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13.3.13 Setting Up the DO Model



Angular Discretization


When you select the Discrete Ordinates model, the Radiation Model panel will expand to show inputs for Angular Discretization (see Figure  13.3.14). In this section, you will set parameters for the angular discretization and pixelation described in Section  13.3.6.

Theta Divisions ( $N_\theta$) and Phi Divisions ( $N_\phi$) will define the number of control angles used to discretize each octant of the angular space (see Figure  13.3.3). Note that higher levels of discretization are recommended for problems where specular exchange of radiation is important to increase the likelihood of the correct beam direction being captured. For a 2D model, FLUENT will solve only 4 octants (due to symmetry); thus, a total of $4 N_\theta N_\phi$ directions ${\vec s}$ will be solved. For a 3D model, 8 octants are solved, resulting in $8 N_\theta N_\phi$ directions ${\vec s}$. By default, the number of Theta Divisions and the number of Phi Divisions are both set to 2. For most practical problems, these settings are acceptable, however, a setting of 2 is considered to be a coarse estimate. Increasing the discretization of Theta Divisions and Phi Divisions to a minimum of 3, or up to 5, will achieve more reliable results. A finer angular discretization can be specified to better resolve the influence of small geometric features or strong spatial variations in temperature, but larger numbers of Theta Divisions and Phi Divisions will add to the cost of the computation.

Theta Pixels and Phi Pixels are used to control the pixelation that accounts for any control volume overhang (see Figure  13.3.7 and the figures and discussion preceding it). For problems involving gray-diffuse radiation, the default pixelation of $1\times1$ is usually sufficient. For problems involving symmetry, periodic, specular, or semi-transparent boundaries, a pixelation of $3\times3$ is recommended and will achieve acceptable results. The computational effort, as a result of increasing the pixelation, is less than the computational effort caused by increasing the divisions. You should be aware, however, that increasing the pixelation adds to the cost of computation.



Defining Non-Gray Radiation for the DO Model


If you want to model non-gray radiation using the DO model, you can specify the Number Of Bands ( $N$) under Non-Gray Model in the expanded Radiation Model panel (Figure  13.3.19). For a 2D model, FLUENT will solve $4 N_\theta N_\phi N$ directions. For a 3D model, $8 N_\theta N_\phi N$ directions will be solved. By default, the Number of Bands is set to zero, indicating that only gray radiation will be modeled. Because the cost of computation increases directly with the number of bands, you should try to minimize the number of bands used. In many cases, the absorption coefficient or the wall emissivity is effectively constant for the wavelengths of importance in the temperature range of the problem. For such cases, the gray DO model can be used with little loss of accuracy. For other cases, non-gray behavior is important, but relatively few bands are necessary. For typical glasses, for example, two or three bands will frequently suffice.

When a non-zero Number Of Bands is specified, the Radiation Model panel will expand once again to show the Wavelength Intervals (Figure  13.3.19). You can specify a Name for each wavelength band, as well as the Start and End wavelength of the band in $\mu$m. Note that the wavelength bands are specified for vacuum ( $n=1$). FLUENT will automatically account for the refractive index in setting band limits for media with $n$ different from unity.

Figure 13.3.19: The Radiation Model Panel (Non-Gray DO Model)
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The frequency of radiation remains constant as radiation travels across a semi-transparent interface. The wavelength, however, changes such that $n \lambda$ is constant. Thus, when radiation passes from a medium with refractive index $n_1$ to one with refractive index $n_2$, the following relationship holds:


 n_1 \lambda_1 = n_2 \lambda_2 (13.3-90)

Here $\lambda_1$ and $\lambda_2$ are the wavelengths associated with the two media. It is conventional to specify the wavelength rather than frequency. FLUENT requires you to specify wavelength bands for an equivalent medium with $n=1$.

For example, consider a typical glass with a step jump in the absorption coefficient at a cut-off wavelength of $\lambda_c$. The absorption coefficient is $a_1$ for $\lambda \leq {\lambda}_c$ $\mu$m and $a_2$ for $\lambda > {\lambda}_c$ $\mu$m. The refractive index of the glass is $n_g$. Since $n \lambda$ is constant across a semi-transparent interface, the equivalent cut-off wavelength for a medium with $n=1$ is $n_g \lambda_c$ using Equation  13.3-90. You should choose two bands in this case, with the limits 0 to $n_g \lambda_c$ and $n_g \lambda_c$ to 100. Here, the upper wavelength limit has been chosen to be a large number, 100, in order to ensure that the entire spectrum is covered by the bands. When multiple materials exist, you should convert all the cut-off wavelengths to equivalent cut-off wavelengths for an $n=1$ medium, and choose the band boundaries accordingly.

The bands can have different widths and need not be contiguous. You can ensure that the entire spectrum is covered by your bands by choosing $\lambda_{\rm min} = 0$ and $n\lambda_{\rm max}T_{\rm min} \geq 50,000$. Here $\lambda_{\rm min}$ and $\lambda_{\rm max}$ are the minimum and maximum wavelength bounds of your wavelength bands, and $T_{\rm min}$ is the minimum expected temperature in the domain.



Enabling DO/Energy Coupling


For applications involving optical thicknesses greater than 10, you can enable the DO/Energy Coupling option in the Radiation Model (Figure  13.3.20) in order to couple the energy and intensity equations at each cell, solving them simultaneously. This approach accelerates the convergence of the finite volume scheme for radiative heat transfer and can be used with the gray or non-gray radiation model.

Figure 13.3.20: The Radiation Model Panel with DO/Energy Coupling Enabled
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This option should not be used when the shell conduction model is enabled.


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