ALTERNATIVE FORMULATIONS
Following from: Discrete ordinates and finite volume methods
The discrete ordinates method (DOM) and the finite volume method (FVM) have been described in the article “Mathematical formulation”. In the FVM, both the spatial and the angular discretization are carried out using the finite volume method (Ferziger and Peric, 2002). In the DOM, the name “discrete ordinates” refers to the angular discretization procedure, the spatial formulation being generally carried out using the FVM. However, other spatial discretization methods are possible, while maintaining the discrete ordinates angular discretization. These alternative formulations of the DOM are addressed in this article, except the so-called modified discrete ordinates method, which will be described in the article “Modified discrete ordinates and finite volume methods”. However, we will first present a few modifications of the DOM and FVM formulation outlined in the article “Mathematical formulation” arising from the grid structure, namely, cell-vertex methods, blocked-off procedure, embedded boundaries, local grid refinement, and multiblock.
Cell-Vertex Methods
In the standard formulation, the radiation intensity is computed at the grid nodes located at the center of the control volumes that define the grid. This option is referred to as a cell-centered grid [see Fig. 1(a)]. Alternatively, the grid nodes may be placed at the vertices of the disjointed subdomains, referred to as elements, that define the mesh. In this case, the radiation intensity is calculated at the vertices of the elements, yielding the so-called vertex-centered grid. The control volume that surrounds a grid node is defined by joining together the centers of the elements that share that grid node [Fig. 1(b)] or by connecting the centers of the elements that share the grid node to the midpoints of the sides of those elements [Fig. 1(c)]. The spatial discretization schemes for unstructured grids described in the article “Spatial discretization schemes” are readily applicable to the control volumes illustrated in Fig. 1(b). In the case of control volumes generated as in Fig. 1(c), the radiation intensities at the cell faces are generally calculated using an upwind or an exponential scheme. Cell-vertex methods were used, for example, by Chui and Raithby (1993), Rousse (2000), Ben Salah et al. (2005a,b), Asllanaj et al. (2007), Grissa et al. (2007, 2010), and Rousse and Asllanaj (2011). Apart from the geometrical definition of the control volumes and the location of grid nodes, the mathematical formulation and the solution algorithm are similar to those described in the articles “Mathematical formulation” and “Solution algorithm.”
(a) | (b) | (c) |
Figure 1. Unstructured 2D cell-centered (a) and vertex-centered (b and c) grids; the dots denote the grid nodes, and the shaded regions identify controls volumes.
The formulation that employs the grid structure depicted in Fig. 1(c) is sometimes referred to as the control volume finite element formulation, due to the work of Baliga and Patankar (1980), who originally employed this grid structure for combined convection-diffusion problems. Strictly speaking, however, the name “control volume finite element” should only be used when the spatial discretization of the governing equations is carried out using a hybrid finite volume/finite element method, as in that work. This means that the RTE is discretized over a control volume defined as in Fig. 1(c), but the dependent variable (radiation intensity) at the cell faces is determined by expressing its variation across an element as a linear combination of shape functions defined according to the finite element method. A control volume finite element method was employed by Fiveland and Jessee (1995).
Blocked-Off Region Procedure
Cartesian or cylindrical coordinates are easier to employ than body-fitted coordinates or unstructured meshes, and they were often preferred in the past. The blocked-off method (Sanchez and Smith, 1992; Chai et al., 1993) was proposed to deal with obstacles and/or to simulate complex geometries, while still using Cartesian coordinates, and was later extended to study the effects of baffles (Adams and Smith, 1993; Chai et al., 1994; Coelho et al., 1998; Borjini et al., 2003, 2007; Guedri et al., 2009; Amiri et al., 2011). The domain contains active regions, where the solution is sought, and inactive regions (see Fig. 2), where the solution is also obtained, even though it is not meaningful. A source term S_{C} + S_{P}I_{b} is added to the final discretized equation for every control volume and direction, i.e., to the right-hand side of Eq. (10) of the article “Mathematical formulation” for the DOM and to the right-hand side of Eq. (25) of the same article for the FVM. The coefficients S_{C} and S_{P} of this source term are defined according to the location of the control volume. In active and internal control volumes, they are both equal to zero. In control volumes located at an interface between active and inactive regions, those source terms are defined in such a way that the boundary conditions for the active regions are enforced (see articles mentioned above for details).
