Autoignition in Non-Homogeneous Mixtures: Conditional Statistics and
Implications for Modeling
Shufen Cao and Tarek Echekki
Department of Mechanical and Aerospace Engineering
North Carolina State University, Raleigh, NC 27695-7910
Conditional statistics associated with the problem of non-homogeneous autoignition are
investigated using Direct Numerical Simulations (DNS). The chemical model is based on a
single-step, second-order, and irreversible reaction mechanism with reaction rate expressed
in Arrhenius law. The mixture is initialized as a random distribution with variable mixture
strength in decaying isotropic turbulence. Both low and high turbulence conditions are
studied and three Lewis number cases are examined for the high turbulence conditions.
Simulation results show that under conditions of non-homogeneous mixture and preheated
air, autoignition is initiated in fuel lean mixture, and evolves by propagation to richer
mixtures. The propagation elements of the autoignition process are found in both statistics
of mean quantities for reactive scalars as well as co-variances and variances of these scalars
with the rate of dissipation. The addition of a second conditioning variable based on a
reduced temperature is investigated. Results show that the addition of a secondconditioning variable that measures the extent of completion of combustion may be a
reasonable choice for non-homogeneous autoignition modeling. However, additional nontrivial closure models are required for both reactive scalars’ phase space equations, and the
transport equations for the second conditioning variable.
Key words: Direct numerical simulations, autoignition, turbulent combustion modeling,
Conditional Moment Closure
2
1. Introduction
Autoignition in non-homogeneous mixtures is a complex process, because it involves
interactions of chemical reactions, molecular diffusion, and turbulent transport over a broad
range of length and time scales. For example, in a non-homogeneous mixture, the rate of
chemistry competes with the rates of turbulent mixing and diffusion resulting in potentially
different modes or regimes of combustion [1]. When mixing is very slow, autoignition is
initiated at the interface between the fuel and oxidizer streams, and may occur primarily in
a non-premixed combustion mode. This ‘diffusive burning’ regime has long been the
standard model for Diesel engine combustion. Another important characteristic of
autoignition is that, it is a transient process. During this process, the dominant chemical
reactions, the mode of chemistry or the combustion regime can evolve in time.
Autoignition in turbulent flows is important in many practical applications, such as diesel
and aircraft jet engines. Recent observations in Diesel engines [2] show that autoignition
occurs primarily in a stratified mixture, and evolves through propagation and mixing to the
remaining regions of the non-homogeneous mixture. In contrast, autoignition under
homogeneous charge compression ignition (HCCI) conditions, occurs in nearly
homogeneous mixtures; although, temperature non-homogeneity can be present. Traditional
modeling approaches based on non-premixed autoignition modes may not work for the
broad range of practical applications involving autoignition.
Recent direct numerical simulations (DNS) of autoignition have also revealed important
characteristics of autoignition under various turbulence and mixture conditions [3-9].
Echekki and Chen [8,9] carried out 2D simulations of autoignition in a stratified hydrogenair mixture. In this study, the mixture contains a random field of fuel and oxidizer regions
(or ‘blobs’) and the mixture fraction varies between zero and one. The results show that
ignition is initiated in lean premixed mixtures and propagates into richer mixtures. The offstoichiometric onset of autoignition is governed by the preheat of the air, and the strong
dependence of ignition reactions on this preheat. They also show that as the kernels of
flames expand into richer mixtures, diffusion flames form along the stoichiometric
isocontours. In Refs. [8,9], Echekki et al. further investigated the coupling between
chemistry and the unsteady scalar dissipation rate in 2D non-homogeneous autoignition.
Because of this coupling, the balance between radical production and dissipation
determines the success or failure of a kernel to ignite. Therefore, the Echekki and Chen
[8,9] simulations illustrate the importance of propagation and the presence of different
modes of combustion to the autoignition process.
Simulation approaches such as the Reynolds-Averaged Navier-Stokes (RANS) and the
Large-Eddy Simulations (LES) have been applied to model autoignition in turbulent media.
Turbulent combustion models are used for closure of chemical sources, in addition to
requirements for the transport of reactive and passive scalars. The transient nature and the
presence of multiple burning modes in non-homogeneous autoignition impose additional
modeling challenges. The Conditional Moment Closure (CMC) model has been proposed in
recent years to provide closure for reactive scalars in turbulent combustion [10]. In its
earliest formulation, the CMC modeling strategy relies on the assumption that fluctuations
in the scalar quantities are primarily correlated with the fluctuation of only one key
variable, the mixture fraction. The CMC governing equations for reactive scalars are
expressed in terms of moments of these scalars conditioned on the value of the mixture
3
fraction. First-order singly-conditioned CMC has been validated for a number of practical
combustion flows [1]-[15]. The studies by Mastorakos et al [3], Mastorakos and Bilger
[16], Sreedhara and Lakshmisha [5], provide validations of the second-order CMC
approach in ‘non-premixed’ autoignition. The principal distinction between the
configurations in Refs. [8] and [9] and those in Refs. [3-7] is that, in the latter set of
simulations, the stratified mixture is represented by fully segregated ‘slabs’ of fuel and
oxidizer. Therefore, the statistics obtained by the DNS studies in Refs. [3-7] reflect
conditions of autoignition that, even though initiated off-stoichiometric conditions, will
evolve into combustion in a non-premixed combustion mode (near-stoichiometric).
However, it is not evident that a similar modeling strategy can be applied for nonhomogeneous autoignition such as the configuration studied by Echekki and Chen [8,9] or
the more general condition of autoignition in stratified mixtures.
Multiple conditioning within the context of CMC has been proposed as an alternative to
higher order conditioning in CMC. Cha et al. [17] proposed the modeling of the dissipation
rate as an additional conditioning variable for the prediction of extinction and re-ignition.
Bilger [18] and later Kronenburg [19] proposed the sensible enthalpy as an additional
conditioning variable for problems involving both extinction and re-ignition. These
problems share fundamental similarities with the problem of non-homogeneous
autoignition.
A principal scope of the present work is to study the important statistical features of
autoignition in non-homogeneous mixture using DNS in a simplified mixture and flow
configurations and using a simple chemical model. The conditions considered in the present
study share similar features to the general problem of autoignition in stratified mixtures.
These statistics will demonstrate the inherent advantages and limitations of conditioning
approaches within the context of the CMC model. It is important to note that additional
modeling considerations must be taken into account when more realistic fuels and
associated chemistries are involved in complex practical configurations. The same
constraints apply to the studies mentioned earlier [3-9]. The present study will address only
the features that are in common with more practical fuels and configurations, including the
transient nature of autoignition, the presence of different modes of burning, combustion off
stoichiometric conditions, and the presence of propagation as an integral mechanism for the
autoignition process. The paper is organized as follows: first, the numerical implementation
of the autoignition process is presented; second, the results are presented and discussed;
finally, the results are summarized and conclusions will be made.
4
2. Numerical Implementation and Post-Processing
2.1. Model Configuration
The present autoignition study is based on 3D DNS of autoignition in a non-homogeneous
mixture using a simple chemical model and a random mixture fraction field in decaying
isotropic turbulence. The mixture fraction field is prescribed initially as a random variation
in space of the mixture strength from fuel-lean to fuel-rich conditions. Therefore, this
mixture fraction field is similar to the one adopted by Echekki and Chen [9] and is
fundamentally different from the configuration adopted in Refs. [3-7]. A principal
consequence of this configuration difference is that a distributed mixture strength enables
the formation of expanding ignition fronts beyond the interface of fuel and oxidizer; while
in the ‘slab’ model, propagation occurs mainly along this interface. This difference also
dictates the dominant modes of combustion during the autoignition process and the
subsequent stages of ignition ‘propagation’.
Different mixing scenarios may yield mixture configurations that are closer to the one
adopted here or the ‘slab’ mixture stratification model. The choice of the present problem
configuration is designed to establish a canonical problem for autoignition in nonhomogeneous spatially distributed mixtures. This configuration may be approximately
reproduced past the injection of a dilute fuel spray in a compressed and heating oxidizer;
but, a similar process may also be present at the leading edge of a lifted flame where both
extinction and re-ignition through broken flames (or flame holes) also may generate a
distributed mixture of reactants and/or products [20].
The present problem is also based on a simplified chemical model. More complex fuels will
inherently exhibit more complex behavior, which can manifest in the evolution of dominant
chemistry present during ignition and subsequent propagation, the dependence of the socalled preferred ignition sites [3] on both temperature and mixture preparation, the
proportion of fuel and oxidizer at stoichiometric conditions, the mixture flammability
limits, the role of preferential and differential diffusion involving primarily intermediate
species, on the structure, dynamics and the fate of ignition kernels, as well as the detailed
mechanism associated with extinction or ignition. Clearly, the present study will target a
subset of modeling issues associated with non-homogeneous autoignition. Instead, we will
establish the common statistical features associated with the onset of off-stoichiometric
ignition and the role of subsequent burning modes on the completion of the combustion
process.
