Radiation Models in CFD
Theory and Physics
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.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, the water comes out spluttering and unstable, but after a while, the flow becomes steady, right? This "period of change" is described by the temporal term. The pulsation of blood flow with each heartbeat, the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost is significantly reduced, the basic CFD strategy is to first try solving it as steady-state.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. The warm air from a heater reaching the far end of a room is also because the air, as a "carrier," transports heat by convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar things" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while, they naturally mix, right? That's molecular diffusion. Now, next question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is high, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelms, and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, the liquid shoots out forcefully from the needle tip, right? Why? Because the plunger side is high pressure, the needle tip is low pressure—this pressure difference becomes the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Where there is a pressure difference, flow is generated"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results become strange immediately after switching to compressible analysis, it might be due to confusion between absolute/gauge pressure.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it is pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force applied to molten metal by an electromagnetic pump in a factory... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—you get a physically impossible result, like turning on a heater in a winter room but the warm air doesn't rise.
Assumptions and Applicability Limits
- Continuum assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian fluid assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressible assumption (for Ma < 0.3): Treat density as constant. For Mach number 0.3 or above, consider compressibility effects
- Boussinesq approximation (Natural convection): Consider density changes only in the buoyancy term, using constant density in other terms
- Non-applicable cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (VOF/Level Set, etc., required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition |
| CFL number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
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.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shock waves and steep gradients.
Finite Volume Method vs. Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when the residuals for the continuity equation, momentum, and energy each drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Pressure: 0.2-0.3, Velocity: 0.5-0.7 are typical initial values. If diverging, lower the relaxation factors. After convergence, increase to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5-20 iterations is a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and then velocity is revised with the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—it's a discretization method reflecting the physics that upstream information determines downstream. Although first-order accurate, it is highly stable because it correctly captures flow direction.
Practical Guide
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
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