Spatial discretization for hybrid grids

The RTE (Eq.) is solved for every discrete direction $\textbf{s}_i$ using a finite volume approach. The integration of the RTE over the volume $V$ of an element limited by a surface $\Sigma$ , and the application of the divergence theorem yields:

$$\int_{\Sigma} I(\textbf{s}_i) . \textbf{s}_i . \textbf{n} d\Sigma = \int_{V} (\kappa I_{b} - \kappa I(\textbf{s}_i)) dV$$ (1.3)

The domain is discretized in three-dimensional control volumes $V$ . It is assumed that $I_b$ and $I(\textbf{s}_i)$ are constant over the volume $V$ and that the intensities $I_j$ at the faces are constant over each face. Considering that $I_j$ is the averaged intensity over the $j^{th}$ face, associated with the center of the corresponding face, that $I_{b,P}$ and $I_P$ are the averaged intensities over the volume $V$ , associated with the center of the cell, and assuming plane faces and vertices linked by straight lines, Eq.() can be discretized as follows :

$$\sum_{j=1}^{N_{face}} I_j(\textbf{s}_i).(\textbf{s}_i.\textbf{n}_j)A_j = \kappa V (I_{b,P}-I_P(\textbf{s}_i))$$ (1.4)

where $\textbf{n}_j$ is the outer unit normal vector of the surface $j$ .
The scalar product of the $i^{th}$ discrete direction vector with the normal vector of the $j^{th}$ face of the considered cell is defined by $D_{ij}$ :

$$D_{ij} = \textbf{s}_i . \textbf{n}_j = \mu_i n_{xj} + \eta_i n_{yj} + \xi_i n_{zj}$$ (1.5)

The discretization of the boundary condition (Eq.()) is straightforward:

$$I_{w} = \epsilon_w I_{b,w} + \frac{1-\epsilon_w}{\pi} \sum_{\textbf{n}.{\textbf{s}_i <0}} w_{i}I(\textbf{s}_i)\mid \textbf{n}.\textbf{s}_i\mid$$ (1.6)

For each cell, the incident radiation $G$ is evaluated as follows:

$$G = \int_{4\pi} I(\textbf{s}) d \Omega \, \simeq \sum_{i=1}^{N_{dir}} w_i I(\textbf{s}_i)$$ (1.7)

and the incident heat flux $H_w$ at the wall surfaces is :

$$H_w = \int_{\textbf{n}.\textbf{s} <0} I(\textbf{s}) \mid \textbf{... ... \sum_{\textbf{n}.\textbf{s}_i <0} w_{i}I_{i} \mid \textbf{n}.\textbf{s}_i \mid$$ (1.8)

For a gray medium, the radiative source term $S_r$ is given by:

$$S_r = \nabla.Q_r \, = \kappa (4\pi I_b - G)$$ (1.9)

where $Q_r$ is the radiative heat flux, and the radiative net heat flux at the wall is:

$$Q_w = \epsilon \pi I_{b,w} - H$$ (1.10)

For the evaluation of the radiative intensity $I(\textbf{s}_i)$ in Eq. () to () Ströhle et al. [#!Str01!#] proposed a simple spatial differencing scheme based on the mean flux scheme that proved to be very efficient in the case of hybrid grids. This scheme relies on the following formulation:

$$I_P = \alpha \overline{I_{out}} + (1-\alpha) \overline{I_{in}}$$ (1.11)

where $\overline{I_{in}}$ and $\overline{I_{out}}$ are respectively the intensities averaged over the entering and the exit faces of the considered cell. $\alpha$ is a weighting number between 0 and $1$ . Substituting $\overline{I_{out}}$ from Eq.() into Eq.() yields (for more details see [#!IJTS!#]):

$$I_P = \frac{\alpha V \kappa I_b - \displaystyle \sum_{\substack{j... ...j I_j}{\alpha\kappa V + \displaystyle \sum_{\substack{j\\ D_{ij}>0}} D_{ij}A_j}$$ (1.12)

The case $\alpha=1$ corresponds to the Step scheme used by Liu et al. [#!Liu00c!#]. The case $\alpha=0.5$ is called Diamond Mean Flux Scheme (DMFS) which is formally more accurate than the Step scheme. After calculation of $I_P$ from Eq.(), the radiation intensities at cell faces such that $D_{ij}>0$ are set equal to $\overline{I_{out}}$ , obtained from Eq.(). For a given discrete direction, each face of each cell is placed either upstream or downstream of the considered cell center (a face parallel to the considered discrete direction plays no role). The control volumes are treated following a sweeping order such as the radiation intensities at upstream cell faces are known. This order depends on the discrete direction under consideration. An algorithm for the optimization of the sweeping order has been implemented [#!IJTS!#]. Note that this sweeping order is stored for each discrete direction, and only depends on the chosen grid and the angular quadrature, i.e. it is independent on the physical parameters or the flow and may be calculated only once, prior to the full computation.