
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 11.31 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.
The implementation in FLUENT uses a conservative variant of the discrete ordinates model called the finitevolume scheme [ 41, 203], and its extension to unstructured meshes [ 179].
The DO Model Equations
The DO model considers the radiative transfer equation (RTE) in the direction as a field equation. Thus, Equation 11.31 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
(11.339) 
where is the fraction of radiant energy emitted by a black body [ 174] 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
(11.340) 
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.
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 11.3.3. 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 11.3.4. For generalized unstructured meshes, however, control volume faces do not in general align with the global angular discretization, as shown in Figure 11.3.5, leading to the problem of control angle overhang [ 179].
Essentially, control angles can straddle the control volume faces, so that they are partially incoming and partially outgoing to the face. Figure 11.3.6 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 [ 179].
Each overhanging control angle is divided into pixels, as shown in Figure 11.3.7.
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 11.314. 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 11.317. The equivalent emission is given by Equation 11.316. The equivalent particle scattering factor , defined in Equation 11.320, 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 GrayDiffuse Walls
For gray radiation, the incident radiative heat flux, , at the wall is
(11.344) 
The net radiative flux leaving the surface is given by
where is the refractive index of the medium next to the wall. The boundary intensity for all outgoing directions at the wall is given by
(11.346) 
For nongray radiation, the incident radiative heat flux in the band at the wall is
(11.347) 
The net radiative flux leaving the surface in the band is given by
where is the wall emissivity in the band. The boundary intensity for all outgoing directions in the band at the wall is given by
(11.349) 
The Diffuse Fraction
FLUENT allows you to specify the fraction of incoming radiation that is treated as diffuse at diffuse boundaries. If is the amount of radiative energy incident on the wall, then
where is the diffuse fraction and is the wall emissivity. See below for more information about specular walls.
For nongray radiation, you can specify the diffuse fraction separately for each band.
Boundary Condition Treatment at SemiTransparent Walls
FLUENT allows the specification of both diffusely and specularly reflecting semitransparent walls. You can prescribe the fraction of the incoming radiation at the semitransparent wall which is to be reflected and transmitted diffusely; the rest is treated specularly.
For nongray radiation, this treatment is applied on a band 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.
Specular SemiTransparent Walls
Consider a ray traveling from a semitransparent medium with refractive index to a semitransparent medium with a refractive index in the direction , as shown in Figure 11.3.8. Side of the interface is the side that faces medium ; similarly, side 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 11.3.8. We also define the direction
shown in Figure 11.3.8.
The interface reflectivity on side [ 174]
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 11.3.9) 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 .
When medium is external to the domain, is given in Equation 11.354 as a part of the problem specification. This boundary specification is usually made by providing the incoming radiative flux and the solid angle over which the radiative flux is to be applied. The refractive index of the external medium is assumed to be unity.
Diffuse SemiTransparent Walls
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 [ 235]
(11.357)  
(11.358) 
The boundary intensity for all outgoing directions on side of the interface is given by
Similarly for side ,
where
(11.361)  
(11.362) 
As before, if medium is external to the domain, is given as a part of the boundary specification.
Beam Irradiation
As mentioned above, FLUENT allows the specification of the irradiation at semitransparent boundaries. The irradiation is specified in terms of an incident radiant heat flux (W/m ). You can specify the solid angle over which the irradiation is distributed, as well as the vector of the centroid of the solid angle. To indicate whether the irradiation is reflected specularly or diffusely, you can specify the diffuse fraction.
For nongray radiation, FLUENT allows you to specify the irradiation at semitransparent boundaries on a band basis. The irradiation is specified as an incident heat flux (W/m ) for each wavelength band. As in the gray case, you can specify the solid angle over which the irradiation is distributed, as well as the vector of the centroid of the solid angle.
The Diffuse Fraction
At semitransparent boundaries, FLUENT allows you to specify the fraction of the incoming radiation that is treated as diffuse. The diffuse fraction is reflected diffusely, using the treatment described above; the transmitted portion is also treated diffusely. The remainder of the incoming energy is treated in a specular fashion.
For nongray radiation, you can specify the diffuse fraction separately for each band.
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 11.350. 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 11.3.3.