques, Nucl. Sci. Eng., vol. 144, pp. 1−22, 2003.
- Raithby, G. D., Evaluation of Discretization Errors in Finite-Volume Radiant Heat Transfer Predictions, Numer. Heat Transfer, Part B, vol. 36, pp. 241−264, 1999.
- Ramankutty, M. A. and Crosbie, A. L., Modified Discrete Ordinates Solution of Radiative Transfer in Two-Dimensional Rectangular Enclosures, J. Quant. Spectrosc. Radiat. Transfer, vol. 57, pp. 107−140, 1997.
- Tan, H.-P., Zhang, H.-C., and Zhen, B., Estimation of Ray Effect and False Scattering in Approximate Solution Method for Thermal Radiative Transfer Equation, Numer. Heat Transfer, Part A, vol. 46, pp. 807−829, 2004.
- Zhang, H.-C. and Tan, H.-P., Evaluation of Numerical Scattering in the Finite volume Method for Solving Radiative Transfer Equation by a Central Laser Incidence Model, J. Quant. Spectrosc. Radiat. Transfer, vol. 110, pp. 1965−1977, 2009.