2.2. Model Formulation and Numerical Implementation
The DNS code is based on the formulation by Mason and Rutland [21]. It solves the nondimensional transport equations for total mass, momentum, temperature, and reactants’
(fuel and oxidizer) mass fractions. The main flow assumptions used here include ideal gas
behavior, constant density, constant transport coefficients, and negligible radiation, Soret,
and Dufour effects. The non-dimensional governing equations are:
The continuity
5
∂ui
= 0,
∂xi
(1)
The momentum
∂ui
∂u
1 ∂p
1 ∂ 2ui
,
+ uj i = −
+
∂t
∂x j
ρ ∂xi ρ Re ∂x j ∂x j
(2)
The temperature
∂T
∂T
1
∂ 2T
+ ui
=
+ Q ω,
∂t
∂xi ρ PrRe ∂xi ∂xi
(3)
and species (fuel and oxidizer mass fractions)
∂YF ,O
∂YF ,O
∂ 2YF ,O
1
+ ui
=
− ω.
∂t
∂xi
ρ LePrRe ∂xi ∂xi
(4)
In the above equations, the subscripts “F” and “O” refer to the fuel and the oxidizer,
respectively; while the other symbols and subscripts carry their usual meaning. The nondimensional parameters, Re, Pr and Le, correspond to the Reynolds number based on the
characteristic scales used for normalizing the physical quantities in the governing equations,
the Prandtl number and the Lewis number for all species, including the fuel (F), the
oxidizer (O) and the product (P), respectively. As listed in the governing equations, the
Lewis number is the same for all species, and, therefore, it measures the relative rates of
heat diffusion to mass diffusion for these species. The low Mach number approximation is
used to solve the transport equations. A third-order Runge-Kutta method is used to integrate
the system of equations in time. A fully consistent fractional-step method is used for the
solution of the momentum equation and a linearly implicit variation of third-order RungeKutta scheme is used to integrate the energy and species equations. Spatial derivatives are
computed using fifth-order compact finite-difference schemes. The chemistry is
characterized by a single-step, second-order irreversible reaction between a fuel and an
oxidizer of equal molecular weights:
F +O⇒ P .
(5)
The molar stoichiometric ratio for the reaction is unity and the reaction rate is given by the
Arrhenius expression:
⎡ − β (1 − T ) ⎤
⎥,
⎣ 1 − α (1 − T ) ⎦
ω = Da ρ YF Yo exp ⎢
(6)
where α is the dimensionless measure of the temperature rise corresponding to a
stoichiometric mixture of fuel and oxidizer, (Tadia − Tu ) / Tadia . The subscripts, “adia” and
“u”, refer to the adiabatic equilibrium and unburnt mixture conditions, respectively. β is the
non-dimensional activation energy expressed as E aα / R̂Tadia , where Ea, R̂ , and Tadia
6
correspond to the activation energy, the universal gas constant, and the adiabatic flame
temperature, respectively.
The velocity and scalar fields are random, and are prescribed using a 3D von Karman-Pao
spectrum [22] for the turbulent kinetic energy and the mixture fraction fluctuations:
( k / ke )
4/3 ⎞
⎛ 3
E (k ) ∼
exp ⎜ − α ( k / kd ) ⎟ ,
2 17 / 6
⎝ 2
⎠
⎡1 + ( k / ke ) ⎤
⎣
⎦
4
(7)
where k is a wave number; ke and kd are the wave numbers associated with the most
energetic eddies and the dissipative eddies, respectively. The two wave numbers reflect the
characteristic scales of the turbulence and the mixture fraction fields. The proportional
constant between the spectral value and the shape reflect the magnitudes of fluctuations of
the velocity and mixture fraction fields. While, the mean velocity components in the
computational domain are zero, a mean value of the mixture fraction is imposed in addition
to the fluctuating mixture field to yield an average value of 0.5.
The initial spatial distribution of the mixture fraction varies from pure fuel to pure oxidizer
over a characteristic length scale, LZ , which is defined as:
LZ
∫
≡
∞
0
QZ ( r )dr
QZ (0)
, where QZ ( r ) ≡ Z ′′( x ) Z ′′( x + r ) .
(8)
In this expression, QZ ( r ) is the two-point correlation function for the local mixture fraction
fluctuation at a distance r. The length scale, LZ, measures the characteristic scale over which
the mixture evolves from fuel to oxidizer, and vice versa. The resulting initial scalar field
(see Fig. 1) is characterized by fuel-centered and oxidizer-centered ‘blobs’ of fluid between
which the mixture fraction changes.
The oxidizer is preheated relative to the fuel temperature to provide a mechanism for the
autoignition of the mixture. Based on the initial random field for the mixture fraction, Z, the
initial fields for the fuel and oxidizer mass fraction and the temperature are prescribed as
follows:
Yf = Z,
Yo = 1 − Z ,
(9)
T = T f + (1 − Z ) (To − T f ) .
2.3. Run Conditions
Four separate simulations are carried out, which correspond to the same random initial
mixture fraction, two different turbulence conditions, and three different Lewis numbers for
the higher turbulence case. The choice of the different simulations is made to address two
pertinent questions related to autoignition in non-homogeneous mixtures. The first is related
to the role of mixing in the autoignition process. Here, mixing is expected to play two key
and competing roles. The first role is associated with the rate of increase of interfaces
7
between reactants and products (in premixed burning mode) and fuel and oxidizer (nonpremixed burning mode). The second role is associated with an increase in the rate of
dissipation, at least initially, when nascent autoignition kernels are trying to form and grow.
Echekki and Chen [8,9] as well as Mastorakos et al [3] have already discussed some of the
consequences of high-dissipation rates in inhibiting autoignition.
The second question is related to the role of preferential diffusion of heat and mass on the
autoignition process. This is achieved by using three different Lewis numbers for the
species, which include 0.5, 1.0 and 2. The different values are expected to accentuate or
dampen the effects discussed above related to the rate of dissipation and the rate of mixing
for the various Lewis number conditions. The variation in the Lewis number is carried out
in the higher turbulence simulation case.
Fig. 1 shows contours of the initial mixture fraction field. Periodic boundary conditions are
imposed in all directions. The grid resolution of 1293 grid points was chosen to sufficiently
resolve the flame structure and the range of flow and scalar scales considered and over the
prescribed non-dimensional domain size of 3.63. This non-dimensionalization is prescribed
by a characteristic Reynolds number of 200, Re (see Eqs. (1)-(4)), which is defined as
Re = U ∞ L∞ ν , where U ∞ and L∞ are the dimensional reference velocity and length scales,
and ν is the kinematic viscosity. The time step, which is set by numerical accuracy and
stability requirements, is 0.001. The Taylor scale based Reynolds number is 405 for the
high turbulence cases and 100 for the low turbulence case. The reaction rate and the heat
release rate parameters, Da, α and β, are 200, 0.75 and 2, respectively. The initial
turbulence intensity, u′′u′′ ( = v′′v′′ = w′′w′′ ) , is 3 for the high turbulence cases and 0.74
for the low turbulence case. The present calculations are implemented with a value of
LZ Lx = 0.1, where Lx is the domain size, 3.6. This value is the same for all cases
considered, since the initial mixture fraction fields are identical. The resulting
homogeneous field with the prescribed value of LZ, results in an average of 6 to 7 of these
structures in a given direction in the computational domain.
2.4. Post-Processing for Conditional Moments
In the present section, we present the singly-conditioned first and second moments of
scalars considered in this study. The conditioning is based on the mixture fraction, which is
the primary variable adopted in the modeling of autoignition problems [5,16]. The mixture
fraction is evaluated at any time in terms of the mass fractions of the fuel and the oxidizer,
based on the equal Lewis number condition for the fuel, the oxidizer and the product:
Z = (1 + Y f − Yo ) 2.
(10)
These computed conditional statistics are of relevance to the first and second order CMC
approach; however, observations may be extended to broader implications to turbulent
combustion models of non-homogeneous autoignition. The first conditional moments
include the conditional means for the fuel and the oxidizer mass fractions, Y f Z
and Yo Z , the temperature, T Z , the dissipation rate,
χ Z , and the reaction rate,
ω Z . Here, Z, is the mixture fraction, which is the conditioning variable; the conditioning
8
operation is expressed in terms of the operator “ < ⋅ | Z > ” for the scalars. The dissipation
rate is expressed as χ ≡ 2 D ( ∇Z ⋅ ∇Z ) , where D is the mass diffusivity associated with the
mixture fraction, which is in this problem identical to the mass diffusivity of the fuel and
the oxidizer.
