
The discrete ordinates (DO) radiation model solves the radiative transfer equation (RTE) for a finite number of discrete solid angles, each associated with a vector direction fixed in the global Cartesian system ( ). The fineness of the angular discretization is controlled by you, analogous to choosing the number of rays for the DTRM. Unlike the DTRM, however, the DO model does not perform ray tracing. Instead, the DO model transforms Equation 13.31 into a transport equation for radiation intensity in the spatial coordinates ( ). The DO model solves for as many transport equations as there are directions . The solution method is identical to that used for the fluid flow and energy equations.
Two implementations of the DO model are available in FLUENT: uncoupled and (energy) coupled. The uncoupled implementation is sequential in nature and uses a conservative variant of the DO model called the finitevolume scheme [ 61, 294], and its extension to unstructured meshes [ 256]. In the uncoupled case, the equations for the energy and radiation intensities are solved one by one, assuming prevailing values for other variables.
Alternatively, in the coupled ordinates method (or COMET) [ 232], the discrete energy and intensity equations at each cell are solved simultaneously, assuming that spatial neighbors are known. The advantages of using the coupled approach is that it speeds up applications involving high optical thicknesses and/or high scattering coefficients. Such applications slow down convergence drastically when the sequential approach is used.
The DO Model Equations
The DO model considers the radiative transfer equation (RTE) in the direction as a field equation. Thus, Equation 13.31 is written as
FLUENT also allows the modeling of nongray radiation using a grayband model. The RTE for the spectral intensity can be written as
Here is the wavelength, is the spectral absorption coefficient, and is the black body intensity given by the Planck function. The scattering coefficient, the scattering phase function, and the refractive index are assumed independent of wavelength.
The nongray DO implementation divides the radiation spectrum into wavelength bands, which need not be contiguous or equal in extent. The wavelength intervals are supplied by you, and correspond to values in vacuum ( ). The RTE is integrated over each wavelength interval, resulting in transport equations for the quantity , the radiant energy contained in the wavelength band . The behavior in each band is assumed gray. The black body emission in the wavelength band per unit solid angle is written as
(13.339) 
where is the fraction of radiant energy emitted by a black body [ 248] in the wavelength interval from 0 to at temperature in a medium of refractive index . and are the wavelength boundaries of the band.
The total intensity in each direction at position is computed using
(13.340) 
where the summation is over the wavelength bands.
Boundary conditions for the nongray DO model are applied on a band basis. The treatment within a band is the same as that for the gray DO model.
Energy Coupling and the DO Model
The coupling between energy and radiation intensities at a cell (which is also known as COMET) [ 232] accelerates the convergence of the finite volume scheme for radiative heat transfer. This method results in significant improvement in the convergence for applications involving optical thicknesses greater than 10. This is typically encountered in glassmelting applications. This feature is advantageous when scattering is significant, resulting in strong coupling between directional radiation intensities. This DO model implementation is utilized in FLUENT by enabling the DO/Energy Coupling option for the DO model in the Radiation Model panel. The discrete energy equations for the coupled method are presented below.
The energy equation when integrated over a control volume , yields the discrete energy equation:
where  =  
=  
=  
=  absorption coefficient  
=  control volume 
The coefficient and the source term are due to the discretization of the convection and diffusion terms as well as the nonradiative source terms.
Combining the discretized form of Equation 13.337 and the discretized energy equation, Equation 13.341, yields [ 232]:
(13.342) 
where
(13.343) 
(13.344) 
(13.345) 
Limitations of DO/Energy Coupling
There are some instances when using DO/Energy coupling is not recommended or is incompatible with certain models:
To find out how to apply DO/Energy coupling, refer to Section 13.3.13.
Angular Discretization and Pixelation
Each octant of the angular space at any spatial location is discretized into solid angles of extent , called control angles. The angles and are the polar and azimuthal angles respectively, and are measured with respect to the global Cartesian system as shown in Figure 13.3.3. The and extents of the control angle, and , are constant. In twodimensional calculations, only four octants are solved due to symmetry, making a total of directions in all. In threedimensional calculations, a total of directions are solved. In the case of the nongray model, or equations are solved for each band.