Figure 2. Spatial domain discretization using a blocked-off grid structure; inactive regions are shaded.
In the case of curved or straight inclined boundaries, these must be approximated in a stepwise fashion, and a loss of accuracy is expected. In this case, it is generally better to use body-fitted coordinates or unstructured meshes (Kim et al., 2001; Sakami et al., 2001; Byun et al., 2003), but embedded boundaries are an alternative. Other disadvantages of the blocked-off procedure are the wasteful computations and memory storage for the inactive regions.
Embedded Boundaries
Embedded boundaries are aimed at an improvement of the blocked-off procedure. A Cartesian (or cylindrical) coordinate system is still employed, but an exact treatment of straight inclined boundaries is used, in contrast to the blocked-off procedure (see Fig. 3). Curved boundaries are approximated as piecewise straight lines, which may be skewed relative to the x and y directions. Hence, irregular polygonal control volumes may appear along the boundaries, and the RTE is integrated over these irregular control volumes, as in the case of arbitrary control volumes. This method was employed by Howell and Beckner (1997) and Byun et al. (2003). In this last work, the results obtained using embedded boundaries and a blocked-off procedure for Cartesian coordinates were compared with the results calculated using a body-fitted mesh. It was concluded that Cartesian meshes with embedded boundaries and body-fitted meshes yield similar results, while the blocked-off procedure produces some errors, especially for radiative heat fluxes on skewed boundaries.
Figure 3. Spatial domain discretization using embedded boundaries.
Local Grid Refinement
Local grid refinement is used to restrict a fine grid to specific locations of the domain, where the spatial discretization error is guessed or estimated to be high, while using a coarse grid elsewhere. The grid structure is characterized by a nested hierarchy of refined subgrids. A coarse mesh covers the entire computational domain. Then, a finer refinement level is placed at the desired locations by dividing the control volumes of the original grid into smaller ones. Typically, a control volume is divided in two, four, or eight equally sized control volumes for 1D, 2D, and 3D problems, respectively. These refined regions do not need to be contiguous. This procedure may be repeated, yielding regions of higher refinement level, as illustrated in Fig. 4.
Figure 4. Spatial domain discretization using local grid refinement.
This approach was used by Jessee et al. (1998) and Howell et al. (1999), who carried out the local grid refinement adaptively, during the course of the solution procedure, based on an estimation of the solution error. A multilevel algorithm was used to obtain the solution of the RTE. The solution algorithm differs substantially from the standard one, and is much more involved.
Multiblock
Spatial-multiblock procedures were used by Chai and Patankar (1997), Byun et al. (2003), and Talukdar et al. (2005). These procedures may be used as an alternative to local refinement by restricting grid refinement to regions where fine grids are needed for numerical accuracy reasons. They are also useful in geometrically complex domains, where a single structured grid would be difficult or even impossible to generate. Neighboring blocks may either overlap or not. The former option is simpler to implement, but the latter is more flexible and saves some memory and computing time. Grid lines at the interface between blocks may be either continuous or not. Continuous grid lines at the interface are straightforward to implement and ensure complete conservation of radiative energy. Discontinuous grid lines at the interface require an interpolation method to exchange data between neighboring blocks. The procedure proposed by Chai and Patankar (1997) and also used by Byun et al. (2003) ensures conservation of heat transfer rate, net radiant power, and other full-range and half-range moments across every block interface. The standard solution algorithm, which is directly applicable to every block, may be easily extended to multiblock grids.
It is worth pointing out that the multiblock strategy may also be applied to the angular discretization, as shown by Chai and Moder (2000), who used a coarse angular discretization in optically thick regions, and a fine angular discretization in optically thin ones. A procedure that ensures integral conservation of heat transferred between neighboring blocks was developed.