The second conditional moments include co-variances and the variances of the species mass
fractions and temperature with the dissipation rate. For convenience, we present the covariances in terms of RMS values,
Y f′′Y f′′ Z
1/ 2
, Yo′′Yo′′ Z
1/ 2
, and
T ′′T ′′ Z
1/ 2
. The
variances are expressed in terms of correlation functions, RTχ, RYfχ, and RYoχ, which are
expressed as:
Rθχ =
θ ′′χ ′′ | Z
θ ′′ | Z
2
1
2
χ ′′ | Z
2
1
2
(11)
,
where θ represents any one of the reactive scalars, T, Yf, and Yo. The variances and covariances are prominent closure terms in the first and second order CMC equations
[16,23,24]. The DNS data was used to compute the conditional moments described above.
The calculations are based on 30 bins of the mixture fraction values ranging between 0 and
1. The statistics are evaluated at different snapshots in time, and cover the process of
autoignition from the formation of autoignition kernels to the completion of the combustion
process.
9
3. Results and Discussions
In the following discussion, we present results of the evolution of conditional statistics for
the passive and reactive scalars. The process of autoignition in non-homogeneous mixtures
involves the formation of autoignition at discrete kernels; and the combustion process
evolves eventually due to the propagation of these kernels through the stratified mixture
[8,9]. The temporal evolution of the process is represented in terms of the non-dimensional
time. The overall time for the low turbulence case is 6.0 and 2.8 for the high turbulence
cases. The computations were carried out on a Linux workstation using a single Xeon
processor of 3.2 MHz. The CPU time ranged from 62 hours for the low-turbulence case to
120 hours for the high-turbulence cases.
3.1. Conditional Means of Scalars
3.1.1. Conditional Means of the Scalar Dissipation Rate
Fig. 2 shows the temporal evolution of the conditional means of the scalar dissipation rate
profiles in the mixture fraction space for the different cases considered. The relatively high
conditions of dissipation rates at lean and rich conditions are consistent with the presence of
local peaks and minima in the mixture fraction. Both turbulence intensity and Lewis number
affect the conditional means of χ . Relatively high initial turbulence intensities introduce
steep initial scalar gradients, which cause the conditional means of χ to increase in the high
turbulence cases (Fig. 2 (b), (c), and (d)) at earlier times. On the other hand, they gradually
decrease in time in the low turbulence case (Fig. 2 (a)). However, at later times, the mixture
fraction approaches a narrow range around the stoichiometric value, which also corresponds
to conditions of a homogeneous mixture. This condition is approached at a faster rate for the
high-turbulence conditions (Fig. 2 (b), (c), and (d)) compared to the low turbulence case
(Fig. 2 (a)). As the mixture becomes more homogeneous, conditional means of χ also
decrease in time regardless of the turbulence conditions. Turbulence effects on the mixture
fraction range are also present in the conditional means of the thermo-chemical scalars and
in the variances and co-variances of these scalars as seen in later sections. Another important
distinction between the low-turbulence conditions and the high-turbulence conditions can be
observed in the evolution of the peaks of the dissipation rates during earlier times. Fig. 2
shows a steady decay in the dissipation rate field for the low turbulence case; in this case, the
role of large-scale mixing is not significant compared to the role of molecular diffusion. In
the high-turbulence cases, (Fig. 2 (b), (c), and (d)), peaks of the dissipation rates increase
initially beyond the pure mixing values before they finally decay. By the common time of
1.6, dissipation rates associated with the high-turbulence conditions are clearly lower than
the corresponding values at low-turbulence conditions.
Lewis number effects also play an important role in the evolution of the mixture fraction
and the dissipation rate fields. Fig. 2 shows that the initial (pure mixing) profiles of the
dissipation rate for the two turbulence conditions at the same Lewis number of unity (Fig. 2
(a) and Fig. 2 (c)) are identical, because the mixture fraction fields of all conditions
considered, including the non-unity Lewis number cases, are also initially identical. The
initial dissipation rate profiles corresponding to Le equal to 0.5 and 2 reflect the different
values of mass diffusivity, such that the case of Le equal to 0.5 (resp. 2) exhibits a factor of
10
2 higher (resp. lower) values of the dissipation rate, compared to the unity Lewis number
conditions. The relative magnitudes of the dissipation rates are reversed at later times; the
higher initial dissipation conditions achieve a more homogeneous mixture at a faster rate
than the lower initial dissipation conditions. Turbulence conditions, as discussed above, and
Lewis number effects, as discussed here, modulate the evolution of the dissipation rate
field; more importantly, this evolution depends on the coupling between large-scale mixing
and molecular mixing.
3.1.2. Conditional Means of the Temperature and the Mass Fractions
Figs. 3 through 5 show the temporal evolution of the conditional means of the temperature,
the fuel mass fraction, and the oxidizer mass fraction profiles, respectively, in the mixture
fraction space for the different cases considered. Note that at t = 0, the profile corresponds
to pure mixing conditions, which also corresponds to the initial state of the mixture.
Because of the initialization stated in Eq. (9), these profiles are linear. They have negative
slopes for the fuel mass fraction and temperature and a positive slope for the oxidizer mass
fraction. Moreover, because the initial mixture distribution is identical for all conditions
considered, the temperature and fuel and oxidizer mass fraction profiles are identical as
well.
The onset of combustion is characterized by the departure of the temperature and the fuel
and oxidizer mass fractions from the linear profile in the mixture fraction space. Because
the oxidizer is preheated, the first departure from pure mixing profiles occurs at fuel lean
conditions, i.e. lower values of the mixture fraction. While leaner mixtures reach their
corresponding adiabatic temperature conditions, the next layers of richer mixtures are
ignited, exhibiting an increasing trend for the temperature in time. The corresponding
values of the fuel and the oxidizer mixture fraction indicate a depletion of the fuel for
mixture fractions below the stoichiometric value of 0.5 and a depletion of the oxidizer at
mixture fractions higher than the stoichiometric value. In both conditions, the deficient
reactant is depleted; while the excess reactants occupy a new linear curve in the mixture
fraction space corresponding to the mixing of reactants with products in the absence of
chemistry. The trend in time clearly indicates a transition from predominantly lean burning
at earlier times to rich burning at later times. The highest temperature is achieved at
stoichiometric conditions. This transition occurs earlier at higher turbulence conditions, as
shown through profiles of conditional means of temperature and fuel and oxidizer mass
fractions at unity Lewis numbers.
A comparison between conditional mean profiles of temperature at high-turbulence
conditions between the different Lewis number cases show that the case with the lowest
Lewis number reaches the highest peak in temperature. This position of this peak in phase
space is also different for the different Lewis number cases considered. For Le equal to 0.5,
this peak is present at fuel lean conditions; while, at Le equal to 1 and 2, this peak is at
stoichiometric conditions. The mechanism, which governs the magnitude and position of
temperature peaks lies in the competition of heat and mass transfer into or out of ignition
kernels. At Lewis numbers below unity, heat diffuses at a lower rate than mass.
Consequently, autoignition kernels remain relatively better shielded from heat loss, than at
higher Lewis numbers. Of course, this remains true before the mixture becomes relatively
homogeneous due to turbulent mixing and molecular diffusion. Because of the important
feedback role of temperature on heat release, autoignition kernels at lower Lewis numbers
tend to be hotter. The same mechanism prevents the preheat of adjacent mixture layers to
11
the ignition kernels, and although the mixture composition is in principle optimum at
stoichiometric conditions, preheat effects from leaner kernels are significantly reduced at
Lewis numbers below unity.
Lewis numbers effects on the evolution of the mass fraction profiles of the fuel and the
oxidizer are also present; but, they are much less pronounced than for temperatures. During
intermediate stages of autoignition, shown here at times of 0.4 and 0.8, the minimum values
of the oxidizer mass fraction are lower at lower Lewis numbers. These values reflect
primarily the role of chemical reaction, as the oxidizer is transported similarly to the
mixture fraction. Beyond the above intermediate stages of autoignition, the higher Lewis
number cases exhibit a mildly higher extent of oxidizer consumption. These trends are
consistent with the evolution of the rates of reaction as discussed below. Finally, and as
shown from dissipation rate profiles, the range of mixture fraction represented for the same
time, but at high-turbulence conditions is broader as the Lewis number is increased. This
range indicates the role of molecular diffusion of mass in generating a more homogeneous
mixture in time.
3.1.3. Conditional Means of the Reaction Rate
Fig. 6 shows the temporal evolution of the conditional means of the reaction rate profiles
for the different cases considered. The transition in combustion from lean to rich conditions
is clearly illustrated in the conditional mean profiles of the reaction rate for the low
turbulence case shown in Fig. 6 (a1) and (a2). An initially single peak of the reaction rate
‘propagates’ from lean mixture fractions at early times to richer conditions at later times
and the transitions occur at Z equal to 0.5. In addition to flame propagation, three different
burning modes are observed for the low turbulence case, and they are clearly shown in Fig.
