This document presents a hybrid cell-vertex method for solving the radiative transfer equation (RTE) using the finite volume method (FVM) in 2D and 3D. It shows results for an L-shaped geometry and cylinder geometry in 2D and 3D using different angular discretizations. The code is written in Fortran with tools in Python for preprocessing, postprocessing, and automation. The hybrid cell-vertex method improves upon the standard cell-vertex formulation for RTE solutions, but the results presented are preliminary.

1. Hybrid cell-vertex ?
In 2D and 3D, version 1
By
Lionel Trovalet
lionel.trovalet@yahoo.fr
Thermal Radiation Simulation 10th July 2016

2. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Plan
Results in two dimensions
Results in three dimensions
Technical part
Potential
Conclusion
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3. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Plan
Cell-center Cell-vertex
Control volume already defined Advantage Value at the node
Value not at the node Inconvenient Control volume have to be defined
Mesh example in considering one direction (RTE with FVM)
same for each direction Computing nodes change every-time due to the re-
ceiving boundary
element belong to one face Boundary Condition corner nodes belong to two faces
in 2D and more in 3D
How to improve cell-vertex ? With hybrid cell vertex
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4. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Results in two dimensions
L-shape
Boundary and medium conditions :
• Black wall and cold, except one
- Twall = 0K
- Thot = 64.8049K at the top
• In the medium
- Tm = 10K
- κ = 0.01m−1
Numerical setup :
• mesh
- Np = 352, Nt = 620, Ne = 82
• two angular discretisation
- 192 directions, Nθ = 64, Nφ = 3
- 384 directions, Nθ = 64, Nφ = 6
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5. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Results in two dimensions
Radiant heat flux
Radiant heat flux
Dimensionless radiant heat flux
along the side A-B-C-D
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6. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Results in two dimensions
Different schema
Dimensionless radiant heat flux
64x3
Dimensionless radiant heat flux
64x6
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7. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Results in three dimensions
L-shape
Boundary and medium conditions :
• Black wall
- Twall = 500K
• In the medium
- Tm = 1000K
- κ = 1, 2 and 10 m−1
Numerical setup :
• Mesh :
- Nt = 10536, Np = 2292,
Ne = 1388
• Angular discretisation
- S4
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8. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Results in three dimensions
Radiant heat flux on AC
Radiant heat flux κ = 1m−1
Relative error
Three value of κ
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9. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Results in three dimensions
Cylinder, height=3m and radius=.5m
Boundary and medium conditions :
• Black wall
- Twall = 300K
• In the medium
- Tm = 1200K
- κ = 0.1, 1 and 10 m−1
Numerical setup :
• Mesh :
- Nt = 7168, Np = 1577,
Ne = 1408
• Angular discretisation
- S8
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10. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Results in three dimensions
Radiant heat flux on the lateral side
Radiant heat flux, κ = .1m−1
Relative error
Three value of κ
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11. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Technical part
The code is written in Fortran 90
• structured in module
• driven by an input file containing the simulation setup
• format result is in vtk and visualised with 1
ParaView
• documented by doxygen
• in order to automatised TheRaSi
- different tools in python was developed
- batch in dos to execute multi simulations
1ParaView is free and can be driven by a python script
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12. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Technical part
Graphical user interface in Python for sensitivity analysis
Python tools to pilot Paraview and plot the results
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13. Plan
Results in two
dimensions
Results in three
dimensions
Technical part
Potential
Conclusion
Conclusion
Why the hybrid cell vertex is developed ?
• new formulation in FVM
• improve the cell vertex formulation "in thermal radiation"
The results are :
• interesting but not complete
TheRaSi is developed in 2D and 3D :
• 2D more interesting to explore the possibility
• 3D to treat real problem
Next ..... ?
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