BOUNDARY CONDITIONS
Following from: Discrete ordinates and finite volume methods
Opaque and Diffuse Walls with Prescribed Temperature
In the article “Mathematical Formulation”, the boundary condition of the radiative transfer equation (RTE) for an opaque surface that emits and reflects diffusely was given (Modest, 2003):
(1) |
In this equation I_{w}(r_{w}, s) is the radiation intensity leaving the boundary surface, I(r_{w}, s′) the radiation intensity in s′ direction arriving at that surface, I_{b}(r_{w}) = I_{bw} the blackbody radiation intensity at the temperature of the boundary surface, ε the surface emissivity, ρ the surface reflectivity, and n_{f} the outward unit vector normal to the surface, pointing from the surface to the medium. Equation (1) holds on a spectral basis. If the medium and the walls are grey, then the radiation intensity and the radiative properties of the wall are independent of the wavelength, and the equation is valid for the total radiation intensity. The equation is discretized as follows for the DOM:
(2) |
where w_{j} is the quadrature weight of the jth direction, and M is the total number of discrete directions. In the FVM, the discrete form of Eq. (1) is expressed as
(3) |
where D^{j}_{cf} is defined as
(4) |
In general, in computational domains of complex shape, the walls are not aligned with the coordinate axes of a Cartesian reference frame. In such a case, body-fitted structured or unstructured meshes are often used, and control angles bisected by the walls are usually found, as illustrated in Fig. 1 for control angle Ω^{m}. This means that s · n_{f} > 0 for some directions and s · n_{f} < 0 for other directions within the control angle. This situation, referred to as control angle overlap or control angle overhang in the FVM literature, is often ignored, and Eq. (3) is employed, yielding reasonable results for most practical problems (Chai et al., 1995, Murthy and Mathur, 1998, Liu et al., 2000). Nevertheless, a few methods to handle this situation are available, as briefly addressed below.
Figure 1. Control angle overhang.
In the case of two-dimensional problems, Chui and Raithby (1993) proposed a treatment for the control angle overhang that splits the evaluation of the integral over Ω^{m} into two parts, corresponding to the integrals over Ω^{m+} and Ω^{m−} (see Fig. 1). In the implementation of the boundary condition, only the integral over Ω^{m+} contributes to the irradiation onto the surface that appears in the last term on the right side of Eqs. (1) and (3). The integral over Ω^{m−} contributes to the radiative heat flux leaving the boundary. A similar procedure has been proposed by Kim et al. (2001).
A sub-solid angle discretization, commonly referred to as pixelation, has been proposed by Murthy and Mathur (1998), and can handle control angle overhang in three-dimensional problems. In this method, the control angle that bisects the boundary is subdivided into smaller control angles (both the azimuthal and the polar angle are subdivided), also referred to as pixels. Then, the calculation of the irradiation is carried out by adding the contributions from all j sub-solid angles that satisfy the condition D^{j}_{cf} < 0. The remaining sub-solid angles are associated to outgoing directions and contribute to the radiative heat flux leaving the boundary.
It is worth to point out that the control angle overhang also occurs at cell faces within the computational domain (see, e.g., Raithby, 1999a). The methods mentioned above for the treatment of control angle overhang at the boundary walls are also applicable to inner cell faces. Murthy and Mathur (1998) concluded that it is not necessary to account for control angle overhang at interior cell faces. At the boundaries, pixelation improves moderately the accuracy, and involves a small computational overhead, but it is not needed for fine angular discretization. A thorough analysis of the errors of the previously mentioned methods has been reported by Raithby (1999a). He concluded that the errors due to control angle overhang decrease with solid angle refinement, and that pixelation is the best option to decrease them.
The control angle overhang also occurs in the DOM, but it is generally ignored. If a quadrature that does not explicitly define the boundaries of the solid angle is used, like the S_{N} quadrature (see article “Angular Discretization Methods”), the control angle overhang must be ignored. However, if those boundaries are defined, like in the polar/azimuthal discretization, then the methods mentioned above to handle the control angle overhang for the FVM could also be used for the DOM.
Control angle overhang may be avoided if the direction of propagation of radiation intensity is defined in a local coordinate system. This option has not been used in the case of the FVM, but it was used by Vaillon et al. (1996) for the DOM, and applied in an orthogonal curvilinear coordinate system. A drawback of this approach is that the RTE includes additional angular redistribution terms that account for the variation of the polar and azimuthal angles with the spatial coordinates.