When Cartesian meshes are used, it is possible to align the global angular discretization with the control volume face, as shown in Figure 13.3.4. For generalized unstructured meshes, however, control volume faces do not in general align with the global angular discretization, as shown in Figure 13.3.5, leading to the problem of control angle overhang [ 256].
Essentially, control angles can straddle the control volume faces, so that they are partially incoming and partially outgoing to the face. Figure 13.3.6 shows a 3D example of a face with control angle overhang.
The control volume face cuts the sphere representing the angular space at an arbitrary angle. The line of intersection is a great circle. Control angle overhang may also occur as a result of reflection and refraction. It is important in these cases to correctly account for the overhanging fraction. This is done through the use of pixelation [ 256].
Each overhanging control angle is divided into pixels, as shown in Figure 13.3.7.
The energy contained in each pixel is then treated as incoming or outgoing to the face. The influence of overhang can thus be accounted for within the pixel resolution. FLUENT allows you to choose the pixel resolution. For problems involving graydiffuse radiation, the default pixelation of is usually sufficient. For problems involving symmetry, periodic, specular, or semitransparent boundaries, a pixelation of is recommended. You should be aware, however, that increasing the pixelation adds to the cost of computation.
Anisotropic Scattering
The DO implementation in FLUENT admits a variety of scattering phase functions. You can choose an isotropic phase function, a linear anisotropic phase function, a DeltaEddington phase function, or a userdefined phase function. The linear anisotropic phase function is described in Equation 13.37. The DeltaEddington function takes the following form:
Here, is the forwardscattering factor and is the Dirac delta function. The term essentially cancels a fraction of the outscattering; thus, for , the DeltaEddington phase function will cause the intensity to behave as if there is no scattering at all. is the asymmetry factor. When the DeltaEddington phase function is used, you will specify values for and .
When a userdefined function is used to specify the scattering phase function, FLUENT assumes the phase function to be of the form
The userdefined function will specify and the forwardscattering factor .
The scattering phase functions available for gray radiation can also be used for nongray radiation. However, the scattered energy is restricted to stay within the band.
Particulate Effects in the DO Model
The DO model allows you to include the effect of a discrete second phase of particulates on radiation. In this case, FLUENT will neglect all other sources of scattering in the gas phase.
The contribution of the particulate phase appears in the RTE as:
where is the equivalent absorption coefficient due to the presence of particulates, and is given by Equation 13.310. The equivalent emission is given by Equation 13.39. The equivalent particle scattering factor , defined in Equation 13.313, is used in the scattering terms.
For nongray radiation, absorption, emission, and scattering due to the particulate phase are included in each wavelength band for the radiation calculation. Particulate emission and absorption terms are also included in the energy equation.
Boundary Condition Treatment at Opaque Walls
The discrete ordinates radiation model allows the specification of opaque walls that are interior to a domain (with adjacent fluid or solid zones on both sides of the wall), or external to the domain (with an adjacent fluid or solid zone on one side, only). Opaque walls are treated as gray if gray radiation is being computed, or nongray if the nongray DO model is being used.
Figure 13.3.8 shows a schematic of radiation on an opaque wall in FLUENT.
The diagram in Figure 13.3.8 shows incident radiation on side a of an opaque wall. Some of the radiant energy is reflected diffusely and specularly, depending on the diffuse fraction for side a of the wall that you specify as a boundary condition.
Some of the incident radiation is absorbed at the surface of the wall and some radiation is emitted from the wall surface as shown in Figure 13.3.8. The amount of incident radiation absorbed at the wall surface and the amount emitted back depends on the emissivity of that surface and the diffuse fraction. For nongray DO models, you must specify internal emissivity for each wavelength band. Radiation is not transmitted through an opaque wall.
