Radiation Models in CFD
Radiation Models in CFD: Theoretical Foundations
Fundamentals of Radiative Heat Transfer
Professor, in what cases is it necessary to handle radiation in CFD?
When wall temperatures are high (as a guideline, 400°C or above), for heat exchange between surfaces with large temperature differences (e.g., furnace walls and heated objects), and when participating media (gases containing smoke, water vapor, CO2) are present. According to the Stefan-Boltzmann law, the radiative heat flux is proportional to the fourth power of temperature.
Here, $\varepsilon$ is the surface emissivity, and $\sigma = 5.67 \times 10^{-8}$ W/(m²K⁴) is the Stefan-Boltzmann constant. Radiation from a 1000K wall is about 100 times that from a 300K wall, so the contribution of radiation becomes overwhelming at higher temperatures.
Radiative Transfer Equation (RTE)
What does the equation look like when there is participating media?
It is necessary to solve the Radiative Transfer Equation (RTE).
The first term on the right side is emission, the second term is attenuation due to absorption and scattering, and the third term is in-scattering. $\kappa$ is the absorption coefficient, $\sigma_s$ is the scattering coefficient, and $\Phi$ is the scattering phase function.
That's quite a complex equation. Solving it directly seems very difficult.
Exactly. The RTE is a 7-dimensional (3 spatial + 2 directional + 1 wavelength + 1 time) integro-differential equation, so several approximation methods have been developed. Let me introduce the typical models used in CFD solvers.
| Model | Abbreviation | Accuracy | Computational Cost | Applicable Range |
|---|---|---|---|---|
| Discrete Ordinates | DO | High | High | General purpose, participating media |
| P1 Approximation | P1 | Medium | Low | Optically thick media |
| Surface-to-Surface | S2S | High | Medium | Transparent media, surface-to-surface radiation |
| Discrete Transfer | DTRM | Medium-High | Medium-High | Participating media |
| Monte Carlo | MC | Highest | Highest | Verification / Reference solutions |
Stefan-Boltzmann Law—The Nonlinear Hell Caused by the Fourth Power
The most distinctive feature of the radiative heat transfer equation is its proportionality to "T⁴," the fourth power of temperature. When the temperature doubles, the radiative heat flux becomes 16 times greater. What this means in numerical analysis is the problem of nonlinear sensitivity, where a small temperature error on a high-temperature wall causes a large error in the radiative flux. For example, in a furnace with a wall temperature of 1000°C, a ±5% temperature deviation can cause the radiative flux to fluctuate by more than ±20%. In practice, a reversal phenomenon can occur where "temperature accuracy is prioritized over mesh convergence." Also, in radiation analysis, the accuracy of view factor calculation is critical; insufficient geometric accuracy for complex shapes leads to integration errors in view factors. It is recommended to check view factors with a coarse mesh in the initial stages and verify if the error is within acceptable limits.
Computational Methods for Radiation Models in CFD
Discrete Ordinates (DO) Model
Please tell me more about the DO model.
The DO model is a method that solves the RTE along a finite number of discrete directions. It divides the full solid angle $4\pi$ into $N$ discrete directions and solves the transport equation for each direction. In Fluent, select it via Radiation Models > Discrete Ordinates, and specify the angular resolution with Theta Divisions and Phi Divisions.
What is an appropriate number of angular divisions?
The default $\Theta \times \Phi = 2 \times 2$ is minimal and may lack accuracy. Increasing to $3 \times 3$ or $4 \times 4$ improves it considerably. However, computational cost is proportional to the square of the number of divisions, so balance is important. If ray effects become problematic, increasing pixelation ($\Theta_p \times \Phi_p$) is good.
Surface-to-Surface (S2S) Model
In what cases is the S2S model used?
When the medium is transparent (no absorption/scattering, like air) and only surface-to-surface radiation exchange needs to be considered. Typical examples are inside electronic device enclosures, automobile cabins, and architectural spaces. The S2S model pre-calculates the view factors between each surface pair and determines radiative heat exchange based on them.
Is view factor calculation a heavy process?
When the number of surfaces is large, storing the $O(N^2)$ view factor matrix can become a memory consumption issue. In Fluent, increasing the Cluster Number allows face clustering, reducing computational load. STAR-CCM+'s S2S model also allows parameter adjustment for View Factor calculation.