Opaque and Diffuse Walls with Prescribed Heat Flux
If the net heat flux, q′′_{w}, defined as the difference between the emitted and the absorbed heat flux, is prescribed at an opaque surface that emits and reflects diffusely, the boundary condition is written as
(5) |
The discretization of this equation for the DOM and FVM is straightforward.
Opaque, Partially Diffuse and Partially Specular Walls
In the case of an opaque surface that emits diffusely and has both diffuse and specular reflection components, the boundary condition is written as follows:
(6) |
where ρ^{d} and ρ^{s} are the diffuse and specular reflectivity components of the boundary surface, respectively, and s_{s} is the direction of the incoming radiation beam that is specularly reflected to direction s, which is given by
(7) |
The boundary condition is discretized as follows for the DOM and FVM, respectively:
(8) |
(9) |
where m_{s} denotes the direction of the incoming radiation beam that is specularly reflected to direction m at the wall. If the net heat flux is prescribed rather than the temperature of the wall, Eq. (5) remains valid.
If Cartesian or cylindrical coordinatesare used, the direction m_{s} belongs to the set of M directions of the angular discretization, provided that this is selected in such a way that the set of directions and weights remains invariant to any rotation of 90° about any one of the coordinate axis. The quadratures employed in the DOM generally satisfy this requirement (see article “Angular Discretization Methods”). In the case of body-fitted structured or unstructured grids, it is not generally feasible to accurately treat partially of fully specular boundaries, because the direction m_{s} is not included in the set of M directions of the quadrature. This difficulty may be overcome using the piecewise quasilinear angular quadrature (Rukolaine and Yuferev, 2001), which allows an analytical representation of the angular dependence of the radiation intensity (see article “Angular Discretization Methods”). Liu et al. (2000) suggested a linear interpolation technique to calculate I^{ms} by means of linear interpolation from the radiation intensities at neighbouring directions that belong to the set of M directions of the angular discretization.
Semitransparent Wall
In the case of a semitransparent wall there is external radiation penetrating into the physical domain, and the boundary condition is written as
(10) |
where I_{ext} denotes the external radiation intensity that penetrates into the domain along direction s, and ε the effective emissivity for the internal emission from the entire semitransparent wall thickness (Modest, 2003). If the wall is fully transparent, then ε = 0. The discretized boundary conditions for the DOM are FVM are given by equations similar to (8) and (9), respectively, except that the term I_{ext} is added to the terms on the right side of those equations.
In the case of external collimated radiation, a small solid angle that contains the direction of propagation may be used in the case of the FVM (Raithby, 1999b). In the DOM, the direction of the collimated beam is generally different from the discrete directions of the quadrature set. To overcome this problem, the direction of the collimated radiation beam may be expressed as an average combination of neighbouring directions (Lacroix et al., 2002). A more elegant approach is to add the direction of the collimated beam to the DOM quadrature set, and set the weight of that direction to be infinitely small (Li et al., 2003).
Adiabatic and Diffuse Wall
The boundary condition for an adiabatic and diffuse wall may be directly obtained from Eq. (5) by setting q′′_{w} = 0. The discretized boundary conditions for the DOM and FVM are obtained from Eqs. (2) and (3), respectively, by setting ε = 0 and ρ = 1.
Open Boundary
Open boundaries, such as the fluid flow inlets and outlets of a combustion chamber, or the free boundary of the computational domain in the simulation of an unconfined flame, are generally treated as black surfaces, since the radiative energy leaving the domain will not reenter the domain, and any external radiative energy reaching the boundary will penetrate into the domain. The temperature of the black surface that simulates the open boundary is equal to the temperature of a blackbody that emits the same amount of radiative energy as the external radiative energy reaching that boundary. In general, the blackbody will be at the temperature of the surrounding surfaces or at the temperature of the fluid at the boundary.
The computational domain should be selected in such a way that the open boundary is placed in a region of the physical domain where there is reliable information about the external radiative energy reaching that boundary, allowing the temperature of the blackbody to be defined. Liu et al. (2003) have shown that large errors may arise if the open boundary condition is placed in a region where there is no reliable information on the incoming radiative energy.
