Premixed Flame Model
Theory and Physics
Overview
Teacher, how is a premixed flame different from a diffusion flame?
A premixed flame is a form where the flame propagates in a state where fuel and oxidizer are sufficiently mixed before combustion. Examples include gasoline engines, lean-burn combustors in gas turbines, and household gas stoves. The flame front has a distinct boundary, separating unburned mixture and burned gas.
The flame propagation speed is an important parameter, right?
Correct. The laminar burning velocity $S_L$ is the fundamental parameter for premixed flames. For methane/air (equivalence ratio 1.0, room temperature and pressure) $S_L \approx 0.36$ m/s, and for hydrogen/air $S_L \approx 2.1$ m/s.
Progress Variable $c$ and Governing Equations
What variables are used in CFD for premixed flames?
For premixed flames, the progress variable $c$ is used to track the flame front. $c=0$ represents unburned mixture, $c=1$ represents burned gas.
Here, $\dot{\omega}_c$ is the reaction source term, which becomes non-zero only near the flame front.
How is the source term $\dot{\omega}_c$ modeled?
This is the core of premixed flame modeling. There are three main approaches.
Major Premixed Combustion Models
| Model | Principle | Advantages | Disadvantages | ||
|---|---|---|---|---|---|
| G-equation (Level Set) | Tracks flame front as iso-surface $G=0$ | Geometrically clear | No internal flame structure | ||
| TFC (Turbulent Flame Closure) | Zimont model. $S_T = A(u'/S_L)^n S_L$ | Easy to implement | Depends on empirical $S_T$ correlation | ||
| FSD (Flame Surface Density) | Transport equation for flame surface density $\Sigma$ | Physics-based | Model constants for $\Sigma$ equation | ||
| c-equation + reaction rate | $\dot{\omega}_c = \rho_u S_L | \nabla c | $ | Direct | Numerical issues with flame thickness |
Turbulent Burning Velocity
What happens to the flame speed in turbulence?
The turbulent burning velocity $S_T$ increases with turbulence intensity $u'$. Zimont's correlation is widely used.
Here, $\alpha$ is the thermal diffusivity, $l_t$ is the turbulent integral scale, and $A$ is a model constant ($A \approx 0.52$).
So, the stronger the turbulence, the more wrinkled the flame front becomes, increasing the apparent burning speed, right?
Exactly. Turbulence wrinkles the flame front, increasing its area, which enhances the burning rate per unit cross-sectional area. This is the classical picture by Damkohler (1940), and modern CFD models are also based on this concept.
So, unlike diffusion flames, the core of premixed flames is "tracking the flame front."
Yes. The fundamental difference is describing the flame with a progress variable rather than a mixture fraction.
"Flame Thickness" is Less Than 1mm—How Difficult It Is to Measure Laminar Burning Velocity
The "laminar burning velocity $S_L$," which is fundamental to premixed flame theory, is actually a very difficult quantity to measure. There are multiple experimental methods like the advanced stagnation flame method, counterflow method, and spherical propagation method, but even for the same gas, values can vary by 10-20% depending on the method. The reason lies in differing opinions among researchers on "how to correct for strain effects." Furthermore, the flame thickness itself is only about 0.1-1mm, and inserting a thermometer to measure it disturbs the flame. This problem of "the measurement itself disturbing the subject" is different from the uncertainty principle in quantum mechanics but shares a similar structure in terms of measurement difficulty. The validation of reaction constants in GRI-Mech 3.0 uses such carefully measured data.
Physical Meaning of Each Term
- Temporal term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, it becomes a steady flow, right? This "state of change" is described by the temporal 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 setting this term to zero. Since this significantly reduces computational cost, 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 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, it naturally mixes, right? 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 in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion dominates. 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, liquid shoots out forcefully from the needle tip, right? Why? The plunger 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. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A common 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 buoyancy pushes it upward. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated in 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 the source term? In natural convection analysis, forgetting to include buoyancy results in the fluid not moving at all—a physically impossible outcome like warm air not rising in a room with the heater on in winter.
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)
- 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
Please tell me about the numerical challenges when solving premixed flames with CFD.
The biggest challenge is resolving the flame thickness. The thickness of a laminar premixed flame is $\delta_L \approx \alpha/S_L$, about 0.5 mm for methane/air and about 0.2 mm for hydrogen/air. Directly resolving this with RANS meshes on the order of millimeters is impossible.
Thickened Flame Model (TFM)
How is that solved?
In LES, the widely used method is the Thickened Flame Model (Colin et al., 2000). It artificially thickens the flame to make it resolvable by the mesh.
Increase the diffusion coefficient by a factor of $F$, and decrease the reaction rate by a factor of $1/F$.
This increases the flame thickness to $F\delta_L$, but $S_L$ remains unchanged. $F = 5-20$ is typical.
But doesn't thickening the flame change its interaction with turbulence?
Sharp observation. A thickened flame cannot resolve small-scale turbulent wrinkling. Therefore, an efficiency function $E$ is introduced for correction.
The Charlette efficiency function is representative, given in the form $E = E(\Delta/\delta_L, u'/S_L)$.
Implementation in Fluent
How do you set up a premixed flame in Fluent?
The following models are available in Fluent.
1. Premixed Combustion (Zimont TFC model): c-equation based. For RANS.
2. Partially Premixed Combustion: Hybrid of premixed + non-premixed.
3. FGM (Flamelet Generated Manifold): Progress Variable + mixture fraction.
Zimont TFC model settings:
- Models > Species > Premixed Combustion
- Turbulent Flame Speed model: Zimont
- Laminar Flame Speed: Input value or calculated value (equivalence ratio dependent)
- Flame Stretch Factor: Default 0.26
Implementation in OpenFOAM
What about in OpenFOAM?
XiFoam is the solver for premixed combustion. It solves a transport equation for the flame wrinkling factor $\Xi$ (= $S_T/S_L$).
| Solver | Target | Model |
|---|---|---|
| XiFoam | Premixed compressible | $\Xi$-equation |
| reactingFoam + PaSR | Premixed/Partially premixed | Species Transport |
| fireFoam | Fire | EDM/Diffusion flame |
Is the Thickened Flame Model included in the standard OpenFOAM distribution?
It is not in the standard distribution, but community versions (like TFM4OpenFOAM) are available. It is widely used in gas turbine LES research.
So the core of the numerical method for premixed flames is the bold idea of "thickening the flame."
Yes. TFM is a physically sophisticated trick and has become the de facto standard for LES premixed combustion.
The True Nature of the "Progress Variable c"—A Story About the Most Troublesome Variable in Premixed Flame Models
In the numerical implementation of premixed flame models, many people stumble over the definition of the reaction progress variable c. c is a variable representing 0 (unburned) to 1 (burned), but "which chemical species' mass fraction is used to define c" varies by model and researcher. There are schools that define it with CO2, schools that use normalized temperature, schools that define it as a linear combination of multiple components—each yielding slightly different results. Fluent's default uses a combination of products, but depending on the fuel or equivalence ratio, other definitions might yield better accuracy. Stories like "the results suddenly matched when I changed the definition of c" are often told at CAE conferences.
Upwind Scheme
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.)
Suppress numerical oscillations while maintaining high accuracy through limiter functions. Effective for capturing shocks 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)
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