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ABSTRACT
The development and comparison of different parallel spatial/angular agglomeration multigrid schemes to accelerate the finite volume method, for the prediction of radiative heat transfer, are reported in this study. The proposed multigrid methodologies are based on the solution of radiative transfer equation with the full approximation scheme coupled with the full multigrid method, considering different types of sequentially coarser spatial and angular resolutions as well as different V-cycle types. The encountered numerical tests, involving highly scattering media and reflecting boundaries, reveal the superiority of the nested scheme along with the V(2,0)-cycle-type strategy, while they highlight the significant contribution of the angular extension of the multigrid technique.
Nomenclature
| AH | = | spatial forcing function |
| AMN | = | angular forcing function |
| c | = | propagation speed of radiation in the medium |
| = | directional weight of solid control angle ΔΩmn and surface i | |
| = | radiative intensity, W/m2 sr | |
| ka | = | absorption coefficient, 1/m |
| = | unit normal vector at i surface | |
| Nθ(Nφ) | = | number of polar (azimuthal) angles |
| = | flux balance of node p | |
| = | position vector | |
| = | unit vector in s direction of a solid control angle | |
| T | = | temperature, K |
| t | = | time |
| Vp | = | volume of the control volume of a node p |
| β | = | extinction coefficient, 1/m |
| ΔAi | = | part i of the surface area of a control volume |
| ΔΩmn | = | discrete control angle |
| ϵw | = | wall emissivity (ϵw = 1 −ρ) |
| θ | = | polar angle |
| σ | = | Stefan–Boltzmann constant (σ = 5.6710−8 W/m2K4) |
| σsσs | = | scattering coefficient, 1/m |
| ρ | = | wall reflectivity |
| φ | = | azimuthal angle |
| Φ | = | scattering phase function |
| Subscripts | = | |
| b | = | blackbody |
| h | = | first nonagglomerated grid |
| H | = | agglomerated grid |
| in | = | ingoing |
| mn | = | first nonagglomerated angular resolution |
| MN | = | agglomerated angular resolution |
| out | = | outgoing |
Nomenclature
| AH | = | spatial forcing function |
| AMN | = | angular forcing function |
| c | = | propagation speed of radiation in the medium |
| = | directional weight of solid control angle ΔΩmn and surface i | |
| = | radiative intensity, W/m2 sr | |
| ka | = | absorption coefficient, 1/m |
| = | unit normal vector at i surface | |
| Nθ(Nφ) | = | number of polar (azimuthal) angles |
| = | flux balance of node p | |
| = | position vector | |
| = | unit vector in s direction of a solid control angle | |
| T | = | temperature, K |
| t | = | time |
| Vp | = | volume of the control volume of a node p |
| β | = | extinction coefficient, 1/m |
| ΔAi | = | part i of the surface area of a control volume |
| ΔΩmn | = | discrete control angle |
| ϵw | = | wall emissivity (ϵw = 1 −ρ) |
| θ | = | polar angle |
| σ | = | Stefan–Boltzmann constant (σ = 5.6710−8 W/m2K4) |
| σsσs | = | scattering coefficient, 1/m |
| ρ | = | wall reflectivity |
| φ | = | azimuthal angle |
| Φ | = | scattering phase function |
| Subscripts | = | |
| b | = | blackbody |
| h | = | first nonagglomerated grid |
| H | = | agglomerated grid |
| in | = | ingoing |
| mn | = | first nonagglomerated angular resolution |
| MN | = | agglomerated angular resolution |
| out | = | outgoing |