Symmetry Plane
A symmetry plane behaves as a specular reflecting wall, and the boundary condition is written as
(11) |
The discretized boundary conditions (8) and (9) still hold by setting ε = ρ^{d} = 0 and ρ^{s} = 1. If the symmetry plane is parallel to a coordinate axis of a Cartesian reference frame, then the direction m_{s} defined by the unit vector s_{s} belongs to the set of M discrete directions. Otherwise, the comments made for opaque, partially diffuse and partially specular walls concerning the calculation of the incoming radiation intensity in s_{s} direction remain valid.
Axis
In the case of axisymmetric geometries, the boundary condition at the axis is written as
(12) |
where θ and ψ are the polar angle measured from the axial direction and the azimuthal angle measured from the local radial direction, respectively (see Fig. 4 of the article “Mathematical formulation”). The direction of the incoming radiation intensity is available in the set of M discrete directions.
Periodic Boundary
The boundary condition for a periodic boundary may be written as
(13) |
where r_{w} and r_{w}* are the locations of points P and P*, respectively, which are periodically coincident, as illustrated in Fig. 2. In the case of translational symmetry, the normals to the periodic boundaries, n_{f} and n_{f}*, are coincident, as well as s and s*. In the case of rotational periodicity, n_{f}* and s* are rotated relatively to n_{f} and s, respectively, and control angle overhang is present. The control angle overhang is handled using the methods described above. Further details on periodic boundary conditions may be found in Mathur and Murthy (1999).
Figure 2. Periodic geometry.
Other kinds of boundary conditions may be found, e.g., surfaces with directionally dependent radiative properties, but they are not so common, and will not be addressed here. In the case of combined heat transfer modes, the boundary conditions are written using the theory provided above together with Fourier´s law for heat conduction, and Newton´s law of cooling for convective heat transfer.
REFERENCES
Chai, J. C., Parthasarathy, G., Lee, H. S., and Patankar, S. V., Finite Volume Radiative Heat Transfer Procedure for Irregular Geometries, J. Thermophysics Heat Transfer, vol. 9, no. 3, pp. 410−415, 1995.
Chui, E. H. and Raithby, G. D., Computation of Radiant Heat Transfer on a Nonorthogomal Mesh Using the Finite-Volume Method, Numerical Heat Transfer, Part B, vol. 23, pp. 269−288, 1993.
Kim, M. Y., Baek, S. W., and Park, J. H., Unstructured Finite-Volume Method for Radiative Heat Transfer in a Complex Two-Dimensional Geometry with Obstacles, Numerical Heat Transfer, Part B, vol. 39, pp. 617−635, 2001.
Lacroix, D., Parent, G., Asllanaj, F., and Jeandel, G., Coupled Radiative and Conductive Heat Transfer in a Non-Grey Absorbing and Emitting Semitransparent Media under Collimated Radiation, J. Quantitative Spectroscopy Radiative Transfer, vol. 75, no. 5, pp. 589−609, 2002.
Li, H.-S., Flamant, G., and Lu, J.-D., An Alternative Discrete Ordinate Scheme for Collimated Irradiation Problems, Int. Comm. Heat Mass Transfer, vol. 30, no. 1, pp. 61−70, 2003
Liu, J., Shang, H. M., and Chen, Y. S., Development of an Unstructured Radiation Model Applicable for Two-Dimensional Planar, Axisymmetric and Three-Dimensional Geometries, J. Quantitative Spectroscopy Radiative Transfer, vol. 66, pp. 17−33, 2000.
Liu, L. H., Ruan, L. M., and Tan, H. P., On the Treatment of Open Boundary Condition for Radiative Transfer Equation, Int. J. Heat Mass Transfer, vol. 46, pp. 181−183, 2003.
Mathur, S. R. and Murthy, J. Y., Radiative Heat Transfer in Periodic Geometries Using a Finite Volume Scheme, J. Heat Transfer, vol. 121, pp. 357−364, 1999.
Modest, M. F., Radiative Heat Transfer, New York: Academic Press, 2003.
Murthy, J. Y. and Mathur, S. R., Finite Volume Method for Radiative Heat Transfer Using Unstructured Meshes, J. Thermophysics Heat Transfer, vol. 12, no. 3, pp. 313−321, 1998.
