Stationary Framelet Model
Theory and Physics
Overview
Professor, what is a flamelet model?
The flamelet model is a model for non-premixed turbulent combustion proposed by Peters in 1984. It views a turbulent diffusion flame as "an ensemble of numerous thin laminar flames (flamelets)" and reduces the internal flame structure to a one-dimensional problem in mixture fraction $Z$.
So you can reduce 3D turbulent combustion to 1D?
Yes. The flamelet assumption is that "the flame thickness is thinner than the smallest turbulent scale (the Kolmogorov scale)". In this case, the internal flame structure becomes locally equivalent to a 1D laminar counterflow flame.
Flamelet Equation
Please explain the flamelet equation.
The flamelet equation for chemical species $Y_i$ can be written as follows.
Here, $\chi$ is the scalar dissipation rate, representing diffusion in $Z$ space.
For a steady flamelet, $\partial/\partial t = 0$, right?
Correct. For a steady flamelet, the time derivative term becomes zero, resulting in an ordinary differential equation with the dissipation rate $\chi_{st}$ (value at the stoichiometric surface) as a parameter. As $\chi_{st}$ increases, the reaction cannot keep up, eventually leading to extinction (quenching dissipation rate $\chi_q$).
S-Curve (S-curve)
What is an S-curve?
Plotting the maximum temperature of the steady flamelet solution against $\chi_{st}$ yields an S-shaped curve. The upper branch represents the burning state, the lower branch the unburned state, and the middle branch is an unstable solution.
What is the typical value of $\chi_q$?
For methane/air, $\chi_q \approx 20-50$ s$^{-1}$; for hydrogen/air, $\chi_q \approx 1000$ s$^{-1}$, which is very large. This means hydrogen is difficult to extinguish.
Flamelet Library
A flamelet library is created by computing multiple steady flamelet solutions with $\chi_{st}$ as a parameter and storing them as a 2D table of $(Z, \chi_{st})$. If the turbulent $\beta$-PDF averaging is applied in advance, it becomes a 3D lookup table of $(\widetilde{Z}, \widetilde{Z''^2}, \widetilde{\chi_{st}})$.
So the core of the flamelet model is the combination of "a 1D laminar flame table + turbulent PDF".
Exactly. The flame structure including detailed chemical reactions is solved offline, and in 3D CFD, only table lookup is required, resulting in very low computational cost.
The Man Who Turned Flames into a "Database" – Norbert Peters' Flamelet Philosophy
The idea proposed by Norbert Peters, the originator of the flamelet model, in 1984 can be summarized in one sentence: "A flame in turbulence is ultimately an ensemble of small laminar flames." If that's the case, then precomputing laminar flames and storing them in a table means that during the main calculation, you only need to reference the mixture fraction. This "precomputation → table lookup" concept dramatically reduced the computational cost of combustion CFD. Peters developed this concept at RWTH Aachen University, creating a lineage that later led to models like FGM (Flamelet-Generated Manifolds) and FPV (Flamelet Progress Variable). Modern gas turbine design cannot be discussed without his legacy.
Physical Meaning of Each Term
- Time 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, it becomes a steady flow, right? This "period of change" is described by the time term. The pulsation of blood flow from a heartbeat, or 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 this term is set to zero. Since computational cost drops significantly, solving first in steady-state is a basic CFD strategy.
- 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. Warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There's 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. That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelmingly dominates, 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 piston side is high pressure, the needle tip is low pressure—this pressure difference provides 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 common point of confusion here: The "pressure" in CFD is often gauge pressure, not absolute pressure. When you switch to compressible analysis and suddenly get strange results, it might be due to mixing up absolute/gauge pressure.
- Source term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's 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 acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget a source term? In a natural convection analysis, if you forget to include buoyancy, the fluid won'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 (mean free path ≪ characteristic length)
- Newtonian fluid assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility assumption (for Ma < 0.3): Treat density as constant. For Mach numbers above 0.3, compressibility effects must be considered.
- Boussinesq approximation (Natural convection): Density variation is considered only in the buoyancy term; constant density is used 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 $\nu = \mu/\rho$ [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
Details of Numerical Methods
How do you create a flamelet library?
Library construction has two steps: (1) Calculation of counterflow diffusion flames, (2) Table creation via PDF integration.
Counterflow Diffusion Flame Calculation
How is it calculated specifically?
Solve 1D counterflow diffusion flames using Cantera, FlameMaster, OPPDIF (CHEMKIN-PRO), etc. Gradually increase the strain rate (strain rate) $a$ and track up to the extinction point. The relationship between $a$ and $\chi_{st}$ is given by $\chi_{st} \approx a \cdot f(Z_{st})$.
Typical library construction parameters:
| Parameter | Recommended Value | Remarks |
|---|---|---|
| Grid points in $Z$ direction | 128-256 | Concentrated near $Z_{st}$ |
| $\chi_{st}$ discretization | 30-50 points | Logarithmic spacing |
| Strain rate range | 1 - $a_q$ s$^{-1}$ | Up to extinction |
| Reaction mechanism | GRI-Mech 3.0, etc. | Use detailed mechanism |
Implementation in Fluent
How do you set it up in Fluent?
Configure the following within Fluent's Non-Premixed Combustion model.
1. Select Flamelet Model (switch from Equilibrium Chemistry)
2. Import reaction mechanism in CHEMKIN format
3. Set Number of Flamelets (default 20, recommended 30-50)
4. Set PDF table resolution
5. After calculation starts, transport equations for $\widetilde{Z}$, $\widetilde{Z''^2}$ are solved, and temperature/species are determined via table lookup.
Relationship with FGM (Flamelet Generated Manifold)
How is FGM different from the flamelet model?
FGM is a method proposed by van Oijen et al. (2000) that constructs a low-dimensional manifold from flamelet solutions. In addition to steady flamelets, it introduces a progress variable $C$, enabling representation of transient processes like ignition and extinction.
Comparison of steady flamelet and FGM:
| Characteristic | Steady Flamelet | FGM |
|---|---|---|
| Table dimension | 2-3D ($Z$, $Z''$, $\chi$) | 3-4D ($Z$, $Z''$, $C$, $C''$) |
| Extinction representation | Possible via S-curve | Naturally represented via Progress Variable |
| Auto-ignition | Difficult | Possible |
| Partial premixing | Difficult | Possible |
| Computational cost | Very low | Low (table lookup) |
So FGM is more versatile.
Yes. FGM is becoming mainstream in STAR-CCM+ and OpenFOAM. Fluent has also strengthened its FGM option since R2. However, steady flamelet remains widely used as the simplest and most stable method.
The key to the flamelet model is "creating a high-quality table."
Exactly. The table resolution and the validity of the reaction mechanism directly determine the model's accuracy.
Results Change with the "Shape" of the PDF – Comparison of β-PDF and Clipped Gaussian
An unavoidable aspect of the numerical implementation of flamelet/PDF models is "the choice of the PDF shape for the mixture fraction." The most common is the β-PDF (beta distribution), which is also the default in ANSYS Fluent. However, while β-PDF is mathematically convenient, it can only represent unimodal shapes and has the problem of overly steep behavior at the two boundaries (pure fuel / pure oxidizer). Clipped Gaussian often matches experimental data better, but table generation takes more time. The choice can change peak temperatures by 50–100 K, and the experience of "casually trying different PDF shapes" becomes a difference in practical skill.
Upwind Scheme (Upwind)
1st-order upwind: Large numerical diffusion but stable. 2nd-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
2nd-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number, diffusion-dominated flows.
TVD Scheme (MUSCL, QUICK, etc.)
Maintains high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shocks and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD.Related Topics
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