6 (a1) and (a2). Those burning modes are lean premixed, rich premixed, and non-premixed
combustion. The premixed flames are indicated by the shifting peaks in the conditional
means of the reaction rate. On the other hand, the non-premixed flame is characterized by a
single fixed peak of the reaction rate around stoichiometric conditions. The peak is visible
at later times as shown in Fig. 6 (a2). The diffusion and rich premixed flames are an order
of magnitude lower than the lean premixed flames. The reduced rates of reaction also
results in the relatively long periods of time associated with fuel-rich combustion in
contrast with fuel-lean combustion. On the other hand, only one burning mode exists in the
high turbulence cases shown in Fig. 6 (b), (c), and (d).
At remaining higher turbulence conditions, the higher rates of mixing result in a different
scenario for the prevailing modes of combustion. Combustion occurs primarily in lean to
stoichiometric premixed modes. A secondary peak in the conditional mean of the reaction
rate, which indicates combustion in diffusion flame mode, is not present at later times.
Instead, the last stages of the completion of the combustion process are more likely
governed by significantly low rates of homogeneous chemistry at stoichiometric conditions.
While the dominant modes of combustion are the same for the different Lewis numbers
considered, important differences in the rates of chemistry can be seen in the conditional
means of the reaction rate. The lower Lewis number case exhibits the highest rates of
reaction during intermediate stages of the autoignition process, as clearly shown at times
0.12, 0.4, and 0.8. These trends are consistent with the observations made based on Figs. 3
to 5, and they reflect the competition between heat and mass transfer within ignition
kernels. At lower Lewis numbers, these kernels are better shielded against the thermal
dissipation of heat. In practical fuels, both preheating and stoichiometry may play an
12
important role in the rate of reaction and heat release. Therefore, the diffusion of reactants
into ignition kernels may also be important. More importantly, the diffusion of radical
species into these kernels was shown to play an important role in the fate of these kernels
[9]. It is also important to note that at later stages of the autoignition process, Lewis number
effects are negligible as the mixture becomes more homogeneous.
3.2. Conditional Co-Variances
Fig. 7 shows the temporal evolution of the conditional RMS of the scalar dissipation rate
profiles in the mixture fraction space for the different cases considered. By comparing Fig.
7 with Fig. 2, the turbulence effects and the Lewis number effects that we have observed
from the conditional means of χ are also present in the conditional RMS of χ . Moreover,
the magnitude of the conditional RMS of χ is nearly twice the magnitude of the conditional
means of χ , reflecting the strong spatial variations in the mixture fraction field.
Fig. 8 show the temporal evolution of the conditional RMS profiles of the temperature for
the different cases considered. Because the conditional RMS of the mass fractions follows
the same trend as that of the temperature, the conditional RMS of the mass fraction profiles
are not included here; similar conclusions can be drawn for the mass fractions. For the low
turbulence case shown in Fig. 8 (a), the peaks of the conditional RMS of temperature shift
from lean mixture conditions (i.e. low mixture fraction values) to richer mixture conditions
as autoignition fronts propagate from fuel-lean to richer mixtures in both physical and
phase spaces. The fluctuations of temperature during rich premixed burning are larger
compared to lean premixed burning. For the high turbulence cases, the peaks of the
conditional RMS of temperature exist at the lean premixed burning side except for time
equal to 1.6 for cases with Lewis numbers of 0.5 and 1.0 and for time equal to 2 for the case
with Lewis number equal to 2. At those times, the peaks appear at the rich premixed side
and rapidly diminish near the stoichiometric mixture conditions. This is again due to the
fact the complete burning is achieved earlier before any transition to burning at fuel-rich
conditions occurs.
Moreover, the overall magnitude of temperature fluctuations shown in the temporal
snapshots is a significant fraction of the mean temperature range. Therefore, temperature
fluctuations during the autoignition process can not be ignored in any model of autoignition
in non-homogeneous mixtures given the strong non-linear dependence of chemical source
terms on temperature. In the low turbulence case, the peak at the stoichiometric mixture
fraction is formed at the later stage of burning due to the burning of excess fuel and
oxidizer from either side of the stoichiometric mixture fraction in the diffusion flame mode.
Turbulence and Lewis number effects that we have addressed earlier are also present here.
Note that the magnitude of the fluctuations of the temperature is smaller for the case of
Lewis number equal to 2 because the magnitude of the reaction rate is smaller in this case,
which is due to the effects of Lewis number.
3.3. Conditional Variances
In this section, we investigate the evolution of the correlation of reactive scalars with the
dissipation rate. To understand this correlation, we briefly discuss this correlation for two
reference problems:
13
(1) The first problem corresponds to pure mixing without chemistry in a configuration
that corresponds exactly to the initial mixture fraction and temperature fields
presented here. Because of the initial random field, there is no correlation between
the mixture fraction and the dissipation rate, except perhaps at the local minima or
maxima of the mixture fraction due to the initial conditions. Accordingly, there is no
correlation between the temperature and the dissipation rate, because under pure
mixing the evolution of the temperature is determined by the evolution of the
mixture fraction field. Therefore, in the pure mixing problem, the correlation, RTχ, is
essentially zero over the bulk of the mixture fraction range. A similar correlation
can be constructed for other reactive scalars, such as the fuel and oxidizer mass
fractions with the dissipation rate.
(2) The second problem is the case of autoignition in fully-segregated mixtures, such as
the cases obtained using the slab model for the mixture fraction field (Mastorakos et
al [3]). For this problem, a correlation between the reaction rate and the dissipation
rate (Fig. 11 in Ref. [16]) and the temperature and the dissipation rate (Fig. 6 in
Sreedhara and Lakshmisha [4]) show a positive correlation with a local peak for rich
mixtures, and negative correlations with a local minimum for lean mixtures. The
overall shape of the correlations remains essentially the same over the course of the
evolution of the autoignition process, and the peak and minimum values of the
correlations do not shift significantly in time.
Correlations of the dissipation rate with other reactive scalars, such as the fuel or oxidizer
mass fraction, in both reference problems are expected to be similar to the temperaturedissipation correlations; although, the signs of these correlations may be inverted as
discussed below. Both correlations will be compared to the findings based on the present
configuration. We will also attempt to explain the mechanisms by which positive and
negative correlations between temperature and dissipation rate are generated.
In contrast to the reference problems, the conditional correlation between the temperature
and the scalar dissipation rate in the present study exhibits different trends. Fig. 9 shows the
conditional profiles of the mean of the dissipation rate-temperature correlation, RTχ, for the
different cases considered. Dissipation rate-temperature correlation profiles for the low
turbulence case are presented separately for earlier and later times (Figs. 9 (a1) and 9 (a2))
for clarity of presentation. For the high-turbulence cases, the correlation profiles are shown
for the entire time range of the simulation. The initial profiles of the correlation, RTχ, for all
cases considered are consistent with the first reference problem of pure mixing, and are
determined primarily by the initialization of both the temperature field and the gradients of
the mixture fraction. As the mixture evolves during to mixing and chemistry, the correlation
evolves closer in shape to the profiles obtained in fully segregated mixtures (i.e. reference
problem 2) as obtained by Mastorakos and Bilger [16]. The correlations exhibit a leading
peak (positively correlated) and a trailing minimum (negatively correlated) around
conditions of maximum chemical activity. In all the cases considered for the present
problem, the peaks of RTχ shift from lean to rich conditions, increase in magnitude initially,
and eventually diminish when the reaction subsides. Once the combustion process subsides,
the pure mixing statistics are recovered with zero correlations in the wake of the burning
conditions. The pure mixing statistics is reflected by the behavior of these correlations at
lean mixture conditions and in the gap separating the stoichiometric conditions and the rich
burning conditions at later times in the simulations. In summary, the peaks present in the
14
positive and negative correlations track closely the active combustion regions in the mixture
fraction space. In contrast with the present observations, these peaks of the correlation, RTχ ,
remain relatively stationary in fully segregated mixtures (Mastorakos and Bilger [16]).
Turbulence intensity also has important effects on the evolution of the dissipation ratetemperature correlation profiles; the low turbulence case exhibits a stronger correlation than
the high turbulence cases. By comparing Fig. 9 (a) with Fig. 9 (b), (c), and (d), we can see
that the transition of the correlation profiles is continuous and the correlation profiles have a
clear shape in the low turbulence case. This again is due to the competing rates of turbulent
mixing and chemistry, which are affected significantly by turbulence conditions. In the low
turbulence case, the turbulent mixing rate is slow and the correlation between the
temperature and the scalar dissipation rate is higher; on the other hand, in the high
turbulence cases, the turbulent mixing is very high, thus, the correlation is much weaker.
This is also consistent with what we have observed from the first reference problem.
Moreover, the Lewis number effects on the mixture fraction range addressed earlier are also
present in the correlation terms.