Raithby, G. D., Evaluation of Discretization Errors in Finite-Volume Radiant Heat Transfer Predictions, Numerical Heat Transfer, Part B, vol. 36, pp. 241−264, 1999a.
Raithby, G. D., Discussion of the Finite-Volume Method for Radiation and its Application Using 3D Unstructured Meshes, Numerical Heat Transfer, Part B, vol. 35, pp. 389−405, 1999b.
Rukolaine, S. A. and Yuferev, V. S., Discrete Ordinates Quadrature Schemes Based on the Angular Interpolation of Radiation Intensity, J. Quantitative Spectroscopy Radiative Transfer, vol. 69, pp. 257−275, 2001.
Vaillon, R., Lallemand, M., andLemonnier, D., Radiative Heat Transfer in Orthogonal Curvilinear Coordinates Using the Discrete Ordinates Method, J. Quantitative Spectroscopy Radiative Transfer, vol. 55, no. 1, pp. 7−17, 1996.
Les références
- Chai, J. C., Parthasarathy, G., Lee, H. S., and Patankar, S. V., Finite Volume Radiative Heat Transfer Procedure for Irregular Geometries, J. Thermophysics Heat Transfer, vol. 9, no. 3, pp. 410−415, 1995.
- Chui, E. H. and Raithby, G. D., Computation of Radiant Heat Transfer on a Nonorthogomal Mesh Using the Finite-Volume Method, Numerical Heat Transfer, Part B, vol. 23, pp. 269−288, 1993.
- Kim, M. Y., Baek, S. W., and Park, J. H., Unstructured Finite-Volume Method for Radiative Heat Transfer in a Complex Two-Dimensional Geometry with Obstacles, Numerical Heat Transfer, Part B, vol. 39, pp. 617−635, 2001.
- Lacroix, D., Parent, G., Asllanaj, F., and Jeandel, G., Coupled Radiative and Conductive Heat Transfer in a Non-Grey Absorbing and Emitting Semitransparent Media under Collimated Radiation, J. Quantitative Spectroscopy Radiative Transfer, vol. 75, no. 5, pp. 589−609, 2002.
- Li, H.-S., Flamant, G., and Lu, J.-D., An Alternative Discrete Ordinate Scheme for Collimated Irradiation Problems, Int. Comm. Heat Mass Transfer, vol. 30, no. 1, pp. 61−70, 2003
- Liu, J., Shang, H. M., and Chen, Y. S., Development of an Unstructured Radiation Model Applicable for Two-Dimensional Planar, Axisymmetric and Three-Dimensional Geometries, J. Quantitative Spectroscopy Radiative Transfer, vol. 66, pp. 17−33, 2000.
- Liu, L. H., Ruan, L. M., and Tan, H. P., On the Treatment of Open Boundary Condition for Radiative Transfer Equation, Int. J. Heat Mass Transfer, vol. 46, pp. 181−183, 2003.
- Mathur, S. R. and Murthy, J. Y., Radiative Heat Transfer in Periodic Geometries Using a Finite Volume Scheme, J. Heat Transfer, vol. 121, pp. 357−364, 1999.
- Modest, M. F., Radiative Heat Transfer, New York: Academic Press, 2003.
- Murthy, J. Y. and Mathur, S. R., Finite Volume Method for Radiative Heat Transfer Using Unstructured Meshes, J. Thermophysics Heat Transfer, vol. 12, no. 3, pp. 313−321, 1998.
- Raithby, G. D., Evaluation of Discretization Errors in Finite-Volume Radiant Heat Transfer Predictions, Numerical Heat Transfer, Part B, vol. 36, pp. 241−264, 1999a.
- Raithby, G. D., Discussion of the Finite-Volume Method for Radiation and its Application Using 3D Unstructured Meshes, Numerical Heat Transfer, Part B, vol. 35, pp. 389−405, 1999b.
- Rukolaine, S. A. and Yuferev, V. S., Discrete Ordinates Quadrature Schemes Based on the Angular Interpolation of Radiation Intensity, J. Quantitative Spectroscopy Radiative Transfer, vol. 69, pp. 257−275, 2001.
- Vaillon, R., Lallemand, M., andLemonnier, D., Radiative Heat Transfer in Orthogonal Curvilinear Coordinates Using the Discrete Ordinates Method, J. Quantitative Spectroscopy Radiative Transfer, vol. 55, no. 1, pp. 7−17, 1996.