The prediction of the turbulent burning velocity plays a crucial role in the modeling of SI engine combustion. The laminar flame speed is one of the most important scaling factors in most of the published correlations for the turbulent flame speed.
Laminar flame speed is an intrinsic property of a premixed fuel/air mixture for given unburned temperature, pressure, and composition (species mass fractions). The composition can be characterized using two derived parameters: equivalence ratio and diluent fraction. Equivalence ratio can be defined using the concentrations of C, H, and O atoms as:
 | (7–17) |
Diluent fraction can be estimated by assuming the inert diluent mixture as complete combustion products. The diluent may consist of CO2 , H2 O, N2 , as well as excess O2 for lean mixtures. It is a function of equivalence ratio, mass fractions of CO2, H2O, and the C/H/O ratio in the fuel species.
In Ansys Forte, two options are available to specify laminar flame speeds. One is using the power-law correlations, the other is through table look-up. For details on using table look-up, see Laminar Flame Speed Through Table Look-up.
The "power law" formula [57] can be written as:
 | (7–18) |
where the subscript ref means the reference condition
(typically at 298 K and 1 atm) and the superscript 0 means the flame is planar and
unstretched. 
is a factor accounting for the diluent’s effect.
The reference flame speed can be specified using two different formulas.
 | (7–19) |
Values for 
, 
and 
for selected fuels are listed in Table 7.1: Values for 
, 
, and in the Metghalchi formula
according to Metghalchi et al. [56].
Table 7.1: Values for , , and in the Metghalchi formula
|
Fuel |
|
|
|
|---|---|---|---|
|
methanol |
36.9 |
-140.5 |
1.11 |
|
propane |
34.2 |
-138.7 |
1.08 |
|
iso-octane |
26.3 |
-84.7 |
1.13 |
|
gasoline |
30.5 |
-54.9 |
1.21 |
Unfortunately, Equation 7–19 gives negative flame speeds for very lean or very rich mixtures. One practical solution to this problem is to follow the expression proposed by Gülder [28] in which the flame speed will never be driven negative, viz.,
 | (7–20) |
where 
, 
, 
, and 
are data-fitting coefficients. For iso-octane, a group
of values for the coefficients in Equation 7–20 can be obtained by correlating
the data of Metghalchi et al. within the range 0.65 < f
< 1.6. For example, then, coefficients obtained for iso-octane's reference flame speed
are:
The exponents 
and 
in Equation 7–18 describe the temperature and
pressure dependencies. These exponents were treated as generic for all fuel types in
Metghalchi et al. [56] and correlated as functions of
equivalence ratio as:
 | (7–21) |
 | (7–22) |
Values obtained using least-squares fits are given as
An additional set of parameters for gasoline were reported by Rhodes et al. [79]:
This effect by the diluent is usually accounted for by a term such as
in Equation 7–18
. The expressions for this factor
take the form:
 | (7–23) |
 | (7–24) |
depending on whether mole fraction or mass fraction is selected, where
is the mole fraction of diluent; 
is the mass fraction of diluent; 
are empirical constants. Fitted values from the literature are:
 | (7–25) |
 | (7–26) |
The laminar flame speed value returned by table look-up is the unstretched
flame speed, which is comparable to the value calculated using Equation 7–18
. It does not contain the stretch
factor 
described in Equation 7–4
. The details of working with
flame speed table lookup is described in Flame-Speed Table Editor in the Ansys Forte User's Guide.
The ratio between turbulent flame speed and laminar flame speed is given as
 | (7–27) |
is the laminar flame speed; 
and 
are the turbulence integral length scale and the laminar flame thickness,
respectively. The term 
in Equation 7–27
is a progress variable, which
takes the form
 | (7–28) |
in Equation 7–28
is the Flame Development
Coefficient, a model constant relevant to spark ignition. Physically, the progress variable
models the increasingly disturbing effect of the surrounding eddies on the
flame front surface as the ignition kernel grows from the laminar flame stage into the fully
developed turbulent stage. 
is needed only when a spark exists. The model constant 
is available in the User Interface under Models > Spark Ignition. It is
provided with an appropriate default value that is applicable in most cases. Increasing the
value of this variable will expedite the transition from the laminar kernel flame to the fully
developed turbulent flame. See Spark Ignition in the Ansys Forte User's Guide.
The other model constants, 
, 
and 
are generic for any turbulent flame (both kernel flame or fully developed
flame). They need to be provided as long as the G-equation model is used. Suggested values were
provided by Peters [69], and options for user
definition are available in the User Interface. Here we present brief explanations of these
constants:
: Coefficient in equation 
, where 
is fully-developed turbulent flame speed, 
is turbulence velocity, defined as sqrt(2/3*k), with
being turbulent kinetic energy. The suggested value is 3.0, which was
obtained by fitting experimental data.
: Coefficient in equation 
, where 
and 
are turbulent diffusivity and laminar diffusivity, respectively. Suggested
value is 1.0, which is based on Damkohler's publication in 1940 [20]. A subsequent DNS study by Wenzel [104] suggested a similar value, 1.07.
: Coefficient in equation 
, where 
is turbulent diffusivity 
, is turbulence velocity, 
is turbulence integral length scale. Suggested value is 0.78.
The turbulent flame speed correlation described above (Equation 7–27) is based on the theory of
Peters [69]. This correlation requires the
turbulence integral length scale, 
, as an input. However, the integral length scale is not well defined in LES
models, and a correlation not dependent on 
is needed when the G-equation model is used with the LES model. For this
purpose, an alternative correlation by Clavin and Williams [18] is
provided:
 | (7–29) |
When ≫
, the Clavin-Williams correlation reduces to 
, which is consistent with the Peters correlation (Equation 7–27
). 
is the only adjustable model input to this correlation. When used with LES,
since is the subgrid turbulence velocity, 
is expected to be larger than the value used with RANS correlations, by
two-to-three times.
The turbulent flame brush thickness 
is defined as the square root of the Favre variance of
G, 
:
 | (7–30) |
For fully developed flames, the turbulent flame brush thickness is correlated with turbulence integral length scale as [58],
 | (7–31) |
In the context of spark ignition, the progress variable 
defined in Equation 7–28
is used to describe the growth of
flame brush thickness from the laminar stage to the fully developed stage. Thus, the flame
brush thickness in spark ignition engine is modeled as
 | (7–32) |