Figs. 10 and 11 show the correlation coefficients of the fuel and the oxidizer mass fraction
with the scalar dissipation rate, respectively, for the different cases considered. These
correlations are opposite in sign to that of the temperature-scalar dissipation correlation; the
fuel and the oxidizer are depleted while the temperature is increased. Moreover, they also
feature a shift of the peaks from lean to rich conditions. Differences between profiles of the
reactants mass fraction correlations with the dissipation rate and those associated with
temperature (Fig. 9) are accentuated at non-unity Lewis number conditions. This difference
presents additional challenges for the modeling of scalar-dissipation variances within the
context of singly-conditioned CMC when more complex chemical models are considered
with important preferential and differential diffusion effects.
3.4. The Progress Variable as a Second Conditioning Variable
The transient evolution of singly-conditioned moments underscores the importance of
adequately predicting the transitions in burning modes from lean premixed flames to rich
premixed flames and combustion in non-premixed mode; this transient evolution is also
characterized by propagation in both physical and phase spaces, and is found in both first
and second order conditional statistics of passive and reactive scalars. Therefore, a critical
modeling element of tracking the transition in combustion mode is still needed with higher
order conditioning. As stated earlier, an alternative strategy to higher order conditioning is
multiple conditioning. Would the choice of a second conditioning variable, which measures
the progress of transition in combustion modes, address the principal deficiency, which
cannot be addressed with higher order modeling? The results presented so far suggest the
following:
(1) For the premixed modes, the mixture exhibits three potential states: an unburned, a
burned and a transition mixture states. Both burned and unburned states are
consistent with pure mixing statistics at conditions of the burned and unburned
mixture, respectively; while the transition state corresponds to active burning, and
covers a range of mixture fractions. The extent of this range is primarily governed
by the extent of stratification of the mixture and the dissipation rate field, which
may allow for conditions of burning to occur at different mixture conditions at the
same time.
15
(2) A secondary diffusion branch is present once the transition to rich premixed burning
occurs. This diffusion branch is essentially similar in nature to the burning mode in
fully-segregated mixtures [3-7] (i.e. the non-premixed autoignition mode). Burning
in the diffusion flame is relatively slow and insignificant relative to the burning in
premixed modes.
(3) A reaction progress variable (e.g. sensible enthalpy or reduced temperature [18,19],
products and reactants’ composition) can provide a measure for the evolution of the
transition zone, and therefore a mechanism for tracking the ‘propagation’ of burning
in mixture fraction space. A reaction progress variable can potentially distinguish
the unburned and burned mixture states as well as the transition states.
In this section, we provide a preliminary assessment of the addition of a second
conditioning variable, which in the present problem is the progress variable, c. The progress
variable is expressed as:
c≡
T − Tu ( Z )
.
Tb ( Z ) − Tu ( Z )
(12)
In this expression, c, represents a temperature normalized using the unburnt and burned
(equilibrium) temperatures, Tu and Tb, which are prescribed for the same mixture fraction.
Therefore, the two temperatures are prescribed a priori and correspond to the asymptotic
values at a given mixture fraction between pure mixing (initial conditions) and final burned
mixtures (the maximum temperatures achieved for a given Z). The value of c varies
between 0 and 1 for any given mixture fraction during the autoignition process.
Figs. 12 and 13 show scatter plots of the fuel mass fraction and the reaction rate as a
function of the reaction progress variable, c, at different times of the autoignition process
and for a narrow range of the mixture fractions between 0.495 and 0.505. This range
corresponds to near stoichiometric conditions. Observations made for this range are
representative of other observations made at other ranges. The presence of extensive scatter
in the plots may indicate that double-conditioning based on the mixture fraction and the
progress variable may not be adequate to represent autoignition in non-homogeneous
mixtures, or that higher order conditioning may be required in addition to double
conditioning. Also note that the fuel mass fraction and the reaction rate shown in Figs. 12
and 13 are already singly-conditioned upon values of the mixture fraction, as indicated by
the narrow range for this variable shown. The extent of scatter of the reactive scalars as a
function of the progress variable provides an indication of the value of second conditioning
using the progress variable, c.
Both low-turbulence and high-turbulence unity Lewis number conditions exhibit the least
scatter in fuel mass fraction and reaction rate data. A superposition of the correlation of fuel
mass fraction and reaction rate data as a function of the progress variable, c, shows that the
correlations are independent of the turbulence intensity. However, this low scatter is more
an indication of the correlation of temperature with other reactive scalars at stoichiometric
conditions than a confirmation of the validity of double conditioning. Indeed, a single
conserved scalar can be constructed at unity Lewis number based on a linear combination of
the temperature and the fuel or oxidizer mass fractions with trivial initial conditions, based
on Eq. (9) to yield a deterministic correlation of temperature with other reactive scalars.
16
This scalar results in a linear profile for the reactants’ mass fractions in terms of
temperature. Therefore, the assessment of the validity of a second conditioning variable
may be made using the non-unity Lewis number cases considered.
For the same range of mixture fractions, the non-unity Lewis number data exhibits a greater
scatter in fuel mass fraction and reaction rate profiles. Under non-unity Lewis number
conditions, the mixture fraction and the temperature fields evolve differently because of
different diffusivities of mass and heat. The case of Lewis number equal to 0.5 exhibits
more scatter in the reactive scalars’ data, especially in the reaction rate profiles. This case is
associated with the highest rates of dissipation and their fluctuations during intermediate
stages of the autoignition process as shown in Figs. 2 and 7; but, at the time the highest
scatter is observed, the rates of dissipation and their fluctuations have already decreased
below the corresponding values at the same time at Lewis numbers of 1 and 2. The scatter
plots of the fuel mass fraction and the reaction rate show the highest scatter during these
stages, as shown at times of 0.8 and 1.2. The figure also shows that a broader range of the
progress variable, c, is obtained for the Lewis number case of 0.5 relative to the Lewis
number case of 2.
In all conditions considered, including low and high turbulence conditions and unity and
non-unity Lewis number effects, and despite the presence of scatter in mass fraction and
reaction rate data, Figs. 12 and 13 show a considerable overlap of the scatter points
corresponding to different times of the autoignition process. A notable exception here is the
case of Lewis number equal to 0.5 when dissipation rates and their fluctuations are high
during intermediate stages of the autoignition process. During these stages, temperature-fuel
mass fraction correlations are significantly reduced, and this process is further amplified by
the reaction rate term in the governing equations. Nonetheless, and despite the presence of
scatter, the results clearly show that a coherent correlation for the singly conditioned (based
on the mixture fraction) reactive scalars at different times as a function of the progress
variable, c. It is important to note that similar features of the correlations of reactive scalars
with the progress variable are observed for different ranges of the mixture fraction; although
these results are not shown here for brevity. An important overlap between different time
ranges suggests that the progress variable may present a reasonable second conditioning
variable to describe the process of autoignition. The important challenge is to model
pertinent variances
The magnitude of the reaction rate and its shape are governed by the relative composition
of the fuel and oxidizer and the preheating effect. The shape of the reaction rate is
determined by the pre-exponential term represented by the reactants’ mass fractions and the
exponential term, which contains the explicit contribution of the temperature. The rates of
reaction at earlier (c = 0) and later times (c =1) approach zero because either the
temperature is low or the reactants are depleted, respectively. Any scatter in the correlation
of temperature (or the reaction progress variable, c) and the fuel and oxidizer mass fractions
is further amplified in the temperature-reaction correlation. To further evaluate the validity
of a second conditioning variable, correlations of the progress variable with other reactive
scalars must be carried out for other ranges of the mixture fraction as done around the
stoichiometric conditions above.
The present results suggest that the use of a second conditioning variable that measures the
progress of completion of reaction may be a reasonable for the prediction of autoignition in
non-homogeneous mixtures. The transport of an additional variable, for example within the
17
context of double conditioning in CMC, also imposes additional challenges for modeling.
For example, in CMC closure is now required for three distinct dissipation rates, including
the mixture fraction dissipation rate, the progress variable or sensible enthalpy dissipation
rate, and the cross-gradient term of mixture fraction and progress variable [19]. A detailed
discussion of the statistics associated with these different dissipation rates is beyond the
scope of the present paper, although some results are already presented here for the
statistics of the mixture fraction dissipation rate. Nonetheless, the statistics of these
dissipation rates will inevitably evolve in time, as shown for the case of the mixture fraction
dissipation rate, and are strongly dependent on initial conditions as well as the coupling
between large-scale mixture and molecular diffusion. For the dissipation rates involving the
progress variable as a whole or in terms of cross-gradients with the mixture fraction, the
evolution of combustion is also a critical component in establishing closure models for
these dissipation rates.
