The steady diffusion flamelet model, described in The Diffusion Flamelet Models Theory and The Steady Diffusion Flamelet Model Theory, models local chemical non-equilibrium due to the straining effect of turbulence. In many combustors the strain is small at the outlet and the steady diffusion flamelet model predicts all species, including slow-forming species like NOx, to be near equilibrium, which is often inaccurate. The cause of this inaccuracy is the disparity between the flamelet time-scale, which is the inverse of the scalar dissipation, and the slow-forming species time-scale, which is the residence time since the species started accumulating after mixing in the combustor.
The unsteady diffusion flamelet model in Ansys Fluent can predict slow-forming species, such as gaseous pollutants or product yields in liquid reactors, more accurately than the steady diffusion flamelet model. Computationally expensive chemical kinetics are reduced to one dimension and the model is significantly faster than the laminar-finite-rate, EDC or PDF Transport models where kinetics are calculated in two or three dimensions. There are two variants of the unsteady flamelet model, namely an Eulerian unsteady flamelet model (described in The Eulerian Unsteady Laminar Flamelet Model) and a diesel unsteady flamelet model for predicting combustion in compression-ignition engines (described in The Diesel Unsteady Laminar Flamelet Model).
Information pertaining strictly to the unsteady flamelet model is presented in the following sections:
The Eulerian unsteady laminar flamelet model can be used to predict slow-forming intermediate and product species that are not in chemical equilibrium. Typical examples of slow-forming species are gas-phase pollutants like NOx, and product compounds in liquid reactors. By reducing the chemistry computations to one dimension, detailed kinetics with multiple species and stiff reactions can be economically simulated in complex 3D geometries.
The model, following the work of Barths et al. [46] and Coelho and Peters [112], postprocesses unsteady marker probability equations on a steady-state converged flow field. The marker field represents the probability of finding a flamelet at any point in time and space. A probability marker transport equation is solved for each flamelet.
In Ansys Fluent, the steady flow solution must be computed with the steady diffusion flamelet model (see The Steady Diffusion Flamelet Model Theory) before starting the unsteady flamelets simulation. Since the unsteady flamelet equations are postprocessed on a steady-state flamelet solution, the effect of the unsteady flamelet species on the flow-field is neglected.
When multiple flamelets are enabled, Ansys Fluent solves Eulerian transport equations representing
the probability of fuel for 
th flamelet, 
, as follows
 | (8–57) |
where 
is laminar thermal conductivity of the mixture, 
is the mixture specific heat, and 
is the Prandtl number.
Each marker probability is initialized as,
 | (8–58) |
where 
is the scalar dissipation, 
are the minimum
and maximum scalar dissipation values for 
th flamelet
marker probability, 
is the mean mixture fraction,
and 
is a user-specified constant
that should be set greater than the stoichiometric mixture fraction.
Hence the marker probabilities are initialized to unity in regions
of the domain where the mean mixture fraction is greater than a user
specified value (typically greater than stoichiometric), and multiple
flamelets sub-divide this initial region by the scalar dissipation.
As previously mentioned, the Eulerian Unsteady Laminar Flamelet
Model is only available for steady-state simulations. However, marker
probability transport equations (Equation 8–57) are always solved time-accurately
as the initial marker probabilities convect and diffuse through the
steady flow field. At the inlet boundaries, 
is set to zero, and hence the 
field decreases to zero with time as it is convected
and diffused out of the domain (for cases with outlet boundaries).
The unsteady flamelet species Equation 8–47) is integrated simultaneously
with the marker probability Equation 8–57 for each marker probability 
. For liquid-phase chemistry, the initial flamelet field is the mixed-but-unburnt
flamelet, as liquid reactions are assumed to proceed immediately upon
mixing. Gas-phase chemistry involves ignition, so the initial flamelet
field is calculated from a steady diffusion flamelet solution. All
of the slow-forming species, such as NOx, must be identified before
solving the unsteady flamelet equations. The mass fractions of all
slow-forming species are set to zero in this initial flamelet profile,
since, at ignition, little residence time has elapsed for any significant
formation.
The scalar dissipation at stoichiometric mixture fraction (
) is required by each flamelet species equation. This is calculated from the
steady-state Ansys Fluent field at each time step as a probability-weighted volume integral [112]:
 | (8–59) |
where 
is defined in Equation 8–45, and 
denotes the fluid volume. Ansys Fluent provides
the option of limiting 
to a user-specified maximum value,
which should be approximately equal to the flamelet extinction scalar
dissipation (the steady diffusion flamelet solver can be used to calculate
this extinction scalar dissipation in a separate simulation).
The unsteady flamelet energy equation is not solved in order
to avoid flamelet extinction for high scalar dissipation, and to account
for non-adiabatic heat loss or gain. For adiabatic cases, the flamelet
temperature, 
, is calculated at each time step
from the steady diffusion flamelet library at the probability-weighted
scalar dissipation 
from Equation 8–59. For non-adiabatic cases, the
flamelet temperature at time 
is calculated from
 | (8–60) |
where
 | (8–61) |
Here, the subscript 
referring to 
th flamelet has been omitted
for simplicity, and 
represents the Ansys Fluent steady-state mean cell
temperature conditioned on the local cell mixture fraction.
Unsteady flamelet mean species mass fractions in each cell are accumulated over time and can be expressed as:
 | (8–62) |
where 
denotes the 
th species unsteady flamelet
mass fraction, and 
is the 
th unsteady flamelet mass fraction of 
th flamelet
and is calculated using a Beta pdf as,
 | (8–63) |
The probability marker equation (Equation 8–57) and the flamelet species equation
(Equation 8–47) are advanced together
in time until the probability marker has substantially convected and
diffused out of the domain. The unsteady flamelet mean species, calculated
from Equation 8–61, reach steady-state
as the probability marker 
vanishes.