Even-Parity Formulation
The methods presented above use the finite volume method to perform spatial discretization of the RTE over a control volume, and are applicable to both the DOM and the FVM. Alternative formulations of the DOM are presented in the remainder of this article, which use other methods to perform the spatial discretization, while the angular discretization is carried out as in the standard DOM.
The even parity formulation of the RTE was originally introduced to solve neutron transport problems, and later brought to the heat transfer community by Song and Park (1992). It is based on the transformation of the RTE (first-order integrodifferential equation) into a second-order integrodifferential equation. In this way, a hyperbolic type equation (RTE) is transformed into an elliptic (or parabolic, depending on the spatial discretization scheme) equation. In other words, an initial value problem is replaced by a boundary value problem. The former may yield physically unrealistic negative intensities when a diamond scheme is used for the discretization of the gradient of the radiation intensity (see article “Spatial discretization schemes”) or ray effects when a step scheme is used. The even-parity equations do not yield unrealistic intensities, since the governing differential equations involve second-order derivatives, yielding a positive definite and self-adjoint system of equations. The even-parity form of the RTE is derived by considering opposite directions s and −s, and by defining the following quantities
(1) |
(2) |
Adding and subtracting the RTE written for directions s and −s, and using Eqs. (1) and (2), yields
(3) |
(4) |
where the integration of the in-scattering terms is only made over 2π. In the case of isotropic scattering (see Song and Park, 1992, for the more general case of anisotropic scattering), Eq. (3) reduces to
(5) |
Inserting this relation into Eq. (4), the following second-order differential equation for ψ is obtained:
(6) |
Note that cross-derivatives appear on the left-hand side of this equation, in contrast to the standard formulation. The boundary conditions for this equation may be derived from the boundary conditions for the RTE. In the case of an opaque and diffuse boundary, the boundary condition of the RTE may be written as (see article “Mathematical formulation”)
(7) |
for n · s > 0, and
(8) |
for n · s < 0. Variable φ may be eliminated and these two expressions merged in a single one by noting that the sign of s · ∇ψ(r, s) depends on the sign of n · s. The following boundary condition is obtained:
(9) |
where sgn(n · s) = n · s / |n · s|. The integration of the even-parity form of the RTE, Eq. (6), is carried out over a solid angle of 2π, i.e., only M/2 unknowns need to be determined instead of M, M being the number of directions considered in the standard formulation. However, the M/2 values of ψ are sufficient to determine the radiation intensities for all M directions. In fact, once ψ is determined for a given direction, s^{m}, φ may be readily calculated for the same direction from Eq. (5), which yields the radiation intensity at directions s^{m} and −s^{m} from Eqs. (1) and (2).
The spatial discretization of Eq. (6) may be carried out using the finite volume method, as in Koch et al. (1996), Liu et al. (1997), and Liu and Chen (1999), while the angular discretization is still carried out using the DOM. Central differences were used in these works to discretize the second-order spatial derivative terms in the governing equation. The nonmixed derivative terms were treated implicitly, while the mixed derivative terms were lumped into the source term and treated explicitly. The discretization of the boundary condition, which involves first-order derivatives, was carried out using a first-order upwind scheme in the work of Koch et al. (1996), and second- or third-order upwind schemes in the papers of Liu et al. (1997) and Liu and Chen (1999) cited above. Further details on the discretization procedure and solution algorithm may be found in these references. Liu et al. (1997) and Liu and Chen (1999) found that the even-parity formulation usually requires more CPU time and more iterations to converge than the standard formulation, especially for optically thin media.
The finite element method was used by Fiveland and Jessee (1994) to spatially discretize the even-parity formulation of the RTE. In a subsequent work (Fiveland and Jessee, 1995), four different formulations were compared, namely, the standard RTE formulation using either the finite volume method or the control volume finite element method, and the even-parity formulation of the RTE using either the finite element method or the control volume finite element method. It was found that, in general, the predictions obtained using the standard RTE are in better agreement with the reference solutions than those calculated using the even-parity formulation. This conclusion is corroborated by Kang and Song (2008), who used the finite element method along with the standard and the even-parity formulations of the RTE.
According to the previous discussion, and despite some advantages, the even-parity formulation of the RTE does not seem to be a good option relatively to the standard RTE. In fact, the computational requirements are higher, and the accuracy is often lower.
