Diffusion Flames and Mixture Fraction
Theory and Physics
Overview
Professor, what kind of combustion mode is a diffusion flame?
A diffusion flame (non-premixed flame) is a mode where fuel and oxidizer are supplied separately, and combustion proceeds simultaneously with mixing. Examples include candle flames, diesel engines, and gas turbine combustors. The flame forms at the surface where fuel and oxidizer meet at the stoichiometric ratio (stoichiometric surface).
So it's fundamentally different from a premixed flame.
Correct. In premixed flames, the burning velocity is the governing parameter, but in diffusion flames, the mixing rate becomes the rate-limiting step. Utilizing this property, describing the flame structure with the variable mixture fraction $Z$ is the core of the Burke-Schumann theory.
Mixture Fraction
Please explain the definition of mixture fraction $Z$.
Mixture fraction is a conserved scalar representing "how much mass from the fuel stream is contained within a fluid element." Bilger's definition is widely used.
Here, $Y_F$ is the fuel mass fraction, $Y_O$ is the oxidizer mass fraction, subscript 1 indicates values from the fuel stream, and subscript 2 indicates values from the oxidizer stream. $s$ is the mass-based stoichiometric ratio.
So $Z=0$ is pure oxidizer and $Z=1$ is pure fuel?
Exactly. The stoichiometric mixture fraction $Z_{st}$ is $Z_{st} = Y_{O,2}/(s\,Y_{F,1} + Y_{O,2})$. For methane/air, $Z_{st} \approx 0.055$. The flame is located at the iso-surface where $Z = Z_{st}$.
Burke-Schumann Solution
What is the Burke-Schumann solution?
Assuming the chemical reaction is infinitely fast ($Da \to \infty$), fuel and oxidizer react instantaneously at the flame surface, and temperature and chemical species become functions of mixture fraction only.
This is the Burke-Schumann solution, where the temperature profile becomes triangular, peaking at $Z_{st}$. This solution is also the starting point for the flamelet model.
Mixture Fraction Transport Equation
Mixture fraction $Z$ obeys a transport equation without a source term (independent of chemical reaction).
Since there's no source term, we can solve for the $Z$ field without knowing the details of the chemical reaction, right?
That's the major advantage of the mixture fraction approach. It encapsulates the complexity of chemical reactions into the flamelet equations in $Z$ space, and in CFD, we only need to transport $Z$ and its variance $\widetilde{Z''^2}$.
So the theory of diffusion flames is entirely based on the concept of mixture fraction.
Yes. It's no exaggeration to say that mixture fraction is the most important variable in CFD for non-premixed combustion.
The Candle Flame Was the "Textbook" Diffusion Flame—The Limits of the Burke-Schumann Solution
The analytical solution for diffusion flames proposed by Burke and Schumann in 1928 is based on the "infinite reaction rate" assumption that mixing of fuel and oxidizer completes instantaneously. A candle flame can be reproduced surprisingly well with this model. However, when the same assumption is used in actual gas turbine combustors, the predicted NOx generation can be over ten times the measured value. Even though mixing should be rate-limiting when mixing is slow, the finite reaction rate becomes apparent in high-temperature regions. This is why field engineers say, "The Burke-Schumann solution is excellent for conceptual understanding, but cannot be used for design."
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, the flow becomes steady, right? This "period 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 computational cost is significantly reduced, trying to solve a steady-state problem first 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" → 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, it naturally mixes, 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 large, the diffusion term becomes strong, and the fluid moves in a "thick, sluggish" manner. 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, liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, and 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 densely 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: "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 confusing absolute and gauge pressure.
- Source term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed upward 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 natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—a physically impossible result where warm air doesn't rise 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): 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 pressure 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 is a mixture fraction-based combustion model implemented in CFD?
In non-premixed combustion models, the transport equations for $Z$ and $\widetilde{Z''^2}$ (mixture fraction variance) are solved in CFD, and chemical reaction information is retrieved from a lookup table.
Transport Equations
The equations for Favre-averaged mixture fraction $\widetilde{Z}$ and variance $\widetilde{Z''^2}$ in a turbulent field are as follows.
What is the last term $\widetilde{\chi}$ in the variance equation?
That's the scalar dissipation rate. It is defined as $\chi = 2D|\nabla Z|^2$ and represents how quickly the fine structure of the mixture fraction dissipates. Under turbulent conditions, it is modeled as $\widetilde{\chi} = C_\chi \frac{\varepsilon}{k}\widetilde{Z''^2}$ (where $C_\chi \approx 2.0$).
PDF (Probability Density Function)
Please explain the role of the PDF.
In a turbulent field, $Z$ fluctuates within a cell, so the average value alone cannot correctly represent the flame structure. We assume a probability density function $P(Z)$ for $Z$ and calculate the average temperature and average chemical species through integration.
The $\beta$ function distribution is widely used for the PDF shape. The shape of the $\beta$ distribution is uniquely determined from the two parameters $\widetilde{Z}$ and $\widetilde{Z''^2}$.
Lookup Table Construction
How is the table created?
We solve the flamelet equations or chemical equilibrium beforehand to create a table of temperature and chemical species with $Z$ and $\chi_{st}$ (stoichiometric dissipation rate) as parameters. The table integrated with the PDF is referenced during CFD runtime.
| Table Variable | Dimensions | Purpose |
|---|---|---|
| $\widetilde{Z}$, $\widetilde{Z''^2}$ | 2D | Equilibrium chemistry / thin flames |
| $\widetilde{Z}$, $\widetilde{Z''^2}$, $\widetilde{\chi_{st}}$ | 3D | Steady flamelet |
| $\widetilde{Z}$, $\widetilde{Z''^2}$, $\widetilde{C}$ | 3D | FGM (with progress variable added) |
How is it set up in Fluent?
In Fluent, select Models > Species > Non-Premixed Combustion, import the reaction mechanism in CHEMKIN format, and let it automatically generate the PDF table. A table resolution (number of divisions in the $Z$ direction) of at least 64 points, preferably 128 points, is recommended.
The mixture fraction + PDF table method is clever because it solves the chemical reactions beforehand.
Yes. The biggest strength is that there's no need to solve chemical reactions during 3D CFD runtime, so computational cost remains almost unchanged even with detailed chemical reactions.
How to Solve for Scalar Dissipation Rate χ—The Most Debated Variable in Diffusion Flame Model Implementation
When implementing a flamelet model for diffusion flames, the variable that troubles implementers the most is the calculation of the scalar dissipation rate χ (chi). χ represents the intensity of mixing at the flame surface (the stretching rate of fuel-oxidizer due to turbulence), and higher χ makes the flame more prone to extinction. However, there are multiple formulas for directly calculating χ in turbulent CFD, and results vary depending on the choice of Favre averaging and model constants. Especially in regions where recirculation zones (burner wake) coexist with areas of low χ near zero and high χ in shear layers, the accuracy of spatial distribution calculation directly affects NOx and soot predictions. "How to calculate χ" often becomes a vendor's secret sauce.
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 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 timestep.
Residual Monitoring
Continuity equation residual should drop by at least 3 orders of magnitude. Convergence criteria: Typically 1e-4 to 1e-6. Monitoring physical quantities (drag coefficient, Nusselt number) is also important.
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