Radiant incident energy that impacts an opaque wall can be reflected back to the surrounding medium and absorbed by the wall. The radiation that is reflected can be diffusely reflected and/or specularly reflected, depending on the diffuse fraction . If is the amount of radiative energy incident on the opaque wall, then the following general quantities are computed by FLUENT for opaque walls:
where is the diffuse fraction, is the refractive index of the adjacent medium, is the wall emissivity, is Boltzmann's Constant, and is the wall temperature.

There is
no emission or absorption in the specular component of reflected energy for an opaque wall surface.

Note that although FLUENT uses emissivity in its computation of radiation quantities, it is not available for postprocessing. Absorption at the wall surface assumes that the absorptivity is equal to the emissivity. For a purely diffused wall, is equal to and there is no specularly reflected energy. Similarly, for a purely specular wall, is equal to and there is no diffusely reflected energy. A diffuse fraction between and will result in partially diffuse and partially reflected energy.

Note that in practice, fully specular surfaces (diffuse fraction =
) do have some emission and absorption occurring, albeit very low. To define a problem that has a purely specular surface with emissivity, you can set the internal emissivity to
and set the diffuse fraction to the intended emissivity. To define a problem that has a partiallyspecular surface, set the internal emissivity to suit the net surface emissivity (not necessarily
) that you expect to achieve (internal emissivity
net emissivity
diffuse fraction).

Gray Diffuse Walls
For gray diffuse radiation, the incident radiative heat flux, , at the wall is
(13.349) 
The net radiative flux leaving the surface is given by
where is the refractive index of the medium next to the wall, is the wall emissivity, is Boltzmann's Constant, and is the wall temperature. This equation is also valid for specular radiation with emissivity = .
The boundary intensity for all outgoing directions at the wall is given by
(13.351) 
NonGray Diffuse Walls
There is a special set of equations that apply uniquely to nongray diffuse opaque walls. These equations assume that the absorptivity is equal to the emissivity for the wall surface. For nongray diffuse radiation, the incident radiative heat flux in the band at the wall is
(13.352) 
The net radiative flux leaving the surface in the band is given by
where is the wall emissivity in the band. provides the Planck distribution function. The boundary intensity for all outgoing directions in the band at the wall is given by
(13.354) 
Boundary Condition Treatment at SemiTransparent Walls
FLUENT allows the specification of interior and exterior semitransparent walls for the DO model. In the case of interior semitransparent walls, incident radiation can pass through the wall and be transmitted to the adjacent medium (and possibly refracted), it can be reflected back into the surrounding medium, and absorbed through the wall thickness. Transmission and reflection can be diffuse and/or specular. You specify the diffuse fraction for all transmitted and reflected radiation; the rest is treated specularly. For exterior semitransparent walls, there are two possible sources of radiation on the boundary wall: an irradiation beam from outside the computational domain and incident radiation from cells in adjacent fluid or solid zones.
For nongray radiation, semitransparent wall boundary conditions are applied on a perband basis. The radiant energy within a band is transmitted, reflected, and refracted as in the gray case; there is no transmission, reflection, or refraction of radiant energy from one band to another.
By default the DO equations are solved in all fluid zones, but not in any solid zones. Therefore, if you have an adjacent solid zone for your thin wall, you will need to specify the solid zone as participating in radiation in the Solid panel as part of the boundary condition setup.

If you are interested in the detailed temperature distribution inside your semitransparent media, then you will need to model a semitransparent wall as a solid zone with adjacent fluid zone(s), and treat the solid as a semitransparent medium. This is discussed in a subsequent section.

SemiTransparent Interior Walls
Figure 13.3.9 shows a schematic of an interior (twosided) wall that is treated as semitransparent in FLUENT and has zero thickness. Incident radiant energy depicted by can pass through the semitransparent wall if and only if the contiguous fluid or solid cell zones participate in radiation, thereby allowing the radiation to be coupled. Radiation coupling is set when a wall is specified as semitransparent. Note that by default, radiation is not coupled and you will need to explicitly specify radiation coupling on the interior wall by changing the boundary condition type to semitransparent in the Wall panel (under the Radiation tab).