Discrete Ordinates Interpolation Method
The discrete ordinates interpolation method (DOIM) was originally developed for the even-parity formulation of the RTE (Cheong and Song, 1995), and later extended to the standard formulation of the RTE (Cheong and Song, 1997). Here, we will briefly describe the method for 2D nonorthogonal meshes, as presented in Cheong and Song (1997). The RTE is written in the form
(10) |
where the source term Ṡ, which includes the radiative emission and the in-scattering terms, is approximated by the first two terms of a Taylor series expansion about a point P as
(11) |
Here, s is the spatial coordinate along the direction of propagation of radiation, and Δs is the distance along which the RTE is integrated. Inserting Eq. (11) into Eq. (10) and treating β as a constant, Eq. (10) can be exactly solved, yielding the radiation intensity at grid node P as a function of the radiation intensity at an upstream location I_{IN}, as shown in Fig. 5,
Figure 5. Grid notation in the discrete ordinates interpolation method.
(12) |
Note that the concept of control volume is not used. Grid nodes are located at the intersection of the grid lines. The radiation intensity at the upstream location is interpolated as follows:
(13) |
where N + 1 is the number of grid nodes used in the interpolation, N being an odd number, and L_{n} is a Lagrange polynomial base of order N. In general, N is taken as 1 (linear interpolation) or 3 (cubic interpolation). This equation needs to be modified in the vicinity of the boundaries. In the case of linear interpolation, and for the situation illustrated in Fig. 1, L_{0} = (Δ_{l} − δ_{l}) / Δ_{l} and L_{1} = δ_{l} / Δ_{l}. The gradient of the source term is evaluated as follows:
(14) |
The source term at the upstream location, IN, is interpolated in the same way as the radiation intensity. Inserting Eqs. (13) and (14) into Eq. (12), the radiation intensity at grid node P is expressed as a linear combination of the radiation intensity at upstream nodes.
Cheong and Song (1997) concluded that the DOIM is more accurate than the standard DOM using the diamond scheme and the FVM of Raithby and Chui (1990), and produces less false scattering. They have also found that cubic interpolation is more accurate than linear interpolation, but undershooting or overshooting may occur. A comparison between the DOIM applied to the standard RTE and to the even-parity formulation is reported in Koo et al. (1997). It was found that the predictions obtained using the standard RTE are more accurate than those determined using the even-parity formulation for most of the tests carried out. This conclusion is in agreement with the results of Fiveland and Jessee (1995) and Kang and Song (2008), who also compared the standard and the even-parity formulations of the RTE using other spatial discretization methods. Koo et al. (2003) compared the DOIM with two other versions of the DOM for 2D curved geometries, namely, a finite volume spatial discretization for orthogonal curvilinear coordinates (Vaillon et al., 1996), and a finite volume spatial discretization for unstructured grids mapped using triangular control volumes (Sakami et al., 1998). All the works mentioned above using the DOIM were restricted to 2D structured grids. An extension to 2D unstructured grids is presented in Cha et al. (1998), while extensions to 3D grids are reported in Seo and Kim (1998) and Cha and Song (2000).
A major drawback of the DOIM is that it is not conservative, i.e., it does not guarantee conservation of energy over a control volume. Moreover, when the DOIM is used together with a finite volume method for coupled fluid flow and heat transfer problems, it is necessary to provide the radiation intensity at the centers of control volumes or cell faces, whichever are missing. An interpolation scheme to provide the missing radiation intensities was proposed in Kim et al. (2009) and tested in 1D problems. The results obtained were accurate and free from physically unrealistic intensities.
Pseudo–Time Stepping and Method of Lines
The discrete ordinates with time stepping (DOTS) (Fiterman et al., 1999) and the method of lines (MOL) solution of the DOM (Selçuk and Kırbaş, 2000; Selçuk and Ayrancı, 2003) transform the original boundary value problem governed by the RTE to an initial-value problem by adding a time derivative term to the RTE. This technique is widely used in computational fluid dynamics. The time is a new independent variable that does not affect the results, since the solution of the original boundary value problem is obtained as the time gets sufficiently large, such that steady state conditions are reached.
