Les bases théoriques

PRISSMA [#!IJTS!#,#!Stanford!#] has been designed to simulate the radiative heat transfer in coupled simulations with flow dynamics, involving unstructured grids. In the following and for sake of clarity, the intensities and radiative properties are expressed for a single wavenumber (monochromatic case) but the formulation can be easily extended to a full spectrum.
Discrete Ordinates Method have been introduced first by [#!Cha50!#] and have been widely used in radiative transfer applications. Considering an absorbing-emitting and non-scattering gray medium, the variation of the radiative intensity $I(\textbf{s})$ along a line of sight can be written as:

$$\frac{dI(\textbf{s})}{ds} = \kappa I_{b} - \kappa I(\textbf{s})$$ (1.1)

where $I(\textbf{s})$ is the radiative intensity along the directional coordinate $\textbf{s}$ , $I_b$ is the blackbody radiative intensity, and $\kappa$ is the absorption coefficient. Boundary conditions for diffuse surfaces are taken from the relation giving the intensity leaving the wall $I_w$ as a function of the blackbody intensity of the wall $I_{b,w}$ and of the incident radiative intensity:

$$I_{w}(\textbf{s})=\epsilon_{w}I_{b,w}+\frac{\rho_{w}}{\pi}\int_{\... ...f{n}.\textbf{s'} <0}I_{w}(\textbf{s'})\vert\textbf{n}.\textbf{s'}\vert d\Omega'$$ (1.2)

where $\epsilon_{w}$ is the wall emissivity, $\rho_{w}$ the wall reflectivity, $\textbf{n}$ the unit vector normal to the wall and $\textbf{s'}$ the direction of propagation of the incident radiation confined within a solid angle $d\Omega'$ .