The key to the premixed combustion model is the prediction of 
, the turbulent
flame speed normal to the mean surface of the flame. The turbulent
flame speed is influenced by the following:
laminar flame speed, which is, in turn, determined by the fuel concentration, temperature, and molecular diffusion properties, as well as the detailed chemical kinetics
flame front wrinkling and stretching by large eddies, and flame thickening by small eddies
Ansys Fluent has two turbulent flame speed models, namely the Zimont turbulent flame speed closure model and the Peters flame speed model.
For more information, see the following sections:
In Ansys Fluent, the Zimont turbulent flame speed closure is computed using a model for wrinkled and thickened flame fronts [736]:
 | (8–77) |
 | (8–78) |
| where | |
 = model constant | |
 = RMS (root-mean-square)
velocity (m/s) | |
 = laminar flame speed (m/s) | |
 = unburnt thermal
diffusivity ( ) | |
 = turbulence length
scale (m) | |
 = turbulence time scale
(s) | |
 = chemical
time scale (s) |
The turbulence length scale, 
, is computed
from
 | (8–79) |
where 
is the turbulence dissipation rate.
The model is based on the assumption of equilibrium small-scale
turbulence inside the laminar flame, resulting in a turbulent flame
speed expression that is purely in terms of the large-scale turbulent
parameters. The default value of 0.52 for 
is recommended [736], and is suitable for most premixed flames. The
default value of 0.37 for 
should also be suitable for
most premixed flames.
The model is strictly applicable when the smallest turbulent
eddies in the flow (the Kolmogorov scales) are smaller than the flame
thickness, and penetrate into the flame zone. This is called the thin
reaction zone combustion region, and can be quantified by Karlovitz
numbers, 
, greater than unity. 
is defined as
 | (8–80) |
| where | |
 = characteristic flame time
scale | |
 = smallest (Kolmogorov)
turbulence time scale | |
 = Kolmogorov velocity | |
 = kinematic viscosity |
For simulations that use the LES turbulence model, the Reynolds-averaged
quantities in the turbulent flame speed expression (Equation 8–77) are replaced by their equivalent
subgrid quantities. In particular, the large eddy length scale 
is
modeled as
 | (8–81) |
where 
is the Smagorinsky constant
and 
is the cell characteristic length.
The RMS velocity in Equation 8–77 is replaced by the subgrid velocity fluctuation, calculated as
 | (8–82) |
where 
is the subgrid scale mixing rate (inverse
of the subgrid scale time scale), given in Equation 7–41.
Since industrial low-emission combustors often operate near
lean blow-off, flame stretching will have a significant effect on
the mean turbulent heat release intensity. To take this flame stretching
into account, the source term for the progress variable (
in Equation 8–70) is multiplied
by a stretch factor, 
[738]. This stretch
factor represents the probability that the stretching will not quench
the flame; if there is no stretching (
), the probability
that the flame will be unquenched is 100%.
The stretch factor, 
, is obtained by integrating the log-normal distribution
of the turbulence dissipation rate, 
:
 | (8–83) |
where 
is the complementary error function, and 
and 
are defined below. For LES, 
is calculated by:
 | (8–84) |
where 
is the time scale and 
is the is subgrid scale (SGS) velocity fluctuations, which depends on
the SGS closure. For details on how 
is calculated, refer to Subgrid-Scale Models.
is the standard deviation of the distribution of 
:
 | (8–85) |
where 
is the stretch factor coefficient for dissipation
pulsation, 
is the turbulent integral length scale, and 
is the Kolmogorov micro-scale.
The default value of 0.26 for 
(measured in turbulent non-reacting flows) is recommended
by [736], and is suitable for most premixed
flames.
is the turbulence
dissipation rate at the critical rate of strain [736]:
 | (8–86) |
By default, 
is set to a very high value (1x108) so no flame
stretching occurs. To include flame stretching effects, the critical rate of strain
should be adjusted based on experimental data for the burner. Numerical
models can suggest a range of physically plausible values [736], or
an appropriate value can be determined from experimental data. A reasonable model for the
critical rate of strain 
is
 | (8–87) |
where 
is a constant (typically 0.5) and 
is the unburnt thermal
diffusivity. Equation 8–87 can be implemented
in Ansys Fluent using a property user-defined function. More information
about user-defined functions can be found in the Fluent Customization Manual.
High turbulent kinetic energy levels at the walls in some problems
can cause an unphysical acceleration of the flame along the wall.
In reality, radical quenching close to walls decreases reaction rates
and therefore the flame speed, but is not included in the model. To
approximate this effect, Ansys Fluent includes a constant multiplier
for the turbulent flame speed, 
,which modifies the
flame speed in the vicinity of wall boundaries:
 | (8–88) |
The default for this constant is 1 which does not change the
flame speed. Values of 
larger than 1 increase
the flame speed, while values less than 1 decrease the flame speed
in the cells next to the wall boundary.
Ansys Fluent will solve the transport equation for the reaction progress variable 
(Equation 8–70), computing the source term,
, based on the theory outlined above:
 | (8–89) |
The Peters model [514] for turbulent flame speed is used in the form proposed by Ewald [167]:
 | (8–90) |
where
 | (8–91) |
The term 
is Ewald's corrector and may be disabled by you, in which case 
and the formulation reduces to that of Peters flame
speed model.
In Equation 8–91:
 is the laminar
flame speed | |
 is the laminar flame thickness | |
 is
the turbulent velocity scale | |
 is the flame brush thickness, given below | |
| |
 is
an algebraic flame brush thickness | |
| |
 is a constant with a default value of 1.0 | |
 is a constant with a default value of 2.0 | |
 is a constant with a default value of 0.66 | |
 is a modeling constant taken
from the  turbulence model with a default value of 0.09 | |
 is a constant with a default value of 2.0 | |
 is the turbulent Schmidt number with a
default value of 0.7 |
If the Blint modifier is being used
 | (8–92) |
| where | |
 = 2.0 | |
 = 0.7 | |
 is the unburned density | |
 is the burned density |
The flame brush thickness is calculated differently for the C-equation and G-equation models.
For the G equation model [514]:
 | (8–93) |
For the C-equation model an algebraic form of this is used giving:
 | (8–94) |
Note that for the C-equation model the Ewald corrector has no affect.
To reduce the flame speed along walls, Ansys Fluent includes a constant
multiplier for the turbulent flame speed, 
, which modifies
the flame speed in the vicinity of wall boundaries by multiplying
the 
expression in Equation 8–90 by a constant between 0 and 1.
For LES the Peters model [514] for turbulent flame speed must be modified to use the subgrid quantities. Here we use the form derived by Pitsch [521].
 | (8–95) |
 | (8–96) |
where 
is the thermal diffusivity.
The turbulent velocity scale 
is the subgrid quantity as given in Equation 8–82.