In the DOTS method of Fiterman et al. (1999), the time derivative of the radiative energy flux and an artificial viscosity term were added to the RTE. The resultant equation was then integrated over a control volume using the central differences scheme. The integration in time was carried out using the Euler forward-difference scheme. A local time step weight function was used for each control volume and direction of propagation. A multigrid method was used to accelerate convergence.
In the MOL solution of the DOM, only the time derivative of the radiation intensity was added to the RTE. The spatial derivatives were calculated using a Taylor series (Selçuk and Kırbaş, 2000) or two- and three-point upwind differencing schemes (Selçuk and Ayrancı, 2003). The resulting set of ordinary differential equations (ODEs) was integrated until steady state using a publicly available solver for ODEs. This solver chooses the time steps in a way that maintains the accuracy and stability of the evolving solution. Selçuk and Kırbaş (2000) claim that the results are significantly better than those computed using the finite volume method when the medium is optically thin.
Other Formulations of the DOM
A second-order formulation of the RTE with the radiation intensity as the dependent variable, in contrast with the even-parity formulation, has been proposed by Zhao and Liu (2007a), who used the finite element method for the spatial discretization and discrete ordinates for the angular discretization. This formulation of the RTE was employed by Hassanzadeh and Raithby (2008), who concluded that, despite the accuracy and smoothness of the results obtained, the computational cost is high, mainly due to the elliptic nature of the governing differential equation, and the large bandwidth and lack of diagonal dominance of the system of discretized equations.
Several other methods for the solution of the RTE have been developed that employ different spatial discretization schemes while retaining the discrete ordinates angular discretization. These methods are generally referenced by the name of the spatial discretization method employed, but they are mentioned here because of their affinity with the DOM, i.e., they use the same angular discretization.
The finite element method was employed by Fiveland and Jessee (1995), Liu (2004), and An et al. (2005) to solve the standard RTE, and by Fiveland and Jessee (1994, 1995) and Kang and Song (2008) to solve the even-parity formulation of the RTE. A discontinuous finite element method, which employs either a discrete ordinates or a finite volume method for the angular discretization, was developed by Cui and Li (2004a,b, 2005a,b). The least-squares finite element method was used by Pontaza and Reddy (2005) and Ruan et al. (2007).
Spectral methods have recently been applied to the spatial discretization of the standard formulation of the RTE. For example, the Chebyshev collocation spectral method was employed by Li et al. (2008). The spectral element method was used by Zhao and Liu (2008), the discontinuous spectral element method was developed by Zhao and Liu (2007b), and the least-squares spectral element method was applied by Zhao and Liu (2006).
Several meshless methods have also been applied to the spatial discretization of the RTE. These include the local Petrov-Galerkin method (Liu, 2006), the moving least-squares collocation method (Sadat, 2006; Liu and Tan, 2007; Tan et al., 2009) and the least-squares radial point interpolation collocation method (Tan et al., 2007). Sadat (2006) considered both the standard and the even-parity formulations of the RTE, and claimed that the former appeared to be unstable and led to oscillatory results, while the latter was always stable and accurate. An extension of this method to complex 2D and 3D geometries is reported in Wang et al. (2010). Tan et al. (2009) compared the moving least-squares collocation meshless method applied to the RTE with a direct collocation meshless method applied to the standard RTE and to the second-order RTE proposed by Zhao and Liu (2007a). They found that the direct collocation meshless method applied to the second-order RTE gives more accurate and stable results than the other options.
Finally, we mention the modified collapsed dimension method (Mishra et al., 2006), whose approach is similar to the DOM after collapsing the radiative information to the 2D solution plane. Recently, the FVM was also applied to the collapsed dimension method (Mishra et al., 2011). The interested reader is referred to the cited references for details on these methods, solution algorithms, and performance.
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Raithby, G. D. and Chui, E. H., A Finite Volume Method for Predicting a Radiant Heat Transfer in Enclosures with Participating Media, J. Heat Transfer, vol. 112, pp. 415−423, 1990.
