Information about the flamelet models are presented in the following sections:
The following restrictions apply to all diffusion flamelet models in Ansys Fluent:
Only a single mixture fraction can be modeled; two-mixture-fraction flamelet models are not allowed.
The mixture fraction is assumed to follow the
-function PDF, and
scalar dissipation fluctuations are ignored.Empirically-based streams cannot be used with the flamelet model.
The flamelet concept views the turbulent flame as an ensemble of thin, laminar, locally one-dimensional flamelet structures embedded within the turbulent flow field [77], [512], [513] (see Figure 8.16: Laminar Opposed-Flow Diffusion Flamelet).
A common laminar flame type used to represent a flamelet in a turbulent flow is the counterflow diffusion flame. This geometry consists of opposed, axisymmetric fuel and oxidizer jets. As the distance between the jets is decreased and/or the velocity of the jets increased, the flame is strained and increasingly departs from chemical equilibrium until it is eventually extinguished. The species mass fraction and temperature fields can be measured in laminar counterflow diffusion flame experiments, or, most commonly, calculated. For the latter, a self-similar solution exists, and the governing equations can be simplified to one dimension along the axis of the fuel and oxidizer jets, where complex chemistry calculations can be affordably performed.
In the laminar counterflow flame, the mixture fraction, 
, (see Definition of the Mixture Fraction) decreases monotonically from unity
at the fuel jet to zero at the oxidizer jet. If the species mass fraction
and temperature along the axis are mapped from physical space to mixture
fraction space, they can be uniquely described by two parameters:
the mixture fraction and the strain rate (or, equivalently, the scalar
dissipation, 
, defined in Equation 8–43). Hence, the chemistry is reduced and completely described by the
two quantities, 
and 
.
This reduction of the complex chemistry to two variables allows the flamelet calculations to be preprocessed, and stored in look-up tables. By preprocessing the chemistry, computational costs are reduced considerably.
The balance equations, solution methods, and sample calculations of the counterflow laminar diffusion flame can be found in several references. Comprehensive reviews and analyses are presented in the works of Bray and Peters, and Dixon-Lewis [77], [143].
A characteristic strain rate for a counterflow diffusion flamelet can be defined as 
, where 
is the relative speed of the fuel and oxidizer jets, and 
is the distance between the jet nozzles.
Instead of using the strain rate to quantify the departure from
equilibrium, it is expedient to use the scalar dissipation, denoted
by 
. The scalar dissipation is defined as
 | (8–42) |
where 
is a representative diffusion coefficient.
Note that the scalar dissipation, 
, varies along the axis
of the flamelet. For the counterflow geometry, the flamelet strain
rate 
can be related to the scalar
dissipation at the position where 
is stoichiometric by [512]:
 | (8–43) |
|
where | |
|
| |
|
| |
|
| |
|
|
Physically, as the flame is strained, the width of the reaction
zone diminishes, and the gradient of 
at the stoichiometric position 
increases. The instantaneous stoichiometric scalar
dissipation, 
, is used as
the essential non-equilibrium parameter. It has the dimensions 
and may be
interpreted as the inverse of a characteristic diffusion time. In
the limit 
the chemistry tends to equilibrium, and
as 
increases
due to aerodynamic straining, the non-equilibrium increases. Local
quenching of the flamelet occurs when 
exceeds a critical value.
A turbulent flame brush is modeled as an ensemble of discrete diffusion flamelets. Since,
for adiabatic systems, the species mass fraction and temperature in the diffusion flamelets are
completely parameterized by 
and 
, density-weighted mean species mass fractions and temperature in the
turbulent flame can be determined from the PDF of 
and 
as:
 | (8–44) |
where 
represents species mass fractions and temperature.
In Ansys Fluent, 
and 
are assumed to be statistically independent, so the joint PDF 
can be simplified as 
. A 
PDF shape is assumed for 
, and transport equations for 
and 
are solved in Ansys Fluent to specify 
. Fluctuations in 
are ignored so that the PDF of 
is a delta function: 
. The first moment, namely the mean scalar dissipation, 
, is modeled in Ansys Fluent as
 | (8–45) |
where 
is a constant with a default value of 2, and 
is the turbulent time scale calculated according to the turbulence model. For
example, for RANS turbulence models,
 | (8–46) |
For non-adiabatic steady diffusion flamelets, the additional parameter of enthalpy is required. However, the computational cost of modeling steady diffusion flamelets over a range of enthalpies is prohibitive, so some approximations are made. Heat gain/loss to the system is assumed to have a negligible effect on the species mass fractions, and adiabatic mass fractions are used
[62], [464]. The temperature is then calculated from Equation 5–8 for a range of mean enthalpy gain/loss, 
. Accordingly, mean temperature and density PDF tables have an extra dimension of mean enthalpy. The approximation of constant adiabatic species mass fractions is, however, not applied for the case corresponding to a scalar dissipation of zero. Such a case is represented by the non-adiabatic equilibrium solution. For 
, the species mass fractions are computed as functions of 
, 
, and 
.