Liquid reactors are typically characterized by:
Near constant density and temperature.
Relatively slow reactions and species far from chemical equilibrium.
High Schmidt number (
) and
hence reduced molecular diffusion.
The Eulerian unsteady laminar flamelet model can be used to model liquid reactions. When the Liquid Micro-Mixing model is enabled, Ansys Fluent uses the volume-weighted-mixing-law formula to calculate the density.
The effect of high 
is to decrease mixing
at the smallest (micro) scales and increase the mixture fraction variance,
which is modeled with the Turbulent Mixer Model [38]. Three transport equations are solved for the
inertial-convective (
), viscous-convective (
), and viscous-diffusive (
) subranges of the turbulent scalar spectrum,
 | (8–64) |
 | (8–65) |
 | (8–66) |
where 
is laminar thermal conductivity of the mixture, 
is the mixture specific heat, and 
is the Prandtl number. The constants 
through 
have values
of 2, 1.86, 0.058, 0.303, and 17050, respectively. The total mixture
fraction variance is the sum of 
, 
and 
.
In Equation 8–66, the cell
Schmidt number, 
, is calculated as 
where 
is the viscosity, 
the density, and 
the mass diffusivity as
defined for the pdf-mixture material.
In diesel engines, fuel sprayed into the cylinder evaporates, mixes with the surrounding gases, and then auto-ignites as compression raises the temperature and pressure. The diesel unsteady laminar flamelet model, based on the work of Pitsch et al. and Barths et al. [522], [45], models the chemistry by a finite number of one-dimensional laminar flamelets. By reducing the costly chemical kinetic calculation to 1D, substantial savings in run time can be achieved over the laminar-finite-rate, EDC or PDF Transport models.
The flamelet species and energy equations (Equation 8–47 and Equation 8–48) are solved simultaneously with the flow. The flamelet equations are advanced for a fractional step using properties from the flow, and then the flow is advanced for the same fractional time step using properties from the flamelet.
The initial condition of each flamelet at the time of its introduction into computational domain is a mixed-but-unburnt distribution. For the flamelet fractional time step, the volume-averaged scalar dissipation and pressure, as well as the fuel and oxidizer temperatures, are passed from the flow solver to the flamelet solver. To account for temperature rise during compression, the flamelet energy equation (Equation 8–48) has an additional term on the right-hand side as
 | (8–67) |
where 
is the specific heat and 
is the volume-averaged pressure
in the cylinder. This rise in flamelet temperature due to compression
eventually leads to ignition of the flamelet.
After the flamelet equations have been advanced for the fractional time step, the PDF Table is created as a Non-Adiabatic Steady Flamelet table (see Non-Adiabatic Steady Diffusion Flamelets). Using the properties from this table, the CFD flow field is then advanced for the same fractional time step.
In certain applications where the ignition in different regions of the combustion domain occurs at different times, the chemistry cannot be accurately represented by a single flamelet. Among the examples are split-injections and sprays with high residence times. In these cases, evaporated spray injected at early stage ignites before spray injected at later stage due to the longer residence time. A single flamelet cannot model the local ignition delay for the late spray as the single flamelet represents a burnt state. This deficiency is overcome with the use of multiple flamelets, which are generated in the reacting domain at user-specified times during the simulation. The new flamelet inherits the preceding flamelet boundary temperature calculated at the time of the new flamelet introduction, and the flamelet species field is initialized as mixed-but-unburnt.
The marker probability equations (Equation 8–57) are solved for 
flamelets, where 
is the total number of unsteady
flamelets. The marker probability of the last flamelet is obtained
as follows:
 | (8–68) |
where 
is the mean mixture fraction,
and 
is the marker probability of the 
th unsteady
flamelet. The scalar dissipation of the 
th flamelet
is calculated using Equation 8–59.
The properties from the PDF tables 
(such as mass fraction, specific heat,
and so on) are calculated using the weighted contribution from each
flamelet:
 | (8–69) |
where 
is the property for 
th flamelet.
The diesel unsteady flamelet approach can model ignition as well as formation of product, intermediate, and pollutant species. The setting of the Diesel Unsteady Flamelet model is described in Using the Diesel Unsteady Laminar Flamelet Model in the Fluent User's Guide.
As the multiple diesel unsteady flamelets ignite, their species and temperature fields tend toward the same chemical equilibrium state as the scalar dissipation (mixing) decreases. To model multiple engine cycles, Ansys Fluent allows flamelet reset events, which resets multiple flamelets to a single flamelet. Additionally, at the end of a cycle, some burnt gases typically remain trapped and mix later with the fresh charge introduced in the next cycle. The presence of the exhaust gas in the engine chamber can be modeled using either a Diesel Unsteady Flamelet Reset or the inert model (via Inert EGR Reset).
For information on how to use and set up the Diesel Unsteady Flamelet Reset and Inert EGR Reset options, see Resetting Diesel Unsteady Flamelets in the Fluent User's Guide.
At the end of combustion stroke and just before the inlet valves open, all existing flamelets, except the first one, are deleted, and the probability of the first flamelet is set as:
where, 
is the probability of the 
th flamelet, and 
is the total number of flamelets.
The mixture composition (and similarly the temperature) of the reset
flamelet is set as:
where 
is the species mass fraction of the 
th species from the 
th flamelet.
After resetting the flamelets, the computations are performed with this single, typically burnt, flamelet. The flamelet start time of subsequent flamelets should be set to just before the fuel injection. Hence, the injected fuel is modeled in the second and higher flamelets. The process is controlled through dynamic mesh events described in Diesel Unsteady Flamelet Reset in the Fluent User's Guide.