Gas Radiation Model
How is radiation from combustion gases modeled?
CO2 and H2O absorb and emit radiation in specific wavelength bands. The Weighted Sum of Gray Gases Model (WSGGM) is the standard approach, approximating the radiative properties of participating gases by a weighted sum of a few gray gases. In Fluent, WSGGM can be automatically applied in combination with the DO model.
For higher accuracy models, there are the Exponential Wide Band Model (EWBM) and Statistical Narrow Band Model (SNB), but they are computationally expensive. Fluent's Full Spectrum k-distribution (FSK) model offers a good balance between accuracy and cost.
Radiation CFD Numerical Schemes—Choosing Between DOM vs. Monte Carlo Method
As CFD solvers for radiative heat transfer, the Discrete Ordinates Method (DOM) and the Monte Carlo Method (MCM) take fundamentally different approaches. DOM discretizes angles with S₄ to S₈ orders, solving deterministically with computational cost O(N³), making it suitable for industrial CFD. On the other hand, MCM uses stochastic ray tracing, is robust to geometric complexity, and naturally handles non-homogeneous scattering of soot particles, but requires tracing 10⁶ or more rays to reduce statistical noise. In practice, a typical division of labor is DOM (computational speed advantage) for glass melting furnaces and combustion chambers, and MCM (accuracy advantage) for solar concentrators and complex geometries.
Radiation Models in CFD in Practice
Radiation Analysis of Industrial Furnaces
How are CFD radiation models used in industrial furnace design?
In heating furnaces (steel heating, glass melting, cement calcination, etc.), radiation from combustion gases is the primary heat transfer path to the heated object. A typical workflow is: (1) Solve for the temperature field and composition of combustion gases using a combustion model (Non-premixed combustion / EDM), (2) Solve for gas radiation using the DO model + WSGGM, (3) Solve for heat exchange with the heated object using CHT.
Is it also necessary to consider the effect of soot?
In rich combustion or diesel combustion, soot particles become a major source of radiation absorption and emission. Coupling Fluent's Soot Model with the DO model allows consideration of soot's radiative contribution. Even a soot volume fraction $f_v$ on the order of $10^{-7}$ can have a non-negligible effect on the absorption coefficient.
Handling Solar Radiation
Is sunlight simulation also done with CFD?
Yes, it is. For building solar radiation analysis, solar thermal concentrator (CSP) design, automobile cabin solar heat load, etc. Fluent has a Solar Load Model that automatically calculates solar direction and intensity based on the sun's position (latitude, longitude, date/time) and building orientation. It is used in combination with the DO model's Solar Ray Tracing function.
Does STAR-CCM+ have equivalent functionality?
STAR-CCM+ has a feature called Solar Load Profile, which similarly calculates solar position and radiation inten
Practical CAE quality notes for Radiation Models in CFD
Radiation Models in CFD should be treated as an engineering model, not as an isolated formula. In fluid simulation, reliable results come from a clear chain of assumptions: governing physics, material data, boundary conditions, numerical discretization, solver settings, and post-processing criteria. Before using this note in a design review, identify which quantities are prescribed, which are solved, and which are only diagnostic indicators.
Model setup checklist
- Define the scope: decide whether Radiation Models in CFD is being used for screening, detailed design, failure investigation, or verification of another simulation.
- Check dimensions and units: keep SI units internally and document every conversion applied to loads, geometry, material constants, and time or frequency scales.
- State assumptions explicitly: record linearity, steady-state or transient behavior, small-deformation limits, continuum assumptions, and any symmetry or ideal boundary conditions.
- Compare with a baseline: use a hand calculation, limiting case, mesh refinement trend, or independent solver result before accepting the final value.
Validation signals
| Review item | What to verify | Typical warning sign |
|---|---|---|
| Inputs | Geometry, material data, loads, and constraints match the intended fluid simulation problem. | Correct-looking plots with unrealistic magnitudes or units. |
| Numerics | Mesh, time step, convergence tolerance, and solver options are adequate for Radiation Cfd. | Large changes after small mesh or tolerance adjustments. |
| Physics | The selected theory remains valid in the expected stress, temperature, velocity, or frequency range. | Results are used outside the assumptions stated in the model. |
For production use, keep the model file, input table, result plots, and review comments together. This makes Radiation Models in CFD traceable and prevents the page from being used as a black-box answer without engineering judgment.
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