In addition to the modeling of reactive scalars’ transport in phase space, the use of a second
conditioning variable also involves the solution of a transport equation for this variable in
physical space. A model that includes the progress variable in addition to the mixture
fraction as measures of the progress of chemistry during the autoignition process has been
proposed recently by Tap et al. [25]. The model is presented in the context of the flamelet
approach. It represents the mean heat release rate in terms of the product of a generalized
flame surface density and a generalized surface average of the reaction rate. A principal
assumption of the model is that the surface average of the reaction rate can be uniquely
expressed in terms of a generalized progress variable, which can be evaluated using laminar
autoignition data. In principle, a similar extension of this model can be adopted to the
present study by using instead turbulent autoignition data, therefore factoring the role of
mixture fraction and progress variable fluctuations in the model.
18
4. Conclusions
We have studied the autoignition process in non-homogeneous mixtures in decaying
isotropic turbulence using DNS and simple chemistry. The mixture is prescribed with a
random distribution of mixture strengths varying from fuel-lean to fuel-rich conditions. The
configuration enables a wide range of autoignition modes, which are governed by the
competition of large-scale turbulent mixing, chemistry and molecular diffusion. Conditional
statistics of reactive scalars are constructed to study the effect of turbulent mixing and
molecular diffusion on the dominant modes of combustion and implications of these modes
in models.
Singly-conditioned moments for reactive scalars show a clear shift from lean burning
following autoignition to richer burning at later times. For the high-turbulence cases, the
rate of mixing was relatively fast compared to chemistry, and the dominant combustion
mode is primarily lean-premixed combustion. At low-turbulence conditions, three modes of
combustion have been identified, which include lean and rich premixed flames and nonpremixed flames. These latter flames are formed in the wake of the lean premixed fronts,
and burn excess fuel and oxidizer from rich and lean premixed fronts.
Preferential diffusion effects have been studied for the high-turbulence conditions. During
intermediate stages of the autoignition process, the case of Lewis number below unity
exhibits higher temperature, corresponding primarily to lean burning. While, at unity and
above unity Lewis numbers, these peak temperatures are lower and are found at nearstoichiometric conditions. The governing mechanism for the temperature profiles, as well
as fuel and mass fraction profiles and reaction rates, may be attributed primarily to the fact
that when heat diffusivity is lower than mass diffusivity autoignition kernels are relatively
shielded from heat dissipation. This effect is less pronounced at later times of autoignition
process when the mixture becomes more homogeneous.
Conditional profiles of the dissipation rate illustrate the complex coupling between largescale mixing and molecular diffusion. Challenges to modeling the dissipation rate are
expected to be greater when cross-gradient terms, involving a second conditioning variable,
are to be modeled. These challenges are related primarily to the addition of a second
conditioning variable. In the present study, the progress variable is evaluated as a second
conditioning variable, within the context of the CMC approach. The results show that the
addition of the progress variable as a second conditioning variable is reasonable, since it
collapses data from different stages of the autoignition process. The lower Lewis number
case reveals the need for additional refinements to double conditioning under strong finiterate chemistry effects.
Finally, we propose the problem of non-homogeneous autoignition as a critical test problem
for turbulent combustion models. The autoignition process is inherently transient and may
involve important transitions during the evolution of the process in dominant chemistries
and burning modes. Moreover, the coupling between chemistry, molecular transport, and
turbulent transport is critical to the fate of ignition kernels and the rate of their evolution in
physical and phase spaces.
19
Acknowledgements
This work has been supported by the Air Force Office of Scientific Research under a
research grant F49620-03-1-0023, monitored by Dr. Julian Tishkoff. We are also grateful to
Prof. Christopher Rutland and to Dr. Scott Mason who kindly provided their DNS code on
which the present results are based.
20
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5. S. Sreedhara, K.N. Lakshmisha, Proc. Combust. Inst. 29 (2002) 2069-2077.
6. S. Sreedhara, K.N. Lakshmisha, Proc. Combust. Inst. 29 (2002) 2051-2059.
7. R. Hilbert, D. Thévenin, Combust. Flame 128 (2002) 22-37.
8. T. Echekki, J.H. Chen, Proc. Combust. Inst. 29 (2002) 2061.
9. T. Echekki, J.H. Chen, Combust. Flame 134 (2003) 169.
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11. M. Fairweather, R.M. Woolley, Combust. Flame 133 (2003) 393.
12. N.S.A. Smith, R.W. Bilger, C.D. Carter, R.S. Barlow, J.-Y. Chen, Combust. Sci.
Technol. 105 (1995) 307.
13. S.H. Kim, K.Y. Huh, Combust. Flame 120 (2000) 75.
14. A. Kronenburg, R.W. Bilger, Combust. Sci. Technol. 166 (2001) 175.
15. A. Kronenburg, R.W. Bilger, J.H. Kent, Twenty-seventh Symposium (International) on
Combustion (The Combustion Institute, Pittsburgh, 1998) 1097.
16. E. Mastorakos, R.W. Bilger, Phys. Fluids 10 (1998) 1246.
17. C. Cha, G. Kosály, H. Pitsch, Phys. Fluids 13 (2001) 3824.
18. R.W. Bilger, in Aerothermodynamics in Combustors, edited by R.S.L. Lee, J.H.
Whitelaw, and T.S. Wang (Springer-Verlag, Heidelberg, 1992).
19. A. Kronenburg, Phys. Fluids 16 (2004) 2640.
20. A. Bourlioux, B. Cuenot, T. Poinsot, Combust. Flame 120 (2000) 143.
21. S.D. Mason, C.J. Rutland, Proc. Combust. Inst. 28 (2000) 505.
22. J.O. Hinze, Turbulence, MacGraw Hill, 1975.
23. N. Swaminathan, R.W. Bilger, Phys. Fluids 11 (1999) 2679.
24. J.D. Li, R.W. Bilger, Phys. Fluids 5 (1993) 3255.
25. F.A. Tap, R. Hilbert, D. Thévenin, D. Veynante, Combust. Theo. Model. 8 (2004) 165.
21
Figure Captions
Figure 1.
Initial Mixture Fraction Field.
Figure 2.
Conditional means of the scalar dissipation rate for different cases
at different stages of the kernels’ evolution. (a) Low turbulence
case with unity Lewis number, (b) High turbulence case with
Lewis number equal to 0.5, (c) High turbulence case with unity
Lewis number, (d) High turbulence case with Lewis number equal
to 2.
Figure 3.
Conditional means of the temperature for different cases at
different stages of the kernels’ evolution. (a) Low turbulence case
with unity Lewis number, (b) High turbulence case with Lewis
number equal to 0.5, (c) High turbulence case with unity Lewis
number, (d) High turbulence case with Lewis number equal to 2.
Figure 4.
Conditional means of the fuel mass fraction for different cases at
different stages of the kernels’ evolution. (a) Low turbulence case
with unity Lewis number, (b) High turbulence case with Lewis
number equal to 0.5, (c) High turbulence case with unity Lewis
number, (d) High turbulence case with Lewis number equal to 2.
Figure 5.
Conditional means of the oxidizer mass fraction for different cases
at different stages of the kernels’ evolution. (a) Low turbulence
case with unity Lewis number, (b) High turbulence case with
Lewis number equal to 0.5, (c) High turbulence case with unity
Lewis number, (d) High turbulence case with Lewis number equal
to 2.
Figure 6.
Conditional means of the reaction rate for different cases at
different stages of the kernels’ evolution. (a) Low turbulence case
with unity Lewis number, (b) High turbulence case with Lewis
number equal to 0.5, (c) High turbulence case with unity Lewis
number, (d) High turbulence case with Lewis number equal to 2.
Figure 7
Conditional RMS of the scalar dissipation rate for different cases at
different stages of the kernels’ evolution. (a) Low turbulence case
with unity Lewis number, (b) High turbulence case with Lewis
number equal to 0.5, (c) High turbulence case with unity Lewis
number, (d) High turbulence case with Lewis number equal to 2.
Figure 8
Conditional RMS of the temperature for different cases at different
stages of the kernels’ evolution. (a) Low turbulence case with unity
Lewis number, (b) High turbulence case with Lewis number equal
to 0.5, (c) High turbulence case with unity Lewis number, (d) High
turbulence case with Lewis number equal to 2.
Figure 9
Conditional profiles of the variance of the temperature and the
dissipation rate. (a) Low turbulence case with unity Lewis number,
(b) High turbulence case with Lewis number equal to 0.5, (c) High
turbulence case with unity Lewis number, (d) High turbulence case
with Lewis number equal to 2.
22
Figure 10
Conditional profiles of the variance of the fuel mass fraction and
the dissipation rate. (a) Low turbulence case with unity Lewis
number, (b) High turbulence case with Lewis number equal to 0.5,
(c) High turbulence case with unity Lewis number, (d) High
turbulence case with Lewis number equal to 2.
Figure 11
Conditional profiles of the variance of the oxidizer mass fraction
and the dissipation rate. (a) Low turbulence case with unity Lewis
number, (b) High turbulence case with Lewis number equal to 0.5,
(c) High turbulence case with unity Lewis number, (d) High
turbulence case with Lewis number equal to 2.