In Ansys Fluent, you can either generate your own diffusion flamelets, or import them as flamelet files calculated with other stand-alone packages. Such stand-alone codes include OPPDIF [401], CFX-RIF [45], [46], [522] and RUN-1DL [516]. Ansys Fluent can import flamelet files in standard flamelet file format.
Instructions for generating and importing diffusion flamelets are provided in Flamelet Generation and Flamelet Import.
The laminar counterflow diffusion flame equations can be transformed
from physical space (with 
as the independent variable) to mixture fraction
space (with 
as the independent variable) [523]. In Ansys Fluent, a simplified set of the mixture
fraction space equations are solved [522]. Here, 
equations are solved for the species mass fractions, 
,
 | (8–47) |
and one equation for temperature:
 | (8–48) |
The notation in Equation 8–47 and Equation 8–48 is as follows: 
, 
, 
, and 
are the 
th species mass fraction, temperature, density, and mixture fraction, respectively. 
and 
are the 
th species specific heat and mixture-averaged specific heat, respectively. 
is the 
th species reaction rate, and 
is the specific enthalpy of the 
th species.
The scalar dissipation, 
, must be modeled across the flamelet. An extension of Equation 8–43 to variable density is used [303], [523]:
 | (8–49) |
where 
is the density
of the oxidizer stream.
Using the definition of the strain rate given by Equation 8–49, the scalar dissipation in the mixture fraction space can be expressed as:
 | (8–50) |
where 
is the stoichiometric mixture fraction, 
is the mixture density, and 
is the scalar dissipation at 
. 
is the user input for flamelet generation. 
and 
are evaluated using Equation 8–49.
Ansys Fluent can import one or more flamelet files, convolute these diffusion flamelets with the assumed-shape PDFs (see Equation 8–44), and construct look-up tables. The flamelet files can be generated in Ansys Fluent, or with separate stand-alone computer codes.
The following types of flamelet files can be imported into Ansys Fluent:
Standard format files described in Standard Files for Diffusion Flamelet Modeling in the User's Guide and in Peters and Rogg [516]
When diffusion flamelets are generated in physical space, the species and temperature vary in one spatial dimension. The species and temperature must then be mapped from physical space to mixture fraction space. If the diffusion coefficients of all species are equal, a unique definition of the mixture fraction exists. However, with differential diffusion, the mixture fraction can be defined in a number of ways.
Ansys Fluent computes the mixture fraction profile along the diffusion flamelet in one of the following ways:
Read from a file (standard format files only)
This option is for diffusion flamelets solved in mixture fraction space. If you choose this method, Ansys Fluent will search for the mixture fraction keyword
Z, as specified in Peter and Roggs’s work [516], and retrieve the data. If Ansys Fluent does not find mixture fraction data in the flamelet file, it will instead use the hydrocarbon formula method described below.Hydrocarbon formula
Following the work of Bilger et al. [61], the mixture fraction is calculated as
(8–51)
where
(8–52)
, 
, and 
are the mass fractions of carbon, hydrogen, and oxygen atoms, and 
, 
, and 
are the molecular weights. 
and 
are the values of 
at the oxidizer and fuel inlets.
The flamelet profiles in the multiple-flamelet data set should vary only in the strain rate imposed; the species and the boundary conditions should be the same. In addition, it is recommended that an extinguished flamelet is excluded from the multiple-flamelet data set. The formats for multiple diffusion flamelets are as follows:
Standard format: If you have a set of standard format flamelet files, you can import them all at the same time, and Ansys Fluent will merge them internally into a multiple-flamelet file. When you import the set of flamelet files, Ansys Fluent will search for and count the occurrences of the
HEADERkeyword to determine the number of diffusion flamelets in the file.CFX-RIF format: A CFX-RIF flamelet file contains multiple diffusion flamelets at various strains and the file should not be modified manually. Only one CFX-RIF flamelet file should be imported.
For either type of file, Ansys Fluent will determine the number of flamelet profiles and sort them in ascending strain-rate order. For diffusion flamelets generated in physical space, you can select one of the four methods available for the calculation of mixture fraction. The scalar dissipation will be calculated from the strain rate using Equation 8–43.