VII.6 Flamelet methods
The main alternative methodology to the one described in the previous section is known as the
laminar flamelet approach. In the correct combustion regimes, the chemical timescale for the
combustion reactions is much shorter than any flow timescale, i.e. the high Damköhler number
regime. In other words the thickness of the flame (lF , l
) is much smaller than any
characteristic flow length, and in particular, much smaller than the Kolmogorov lengthscale
which is the shortest turbulent length scale in the flow. This implies that the
flame can be considered as a 2-d sheet separating regions of burnt and unburnt gas
(for premixed combustion) or regions of fuel and air (non-premixed combustion).
This flame front propagates through the flow the laminar flame speed sL, which
is relatively easy to specify (from experiment or modelling), and it also interacts
passively with the flow, being transported by the mean flow and wrinkled by the
turbulence. This motion can be linked to the flow characteristics as computed from, say, a
standard k - 
model, thus providing a complete combusting flow simulation. The
detailed modelling will vary depending on case, but the framework works very well for
premixed, non-premixed and partially premixed combustion. Figure 2 shows this
diagramatically.
In detail, the location of the flame sheet is specified by some form of indicator function. For
instance in premixed combustion, a progress variable c can be defined as a normalised
temperature or normalised mass fraction
Putting in appropriate values we see that c varies from 0 in the unburned gas to 1 in the
burned gas. Such a variable can be defined at all points in the flow domain, and the
location of the flame itself is defined by the isosurface where c has some constant value
(say c = 0.5). In any realistic turbulent flow the small turbulent scales will be too
small to explicitly simulate, so we will want to provide a turbulence model : similarly,
we will in practice be calculating the average location of the flame front, and thus
be dealing with the Favre averaged progress variable 
. We can create transport
equations describing the way in which it propagates by convection by the flow and by
combustion,
This requires modelling of the turbulent transport term 
and of the reaction term 
.
However we have reduced the complexities of the previous section to a much more manageable
set of equations. Several other related formulations are also possible, for example using a
quantity called the flame surface density 
, related to the degree of wrinkling. For
non-premixed combustion a suitable progress variable is the mixture fraction Z. For that
matter the progress variable does not even have to be physically motivated, and there is a class
of combustion models called the G-equation models where the location of the flame is
marked by an isosurface (eg. G = 0) of a function G which is otherwise entirely
arbitary.
One final point concerns the modelling of the correlation term 
. Correlation
terms of this form crop up frequently in turbulent combustion modelling, and are
frequently modelled using the gradient transport assumption. This is the assumption that
this sort of term is essentially a diffusive effect and can be modelled in this way, as
 | (VII.9) |
This is a simple assumption to make. It is also frequently wrong. In this case it is possible to
show (both analytically and experimentally) that 
> 0 in some parts of the flame, in
contradiction to this assumption (equation (VII.9) would make it negative). This phenomenon
is known as countergradient diffusion. Turbulent combustion models are often constructed to
split this correlation into diffusion and countergradient diffusion terms, in order to model the
two effects differently.