Incident radiant energy that is transmitted through a semitransparent wall can be transmitted specularly and diffusely. Radiation can also be reflected at the interior wall back to the surrounding medium if the refractive index for the fluid zone that represents medium is different than the refractive index for medium . Reflected radiation can be reflected specularly and diffusely. The fraction of diffuse versus specular radiation that is transmitted and reflected depends on the diffuse fraction for the wall. The special cases of purely diffuse and purely specular transmission and reflection on semitransparent walls is presented in the following sections.
If the semitransparent wall has thickness, then the thickness and the absorption coefficient determine the absorptivity of the 'thin' wall. If either the wall thickness or absorption coefficient is set to , then the wall has no absorptivity. Although incident radiation can be absorbed in a semitransparent wall that has thickness, note that the absorbed radiation flux does not affect the energy equation. FLUENT does not consider emission from semitransparent walls except for the case when a specified temperature boundary condition is defined.
Specular SemiTransparent Walls
Consider the special case for a semitransparent wall, when the diffuse fraction is equal to and all of the transmitted and reflected radiant energy at the semitransparent wall is purely specular.
Figure 13.3.10 shows a ray traveling from a semitransparent medium with refractive index to a semitransparent medium with a refractive index in the direction . Surface of the interface is the side that faces medium ; similarly, surface faces medium . The interface normal is assumed to point into side . We distinguish between the intensity , the intensity in the direction on side of the interface, and the corresponding quantity on the side , .
A part of the energy incident on the interface is reflected, and the rest is transmitted. The reflection is specular, so that the direction of reflected radiation is given by
The radiation transmitted from medium to medium undergoes refraction. The direction of the transmitted energy, , is given by Snell's law:
where is the angle of incidence and is the angle of transmission, as shown in Figure 13.3.10. We also define the direction
shown in Figure 13.3.10.
The interface reflectivity on side [ 248]
represents the fraction of incident energy transferred from to .
The boundary intensity in the outgoing direction on side of the interface is determined from the reflected component of the incoming radiation and the transmission from side . Thus
where is the transmissivity of side in direction . Similarly, the outgoing intensity in the direction on side of the interface, , is given by
For the case , the energy transmitted from medium to medium in the incoming solid angle must be refracted into a cone of apex angle (see Figure 13.3.11) where
Similarly, the transmitted component of the radiant energy going from medium to medium in the cone of apex angle is refracted into the outgoing solid angle . For incident angles greater than , total internal reflection occurs and all the incoming energy is reflected specularly back into medium . The equations presented above can be applied to the general case of interior semitransparent walls that is shown in Figure 13.3.9.
When medium is external to the domain as in the case of an external semitransparent wall (Figure 13.3.12), is given in Equation 13.359 as a part of the boundary condition inputs. You supply this incoming irradiation flux in terms of its magnitude, beam direction, and the solid angle over which the radiative flux is to be applied. Note that the refractive index of the external medium is assumed to be .
Diffuse SemiTransparent Walls
Consider the special case for a semitransparent wall, when the diffuse fraction is equal to and all of the transmitted and reflected radiant energy at the semitransparent wall is purely diffuse.
In many engineering problems, the semitransparent interface may be a diffuse reflector. For such a case, the interfacial reflectivity is assumed independent of , and equal to the hemispherically averaged value . For , and are given by [ 333]
(13.362)  
(13.363) 
The boundary intensity for all outgoing directions on side of the interface is given by
Similarly for side ,
where
(13.366)  
(13.367) 
When medium is external to the domain as in the case of an external semitransparent wall (Figure 13.3.12), is given as a part of the boundary condition inputs. You supply this incoming irradiation flux in terms of its magnitude, beam direction, and the solid angle over which the radiative flux is to be applied. Note that the refractive index of the external medium is assumed to be .