Figure 12
Scatter plot of the fuel mass fraction at different times vs. the
reaction progress variable, c, for a range of mixture fractions
between 0.495 and 0.505. (a) Low turbulence case with unity
Lewis number, (b) High turbulence case with Lewis number equal
to 0.5, (c) High turbulence case with unity Lewis number, (d) High
turbulence case with Lewis number equal to 2.
Figure 13
Scatter plot of the reaction rate at different times vs. the reaction
progress variable, c, for a range of mixture fractions between 0.495
and 0.505. (a) Low turbulence case with unity Lewis number, (b)
High turbulence case with Lewis number equal to 0.5, (c) High
turbulence case with unity Lewis number, (d) High turbulence case
with Lewis number equal to 2.
23
Figure1: Initial Mixture Fraction Field
24
t=0
t=0.4
t=0.8
t=1.2
t=1.6
t=2.8
t=3.6
<χ|Z>
0.02
Zst
0.3
0.5
0.7
0.9
Pure
Mixing
0
0.1
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
Zst
Mixing
0.01
0.3
0.5
Z
0.7
0
0.1
0.9
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
0.03
Z st
0.5
Z
0.7
0.9
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
Z st
0.02
<χ|Z>
<χ|Z>
0.02
Pure
Mixing
0.01
0
0.1
0.3
(b) uturb = 3.0 , Le = 0.5
(a) uturb = 0.74 , Le = 1.0
0.03
Z st
0.02 Pure
0
0.1
0.01
0.03
0.01
<χ|Z>
0.03
Pure
Mixing
0.01
0.3
0.5
Z
0.7
0
0.1
0.9
0.3
0.5
Z
0.7
0.9
(d) uturb = 3.0 , Le = 2
(c) uturb = 3.0 , Le = 1.0
Figure 2. Conditional means of the scalar dissipation rate for different cases at different
stages of the kernels’ evolution. (a) Low turbulence case with unity Lewis number, (b)
High turbulence case with Lewis number equal to 0.5, (c) High turbulence case with unity
Lewis number, (d) High turbulence case with Lewis number equal to 2.
25
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.8
t=4.4
t=6.0
Pure
Mixing
<T|Z>
0.3
Direction
of Time
Evolution
0.6
0.3
Direction
of Time
Evolution
0
0
Z st
-0.3
0.1
0.3
0.5
Z
0.7
0.9
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
Pure
Mixing
0.3
Direction
of Time
Evolution
Z st
0.5
Z
0.7
0.7
0.9
0.9
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
Pure
Mixing
0.3
0
0.3
0.5
Z
0.6
0
-0.3
0.1
0.3
(b) uturb = 3.0 , Le = 0.5
<T|Z>
0.6
Z st
-0.3
0.1
(a) uturb = 0.74 , Le = 1.0
<T|Z>
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
Pure
Mixing
<T|Z>
0.6
Direction
of Time
Evolution
-0.3
0.1
Z st
0.3
0.5
Z
0.7
0.9
(d) uturb = 3.0 , Le = 2.
(c) uturb = 3.0 , Le = 1.0
Figure 3. Conditional means of the temperature for different cases at different stages of
the kernels’ evolution. (a) Low turbulence case with unity Lewis number, (b) High
turbulence case with Lewis number equal to 0.5, (c) High turbulence case with unity
Lewis number, (d) High turbulence case with Lewis number equal to 2.
26
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.8
t=4.4
t=6.0
<Y f|Z>
0.6
0.9
0.6
Pure
0.3 Mixing
0
0.1
0.5
Z
0.7
Direction
of Time
Evolution
0
0.1
0.9
<Y f|Z>
0.6
0.9
0.6
Pure
0.3 Mixing
0
0.1
0.5
Z
0.7
0.5
Z
0.7
0.9
Zst
Pure
0.3 Mixing
Direction
of Time
Evolution
0.3
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
Zst
<Y f|Z>
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
0.3
(b) uturb = 3.0 , Le = 0.5
(a) uturb = 0.74 , Le = 1.0
0.9
Zst
Pure
0.3 Mixing
Direction
of Time
Evolution
0.3
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
Zst
<Y f|Z>
0.9
Direction
of Time
Evolution
0
0.1
0.9
0.3
0.5
Z
0.7
0.9
(d) uturb = 3.0 , Le = 2.
(c) uturb = 3.0 , Le = 1.0
Figure 4. Conditional means of the fuel mass fraction for different cases at different
stages of the kernels’ evolution. (a) Low turbulence case with unity Lewis number, (b)
High turbulence case with Lewis number equal to 0.5, (c) High turbulence case with
unity Lewis number, (d) High turbulence case with Lewis number equal to 2.
27
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.8
t=4.4
t=6.0
Z st
Direction
of Time
Evolution
<Y o|Z>
0.6
Pure
Mixing
0.3
0.9
0.6
Pure
Mixing
0.3
0
0.1
0.3
0.5
Z
0.7
0
0.1
0.9
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
Z st
Direction
of Time
Evolution
0.6
Pure
Mixing
0.3
0
0.1
0.5
Z
0.7
0.9
0.9
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
Z st
Direction
of Time
Evolution
0.6
<Y o|Z>
0.9
0.3
(b) uturb = 3.0 , Le = 0.5
(a) uturb = 0.74 , Le = 1.0
<Y o|Z>
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
Z st
Direction
of Time
Evolution
<Y o|Z>
0.9
Pure
Mixing
0.3
0.3
0.5
Z
0.7
0
0.1
0.9
0.3
0.5
Z
0.7
0.9
(d) uturb = 3.0 , Le = 2.
(c) uturb = 3.0 , Le = 1.0
Figure 5. Conditional means of the oxidizer mass fraction for different cases at
different stages of the kernels’ evolution. (a) Low turbulence case with unity Lewis
number, (b) High turbulence case with Lewis number equal to 0.5, (c) High turbulence
case with unity Lewis number, (d) High turbulence case with Lewis number equal to 2.
28
1.2
1.2
Z st
1
1
Lean Premixed
Flames
0.8
0.6
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
0.4 Direction
of Time
Evolution
0.2
0
0.1
0.3
0.5
Z
0.7
<ω|Z>
<ω|Z>
0.8
1.2
t=4.4
t=5.2
t=6.0
0.005
0
0.4
Z st
0.5
0.6
Rich Premixed
Flames
0.2
0
0.1
0.9
0.3
0.5
Z
0.7
0.9
(a2) uturb = 0.74 , Le = 1.0, later times
1.2
Z st
1
Lean Premixed
Flames
Lean Premixed
Flames
0.8
0.6
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
0.4
Direction
of Time
Evolution
0.3
0.5
Z
0.7
<ω|Z>
0.8
0
0.1
0.01
0.6
Z st
1
0.2
Diffusion
Flames
Z st
0.4
(a1) uturb = 0.74 , Le = 1.0, earlier times
<ω|Z>
t=2.8
t=3.6
t=4.4
t=5.2
t=6.0
0.6
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
0.4 Direction
of Time
Evolution
0.2
0
0.1
0.9
(b) uturb = 3.0 , Le = 0.5
0.3
0.5
Z
0.7
0.9
(c) uturb = 3.0 , Le = 1.0
1.2
Z st
1
Lean Premixed
Flames
<ω|Z>
0.8
0.6
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
0.4 Direction
of Time
Evolution
0.2
0
0.1
0.3
0.5
Z
0.7
0.9
(d) uturb = 3.0 , Le = 2.
Figure 6. Conditional means of the reaction rate for different cases at different stages
of the kernels’ evolution. (a) Low turbulence case with unity Lewis number, (b) High
turbulence case with Lewis number equal to 0.5, (c) High turbulence case with unity
Lewis number, (d) High turbulence case with Lewis number equal to 2.
29
Z st
0.04
0.01
Z st
"2
<χ |Z>
1/2
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.8
t=3.6
0.02
0
0.1
0
0.1
0.3
0.5
Z
0.3
0.5
0.7
0.7
0.9
<χ"2|Z> 1/2
0.04
0
0.1
0.9
0.04
Z st
0.02
0
0.1
0.3
0.5
Z
0.7
0.3
0.5
Z
0.7
0.9
(b) uturb = 3.0 , Le = 0.5
<χ"2|Z> 1/2
<χ"2|Z> 1/2
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
Z st
0.02
(a) uturb = 0.74 , Le = 1.0
0.04
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
Z st
0.02
0
0.1
0.9
0.3
0.5
Z
0.7
0.9
(d) uturb = 3.0 , Le = 2.
(c) uturb = 3.0 , Le = 1.0
Figure 7. Conditional RMS of the scalar dissipation rate for different cases at different
stages of the kernels’ evolution. (a) Low turbulence case with unity Lewis number, (b)
High turbulence case with Lewis number equal to 0.5, (c) High turbulence case with
unity Lewis number, (d) High turbulence case with Lewis number equal to 2.