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Sadat, H., On the Use of a Meshless Method for Solving Radiative Transfer with the Discrete Ordinates Formulation, J. Quant. Spectrosc. Radiat. Transfer, vol. 101, pp. 263−268, 2006.
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Sanchez, A. and Smith, T. F., Surface Radiation Exchange for Two-Dimensional Rectangular Enclosures Using the Discrete-Ordinates Method, J. Heat Transfer, vol. 114, pp. 465−472, 1992.
Selçuk, N. and Ayrancı, I., The Method of Lines Solution of the Discrete Ordinates Method for Radiative Heat Transfer in Enclosures Containing Scattering Media, Numer. Heat Transfer, Part B, vol. 43, pp. 179−201, 2003.
Selçuk, N. and Kırbaş, G., The Method of Lines Solution of the Discrete Ordinates Method for Radiative Heat Transfer in Enclosures, Numer. Heat Transfer, Part B, vol. 37, pp. 379−392, 2000.
Seo, S.-H. and Kim, T.-K., Study on Interpolation Schemes of the Discrete Ordinates Interpolation Method for Three-Dimensional Radiative Transfer with Nonorthogonal Grids, J. Heat Transfer, vol. 120, pp. 1091−1094, 1998.
Song, T. H. and Park, C. W., Formulation and Application of the Second Order Discrete Ordinate Method, Transport Phenomena Science and Technology, B.-X. Wang, Ed., Beijing: Higher Education Press, 1992.
Talukdar, P., Steven, M. Isendorff, F. V., and Trimis, D., Finite Volume Method in 3-D Curvilinear Coordinates with Multiblocking Procedure for Radiative Transport Problems, Int. J. Heat Mass Transfer, vol. 48, pp. 4657−4666, 2005.
Tan, J. Y., Liu, L. H., and Li, B. X., Least-Squares Radial Point Interpolation Collocation Meshless Method for Radiative Heat Transfer, J. Heat Transfer, vol. 129, pp. 669−673, 2007.
Tan, J. Y., Zhao, J. M., Liu, L. H., and Wang, Y. Y., Comparative Cost on Accuracy and Solution Cost of the First/Second-Order Radiative Transfer Equations Using the Meshless Method, Numer. Heat Transfer, Part B, vol. 55, pp. 324−337, 2009.
Vaillon, R., Lallemand, M., and Lemonnier, D., Radiative Heat Transfer in Orthogonal Curvilinear Coordinates Using the Discrete Ordinates Method, J. Quant. Spectrosc. Radiat. Transfer, vol. 55, no. 1, pp. 7−17, 1996.
Wang, C.-A., Sadat, H., Ledez, V., and Lemonnier, D., Meshless Method for Solving Radiative Transfer in Complex Two-Dimensional and Three-Dimensional Geometries, Int. J. Thermal Sci., vol. 49, pp. 2282−2288, 2010.
Zhao, J. M. and Liu, L. H., Least-Squares Spectral Element Method for Radiative Heat Transfer in Semitransparent Media, Numer. Heat Transfer, Part B, vol. 50, no. 5, pp. 473−489, 2006.
Zhao, J. M. and Liu, L. H., Second-Order Radiative Transfer Equation and its Properties of Numerical Solution Using the Finite-Element Method, Numer. Heat Transfer, Part B, vol. 51, pp. 391−409, 2007a.
Zhao, J. M. and Liu, L. H., Discontinuous Spectral Element Method for Solving Radiative Heat Transfer in Multidimensional Semitransparent Media, J. Quant. Spectrosc. Radiat. Transfer, vol. 107, pp. 1−16, 2007b.
Zhao, J. M. and Liu, L. H., Spectral Element Method with Adaptive Artificial Diffusion for Solving the Radiative Transfer Equation, Numer. Heat Transfer, Part B, vol. 53, no. 6, pp. 536−554, 2008.
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- Zhao, J. M. and Liu, L. H., Spectral Element Method with Adaptive Artificial Diffusion for Solving the Radiative Transfer Equation, Numer. Heat Transfer, Part B, vol. 53, no. 6, pp. 536−554, 2008.