Partially Diffuse SemiTransparent Walls
When the diffuse fraction that you enter for a semitransparent wall is between and , the wall is partially diffuse and partially specular. In this case, FLUENT includes the reflective and transmitted radiative flux contributions from both diffuse and specular components to the defining equations.
SemiTransparent Exterior Walls
Figure 13.3.12 shows the general case of an irradiation beam applied to an exterior semitransparent wall with zero thickness and a nonzero absorption coefficient for the material property. Refer to the previous section for the radiation effects of wall thickness on semitransparent walls.
An irradiation flux passes through the semitransparent wall from outside the computational domain (Figure 13.3.12) into the adjacent fluid or solid medium a. The transmitted radiation can be refracted (bent) and dispersed specularly and diffusely, depending on the refractive index and the diffuse fraction that you provide as a boundary condition input. Note that there is a reflected component of when the refractive index of the wall ( ) is not equal to , as shown.
There is an additional flux beyond that is applied when the Mixed or Radiation boundary conditions are selected in the Thermal tab. This external flux at the semitransparent wall is computed by FLUENT as
(13.368) 
The fraction of the above energy that will enter into the domain depends on the transmissivity of the semitransparent wall under consideration. Note that this energy is distributed across the solid angles (i.e., similar treatment as diffuse component.)
Incident radiation can also occur on external semitransparent walls. Refer to the previous discussion on interior walls for details, since the radiation effects are the same.
The irradiation beam is defined by the magnitude, beam direction, and beam width that you supply. The irradiation magnitude is specified in terms of an incident radiant heat flux (W/m ). Beam width is specified as the solid angle over which the irradiation is distributed (i.e., the beam and extents). The default beam width in FLUENT is degrees which is suitable for collimated beam radiation. Beam direction is defined by the vector of the centroid of the solid angle. If you select the feature Apply Irradiation Parallel to Beam in the Wall boundary condition panel, then you supply for irradiation (Figure 13.3.12) and FLUENT computes and uses the surface normal flux in its radiation calculation. If this feature is not checked, then you must supply the surface normal flux for irradiation.
Figure 13.3.13 shows a schematic of the beam direction and beam width for the irradiation beam. You provide these inputs (in addition to irradiation magnitude) as part of the boundary conditions for a semitransparent wall.
The irradiation beam can be refracted in medium a depending on the refractive index that is specified for the particular fluid or solid zone material.
Limitations
The thin wall treatment should be used for semitransparent walls only where absorption or emission by the walls is not significant. In cases with significant emission or absorption of radiation in a participating solid material, such as the absorption of long wavelength radiation in a glass window, the use of semitransparent thin walls can result in the prediction of unphysical temperatures in the numerical solution. To avoid potentially erroneous results, one or more solid cell zones should be used to represent the solid material in such a case.
Solid SemiTransparent Media
The discrete ordinates radiation model allows you to model a solid zone that has adjacent fluid or solid zones on either side as a "semitransparent'' medium. This is done by designating the solid zone to participate in radiation as part of the boundary condition setup. Modeling a solid zone as a semitransparent medium allows you to obtain a detailed temperature distribution inside the semitransparent zone since FLUENT solves the energy equation on a percell basis for the solid and provides you with the thermal results. By default however, the DO equations are solved in fluid zones, but not in any solid zones. Therefore, you will need to specify the solid zone as participating in radiation in the Solid panel as part of the boundary condition setup.
Boundary Condition Treatment at Specular Walls and Symmetry Boundaries
At specular walls and symmetry boundaries, the direction of the reflected ray corresponding to the incoming direction is given by Equation 13.355. Furthermore,
Boundary Condition Treatment at Periodic Boundaries
When rotationally periodic boundaries are used, it is important to use pixelation in order to ensure that radiant energy is correctly transferred between the periodic and shadow faces. A pixelation between and is recommended.
Boundary Condition Treatment at Flow Inlets and Exits
The treatment at flow inlets and exits is described in Section 13.3.5.