30
0.15
Z st
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.8
t=4.4
t=6.0
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
1/2
0.1
"2
"2
<T |Z>
1/2
0.1
Z st
<T |Z>
0.15
0.05
0.05
Direction
of Time
Evolution
0
0.1
0.3
0.5
Z
0.7
0
0.1
0.9
(a) uturb = 0.74 , Le = 1.0
0.3
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
0.9
Z st
t=0.12
t=0.4
t=0.8
t=1.2
t=1.6
t=2.0
t=2.4
t=2.8
0.1
"2
"2
<T |Z>
1/2
0.1
0.7
1/2
Z st
0.5
Z
(b) uturb = 3.0 , Le = 0.5
0.15
<T |Z>
0.15
Direction
of Time
Evolution
0.05
0
0.1
0.05
Direction
of Time
Evolution
0.3
0.5
Z
0.7
0
0.1
0.9
Direction
of Time
Evolution
0.3
0.5
Z
0.7
0.9
(d) uturb = 3.0 , Le = 2.
(c) uturb = 3.0 , Le = 1.0
Figure 8. Conditional RMS of the temperature for different cases at different stages of
the kernels’ evolution. (a) Low turbulence case with unity Lewis number, (b) High
turbulence case with Lewis number equal to 0.5, (c) High turbulence case with unity
Lewis number, (d) High turbulence case with Lewis number equal to 2.
31
1
1
Z st
0.5
R Tχ
R Tχ
0.5
0
t=0
t=0.4
t=0.8
t=1.2
t=2.0
-0.5
-1
0.1
0.3
0.5
Z
0.7
1
t=2.8
t=4.4
t=6.0
-1
0.1
0.9
0.3
0.5
Z
0.7
0.9
(a2) uturb = 0.74 , Le = 1.0, later times
1
Z st
Z st
0.5
R Tχ
0.5
R Tχ
0
-0.5
(a1) uturb = 0.74 , Le = 1.0, earlier times
0
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=2.0
t=2.8
-0.5
-1
0.1
Z st
0.3
0.5
Z
0.7
0
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=2.0
t=2.8
-0.5
-1
0.1
0.9
0.3
0.5
Z
0.7
0.9
(c) uturb = 3.0 , Le = 1.0
(b) uturb = 3.0 , Le = 0.5
1
Z st
R Tχ
0.5
0
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=2.0
t=2.8
-0.5
-1
0.1
0.3
0.5
Z
0.7
0.9
(d) uturb = 3.0 , Le = 2.
Figure 9. Conditional profiles of the variance of the temperature and the dissipation
rate. (a) Low turbulence case with unity Lewis number, (b) High turbulence case with
Lewis number equal to 0.5, (c) High turbulence case with unity Lewis number, (d)
High turbulence case with Lewis number equal to 2.
32
1
1
Z st
t=0
t=0.4
t=0.8
t=1.2
t=2.0
0.5
R Yfχ
R Yfχ
0.5
0
-0.5
-1
0.1
0.3
1
0.5
Z
0.7
Z st
0.9
t=2.8
t=4.4
t=6.0
0.1
1
0.5
Z
0.7
Z st
0.9
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=2.0
t=2.8
0.5
R Yfχ
0
0.3
(a2) uturb = 0.74 , Le = 1.0, later times
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=2.0
t=2.8
0.5
R Yfχ
0
-0.5
(a1) uturb = 0.74 , Le = 1.0, earlier times
-0.5
-1
0.1
Z st
0
-0.5
0.3
0.5
Z
0.7
-1
0.1
0.9
0.3
0.7
0.9
(c) uturb = 3.0 , Le = 1.0
(b) uturb = 3.0 , Le = 0.5
1
Z st
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=2.0
t=2.8
0.5
R Yfχ
0.5
Z
0
-0.5
-1
0.1
0.3
0.5
Z
0.7
0.9
(d) uturb = 3.0 , Le = 2.
Figure 10. Conditional profiles of the variance of the fuel mass fraction and the
dissipation rate. (a) Low turbulence case with unity Lewis number, (b) High turbulence
case with Lewis number equal to 0.5, (c) High turbulence case with unity Lewis
number, (d) High turbulence case with Lewis number equal to 2.
33
1
1
Z st
t=0
t=0.4
t=0.8
t=1.2
t=2.0
0.5
R Yoχ
R Yoχ
0.5
0
-0.5
-1
0.1
0.3
1
0.5
Z
0.7
Z st
0.9
t=2.8
t=4.4
t=6.0
0.1
1
0.5
Z
0.7
Z st
0.9
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=2.0
t=2.8
0.5
R Yoχ
0
0.3
(a2) uturb = 0.74 , Le = 1.0, later times
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=2.0
t=2.8
0.5
R Yoχ
0
-0.5
(a1) uturb = 0.74 , Le = 1.0, earlier times
-0.5
-1
0.1
Z st
0
-0.5
0.3
0.5
Z
0.7
-1
0.1
0.9
0.3
0.7
0.9
(c) uturb = 3.0 , Le = 1.0
(b) uturb = 3.0 , Le = 0.5
1
Z st
t=0
t=0.12
t=0.4
t=0.8
t=1.2
t=2.0
t=2.8
0.5
R Yoχ
0.5
Z
0
-0.5
-1
0.1
0.3
0.5
Z
0.7
0.9
(d) uturb = 3.0 , Le = 2.
Figure 11. Conditional profiles of the variance of the oxidizer mass fraction and the
dissipation rate. (a) Low turbulence case with unity Lewis number, (b) High turbulence
case with Lewis number equal to 0.5, (c) High turbulence case with unity Lewis
number, (d) High turbulence case with Lewis number equal to 2.
34
0.6
0.6
t=0.12
t=t=0.12
0 .4
0.4
.4
0.4
.8
t=0.8
.2
Yf
t=
1
Yf
t=
0
t=
0
0.2
t=
1
.6
t=
2
.0
t=
2
0
0
0.25
0.5
C
t=4.4
t=6.0
t=2
t=
1
t=
1
.8
.4
0.75
t=2.4 & 2.8
0
0
1
0.25
0.5
C
0.75
1
(b) uturb = 3.0 , Le = 0.5
0.6
0.6
t=0.12
t=
0
t=
0
.4
0.4
.8
Yf
0.2
.2
t=
1
t=
1
t=
1
0.2
t=
2
0.5
C
.4
.8
.6
t=
2.
4
t=2.8
0.25
t=
0
.2
Yf
t=
0
t=0.12
t=
1
0
0
.6
t=2.0
(a) uturb = 0.74 , Le = 1.0
0.4
.2
0.2
0.75
.6
t=
2
.0
.0
t=2.4 & 2.8
0
0
1
0.25
0.5
C
0.75
1
(d) uturb = 3.0 , Le = 2.
(c) uturb = 3.0 , Le = 1.0
Figure 12. Scatter plot of the fuel mass fraction at different times vs. the reaction
progress variable, c, for a range of mixture fractions between 0.495 and 0.505. (a) Low
turbulence case with unity Lewis number, (b) High turbulence case with Lewis number
equal to 0.5, (c) High turbulence case with unity Lewis number, (d) High turbulence
case with Lewis number equal to 2.
35
1
1
0.8
0.8
0.6
ω
ω
0.4
0.6
t=1.6
t=1.2
t=1.2
t=0.8
t=2.0
t=0.8
0.4
t=2.4
t=0.4
t=1.6
t=0.4
0.2
t=0.12
0.2
t=4.4
t=6.0
t=2.0
t=2.8
0
0
0.25
0.5
C
t=0.12
0.75
t=2.4 & t=2.8
0
0
1
0.25
0.5
C
0.75
1
(b) uturb = 3.0 , Le = 0.5
(a) uturb = 0.74 , Le = 1.0
1
1
0.8
0.8
t=1.6
t=1.2
0.6
0.6
t=1.6
0.4
ω
ω
t=1.2
t=0.8
0.4
t=2.0
t=0.8
t=0.4
0.2
t=2.4
t=2.8
t=0.4
0.2
t=2.0
t=0.12
t=0.12
0
0
t=2.4
0.25
0.5
C
0.75
1
0
0
t=2.8
0.25
0.5
C
0.75
1
(d) uturb = 3.0 , Le = 2.
(c) uturb = 3.0 , Le = 1.0
Figure 13. Scatter plot of the reaction rate at different times vs. the reaction progress
variable, c, for a range of mixture fractions between 0.495 and 0.505. (a) Low
turbulence case with unity Lewis number, (b) High turbulence case with Lewis number
equal to 0.5, (c) High turbulence case with unity Lewis number, (d) High turbulence
case with Lewis number equal to 2.
36